DYNAMIC TUNING OF HYDRAULIC ENGINE MOUNT USING MULTIPLE INERTIA TRACKS. A Thesis

Transcription

1 DYNAMIC TUNING OF HYDRAULIC ENGINE MOUNT USING MULTIPLE INERTIA TRACKS A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Benjamin Barszcz, B.S. Graduate Program in Mechanical Engineering The Ohio State University 2010 Thesis Committee: Prof. Rajendra Singh, Advisor Prof. Ahmet Kahraman

2 Copyright by The Ohio State University 2010

3 ABSTRACT Passive hydraulic engine mounts are commonly employed for motion control and vibration isolation in vehicle powertrain systems. Such devices are often tuned in terms of their low frequency resonance and damping ratio (say corresponding to the engine bounce mode) to control noise and vibration and improve the ride comfort, quality, and safety of the vehicle. Mount tuning concepts with one inertia track and one decoupler using the track length or diameter are well understood, but the dynamic response with multiple tracks, orifices, or decouplers is not. To overcome this void in the literature, dynamic tuning concepts of hydraulic engine mounts, with emphasis on multiple (n-) inertia tracks, fixed decoupler-type designs, are analytically and experimentally examined in this thesis. Since a wide variety of n-inertia track configurations is possible, dynamic stiffness models are developed to explain a family of such configurations, based on linear time-invariant lumped fluid system theory. Furthermore, a new n-track prototype mount concept is designed, built, and tested in a controlled manner, with the capability of varying the type (capillary tube, orifice) and number (n) of inertia tracks, in addition to length and diameter of each. This prototype is used to examine several designs with alternate n-track configurations for improving performance compared to the n = 1 track case. Three narrowband devices are designed and tested to refine existing theory for predicting peak frequency of loss angle, in addition to examining and validating an n = 3 ii

4 track mount for the first time. Two broadband devices are designed and tested successfully by tuning damping ratios of the mount with orifice-type tracks for the first time. Several n-track mount designs with orifice-type tracks are also proposed, which successfully describe a special broad-tuned design utilizing a controlled leakage path flow area for the first time. Lastly, a quasi-linear dynamic stiffness model is developed to study excitation amplitude- and frequency-dependent behavior of equivalent inertia track resistance, which should lead to nonlinear models of n-track devices and improved adaptive or active mounts in future studies. Chief contributions of this work include experimentally validated extensions of prior lumped parameter, linear time-invariant dynamic stiffness models, which are now applicable to predictions for narrow-tuned and/or broad-tuned mounting devices with n 2. iii

5 DEDICATION To my family, for their unconditional love and support. To my brother, for his patience the past five and a half years we ve been living together at The Ohio State University. To my friends, for helping me grow as a person and being a part of my college experience (for better or worse). iv

6 AKNOWLEDGEMENTS I would like to first and foremost thank those at The Ohio State University who made my experience in the Graduate School so great. The utmost thanks goes to my undergraduate and graduate advisor Dr. Singh for his knowledge, advice, trust, recommendations, and faith during my tenure. I have grown to know him on both a personal and professional level the past 5 years, and his guidance has been an invaluable contribution to my accomplishments here. Thanks to Dr. Kahraman for providing a recommendation which helped me land a great job right out of college and taking time from his busy schedule to be on my defense committee. I would also like to thank the members of Acoustics and Dynamics Lab for their insights, companionship, and sense of humor during my time as a Graduate Research Associate. A special thanks to Dr. Jason Dreyer for his help with my experimental design and testing, and Neil Gardener and the student machine shop for their help in fabricating my prototype. Some thank you s I d like to extend to the corporate world include the Smart Vehicle Concept Center and Transportation Research Center for sponsoring my project and giving me the opportunity to be funded as a Graduate Research Associate. Special thanks to Charlie Gagliano and Honda R&D Americas for granting me clearance to test at their facilities. v

7 VITA April Born, Parma OH December B.S., Mechanical Engineering, The Ohio State University January 2009 to Present..Graduate Research Associate, Department of Mechanical Engineering, The Ohio State University Publications [1] Barszcz, B. Design and Experimental Study of a Mechanical System with Air Springs. Honors Undergraduate Thesis. The Ohio State University, Major Field: Mechanical Engineering Fields of Study Concentration: Automobile Noise, Vibration, and Harshness Control; System Dynamics vi

8 TABLE OF CONTENTS ABSTRACT... ii DEDICATION... iv AKNOWLEDGEMENTS... v VITA... vi LIST OF TABLES... ix LIST OF FIGURES... xi LIST OF SYMBOLS... xiii CHAPTER 1: INTRODUCTION Motivation Problem Formulation Literature Review Scope and Objectives... 6 CHAPTER 2: LUMPED PARAMETER MODELING OF LINEAR HYDRAULIC ENGINE MOUNT WITH MULTIPLE INERTIA TRACKS Hydraulic Engine Mount Design Assumptions for Lumped Parameter Linear Model and Mechanical Analog Linear Modeling of Passive Hydraulic Engine Mount with Decoupler and Two Inertia Tracks Driving-Point Dynamic Stiffness Model Cross-Point Dynamic Stiffness Model Reduced Order Form of Cross-Point Dynamic Stiffness Model Equivalent Inertia Track Dynamic Stiffness Model Equivalent Hydraulic Element Dynamic Stiffness Model Fixed Decoupler Dynamic Stiffness Models vii

9 2.4 Linear K * 22(jω) Model for Multiple (n-number) of Inertia Tracks CHAPTER 3: EXPERIMENTAL STUDIES AND DYNAMIC TUNING OF MULTIPLE INERTIA TRACK MOUNT USING PHYSICAL MODELS Validation of Linear K * 22(jω) Model Design of Multiple Inertia Track Mount Experiment Linear K * 22(jω) Predictions Using Physical Damping Models Physical Description of Prototype Hydraulic Engine Mount Experimental Tuning Using Multiple Inertia Track Prototype Mount Narrow-Tuned Designs Quasi-Broadband Design Broadband Design Controlled Leakage Path Design Quasi-linear Mathematical Damping Model Simulations CHAPTER 4: CONCLUSION Summary and Contributions Recommendations for Future Work REFERENCES APPENDIX A: PHYSICAL DESCRIPTION OF HYDRAULIC ENGINE MOUNT DESIGN A.1 Elastomeric Casing A.2 Chambers A.3 Midplate A.4 Inertia Track A.5 Decoupler viii

10 LIST OF TABLES Table 2.1. System parameters of K * 22(jω) model, free decoupler 37 Table 2.2. Mount parameters for a typical narrowband dynamic stiffness response..39 Table 2.3. Comparison of mount dynamics using K 22 * (jω) prediction for a narrowband device 44 Table 3.1. Typical lumped mount parameters used during order reduction of K * (jω) models...50 Table 3.2. System parameters of narrowband K * (jω) models, free decoupler.51 Table 3.3. System parameters of narrowband K * (jω) models, fixed decoupler...52 Table 3.4. Comparison of system parameters using broadband K 22 * (jω) prediction...54 Table 3.5. System parameters of narrowband K * (jω) models, free decoupler.57 Table 3.6. System parameters of narrowband K * (jω) models, fixed decoupler...57 Table 3.7. Inertia track geometries predicted by K * 22(jω) model for a sample broadband device, k r 2.27 x 10 5 N/m Table 3.8. Linear system parameters predicted by K * 22(jω) model for a sample broadband device, k r 2.27 x 10 5 N/m.61 Table 3.9. Multiple inertia track designs tested using prototype hydraulic engine mount.70 Table Mount upper chamber stiffness predicted by K * 22(jω) model for narrowband designs (without empirical coefficients)...73 Table Mount upper chamber stiffness predicted by K * 22(jω) model for narrowband designs (with empirical coefficients) 73 Table Linear system parameters predicted by K * 22(jω) model for narrowband designs (without empirical coefficients)...79 ix

11 Table Linear system parameters predicted by K * 22(jω) model for narrowband designs (with empirical coefficients)...79 Table Mount upper chamber stiffness predicted by K * 22(jω) model for broadband designs (without empirical coefficients) Table Mount upper chamber stiffness predicted by K * 22(jω) model for broadband designs (with empirical coefficients).. 85 Table Linear system parameters predicted by K * 22(jω) model for broadband designs (without empirical coefficients) Table Linear system parameters predicted by K * 22(jω) model for broadband designs (with empirical coefficients)..86 Table Orifice-type inertia track geometries predicted by K * 22(jω) model used to explain controlled leakage path device response, #i,1 geometry of l t = 21.4 cm, d t = 4.93 mm 90 Table Linear system parameters predicted by K * 22(jω) model for controlled leakage path design 92 Table Error norms of quasi-linear K * 22(jω) model..96 x

12 LIST OF FIGURES Figure 1.1. Representations of hydraulic engine mounting device.4 Figure 2.1. Predicted mechanical properties of common engine mounts..9 Figure 2.2. Schematic of lumped fluid element 14 Figure 2.3. Schematic of lumped equivalent mechanical models for a passive hydraulic engine mount with one decoupler and two inertia tracks, fixed base..16 Figure 2.4. Cross-point dynamic stiffness model of hydraulic engine mount.23 Figure 2.5. Physical interpretation of K * 22(jω).37 Figure 2.6. Typical narrowband frequency response of a hydraulic engine mount using K * 22(jω) model, free decoupler...39 Figure 2.7. K * 22(jω) frequency response prediction with respect to variation in n-identical inertia tracks, narrowband device..45 Figure 2.8. Engine mount parameters f n,num, f n,den, ζ num, and ζ den with respect to variation in n-identical inertia tracks Figure 3.1. Validation of K * (jω) order reduction for a narrowband device, free decoupler...50 Figure 3.2. Validation of K * (jω) order reduction for a narrowband device, fixed decoupler Figure 3.3. Validation of K * (jω) order reduction for a broadband device, free decoupler...53 Figure 3.4. Validation of K * (jω) order reduction for a broadband device, fixed decoupler. 54 Figure 3.5. Validation of K * 22(jω) accuracy for a narrowband device, free decoupler xi

13 Figure 3.6. Validation of K * 22(jω) accuracy for a narrowband device, fixed decoupler Figure 3.7. Dynamic stiffness comparison to a sample broadband mount response...60 Figure 3.8. Views of solid midplate design geometry Figure 3.9. Examples of inertia tracks on final prototype mount device.66 Figure Sample assembly of take apart prototype hydraulic mount consisting of 1 external capillary-type track and 1 internal orifice-type track Figure Views of tested prototype mount designs 70 Figure Bounded dynamic stiffness comparisons for design Figure Bounded dynamic stiffness comparisons for design Figure Bounded dynamic stiffness comparisons for design Figure Static stiffness behavior, k r, of #r for prototype hydraulic engine mount..75 Figure Dynamic stiffness comparison of narrowband designs, X r = 1.0 mm...77 Figure Dynamic stiffness comparison of narrowband designs, X r = 0.3 mm...77 Figure Bounded dynamic stiffness comparisons for design Figure Bounded dynamic stiffness comparisons for design Figure Bounded dynamic stiffness comparisons for design Figure Results of quasi-linear K * 22(jω) model for a sample broadband mount response..96 Figure A.1. Example of hydraulic engine mount inertia track 108 Figure A.2. Example of hydraulic engine mount decoupler xii

14 LIST OF SYMBOLS 1 Upper chamber of mount 2 Lower chamber of mount A Cross-sectional area b Mechanical viscous damping coefficient of an element c Discharge coefficient C Fluid compliance d Decoupler of mount d Hydraulic diameter f Frequency (Hz) F Force H Hydraulic path of force transmission i Inertia track of mount i,n n th inertia track of mount I Fluid inertance j Imaginary number -1 k Mechanical stiffness of an element K Dynamic stiffness l Length m Mechanical mass of an element M Magnitude of dynamic stiffness n Number of inertia tracks N Size of sample set p Absolute pressure q Volumetric flowrate r Upper rubber of mount Re Reynold s number R Fluid resistance s Laplace variable S Norm size t Time variable u Measured value U Actual value v Error norm V Volume x Displacement X Displacement amplitude Z Ideal dynamic mechanical impedance xiii

15 α Α β Β γ Γ ε ζ ι κ λ μ ρ τ φ ω Ω Midplate (hydraulic elements collectively) Real part of complex number Empirical coefficient of orifice fluid inertance Imaginary part of complex number Empirical coefficient of capillary tube fluid resistance Force amplitude Empirical coefficient of orifice fluid resistance Damping ratio Index Speed of propagation Wavelength Dynamic viscosity Density Inertia tracks (collectively) Phase angle Frequency (rad/s) Engine speed (RPM, rev/min) Superscripts t * Total amplitude (static plus dynamic) First derivative with respect to time, d/dt Second derivative with respect to time, d 2 /dt 2 Complex-valued quantity Mean, static, or time-averaged value Subscripts 1 Quantity associated with upper chamber 2 Quantity associated with lower chamber 22,44,54,64,66,76,86 Refers to the order of dynamic stiffness model atm Quantity associated with ambient or atmospheric c Chassis d Quantity associated with decoupler damped Denotes a damped natural frequency damper Quantity associated with a viscous damper element den Quantity associated with denominator of rational expression eq Effective or equivalent value i Quantity associated with inertia track in Entering a control volume i,n Quantity associated with n th inertia track K Relating to dynamic stiffness m Motor or engine Quantity associated with a mass element mass xiv

16 max min n num o out r spring t T Maximum value Minimum value Denotes a natural frequency Quantity associated with numerator of rational expression Orifice Exiting a control volume Quantity associated with mount rubber Quantity associated with a spring element Capillary tube Transmitted Operators Norm Abbreviations DOF LTI Degree of freedom Linear time-invariant xv

17 CHAPTER 1: INTRODUCTION 1.1 Motivation Passive hydraulic engine mounts are advanced isolators commonly used for vertical motion control and vibration isolation of the engine bounce mode in automobile and other vehicle powertrain systems. In specific, engine kinematics and road-profiles typically subject the engine-body-chassis system of an automobile to vibrations of up to f = 250 Hz in frequency. This is especially true for three common types of excitation in particular. An automobile chassis is generally excited in the low frequency regime (at or below about f = 25 Hz) due to (i) high amplitude road excitations from the suspension and (ii) low amplitude engine excitations due to engine idle. Above f = 25 Hz, vibration transmission is often low amplitude but higher in energy content than road or idle excitations. These motions can often be attributed to (iii) engine firing kinematics and associated inertial forces from piston reciprocation and higher-order crankshaft torques at these higher engine speeds [1]. Structure- and air-borne noise is often the consequence of these excitations. It is therefore important to properly tune the low frequency resonance ω n and corresponding damping ratio ζ of engine mounting devices in such a manner that reduction/control of the noise and vibration improves the ride comfort, quality, and safety of the vehicle [1-9]. During the design stage, each mount is currently tuned individually prior to system integration in order to satisfy both component (i.e. resonance) and system (i.e. 1

18 torque-roll axis decoupling methods, cylinder deactivation schemes, powertrain-chassis coupling in 1/4, 1/2, or full vehicle models) performance criteria. Due to the large number of parameters governing the hydraulic mount s response, trial and error techniques are often utilized. Furthermore, design constraints such as locations on the automobile chassis or manufacturing and material imperfections further complicate dynamic tuning [1-13]. More refined models and tuning techniques could facilitate hydraulic engine mount design and quality, in addition to saving time and cost during development. Prior literature on the topic motivates further examination of hydraulic engine mounts with n 2 inertia tracks, both with or without decoupler. Tuning a one inertia track, one decoupler hydraulic engine mount using l i or d i is well understood, but behavior with multiple tracks, decouplers, or orifices is not. Despite this, many inventors have patented mount designs utilizing the multiple inertia track tuning concept to improve vibration isolation. A number of patents document ideas regarding the use of two or more independent parallel paths connecting the upper and lower chambers of a mounting device [14-21]. Other inventors present comparable ideas involving mounting devices with single inertia tracks having one or more tracks contained within [22-28]. However, there is no common basis for looking at such multiple inertia track mount designs analytically. For this reason, there is a need for investigating the dynamic tuning of hydraulic engine mounts with n-inertia tracks. In specific, knowledge of the influence of multiple inertia track geometry (length l i, diameter d i ) on mount dynamics is desired for tuning purposes. Furthermore, investigation of narrow-tuned, broad-tuned, and amplitude (in)sensitive behavior in the multiple inertia track case is of interest. 2

19 1.2 Problem Formulation Component design and testing of hydraulic engine mounts is typically conducted in the frequency domain given steady-state harmonic input [1-7,12,13]. Accurate modeling and tuning techniques can facilitate mount design for motion control or vibration isolation. In general, a passive hydraulic engine mount has one decoupler and one inertia track. However, preliminary investigations have shown that a hydraulic engine mount with multiple inertia tracks is promising for passive, adaptive, and active control [2-4]. For this reason, emphasis will be placed on devices with a fixed decoupler. As seen in the schematic of Figure 1.1, the mount is excited at the top through its upper rubber #r of mass m r with a displacement x t r (t) corresponding to an engine force F t m (t), and transmits a force F t T (t) through its bottom (via a baseplate or bracket) to a fixed base (the chassis or frame). The mount is comprised of an elastomeric outer casing (represented by stiffness and viscous damping coefficient, k r, b r, respectively) encompassing upper (#1) and lower (#2) fluid-filled chambers, which communicate with one another via hydraulic paths (#d, #i,1 and #i,2) on a midplate. Each of the fluid control volume elements can be described with lumped fluid elements of compliance C, resistance R, and inertance I. The hydraulic mount of Figure 1.1b is atypical in that although it has only one decoupler (#d) it has multiple inertia tracks; this specific example having n = 2 (#i,1 and #i,2). The parameters of each of these common mount elements can be computed using theoretical, computational, or experimental methods, based on the degree of nonlinearity existing in each and/or the degree of difficulty associated with approximation. Each common mount element is briefly discussed in more detail in Appendix A. 3

20 (a) (b) Figure 1.1. Representations of hydraulic engine mounting device: (a) Schematic of hydraulic engine mount as seen from an experimental context using a blackbox concept; (b) Schematic of lumped fluid model for a passive hydraulic engine mount with one decoupler (#d) and two inertia tracks (#i,1, #i,2) 1.3 Literature Review It has been shown that changing length l i or cross sectional area A i of a single inertia track mount can be used to tune the dynamic performance via peak frequency of the loss angle within the low frequency regime (f = 1 to 50 Hz). However, little work in the literature has been performed on the dynamic tuning of hydraulic engine mounts using multiple (n-) inertia tracks. Zhang and Shangguan et al [2,3] present two nonlinear lumped parameter state-space models (with an assumed nonlinear track resistance R i ) for a passive hydraulic engine mount; one model having n = 3 non-identical inertia tracks and the other having two groups of identical inertia tracks. These models are validated experimentally using two mounts, each manufactured to have n = 2 tracks; one of which has two identical inertia tracks and the other having two unequal length inertia tracks. The proposed models are seen to agree well with dynamic stiffness measurements at 1 mm excitation amplitude. Further, analytical results are presented 4

21 which approximate for both of the models when R i s are assumed to be linear. The dynamic stiffness magnitude K * (jω), loss angle φ K* (jω), and are seen to increase in the n = 2 or 3 inertia track configuration when compared to the conventional one track configuration. Lu and Ari-Gur [4] have conducted similar studies on the effects of n = 2 inertia tracks, but on both passive hydraulic engine mounts and hydrobushing devices. They present linear lumped parameter models for each device assuming n = 2 and use them to derive an analytical natural frequency result to approximate of a mount with multiple tracks that is comparable to that done by Shangguan and Zhang et al [2,3]. The mount model is validated via dynamic tests of a mount manufactured with n = 2. They extend the two inertia track model to simulations for mounts with n 2 and conclude that K * (jω), φ K* (jω), and increase with respect to n as well. Furthermore, Kim and Singh [5] suggest that a typical hydraulic engine mount with a single inertia track can be modeled with two independent flow paths by virtue of a secondary leakage path existing between #1 and #2 within the orifice plate assembly of the midplate. This is justified by identifying a small clearance gap existing between the blocking marker on the upper and lower orifice plates. The blocking marker is designed with the intent to restrict flow in this direction between the inlet and outlet of #i. However, the clearance allows flow to leak through this secondary path of equal A i. The steady p-q characteristics of #i,1 and #i,2 are measured with a specially prepared hydraulic circuit, and then curve-fitted using the orifice square-root relation to find Δp across each. These empirical pressure drop expressions were then used in a formulation of a nonlinear, numerical K * (jω) model via the momentum equations of each respective 5

22 fluid path between #1 and #2. K * (jω) is seen to approximate experimental results for a mount with n = 1 (assumed to also have the inherent leakage path) well for low frequencies (f = 1 to 50 Hz). To the best of our knowledge, no other formal investigation has analytically examined the influence of multiple inertia tracks on mount response. Several deficiencies and unresolved issues in the literature still remain on the topic, however. Closed-form K * (jω) models have been developed in limited number for very specific multiple inertia track cases of n 3, for narrow-tuned mounting devices only. Existing K * (jω) models have not been experimentally validated for any case of n 2 tracks, and often must be supplemented with numerical solutions. Analytical approximation methods for are limited and could be improved. Furthermore, no direction has been established for nonlinear modeling and/or adaptive/active control concepting of n- track devices. Lastly, decoupler dynamics have not been considered for mount designs when n Scope and Objectives This research will focus on the vertical motion control and vibration isolation of an automobile powertrain from the chassis due to the tuning of a hydraulic engine mount using the multiple (n-) inertia track concept. Emphasis will be placed on mount characterization and the ability to tune the mount accurately for the design of ideal mounting systems. It is hypothesized that the geometries of the n-inertia tracks (specifically l i and d i ) can be tuned to obtain a desired response performance (narrowband, broadband, amplitude (in)sensitive). The hydraulic engine mount is a 6

23 highly nonlinear device, in particular due to significant nonlinearities in its upper chamber compliance C 1, R i, decoupler switching action, and nonlinear material properties/geometry. However, the majority of this work will focus on development of lumped parameter LTI K * (jω) models, n 2, with application to tuning hydraulic engine mounts over the low frequency regime (f = 1 to 50 Hz, or Ω = 60 to 3000 RPM). Fluid resistance models will be investigated to better understand the role of l i and d i on tuning and device damping of a hydraulic engine mount with n 2, especially in broad-tuned designs. This should yield a more tunable device that could improve ride quality and vibration isolation. Specific research objectives include the following: (i) Develop and refine analytical models for the hydraulic engine mount such as that in Figure 1.1 for the case of n > 1 using mathematics, with emphasis on n 2. Present lumped parameter, frequency domain, LTI models which predict steady-state K * (jω) and force transmitted F T (t) of an n-track mount, for fixed or free decoupler, narrow-tuned or broad-tuned devices. Validate these models with previous and/or new experimental results. (ii) Design and build a controlled experiment which can be used to evaluate the impact of different types and numbers of track combinations (capillary tube, orifice) [29] on the dynamic tuning of a hydraulic engine mount, and validate proposed theory. (iii) Suggest n-track mount designs (narrow-tuned, broad-tuned, amplitude (in)sensitive) which could improve hydraulic engine mount performance and tunability when compared to the conventional n = 1 track case. 7

24 CHAPTER 2: LUMPED PARAMETER MODELING OF LINEAR HYDRAULIC ENGINE MOUNT WITH MULTIPLE INERTIA TRACKS 2.1 Hydraulic Engine Mount Design An automobile engine mounting system typically consists of three to four engine mounts, one or two of which are passive, adaptive, or active. When attempting to control say the engine bounce mode of powertrain vibrations, trade-offs must be made between mounting devices with superior motion control (stiff and highly damped at lower frequencies) and superior vibration isolation (compliant and low damping at larger frequencies). It is well known that the passive hydraulic engine mount is a device capable of providing superior frequency and amplitude response characteristics when compared to the behavior of the conventional elastomeric-type mounting component [1-13]. Pictured in Figure 2.1 below are typical predicted equivalent stiffness k eq and damping b eq profiles of these two most common engine mounts using a Voigt-type model [1]. It can be seen that the elastomeric mount exhibits frequency insensitive compliance and low damping, while the highly stiff and damped hydraulic mount is very dependent on frequency. The particular hydraulic engine mount here is typical in that its response is considered narrowband, i.e. k eq and b eq are very large over a narrow frequency band. An engine mount design which is some combination of these two is particularly desirable, since motion control and vibration isolation are defined by different frequency regimes. 8

25 Figure 2.1. Predicted mechanical properties of common engine mounts:, typical passive hydraulic engine mount (narrow-tuned); ---, elastomeric mount 2.2 Assumptions for Lumped Parameter Linear Model and Mechanical Analog As seen in the fluid schematic of Figure 1.1, the mount is excited in the vertical direction at upper rubber #r with displacement x t r (t) corresponding to force F t m (t) due to engine firing kinematics. This will result in some overall force transmitted F t T (t) to the assumed fixed chassis base. For a linear system, the principle of superposition will apply, and these total time-varying forces and motion can be decomposed into the summation of static and dynamic terms, (2.2a) (2.2b) (2.2c) 9

26 where is the compressive mean preload of the engine due to its weight, F m (t) the dynamic engine force, the static displacement of the mount associated with and x r (t) the dynamic response associated with F m (t), the mean force transmitted to the chassis, and F T (t) the dynamic force transmitted. Force generated by the engine motion is transmitted to the chassis base via F t T (t) through both the static path represented by the stiffness and damping of the rubber, k r and b r, and the parallel hydraulic path represented by the internal fluid elements of the mount. The assumption of a fixed chassis implies that the mass of the chassis is at least an order of magnitude larger than that of the engine. A typical 4-cylinder internal combustion engine will have a mass of anywhere from 325 to 365 kg ( 3,200 to 3,600 N in weight), which determines the preload on the mount ( 1/3 or 1/4 of the engine weight, depending on whether 3 or 4 mounts are used for the powertrain, respectively) [9]. While this value is smaller in magnitude than that of the chassis of the vehicle, it is still comparable. However, forces transmitted to the chassis are not only due to engine motions but also road inputs through the tire-wheel-suspension path. Since the suspension is designed to deal directly with the road excitations, it is assumed that the mount will primarily see excitations due to only the engine. On this basis, the chassis will be viewed as rigid and fixed and excitation amplitudes/forces will reflect nominal values as seen in practice, even though the chassis will have some compliance associated with it which would otherwise affect the mount response. It is desirable to relate the motion of the mount x t r (t) to the force F t m (t) and/or force transmitted F t T (t) for motion control and vibration isolation purposes. For this reason, the input is assumed harmonic, and the resulting mount response is therefore of the same form. For the purpose of modeling, F t m (t) is considered the input for driving- 10

27 point dynamic stiffness models, whereas x r t (t) is considered the input for cross-point dynamic stiffness models, as will be shown later. Typically in industrial testing practices for hydraulic engine mounts, designs are dynamically tested to measure output F T t (t) given an assumed sinusoidal displacement input x r t (t) = + X r sin(ωt), X r being the input displacement amplitude of #r, ω the driving frequency, and t the time. This type of nonresonant testing assumes m r = 0 and attempts to dynamically characterize the mount in terms of its k eq and b eq mechanical properties using standard procedures [30]. This approach is consistent with the cross-point dynamic stiffness models developed later in this chapter. The total amplitude of this displacement will vary depending on the displacement preload, but X r will generally range between 0.05 to 1.5 mm for low ω (f = 1 to 50 Hz) automotive mount tests [8]. Since the input is assumed to be a pure harmonic in such test protocols, super- and sub-harmonics are ignored, and design is based solely on response to the first harmonic frequency, as is all subsequent modeling [7]. As an example, consider some harmonic force F t (t) as an input to an arbitrary lumped LTI mechanical system. The corresponding arbitrary responses due to the dynamic part F(t) of this total input signal will take the following form, (2.2d) (2.2e) (2.2f) where x(t) = displacement response of the arbitrary lumped mass, is its velocity, its acceleration, and j is the imaginary number -1. The response is interpreted as complex due to X(jω) amplitude term, which is just the product of the linear amplitude of 11

28 x(t) and phase angle between response and excitation, φ. In other words, X(jω) = Xe jφ (in this manner, the phase angle can be kept implicit to response formulations). This complex representation of system response can be used to define the dynamic impedance of an ideal mechanical element as. Z can be thought of as resistance to motion due to arbitrary force F(t). Accordingly, the dynamic impedance associated with each ideal mechanical element can be expressed as so: (2.2g) (2.2h) (2.2i) The dynamic impedance is related to the forcing frequency ω in this manner. With respect to the mount under consideration, the masses of #1 and #2 can then be approximated as pure stiffness elements k 1 and k 2 respectively at low frequencies, since Z mass will be negligible at low frequencies when compared to Z spring, and damping b of the fluid in #1 and #2 is negligible (assumed to have low viscosity). This is the fluid equivalent of each chamber carrying a uniform pressure p 1 (t) and p 2 (t), and storing potential energy in the form of compliances C 1 and C 2, respectively, as shown in Figure 1.1. Note that this is a frequency domain interpretation of the system, so modeling will reflect this accordingly. The hydraulic behavior of the mount can be characterized on the premise of linear fluid system parameters and one-dimensional, steady flow assumptions. As previously indicated, the hydraulic path (namely the hydraulic mechanisms of #α) strongly influences the dynamics of the mount. The decoupler #d and inertia tracks #i,1 and #i,2 12

29 can be assumed to be lumped inertance and resistance elements as shown in Figure 1.1 on the premise of control volume analysis. Consider the fundamental lumped fluid elements shown below in Figure 2.2 with steady, incompressible flow q due to applied pressure Δp across #1 and #2. The inertance I can be thought of as the resistance to changes in flow rate to the applied pressure gradient Δp on the fluid element. Since it is difficult to measure, it is often expressed using one dimensional, steady flow theory in a pipe as I = ρl/a, where ρ is the density of the mount fluid, l the effective length of the hydraulic element, and A its cross-sectional area [29]. The effective length can typically range anywhere from 1 to 4/3 of the geometric length, and will generally depend on end effects due to the 1-D flow assumption [29]. The resistance R of these elements is fundamentally the energy dissipation of fluid flow q over the pressure difference Δp of the element, which is inherently difficult to describe with analytical expressions for #d, #i,1, and #i,2 since damping is frequently a nonlinear effect. While R d is highly nonlinear and there is no good theory for describing it in a closed-form fashion, R i,1 and R i,2 can be described with a variety of different linear damping models based on geometric considerations. Many of these models are based on the steady flow assumption and can be manifested for either laminar or turbulent flows, including capillary tube (circular or non-circular cross-section) or orifice models. A linear capillary tube fluid resistance can be assumed when #i,1, or #i,2 is sufficiently long. Namely, when l d, fluid resistance R is concentrated over a long distance for the given inertia track. Capillary tube models are appropriate selections for this type of inertia track geometry. In this case, R will be assumed to have the form R t = 128μl t /πd 4 t, where μ is the dynamic viscosity of the mount fluid, and d t the circular diameter of the capillary tube cross section. In general, the cross 13

30 section need not be circular; d t can represent the hydraulic diameter of a different crosssectional geometry as well (such as a square, rectangular prism, oval, triangle), which would then depend on the perimeter. Thus, variations in cross-sectional area will yield alternative forms of R t which can be found in texts or fluid mechanics handbooks [29]. Orifice-type models can also be used to describe the damping of #i,1 and #i,2 for R concentrated over a short distance (l using the square-root relation, d). The resistance of an orifice can be found, where c o is the discharge coefficient of the orifice, and A o the cross-sectional area of the orifice. This form includes sharpedged orifices (most predictable) and short-tube orifices (easiest to fabricate). c o is experimental, and will depend on the Reynolds number (Re) and temperature of the flow. Note that the square-root relation is a nonlinear relation, so linearization of this expression about an operating point flow or pressure will be necessary to find R o for application to lumped parameter LTI theory [29]. (a) (b) Figure 2.2. Schematic of lumped fluid element: (a) Capillary-tube type; (b) Orifice-type One dimensional flow is assumed valid for the low frequency analysis (f = 1 to 50 Hz) considered in the scope of this study. Namely, since the chambers #1 and #2 are modeled as compliance elements C 1 and C 2, flow is assumed to take place only in the direction through the hydraulic elements of the midplate (#d, #i,1, #i,2). Based on 14

31 acoustic theory, the cross-dimensions of the decoupler and inertia tracks (represented by d, circular or hydraulic diameter) must be much smaller than the wavelength of the propagation, d λ. For a given medium, propagation speed κ = fλ. It can be seen in this manner that for a certain propagation medium with constant κ, λ must be large for the small frequencies of interest, therefore satisfying the 1-D flow assumption. In specific, κ is of the order of 10 3, which corresponds to λ of the order of 10 1, around 3 orders of magnitude larger than a typical inertia track or decoupler, which will nominally measure of the order d = 10-2 m. The lumped system is assumed to consist of both mechanical and fluid elements. Since the dynamics of the fluid system can be difficult to model, and the fluidmechanical interactions within the system can create complexities and nonlinearities which are difficult to understand, it is often easier to use mechanical analogs to describe mount behavior. Namely, transforming the hydraulic engine mount system into an equivalent mechanical system will greatly simplify design, analysis, and understanding of system behavior, especially with respect to force transmissibility considerations. Consider the following ideal fluid elements and corresponding mechanical analogs describing the dynamics of an arbitrary system: (2.2j) (2.2k) (2.2l) The resulting mass-damper-spring system, using the above analogies and assumptions for the fluid system, is shown below in Figure 2.3 for the fixed chassis base. 15

32 (a) (b) Figure 2.3. Schematic of lumped equivalent mechanical models for a passive hydraulic engine mount with one decoupler and two inertia tracks, fixed base: (a) Driving-point model (F m t (t) as input, x r t (t) as output); (b) Cross-point model (x r t (t) as input, F T t (t) as output) 2.3 Linear Modeling of Passive Hydraulic Engine Mount with Decoupler and Two Inertia Tracks Driving-Point Dynamic Stiffness Model The fluid path of force transmission can be described by using the continuity equations on each chamber and the momentum equations on each hydraulic element. In this manner it is possible to find an equivalent mechanical system such as that shown in Figure 2.3 described by a system of force equations. The force equations can be solved for a transfer function called the dynamic stiffness of the mount, which is defined as F(jω)/x(jω) of the mount. In specific, dynamic stiffness can be used to describe the mount using either F m (jω) or F T (jω) as so: (2.3.1a) (2.3.1b) 16

33 Namely, K * (jω) in terms of is referred to as drivingpoint dynamic stiffness, and K * (jω) in terms of is called the cross-point stiffness. These are both useful tools for assessing the vibration isolation and force transmission of the mount with respect to the frame of the vehicle. Note again that driving-point K * (jω) assumes the input as F m (t), whereas cross-point K * (jω) will assume x r (t) as the input (shown later). To develop the mechanical analog shown in Figure 2.3, Newton s 2 nd law can be applied to upper rubber #r, and the momentum equations to decoupler #d and inertia tracks #i,1 and #i,2 in the vertical direction as so, (2.3.1c) (2.3.1d) (2.3.1e) (2.3.1f) where A r is the effective piston cross-sectional area of the upper rubber in contact with the fluid of #1. F t m (t) is the input here, as x t r (t) is implied by Newton s 3 rd Law between the powertrain and #r. Similarly, the continuity equations to upper and lower chambers #1 and #2 can be applied using the definition of fluid compliance and 1-D flow assumptions on the control volumes as so: (2.3.1g) (2.3.1h) (2.3.1i) (2.3.1j) (2.3.1k) 17

34 Note that incompressible flow is assumed here (ρ = constant for fluid of #1 and #2). The static components of each DOF must be eliminated so as to express each in terms of its dynamic components for application of the K * (jω) model. Considering the static case of the equations of motion of the mount, we have the following: (2.3.1l) (2.3.1m) Since there is no flow in the static case, we can assume that q t (t) = q(t). Similarly, there will also be no static terms associated with,,,,,, or. Furthermore, the volume stored in #d, #i,1, and #i,2 is minimal in the static case, so these are ignored (since they will have an assumed negligible fluid compliance). Applying the static conditions the equations of motion simplify as so: (2.3.1n) (2.3.1o) (2.3.1p) (2.3.1q) (2.3.1r) (2.3.1s) The dynamic pressures p 1 (t) and p 2 (t) can be eliminated using the continuity expressions via integration and substituting the expressions into the momentum equations. For 1-D flow, we can transform the fluid degrees of freedom of #d, #i,1, and #i,2 to mechanical degrees of freedom by assuming them to be lumped masses with velocities = q d (t)/a d, = q i,1 (t)/a i,1, and = q i,2 (t)/a i,2, respectively, where A d, A i,1, A i,2 are the constant cross-sectional areas of the decoupler and inertia 18

35 tracks and,, and their linear velocities. The continuity equations simplify in the following manner: (2.3.1t) (2.3.1u) The dc components via constants of integration are ignored here since we are only concerned with the dynamic (and not total) response (negligible volume V will be stored in #d, #i,1, and #i,2). The continuity equations and 1-D flow assumption can be used in this manner to eliminate the pressure terms in each of the equations of motion as so: (2.3.1v) (2.3.1w) (2.3.1x) (2.3.1y) The momentum equations for #d, #i,1, and #i,2 must be multiplied by an area to transform pressures to forces and yield a mechanical equivalent system of equations, i.e. a mechanical analog. This must be done with care, however, as it must be done on a consistent basis. It can be seen that since p 1 (t) is assumed uniform, F m (t) will be magnified due to the area reduction from the piston A r with respect to A d, A i,1, and A i,2, on the premise of hydraulic theory. In specific, the mechanical force across each of the hydraulic mechanisms due to Δp(t) can be related to F m (t) via normalization of the crosssectional areas of #d, #i,1, and #i,2 by the ratios A r /A d, A r /A i,1, and A r /A i,2, respectively. This is identified as a phenomenon called inertia-augmented damping in the engine 19

36 mount literature [10]. In this manner, the momentum equations can then be multiplied by A r to get forces in terms of the input F m (t): (2.3.1z) (2.3.1ab) (2.3.1ac) (2.3.1ad) It can be seen that the above system of equations is the mechanical analog of the hydraulic engine mount, where A 2 r /C 1 can be thought of as the stiffness of the upper chamber k 1, A 2 r /C 2 the stiffness of the lower chamber k 2, I d A 2 r, I i,1 A 2 r, I i,2 A 2 r the masses m d, m i,1, and m i,2 of lumped #d, #i,1, and #i,2, and R d A 2 r, R i,1 A 2 r, R i,2 A 2 r the viscous damping coefficients b d, b i,1, and b i,2, as shown in Figure 2.3. The driving-point dynamic stiffness model can be developed by considering the outputs of the system to be harmonic in response to assumed harmonic input F m (t), (2.3.1ae) (2.3.1af) (2.3.1ag) (2.3.1ah) (2.3.1ai) where Γ m is the force amplitude of the input, and X r (jω), X d (jω), X i,1 (jω), and X i,2 (jω) the complex amplitudes of the respective displacement responses. As assumed earlier, phase 20

37 information will remain contained within the complex amplitudes of the responses. Assuming the system is causal and defined in a physical sense (ω 0), the Laplace variable s = jω can be substituted into the above responses for zero initial conditions. The resulting system of equations in matrix form can be expressed as so: (2.3.1aj) The characteristic equations of each lumped mass of the mount model are represented by the diagonal terms of the coefficient matrix in (2.3.1aj) above. Namely, r(jω) = m r (jω) 2 + b r (jω) + k r + k 1, d(jω) = m d (jω) 2 + b d (jω) + k 1 + k 2, i,1(jω) = m i,1 (jω) 2 + b i,1 (jω) + k 1 + k 2, and i,2(jω) = m i,2 (jω) 2 + b i,2 (jω) + k 1 + k 2 are the characteristic equations of upper rubber #r, decoupler #d, and inertia tracks #i,1 and #i,2, respectively. Using Kramer s method, the linear system of four equations in four unknowns x r (jω), x d (jω), x i,1 (jω), and x i,2 (jω) can be solved for any of the displacements in terms of F m (jω) and the coefficient matrix. Note that although these outputs will be complex using this approach, they can be manipulated to find the time domain responses since they are simply harmonics with some phase shifts with respect to input F m (t). This will be true for all dynamic stiffness models to follow similarly. Solving for x r (jω): (2.3.1ak) Simplifying the determinants of the numerator and denominator, dividing both sides by F m (jω), and taking the reciprocal of both sides yields the following dynamic 21

38 stiffness expression for a hydraulic engine mount with decoupler #d and inertia tracks #i,1 and #i,2: (2.3.1al) K * 86(jω) is the driving-point dynamic stiffness of the hydraulic engine mount due to input F m (jω), and is the ratio of the driving-point dynamic force input F m (jω) to the dynamic motion output of the engine mount x r (jω). The subscript 86 represents the order of the dynamic stiffness transfer function model. For example, the numerator of K * 86(jω) is 8 th order in ω and the denominator is 6 th order in ω. In the general case, however, the dynamic stiffness model K * (jω) of the mount can have an arbitrary order numerator and denominator based on the number of hydraulic elements and the assumptions made. The higher the order of the expression, the more mount dynamics (and modes) under consideration Cross-Point Dynamic Stiffness Model The K * 86(jω) driving-point dynamic stiffness model is useful, in that it relates the dynamic force at the engine side F m (t) to a dynamic output displacement x r (t) of the mount at low driving frequencies ω. However, an expression relating F T (t) and x r (t) is more useful, as it is instead a function of the force transmitted to the frame of the vehicle and is more standard when characterizing elastomers in the materials industry. A crosspoint dynamic stiffness of the mount is also a direct function of motion control/vibration isolation capabilities of the device. The cross-point dynamic stiffness can be derived by solving for F m (t) as a function of F T (t) when the base is assumed to be fixed. When this 22

39 is done, it will become clear why the cross-point K * (jω) model assumes x r (t) as the input and F T (t) as the output. (a) (b) Figure 2.4. Cross-point dynamic stiffness model of hydraulic engine mount: (a) Quarter vehicle system representation of hydraulic engine mount; (b) Hydraulic (fluid) path of force transmission To relate F m (t) to F T (t), consider the depiction of the hydraulic engine mount in the context of the quarter vehicle system as shown above in Figure 2.4. Figure 2.4a shows the engine mount as it interacts in general with a lumped chassis mass-springdamper vehicle system as m c, k c, and b c with motion x t c (t). Motion x t c (t) is only used for illustrative purposes here and will be assumed zero as was done in prior fixed base analysis upon completion. As mentioned previously, the mount can transfer force hydraulically (path denoted as H t (t) here in Figure 2.4a) and through the path of the rubber via k r and b r. Figure 2.4b shows to some detail the manner in which the hydraulic path H t (t) transfers force via absolute pressures p t 1 (t), p t 2 (t), and p atm. The hydraulic elements #d, #i,1, and #i,2 are implicit to these pressures through the momentum 23

40 equations. Free body analysis of the lumped masses m r and m c yields the following equations: (2.3.2a) (2.3.2b) The total response terms can be broken down into a summation of their dynamic and static terms, as was done earlier. In specific, we are primarily concerned with the dynamic part of the total response once again, and can use the static condition of the system to simplify the equations of motion above: (2.3.2c) (2.3.2d) Applying the static condition, and using the fact that,,,, the equations of motion reduce to the following dynamic expressions: (2.3.2e) (2.3.2f) Upon observation, it can be seen that the dynamic equations of motion share the common terms k r [x r (t) x c (t)],, and A r p 1 (t). These common terms can be thought of as the force transmitted between the masses, or using the previous notation, the force transmitted from the mount to the chassis base, F T (t): (2.3.2g) 24

41 This interpretation is a physical reflection of the force being transmitted in a positive sense to m c, as indicated by the sign convention of each of the terms. Substituting this into the equation of motion of the mount above reveals a direct relationship between the driving point force F m (t) and the force transmitted F T (t): (2.3.2h) For all practical purposes, m r will be small and F m (t) F T (t). Also, note that since low frequency excitations are assumed, the impedance of the upper rubber is negligible and will therefore have little effect on dynamic motion. When m r 0, x r (t) becomes interpreted as a base input excitation at the engine side, since force input F m (t) requires application to some finite inertia m r. It is therefore implied that load is carried by b r, k r, and p 1 (t) due to the displacement input x r (t) instead of being due to F m (t) being applied at a finite m r. Assuming the chassis base is fixed as had been done previously, and the above equation of motion of the mount (2.3.2e) will reduce to the previous dynamic force equation used for the upper rubber #r of the K * 86(jω) model, but now in terms of output F T (t) due to excitation x r (t): (2.3.2i) Once again, the continuity and momentum equations can be utilized to describe the behavior of #d, #i,1, and #i,2. The equivalent mechanical analog of the engine mount can be found the same way in which it had via the formulation of the K * 86(jω) model via considering magnification at #α and normalizing the hydraulic elements with the piston area A r. This time, however, the system will effectively be described with one less degree of freedom than before, due to the elimination of m r in the force equation of the upper rubber #r. While it is not entirely correct to say that the degree of freedom 25

42 associated with m r is eliminated, for all intensive purposes it is, since the natural mode of #r will take place at very high frequencies when m r is sufficiently small. The resulting equations of motion for the cross-point model of dynamic stiffness are as so: (2.3.2j) (2.3.2k) (2.3.2l) (2.3.2m) Since x r (t) is now the assumed mount input, the output F T (t) will take the following form, (2.3.2n) (2.3.2o) where Γ T (jω) is the complex amplitude of the force transmission output. The response outputs x d (t), x i,1 (t), and x i,2 (t) will remain unchanged from the expressions shown in (2.3.1ag-ai). As assumed earlier, Laplace transformation can be applied to yield the steady state responses for each DOF. The cross-point model can be rearranged to matrix form as shown below (2.3.2p) 26

43 where r T (jω) = b r (jω) + k r + k 1 is the new characteristic equation of upper rubber #r in terms of force transmitted F T (jω), and d(jω), i,1(jω), i,2(jω) are the same characteristic equations of decoupler #d, and inertia tracks #i,1 and #i,2, as those previously defined in the K * 86(jω) model, respectively. It is worth noting that based on the way they are defined, the matrix forms of the equations of motion for both the driving-point and crosspoint models will take the same exact form. This can be seen above in (2.3.2p) when comparing it to the matrix form of equations of (2.3.1aj) in the previous section. Kramer s method is once used to solve for the motion of #r to yield the following expression for x r (jω): (2.3.2q) Note that here x r (jω) is the assumed (known) complex harmonic input for crosspoint K * (jω), but the same procedure used previously will stiff apply here. Using linear algebra to simplify the above expression, the new dynamic stiffness expression for a hydraulic engine mount is the following: (2.3.2r) K * 76(jω) is therefore defined as the cross-point dynamic stiffness of a hydraulic engine mount due to input x r (jω) with decoupler #d and inertia tracks #i,1 and #i,2. The cross-point model now relates x r (jω) to F T (jω) instead of F m (jω). Its numerator is 7 th order in ω, while its denominator is 6 th order in ω. Physically and mathematically speaking, this model essentially assumes that the eigenvalue of the upper rubber mode 27

44 corresponds to a very large frequency. Due to the previous low frequencies assumption made, the natural frequency of #r is therefore not expected to affect the dynamic response in the f = 1 to 50 Hz frequency range of interest, since m r will be negligible (very small) Reduced Order Form of Cross-Point Dynamic Stiffness Model As is seen in the derivation of the K * 86(jω) and K * 76(jω) models, linear modeling of a hydraulic engine mount can yield complicated expressions for dynamic stiffness. Accordingly, the order of the K * 76(jω) transfer function expression can be reduced greatly by making several more reasonable assumptions about the behavior of the mount. In doing so, the number of effective DOF s will be reduced further, thus simplifying the understanding, application, and analysis of the cross-point dynamic stiffness via substitution of K * 76(jω) with more compact and robust models. Order reduction of K * 76(jω) can begin by considering the equation of motion of the rubber for this system in the Laplace domain: (2.3.3a) As seen above, force transmission occurs through the parallel paths of the outer rubber (first term) and the hydraulic path (second term). Since s = jω, it can be seen that the dynamic force transmission due to the path of the rubber is dominated by its stiffness k r at low frequencies ω. This is equivalent to stating that the impedance due to the damping of the rubber will be much smaller than that of the stiffness at low frequencies, b r k r /jω. Using this assumption, b r 0. While this is not completely true, b r will be small enough in magnitude (as asserted earlier) that it can be neglected for all intensive purposes. In other words, b r is not expected to contribute significantly to the dynamic 28

45 response of the mount within the low frequency range of interest [1]. The resulting equation of motion of upper rubber #r reduces as so: (2.3.3b) Furthermore, the lower chamber compliance can be assumed to be much greater than the upper chamber compliance, C 2 C 1 (C 2 150C 1, nominally two orders of magnitude larger). This is due to the fact that p t 2 (t) is approximately equal to p atm at low mount preloads ( 1200 N or less). Physically speaking, #2 essentially accumulates all fluid from #1 due to external excitations at preload of approximately this magnitude and lower. A 1200 N preload corresponds to a typical 4-cylinder engine, however, and the dynamic component of the lower chamber pressure p 2 (t) starts to become significant as the preload increases from this point and/or excitation amplitude is sufficiently large (X r 1.0 mm). When one or both of these conditions occurs, the lower chamber pressure p 2 (t) cannot be considered negligible, and C 2 may no longer be much greater than C 1 [7,9]. Since C 2 C 1 and the mechanical analogs of chamber stiffness for #2 and #1 are inversely proportional to these respective compliances,. In this manner, the absolute pressure drop across a given hydraulic element Δp(t) = p 1 (t) p 2 (t) p 1 (t). Similar to previous assumptions for m r and b r, k 2 0 is not physically correct, but due to the assumed operating conditions of the mount ( 1200 N, X r 1.0 mm), the dynamics of lower chamber #2 due to p 2 (t) are insignificant for all intensive purposes when compared to upper chamber dynamics associated with p 1 (t). In this manner, the equations of motion of #d, #i,1, and #i,2 from K * 76(jω) will simplify as follows upon dropping k 2 : 29

46 (2.3.3c) (2.3.3d) (2.3.3e) Using these assumptions, the dynamic stiffness of the mount will change compared to K * 76(jω) not only due to the characteristic equation of #r, but this time due the characteristic equations of #d, #i,1, and #i,2 as well. Now, r T (jω) = k r + k 1, d(jω) = m d (jω) 2 + b d (jω) + k 1, i,1(jω) = m i,1 (jω) 2 + b i,1 (jω) + k 1, and i,2(jω) = m i,2 (jω) 2 + b i,2 (jω) + k 1. Using the same matrix form shown in (2.3.2p) for the K * 76(jω) model, x r (t) will take the same form as (2.3.2q) but with the new characteristic equations. Linear algebra can be used to yield a now reduced order cross-point dynamic stiffness model as so using the simplified equations of motion above from (2.3.3c-e): (2.3.3f) K * 66(jω) is the reduced-order cross-point dynamic stiffness of order 6 th /6 th of a hydraulic engine mount with decoupler #d and inertia tracks #i,1 and #i,2. This expression is still sufficiently complicated, as its dynamic behavior is influenced by a number of parameters and terms which cannot be rearranged or simplified in a compact or standard form. More specifically, via the low frequency mount resonance is still implicit to the dynamic stiffness formulation Equivalent Inertia Track Cross-Point Dynamic Stiffness Model Without an explicit representation of the low frequency resonance of the mount appearing in the K * 66(jω) model, it can still be difficult to effectively tune the mount 30

47 using such an analysis tool during design. Namely, will vary with several mount parameters, however it is unknown as to exactly how this will happen. Of course, qualitative comparisons can still be made, but a closed-form solution is much more useful. For this reason, the aforementioned assumptions used to obtain K * 66(jω) will not be sufficient alone for tuning purposes of the multiple inertia track mount shown Figure 1.1b. The low frequency assumption can be used once again, this time to take advantage of the physics of the hydraulic elements on the midplate. In specific, the control volumes represented by #d, #i,1, and #i,2 are all subjected to the same absolute pressure differential p 1 (t) p 2 (t). From the mechanical analog of Figure 2.3b, it can be seen that this is equivalent to lumped masses m d, m i,1, and m i,2 all being subjected to the same net force due to springs k 1 and k 2. In other words, #d, #i,1, and #i,2 can be interpreted as having an effective lumped mass m eq and effective lumped damping coefficient b eq that can be represented by the parallel combination of the individual masses and damping coefficients m d, m i,1, m i,2, and b d, b i,1, b i,2, respectively. However, since the decoupler and inertia track are very physically dissimilar in nature, it will be assumed that only the inertia tracks can be combined in parallel at this point. Accordingly: (2.3.4a) (2.3.4b) R eq, I eq are the equivalent resistance and inertance of the n-inertia tracks of the engine mount, where ι is the product or summation index., are the 31

48 binomial coefficients which represent the set of (n - 1)-sized combinations of the set of linear inertia track resistances and inertances, R i,ι and I i,ι, respectively. Note that here n = 2 represents the number of lumped inertia tracks in this particular mount, and that R eq, I eq are comprised of linear lumped parameter R ι and I ι terms without consideration to physical resistance or inertance models at this point. However, it will be seen later that linear capillary tube and the linearized orifice model (discussed in Section 2.2) will be appropriate choices for R ι and I ι. Also note that the mechanical analogs are simply related to the fluid elements via A 2 r, as formulated previously. In performing this parallel combination, the two inertia tracks are essentially lumped into an equivalent single mass m eq of motion x eq (t) = x i,1 (t) + x i,2 (t), effectively reducing a DOF of the mount. This is analogous to q eq (t) = q i,1 + q i,1 in the fluid sense by considering that for 1-D flow in the mechanical system, A eq being the equivalent crosssectional area of tracks #i,1 and #i,2. In this manner, the equations of motion of the hydraulic engine mount have been reduced to the following: (2.3.4c) (2.3.4d) (2.3.4e) The motion of the equivalent inertia track is purely dynamic here. This is because it is not necessary to define a x t eq (t), since the static portions of the individual tracks were removed in the original driving-point formulation of K * 86(jω) and will not be needed here. Since the DOF s of the system have been reduced from 4 to 3, the linear algebra used to express the new dynamic stiffness model will be substantially simplified. Using the matrix form of the equations of motion: 32

49 (2.3.4f) τ(jω) = m eq (jω) 2 + b eq (jω) + k 1 is the new characteristic equation representing the equivalent inertia track, while r T (jω) and d(jω) remain unchanged from the previous K * 66(jω) model. While this is not the same matrix form as that which appeared in all of the previous dynamic stiffness models, we can solve for any of the displacements x r (jω), x d (jω) and x eq (jω) using Kramer s method in a much simpler manner. Solving for x r (jω): (2.3.4g) It can be seen that the determinant calculations for the 3x3 matrices above in (2.3.4g) will be substantially easier than those performed for the 4x4 matrices contained within the K * 86(jω), K * 76(jω), and K * 66(jω) models. Simplifying the determinants yields the following dynamic stiffness expression: (2.3.4h) K * 44(jω) is the new cross-point dynamic stiffness of the hydraulic engine mount with equivalent inertia track #τ and decoupler #d. The order of both the numerator and denominator is seen to be reduced by 2 when using the absolute pressure differential assumption of above, which is much more substantial than any of the previous assumptions made, thus significantly simplifying the current K * 44(jω) model when comparing it to its predecessors. Physically speaking, the low frequency modes characteristic of each of the inertia tracks have been condensed into a single equivalent 33

50 mode which is representative of of the mount. The eigenvalue associated with this equivalent mode will be larger than that of just one of the inertia tracks #i,1 or #i,2 alone. This is due to the fact that the equivalent inertance I eq of the two track mount after the parallel combination operation will be smaller that the individual track inertances I i,1 or I i,2. Note that Δp(t) is constant across each of the hydraulic elements only when two important conditions are met for the hydraulic mount: (i) Flow can be classified as incompressible (ρ of fluid in #1 and #2 does not vary with time or space as it enters/leaves control volumes #d, #i,1, #i,2) and (ii) the mount is excited only within the low frequency range of f = 1 to 50 Hz (as f gets too large, one dimensional flow theory assumptions will be violated) Equivalent Hydraulic Element Cross-Point Dynamic Stiffness Model Although the decoupler #d and inertia tracks #i,1 and #i,2 are very different in physical nature and behavior, their absolute pressure drops will still be similar. In the same manner as the previous section, the principle of parallel combination and its assumptions can be used to take advantage of this behavior to further simplify K * 44(jω). Namely, it is reasonable to assume that #d, #i,1, and #i,2 can all be combined in parallel to form an equivalent lumped fluid mass for all of the hydraulic elements of the midplate, having a resulting equivalent dynamic motion x eq (t) = x d + x i,1 + x i,2. The equivalent hydraulic flow q eq (t) is analogous to this expression via A eq as done in K * 44(jω), this time including #d as well. The new expressions for the equivalent resistance R eq and inertance I eq and their mechanical analogs are as so using parallel combination: 34

51 (2.3.5a) (2.3.5b) This formulation is composed of generic resistance terms R ι to define the equivalent parameter R eq in such a manner to include the additional term due to #d. As a result, the index ι now goes to n + 1 and the binomial coefficient has changed according to the new set size of resistances. R eq, I eq once again consist of linear lumped parameter R ι and I ι terms without consideration to actual physical resistance or inertance models at this point. Also note that the equivalent inertance I eq remains the parallel combination of only I i,1 and I i,2. Accordingly, this assumes that I d I i,1, I i,2. The fluid mass of #d at any given time is negligible when compared to that of #i,1 or #i,2. Physically speaking, the eigenvalue associated with #d is typically represented only in high frequency response content due to the small I d (f 250 Hz) [1]. While it is not correct to say that #d will not affect the mount at low frequency excitations, it is correct to say that the dynamics due to its fluid mass I d will be negligible in this regime (as has been done with previous assumptions). However, the decoupler can still switch between the open and closed state based on the magnitude of X r. Thus its contribution to mount dynamics at low frequencies will be due to R d for either (i) the free state (finite R d ) or (ii) the fixed state (q d 0, R d ). The DOF of the system is now further reduced, and the equations of motion are now even more manageable due to elimination of the decoupler equation from the K * 44(jω) model: (2.3.5c) (2.3.5d) 35

52 As seen above, the system is now 2-DOF in x r (t) and x eq (t). In matrix form: (2.3.5e) r T (jω) will remain unchanged from the previous K * 44(jω) model, and α(jω) = m eq (jω) 2 + b eq (jω) + k 1 is the characteristic equation of all hydraulic elements of the midplate #d, #i,1, and #i,2 collectively, as defined by the new mount parameter b eq. Solving for x r (jω) to obtain K * (jω): (2.3.5f) Algebraic manipulation yields the following cross-point dynamic stiffness representation of the mount: (2.3.5g) K * 22(jω) is the new cross-point dynamic stiffness of the hydraulic engine mount with equivalent hydraulic element #α comprised of #d, #i,1, and #i,2. It can be seen that this model is 2 nd order in ω for both numerator and denominator. The expression is relatively simple, and can be rearranged to be written in standard form in terms of second-order system parameters as so: (2.3.5h) The analytical natural frequency and damping ratio of the numerator and denominator of K * 22(jω) are summarized below in Table 2.1. Utilizing all previous assumptions, K * 22(jω) can physically be interpreted as a viscously damped lumped mechanical oscillator in the numerator and a viscously damped lumped Helmholtz resonator in the denominator, as illustrated in Figure 2.5 below. 36

53 Table 2.1. System parameters of K * 22(jω) model, free decoupler Parameter Expression ω n,num ω n,den ζ num ζ den (a) (b) Figure 2.5. Physical interpretation of K * 22(jω): (a) Mechanical oscillator, numerator of K * 22(jω); (b) Helmholtz resonator, denominator of K * 22(jω) From observation of Figures 2.5a and 2.5b, it is worth noting a couple inherent assumptions to these physical interpretations. The numerator of K * 22(jω) can be viewed as a mechanical oscillator via Newton s 2 nd law of lumped mass m eq subjected to some 37

54 arbitrary displacement input x(t). x(t) implies some arbitrary load F(t) carried by k r and k 1. This is normalized by the stiffness term (k r + k 1 ) to get the following: (2.3.5i) Normalization of F(t) by stiffness (k r + k 1 ) is done in order to keep the left hand side of (2.3.5i) physically consistent with the units on the right hand side (the exact expression appearing in the numerator of K * 22(jω)). In this manner, (k r + k 1 ) can be canceled altogether, and the equation is representative of a conventional 1-DOF mechanical oscillator. With respect to the mount, arbitrary input force F(t) is then equivalent to that introduced at the engine side, F m (t), associated with motion x r (t), as defined previously. This is because stiffness k 1 is the only parameter which couple the motions x r (t) and x eq (t), and k r can only appear due to x r (t), as shown in the equation of motion of #r in (2.3.5c). Furthermore, damping in the mount can only occur due to the fluid resistance b eq = A 2 r R eq imposed by the equivalent hydraulic element #α given the current set of assumptions, since #1 is assumed to be a pure stiffness element k 1 = A r 2 /C 1 and b r is negligible. b eq is due to losses in a frictional sense as described by boundary layer fluid mechanics, end effects of the hydraulic elements, and flow conditions described by Re, since the cross-sectional area of the individual tracks A d, A i,1, and A i,2 (and thus equivalent track area A eq ) will be constant through their respective lengths l d, l i,1, and l i,2. The Helmholtz resonator shown in Figure 2.5b representing the denominator of K * 22(jω) is then simply interpreted as a fluid mass of I eq oscillating in the neck (#α) of a volume cavity (#1) with cross-sectional area A r acting as a fluid spring due to C 1, as in the conventional sense of acoustical systems theory. 38

55 For the mount parameters shown in Table 2.2, a typical frequency response can be found using K * 22(jω). The results are shown below in Figure 2.6 for a mount with #d and identical #i,1, #i,2. Table 2.2. Mount parameters for a typical narrowband dynamic stiffness response Parameter k r A r C 1 R i I i R d Value 210,000 N/m 9.25 x 10-4 m x m 5 /N 8.81 x 10 9 (Pa-s)/m x 10 8 kg/m x (Pa-s)/m 3 Figure 2.6. Typical narrowband frequency response of a hydraulic engine mount using K * 22(jω) model, free decoupler:, K * 22(jω) prediction;, ω n,num ; ---, ω n,den 39

56 The above dynamic stiffness response is considered typical for a production hydraulic engine mount, since is usually in the range of 8 to 10 Hz with φ K*,max = 50 to 65. A K * (jω) /k r and φ K* (jω) distribution with the above shape is often classified as a narrowband or narrow-tuned mount response, since dynamic stiffness magnitude and loss angle content are particularly large (peaks) within a relatively small bandwidth of f. Upon comparison of the shape of K * 22(jω) /k r and φ K* (jω) to k eq and b eq of Figure 2.1, respectively, it is observed that the stiffness effects of the mount will dominate K * 22(jω) /k r, while damping effects will dominate φ K* (jω). This will be true for both narrow-tuned and broad-tuned devices alike. is seen to lie somewhere between ω n,num /2π and ω n,den /2π for such a device. For this reason, ω n,num /2π and ω n,den /2π can serve as approximate bounds of. Note that the damped natural frequencies will be less than the values of ω n and provide a more accurate approximation of using a ω damped,num -ω damped,den type bound when the system cannot be classified as lightly damped. This will not be true when the device becomes more broadband in nature, as will be seen later. At this point, however, ω damped,num /2π and ω damped,den /2π serve as reasonable bounds for of typical narrow-tuned devices, and the values of ω n,num /2π, ω n,den /2π, ζ num, and ζ den can be used to approximate these respective values. Taking the average value of ω damped,num /2π and ω damped,den /2π will yield the approximate value of. Namely, [(ω damped,num /2π + ω damped,den /2π)/2] = [( )/2] Hz = 7.89 Hz = 8.02 Hz. This example illustrates why designers try to introduce the most damping at in practice, since the low frequency resonance is expected to occur nearby. 40

57 2.3.6 Fixed Decoupler Dynamic Stiffness Models While K * 22(jω) is developed in the preceding sections assuming the free decoupler state, it is not practical to lump #d into I d and R d. Free decoupler dynamics are complicated and highly nonlinear in nature, and lumped parameter LTI models do not describe its switching action well (where this is done on an arbitrary basis rather than device specific). Furthermore, the emphasis of this research is placed on the study of mount behavior due to the influence of multiple inertia tracks. This motivates the extension of analogous fixed decoupler K * (jω) models from the previous free decoupler formulations. To assume #d to be in a fixed state, R d R i,1, R i,2, therefore q d 0 and the momentum equation across #d can be ignored for modeling purposes. Starting with the original driving-point K * (jω) model of Section 2.3.1, where F m (t) has been defined as the input and x r (t) the output: (2.3.6a) (2.3.6b) (2.3.6c) Taking the Laplace Transform of the system of equations, setting s = jω and rearranging in matrix form yields the following, (2.3.6d) where the characteristic equations r(jω) = m r (jω) 2 + b r (jω) + k r + k 1, i,1(jω) = m i,1 (jω) 2 + b i,1 (jω) + k 1 + k 2, and i,2(jω) = m i,2 (jω) 2 + b i,2 (jω) + k 1 + k 2 will remain unchanged from the driving-point K * (jω) model for the free decoupler state. Solving for x r (jω) and 41

58 simplifying yields the new driving-point K * (jω) model for a hydraulic engine mount with inertia tracks #i,1 and #i,2 and fixed decoupler: (2.3.6e) The fixed decoupler equivalent of the cross-point K * (jω) model assumes m r 0 to relate F m (t) to F T (t), making x r (t) the input and F T (t) the output. Accordingly, this only changes the characteristic equation of #r to r T (jω) = b r (jω) + k r + k 1. Upon inspection of (2.3.6e) above, it is therefore not necessary to repeat the same procedure once again. The cross-point K * (jω) model for a hydraulic engine mount with #i,1 and #i,2 will be the following, (2.3.6f) where now r T (jω) = b r (jω) + k r + k 1. Using the assumptions of the free decoupler reduced-order cross-point K * (jω) model, k 2, b r 0. This will reduce the characteristic equation of #r further to r T (jω) = k r + k 1, as well as change those for #i,1 and #i,2 to i,1(jω) = m i,1 (jω) 2 + b i,1 (jω) + k 1, and i,2(jω) = m i,2 (jω) 2 + b i,2 (jω) + k 1, respectively. Using these new characteristic equations for the mount, (2.3.6f) simplifies to the following reduced-order cross-point K * (jω) model for a mount with fixed #d and inertia tracks #i,1 and #i,2: (2.3.6g) Since #d is fixed, the fixed decoupler versions of equivalent inertia track crosspoint K * (jω) model and equivalent hydraulic element cross-point K * (jω) model will be the same. Previously I d I i,1, I i,2, and inclusion of R d is now neglected on the premise of q d 0 in the fixed state having no assumed contribution to the dynamics of the mount. Thus, x eq (t) = x i,1 (t) + x i,2 (t), and the characteristic equation of the equivalent hydraulic 42

59 element is now represented by only #i,1 and #i,2 as α(jω) = m eq (jω) 2 + b eq (jω) + k 1. R eq is simply the expression used for the equivalent inertia track cross-point model of Section 2.3.4,, with still and m eq, b eq, k 1 having the same definition as they normally do. It then follows that the cross-point K * (jω) model for fixed #d and equivalent hydraulic element #α composed of #i,1 and #i,2 will be the same K * 22(jω) expression as that in Section in equation (2.3.5g), but now with R d excluded in the R eq computation and x eq (t) = x i,1 (t) + x i,2 (t) in the equations of motion of (2.3.5c-d). It can be seen in this manner that an arbitrary number of inertia tracks can be considered using either the fixed or free K * 22(jω) models through mount parameters I eq and R eq. The effect of R d on K * 22(jω) can be seen in Table 2.3 below. Using the same nominal parameters listed in Table 2.2, it can be seen that when R d in the free decoupler case, the fixed decoupler mount parameters will be very similar. For this reason, it is reasonable to assume a fixed decoupler state for the remainder of analysis. Note that damping ratios of 0.3 ζ 0.6 are common for production narrowband devices, which agrees with the values of ζ num and ζ den listed in Table

60 Table 2.3. Comparison of mount dynamics using K 22 * (jω) prediction for a narrowband device Parameter Free Decoupler Fixed Decoupler ω n,num /2π 6.44 Hz 6.44 Hz ω n,den /2π 11.0 Hz 11.0 Hz ζ num ζ den Linear K * 22(jω) Model for Multiple (n-number) of Inertia Tracks For an assumed mount design (with known k r, C 1, A r ), it is well known that tuning the dynamic stiffness behavior of the mount can be achieved by changing the geometry (l i, d i ) of the inertia tracks. This will be true for common R and I models such as linear capillary tube or linearized orifice, as will be shown in Chapter 3. Upon observation of 2 nd order parameters for K * 22(jω) in Table 2.1, a simple means to tune the values of ω n,num and ω n,den is changing I eq (which will be some combination of linear I i s for n 2) via geometry (l i s and d i s). However, this will also have an impact on the damping ratios ζ num and ζ den. It can be seen in this simple manner that since linear R i is very sensitive to d i, it is necessary to tune both l i and d i simultaneously to obtain a desired device performance and have control over both sets of mount system parameters. Furthermore, the greater the number of inertia tracks n, the more flexible the device is with respect to tunability. The K * 22(jω) model can be used to illustrate the benefits of n 2 inertia tracks in a hydraulic engine mount. For calculation of linear I eq and R eq, the smallest I i and R i will dominate, respectively, and two limiting cases will bound these equivalent track 44

61 parameters. When I i,1, R i,1 are much smaller than I i,2, I i,3,, I i,n and R i,2, R i,3,, R i,n, respectively, I eq I i,1 and R eq R i,1. This limiting case is not particularly interesting, since the multiple-track mount will therefore effectively behave the same way as it would for n = 1. However, the second limiting case is particularly interesting. For a device with n-identical, lumped, linear inertia tracks, I i,1 = I i,2 = I i,3 = = I i,n, R i,1 = R i,2 = R i,3 = = R i,n, and it follows that I eq = I i,1 /n and R eq = R i,1 /n, respectively. This type of device will have the largest change on mount dynamics when compared to an n = 1 track case with I eq = I i,1 and R eq = R i,1. Using this fact, the dynamic stiffness for 1 n 7 is shown below in Figure 2.7 using the K * 22(jω) model and the parameters of Table 2.2. Figure 2.7. K * 22(jω) frequency response prediction with respect to variation in n-identical inertia tracks, narrowband device 45

62 In general, not only is a mount with n 2 more versatile for tuning since there are now many l i s and d i s to change, but it can also be seen in Figure 2.7 that as n, K * 22(jω), φ K* (jω) also increase, which is good for motion control of say the engine bounce mode. This is expected, as an increase in n will introduce more reciprocating fluid mass to contribute to F T (jω). When comparing the same n = 2 result shown in Figure 2.6 to the largest inertia track case of n = 7 above, it can be seen that ( K * /k r ) max increased from approximately 4.5 to 6.5. In addition, φ K*,max increased from approximately 67 to 100. As a result, the sensitivity of the mount is increased as well near ( K * /k r ) max and φ K*,max as n is increased. Furthermore, the number of tracks can be used to tune the device over a wider range of frequencies via the increasing. For the n = 2 case, = 8.02 Hz is bounded by and, whereas for the largest track case of n = 7,, bounds the of 16.0 Hz. Lastly, it can be seen that increasing n makes the device response more narrowband in nature. Expressions relating the mount parameters ω n,num, ω n,den, ζ num, and ζ den to change in n can also be developed for the n-identical track case from the expressions in Table 2.1 using the fact that I eq = I i,1 /n and R eq = R i,1 /n. The results are summarized in Figure 2.8 for 1 n 7 using the following analogs to a 1-track case: (2.4a) (2.4b) 46

63 (2.4c) (2.4d) Figure 2.8. Engine mount parameters f n,num, f n,den, ζ num, and ζ den with respect to variation in n-identical inertia tracks:, K * 22(jω) prediction;, mount parameter relations to n = 1 case, (2.4a-d) The result for f n,den is comparable to analytical approximations for presented by Shangguan and Zhang [2,3], and Lu and Ari-Gur [4]. However, f n,num, and the expressions for ζ num, and ζ den in (2.4a-d) are refinements of existing theory based on the linear K * 22(jω) model which can better approximate for a narrow-tuned device. 47

64 ζ num, and ζ den will decrease with n even though φ K* (jω) generally grows. This is because the number of parallel paths of communication between #1 and #2 are increasing, and R eq, the only modeled source of viscous damping in the linear mount model, will decrease with n so long as the mount inertia tracks configuration does not satisfy the first limiting case. Note that the predictions and theory used in Figures 2.7 and 2.8 can be applied in concept to any case of multiple inertia tracks, not just the limiting case of n-identical tracks. These cases will exhibit system parameters that will fall somewhere between the n-identical track case and the n = 1 track case. This will always be true as long as I eq and R eq can be expressed in terms of I i,1 and R i,1, respectively, of any one of the linear tracks (#i,1 is preferably the track with the smallest values of I and R). Sensitivity analysis of ω n,num, ω n,den for f = 1 to 50 Hz reveals that a good practical limit on the number of inertia tracks is around n = 3 to 4. This most importantly takes into consideration the tunability of ω n,num, ω n,den within this regime, in addition to manufacturing, size, and design constraints for a passive device such as the hydraulic engine mount under consideration. 48

65 CHAPTER 3: EXPERIMENTAL STUDIES AND DYNAMIC TUNING OF MULTIPLE INERTIA TRACK MOUNT USING PHYSICAL MODELS 3.1 Validation of Linear K * 22(jω) Model Frequency response techniques are most popular and useful when designing and tuning a hydraulic engine mount due to their ability for approximating and studying narrowband, broadband, and amplitude sensitive behaviors analytically. While a lumped LTI model such as K * 22(jω) cannot be used to study amplitude-sensitive response spectrums directly, it can still be used for narrowband and broadband predictions. However, up to this point there has been no validation method applied to evaluate the accuracy of K * 22(jω). First, the correctness of the assumptions applied during order reduction of K * 86(jω) and K * 64(jω) is considered. Namely, Figures 3.1 and 3.2 (free and fixed decoupler, respectively) show the frequency responses K * (jω) as they are reduced from the driving-point models (K * 86(jω) and K * 64(jω), respectively) to K * 22(jω) in Chapter 2. These spectrums use the same lumped parameters as those for a typical mount response listed in Table 2.2, in addition to the mount parameters of Table 3.1 required for the higher-order K * (jω) models. Tables 3.2 and 3.3 list system parameters ω n,num, ω n,den, ζ num, and ζ den of the mount for each model during this order reduction. For the higher-order K * (jω) models, values in Tables 3.2 and 3.3 are reported using numerical zero-pole analysis methods. 49

66 Table 3.1. Typical lumped mount parameters used during order reduction of K * (jω) models Parameter Value I d b r m r 100 (N-s)/m 0.1 kg C 2 150C 1 Figure 3.1. Validation of K * (jω) order reduction for a narrowband device, free decoupler:, K * 86(jω) prediction;, K * 76(jω) prediction;, K * 66(jω) prediction;, K * 44(jω) prediction;, K * 22(jω) prediction 50

67 Table 3.2. System parameters of narrowband K * (jω) models, free decoupler Model ω n,num /2π ω n,den /2π ζ num ζ den K * 86(jω) 6.53 Hz 11.1 Hz K * 76(jω) 6.53 Hz 11.1 Hz K * 66(jω) 6.47 Hz 11.1 Hz K * 44(jω) 6.47 Hz 11.1 Hz K * 22(jω) 6.44 Hz 11.0 Hz Figure 3.2. Validation of K * (jω) order reduction for a narrowband device, fixed decoupler:, K * 64(jω) prediction;, K * 54(jω) prediction;, K * 44(jω) prediction;, K * 22(jω) prediction 51

68 Table 3.3. System parameters of narrowband K * (jω) models, fixed decoupler Model ω n,num /2π ω n,den /2π ζ num ζ den K * 64(jω) 6.50 Hz 11.1 Hz K * 54(jω) 6.50 Hz 11.1 Hz K * 44(jω) 6.44 Hz 11.0 Hz K * 22(jω) 6.44 Hz 11.0 Hz It can be seen that the frequency response of each model, for both free and fixed decoupler cases, is nearly the same for the assumed set of typical mount parameters. Note the singularity at K * 66(0) /k r, and the loss angle of = 180. These discrepancies are the result of numerical transfer function calculations, as evaluating the rational expression K * (jω) where s = jω 0 in its denominator can cause the real part of complex-valued K * (jω) to approach -. These very low frequencies of f 1 Hz are out of the dynamic range of most transducers anyhow. Of particular interest are the natural frequencies and damping ratios of the models in Tables 3.2 and 3.3. It can be seen that ω n,num, ω n,den, ζ num, and ζ den of the K * 22(jω) model are very similar to the respective fullorder models. For this reason, the K * 22(jω) model is seen to represent the low frequency dynamics of the mount as well as the higher-order models when the assumptions of Chapter 2 are used. Order reduction still applies when the mount exhibits a broadband dynamic stiffness response. While there is no typical broad-tuned device response as there is in the narrow-tuned case, these devices are known to be highly damped, and increasing ζ num, ζ den compared to the narrow-tuned case will result in a broad-tuning. Shown below in 52

69 Figures 3.3 and 3.4 are the same order reductions performed for the narrowband device in Figures 3.1 and 3.2, simply by increasing the previous by a factor of 5 (half an order of magnitude). All other mount parameters remain unchanged from this previous order reduction. Table 3.4 summarizes the new damping ratios ζ num, and ζ den for these broadband responses for the K * 22(jω) model only, for the sake of avoiding redundancy. Figure 3.3. Validation of K * (jω) order reduction for a broadband device, free decoupler:, K * 86(jω) prediction;, K * 76(jω) prediction;, K * 66(jω) prediction;, K * 44(jω) prediction;, K * 22(jω) prediction 53

70 Figure 3.4. Validation of K * (jω) order reduction for a broadband device, fixed decoupler:, K * 64(jω) prediction;, K * 54(jω) prediction;, K * 44(jω) prediction;, K * 22(jω) prediction Table 3.4. Comparison of system parameters using broadband K 22 * (jω) prediction Decoupler State ζ num ζ den Free Fixed For a broad-tuned device, the dynamic stiffness spectrum also remains almost unchanged as the order of K * (jω) is reduced. As a result of increasing by half an order of magnitude, ζ num and ζ den have increased by the same factor. Some of the same issues near f = 0 Hz occur once again due to numerical calculations, as expected. One very important note to make is the inability to approximate for broadband 54

71 responses. ω n,num, ω n,den of the mount will remain the same as those values in narrowband free and fixed decoupler cases (6.44 Hz and 11.0 Hz, respectively), since R eq was the only parameter changed. However, is seen to be decrease significantly from 8.00 Hz to around 2.15 Hz, since ζ and the system becomes overdamped in broadband devices. As a result, occur somewhere to the right of 0 as ζ 1, and the ω n,num -ω n,den bound will in broadband device predictions using K 22 * (jω). This illustrates the errors in approximating for broad-tuned devices using the ω damped,num -ω damped,den bounding method, whereas this was a reasonable approach for narrow-tuned devices. Furthermore, since broad-tuned behavior is very dependent on ζ and n, broadband responses are often the result of very specific mount parameters. For this reason, simply increasing n could cause a broad-tuned device to transition to narrowband behavior, since ζ decreases with n. Therefore, although it is difficult to approximate for a broad-tuned device, it is possible to tune broadband behavior with ζ by introducing highly-damped inertia tracks to the system. This will be shown in the later sections of Chapter 3. While K * 22(jω) is seen to capture the same dynamics as the full-order models, it is important to demonstrate its accuracy as well. It can therefore be validated with previous dynamic stiffness models. K * 22(jω) was developed in the same fundamental manner as the LTI dynamic stiffness models formulated by Singh et al. [1] for n = 1. Singh et al. [1] match their comparable reduced-order form of model II K * (jω) model to experiment with good agreement. By normalizing the K * 22(jω) model here appropriately to behave like an n = 1 track mount and comparing it to the LTI model of Singh et al. [1], the accuracy of K * 22(jω) can be illustrated. This is attained by considering equations the 55

72 equations of motion of (2.3.5c-d) for K * 22(jω), where x eq (t) = x i,1 (t) + x d (t), R d = 100R i,1, R eq = for the free decoupler case, x eq (t) = x i,1 (t), R eq = R i,1 for the fixed decoupler case, I eq = I i,1, and b eq, m eq have their usual meanings. This is shown below for both the free and fixed decoupler states in Figures 3.5 and 3.6, and Tables 3.5 and 3.6, respectively, using the mount parameters of Table 2.2 (where R i,1 and I i,1 have been normalized by a factor of 0.5 to yield the same typical narrowband mount response seen previously for n = 2 tracks). Figure 3.5. Validation of K * 22(jω) accuracy for a narrowband device, free decoupler:, K * 22(jω) prediction;, Singh et al. [1] prediction 56

73 Table 3.5. System parameters of narrowband K * (jω) models, free decoupler Model ω n,num /2π ω n,den /2π ζ num ζ den K * 22(jω) 6.44 Hz 11.0 Hz Singh et al. [1] 6.47 Hz 11.1 Hz Figure 3.6. Validation of K * 22(jω) accuracy for a narrowband device, fixed decoupler:, K * 22(jω) prediction;, Singh et al. [1] prediction Table 3.6. System parameters of narrowband K * (jω) models, fixed decoupler Model ω n,num /2π ω n,den /2π ζ num ζ den K * 22(jω) 6.44 Hz 11.0 Hz Singh et al. [1] 6.44 Hz 11.0 Hz

74 The K * (jω) spectrums are nearly identical when comparing K * 22(jω) to the model of Singh et al. [1] in both free and fixed decoupler states. System parameters ω n,num, ω n,den, ζ num, and ζ den match exactly for a fixed decoupler when comparing the two models, with some negligible differences in ζ num, ζ den when the decoupler is free. K * 22(jω) is therefore not only seen to be accurate, but also be much easier to use, apply, and interpret than its higher-order predecessors here. For these reasons, the fixed decoupler version of K * 22(jω) is used for all subsequent analyses and predictions. Note that since all published results focus on narrow-tuned devices, it is difficult to validate the accuracy of broad-tuned K * 22(jω) response spectrums with previous models. However, it will be seen later in Chapter 3 that K * 22(jω) does reasonably well at approximating such behavior from experimental results. 3.2 Design of Multiple Inertia Track Mount Experiment Subsequent response predictions using the K * 22(jω) model have been made under the assumption of lumped parameter LTI analysis for the fluid elements of the mount. Accordingly, existing physical models will be applied to dynamic stiffness spectra in order to consider the effects of geometry and fluid properties on tuning the device. Viscous damping models commonly used to describe R i include the capillary tube and orifice models summarized in Section 2.2, where the linearized orifice resistance about an operating point flow q o is used for all K * 22(jω) predictions utilizing orifice-type tracks. This is found by rearranging the square root relation of Section 2.2 and taking the derivative with respect to q o : (3.2a) 58

75 3.2.1 Linear K * 22(jω) Predictions Using Physical Damping Models It is well known that a single capillary tube-type inertia track mount will allow for a narrow-tuned device. For instance, capillary-type inertia track geometries of d i = 4 to 6 mm and l i = 10 to 25 cm are typical for narrow-tuned devices [1-13]. For this reason, it is not very difficult to design a narrow-tuned device given the ability to change d i and/or l i of a single capillary tube-type track. However, broad-tuned designs are not very well understood, so design predictions will focus on these types of devices here. Some production mounts with n = 1 capillary tube-type track exhibit responses that are broadband in nature, evidencing some high resistance leakage path described by a constant or load-dependent fluid flow area A between #1 and #2. Orifice-type flow paths between #1 and #2 are speculated as possible explanations of these broadband mount dynamics in a physical sense here. For this reason, consider the linearized orifice model of (3.2a) as a controlled type of inertia track geometry of A o and l o capable of influencing the force transmission spectrum as well. Using the K * 22(jω) model, broadband dynamic stiffness predictions can be made via specification of the mount stiffnesses k r and k 1, properties of the fluid in #1 and #2, and the lengths and diameters of the inertia tracks l i and d i. This is done by best-fitting to a set of measured broadband dynamic stiffness data from a sample production mount. Namely, linear simulations using the K * 22(jω) model utilizing different combinations of linear capillary tube and orifice-type tracks in n = 2, 3, and 4 track configurations have been completed in order to match the dynamic stiffness of a sample broad-tuned device of unknown track configuration at both X r = 1.00 mm and X r = 0.25 mm excitations. It has been found that designs utilizing one capillary tube-type track and the rest orifice- 59

76 type tracks can describe a broad-tuned device particularly well. Assuming ρ = 1059 kg/m 3, μ = x 10-3 Pa-s for a 50%-50% ethylene glycol-water antifreeze solution by volume [9], steady volumetric flow q eq = q i,1 + q i,2 + + q i,n = 2.25 x 10-4 m 3 /s to ensure that adequate φ K*,max can be represented near low frequency resonance [1], and c o = 0.61 for a sharp-edged orifice, Figure 3.7 summarizes the results of broadband response approximations using K * 22(jω) to match the sample broadband production mount data. The total volumetric flow within a given orifice-type track is approximated using its area percentage, is equal, i.e.. This assumes the flow velocity in each of the tracks. Table 3.7 lists the mount parameters and track geometries used for these dynamic stiffness approximations, with the accompanying system parameters shown in Table 3.8. (a) (b) Figure 3.7. Dynamic stiffness comparison to a sample broadband mount response: (a) X r = 1.00 mm; (b) X r = 0.30 mm:, measured production mount;, K * 22(jω) prediction, n = 2 tracks (1 capillary tube, 1 sharp-edged orifice);, K * 22(jω) prediction, n = 3 tracks (1 capillary tube, 2 identical sharp-edged orifices);, K * 22(jω) prediction, n = 4 tracks (1 capillary tube, 3 identical sharp-edged orifices) 60

77 Table 3.7. Inertia track geometries predicted by K * 22(jω) model for a sample broadband device, k r 2.27 x 10 5 N/m X r = 1.00 mm, k x 10 5 N/m X r = 0.25 mm, k x 10 4 N/m Parameter n = 2 n = 3 n = 4 n = 2 n = 3 n = 4 d t 3.00 mm 4.15 mm 4.60 mm 2.70 mm 3.20 mm 3.70 mm l t 11.5 cm 30.0 cm 40.0 cm 14.0 cm 25.0 cm 41.0 cm d o 3.00 mm 2.50 mm 1.95 mm 2.40 mm 1.70 mm 1.45 mm l o 1.75 mm 1.60 mm 1.40 mm 1.25 mm 1.00 mm 1.00 mm Table 3.8. Linear system parameters predicted by K * 22(jω) model for a sample broadband device, k r 2.27 x 10 5 N/m X r = 1.00 mm, k x 10 5 N/m X r = 0.25 mm, k x 10 4 N/m Parameter n = 2 n = 3 n = 4 n = 2 n = 3 n = 4 ω n,num /2π 13.9 Hz 14.0 Hz 13.6 Hz 19.2 Hz 19.6 Hz 19.8 Hz ω n,den /2π 25.6 Hz 25.6 Hz 24.8 Hz 20.9 Hz 21.3 Hz 21.6 Hz ζ num ζ den The designs of Table 3.7 match the broadband response of the sample production mount of unknown track configuration well, especially for the shapes and absolute values of K * (jω) /k r. However, LTI theory is limited in predicting φ K* (jω), especially at low excitation amplitude. When X r = 1.00 mm, K * 22(jω) does a reasonable job at matching and φ K*,max ; however, ω damped,num, ω damped,den do a poor job of bounding. This approximation error is the result of the large amounts of damping from ζ num and ζ den 61

78 in broad-tuned devices as asserted earlier in this chapter. This will be supported by experiment and physics later. Furthermore for X r = 0.25 mm, the shape of the φ K* (jω) could not be matched well at all within the f = 1 to 50 Hz tuning range. Note there is no distinct present for this particular device. In specific, the sample production mount resembles the response yielded by #r (with fluid drained) using a Voigt-type model [1], which evidences that the hydraulic path of force transmission is nearly negligible at this excitation amplitude X r. Physically speaking, this suggests that the inertia tracks have a higher equivalent resistance R eq at this smaller excitation amplitude, as evidenced by the large damping ratios ζ num and ζ den needed to reproduce K * /k r. Furthermore, k 1 is also dependent on X r in these designs. This suggests that the upper chamber compliance C 1 is nonlinear with respect to X r as well. As a result, it is clear that the production broadband device exhibits significant X r -dependent nonlinear dynamics which must be considered for more accurate analysis. A relatively comprehensive set of practical geometries can be considered using these n-track designs. Generally speaking, increasing n will increase l t and d t of the capillary-type track, while decreasing d o and increasing l o for the orifice-type tracks in the designs. The geometries of all track designs reflect inertia track dimensions which could be applied in practice. However, the n = 2 designs would be easiest to implement in a controlled manner via prototype since only 2 parallel paths would exist here. For this reason, the n = 2 track designs of one capillary tube-type and one orifice-type track in tandem are particularly appealing. Note how the aspect ratio of d o /l o 1 as n increases and more orifice-type inertia tracks are introduced to the mount. When this condition is approximately satisfied, the 62

79 orifice can be modeled using the short-tube orifice-type track model [29], which tends to have a larger c o than the sharp-edged orifice value of c o = For this reason, it is appropriate to consider either orifice model for modeling R o based on the predicted geometries of Table Physical Description of Prototype Hydraulic Engine Mount Having predicted that broadband response behavior is feasible using K * 22(jω) and a capillary tube-type track in tandem to an orifice-type track, it is desirable to prove this experimentally. Furthermore, validation of K * 22(jω) is desired to study the effects of n > 2 with respect to improving tuning capabilities in narrowband devices given the analytical framework provided for lumping an arbitrary number of tracks #τ into an equivalent fluid element. Given this, a new prototype hydraulic engine mount with a possible 3 external capillary tube-type tracks and 3 internal orifice-type tracks has been built. The mount has been created in such a manner that the tester has the flexibility of changing not only the total number of n-possible inertia tracks from the 6 possible, but also the l i and d i of each as well. In this manner, 15 distinct track configurations and countless geometries can be tested to study mount responses with the physics of these tracks. A custom midplate was designed and built to accommodate multiple inertia track configurations. Shown in Figure 3.8 below, the solid geometry of the midplate is a onepiece design that incorporates 3 internal openings for orifice-type tracks and 3 external openings for capillary tube-type tracks for #1 and #2 to communicate hydraulically with one another. Physical representations illustrating an example of each type in the final 63

80 device are highlighted in Figure 3.9. The inlets and outlets of these tracks are located about its outer circumference using the radial symmetry of the midplate. Figure 3.8a illustrates the 3 openings used to accommodate orifice-type tracks in device designs. By fastening a thin disk to the 3 ribs in this internal basket-like area of the midplate, an orifice-type track can be introduced by drilling holes through the disc at any of the 3 opening locations. Fluid leakage from #1 to #2 around the outer edges of this orifice disc can be prevented via the use of gasket material between the disc and the midplate ribs, in addition to a silicone caulk applied about the circumference of the orifice disc. The disc can be interchanged to modify the number or geometry of orifices in the device. Figures 3.8b and 3.8c illustrate the locations for the 3 possible capillary-type tracks. From Figure 3.8b, the 3 milled-out recessions along the inner diameter of the midplate are the locations of cross-drilled holes leading to external 1/8 NPT pipe fittings. These pipe fittings are then fitted with a given length of 1/4 steel brake line using compression fittings and then mated with a similar pipe fitting leading to #2 at the other end. The locations of these entrances to #2 are shown in Figure 3.8c, where similar holes have been cross-drilled on the actual device to accommodate the pipe fittings. Depending on the design, these external flow paths can be blocked using dowel pins with the same compression-pipe fitting assembly. A standing wave of λ = 1/4 wavelengthtype is not expected to occur, as the fluid entrained in the 0.02 m pipe fitting sidebranch will have a resonant frequency of f = 1500/(0.08) 20 khz. For this reason, the effects of a sidebranch resonator on mount dynamics due to a blocked external track are not expected in f = 1 to 50 Hz range of interest. In fact, the sidebranch resonator would need to have a length of about 7.5 m for any nominal 1/4-wavelength resonator effects to be 64

81 expected with this design consideration. d i of the external tracks can be modified by using different thicknesses of 1/4 break line, and l i by cutting the desired length and fastening it appropriately between pipe fittings which couple #1 to #2. Longer external track geometries can also be accommodated by using a #1 to #2 pipe fitting combination that is non-adjacent. (a) (b) (b) Figure 3.8. Views of solid midplate design geometry: (a) Top view showing location for surface for orifice-type tracks; (b) Isometric view illustrating the locations of external capillary-type track inlets from #1; (c) Flipped isometric view illustrating the locations of external capillary-type track outlets to #2. 65

82 (a) (b) (c) Figure 3.9. Examples of inertia tracks on final prototype mount device: (a) Internal orifice-type track; (b) External capillary tube-type track with plugs; (c) External capillary tube-type track with steel brake line This custom midplate was used with a modified take-apart Delphi mount. The overall assembly of the prototype mount includes a modified shell to hold together the upper rubber with #1, the midplate, and the lower chamber with #2 as shown in Figure Pre-existing rubber seals are located on the mating surfaces of the upper and lower chambers to ensure a good fluid seal with the midplate part when clamped together. Fixturing devices have been included to couple the base of the shell and the upper rubber to an elastomer test machine. These members are lightweight (Al-6061) to avoid introducing any extraneous dynamic mass, in addition to being much stiffer and of negligible material damping when compared to the mounting system or test machine. 66

83 The stiffnesses of the test machine and these fixturing devices have been calibrated in order to adjust the absolute values of mount dynamic stiffness appropriately. In order to fill the device with antifreeze solution and remove entrained air, a brake bleed system is incorporated into the prototype. 1/8 NPT bleed valves are adapted to fit to #1 and #2. By connecting a reservoir of fluid to the bleed orifice of #2 and pulling it through the device using vacuum pressure via a bleed pump through #1 and the upper bleed valve, it is possible to fill the device and remove most of the entrained air within the device. Entrained air will create a nonlinear C 1 due to vacuum effects of p 1 (t), so this is avoided as best possible [7-9]. The box-type fixturing at the top of #1 is designed to accommodate the upper bleed valve while having the functionality of coupling to the crosshead of the test machine. 67

84 (a) (b) (c) (d) Figure Sample assembly of take apart prototype hydraulic mount consisting of 1 external capillary-type track and 1 internal orifice-type track: (a) View of midplate mated with #2 consisting of external capillary tube-type pipe fittings and orifice disc; (b) Midplate and #2 assembly (foreground) with #r and #1 of mount (background); (c) #2, midplate, #1 assembly within modified shell; (d) Profile view of assembled prototype including upper and lower fixturing devices. 68

85 3.3 Experimental Tuning Using Multiple Inertia Track Prototype Mount The switchable configuration prototype hydraulic mount can be used to study many different dynamic behaviors based on the types, configurations, and geometry of inertia tracks present. However, the experimental scope will be narrowed to emphasize the effect of n 2 and orifice-type tracks on tuning the mount. Once the mount is tested dynamically for a given inertia track design configuration, the K * 22(jω) model will be used to characterize it in the f-domain. Each configuration was tested sinusoidally using an MTS Landmark elastomer test machine at a preload of = 1200 N (typical preload due to 4 cylinder engine) given inputs of X r = 0.3 mm and X r = 1.0 mm excitation amplitude. Typically, X r will nominally be around 0.1 mm in amplitude, but these larger input magnitudes were chosen with the hopes of exciting some nonlinearities due to the dynamics of the tracks #τ [7-9]. Table 3.9 below summarizes each of the chosen mount design configurations, with Figure 3.11 displaying some pictures which display the mount physical track arrangements and geometries used during testing. Note that the only difference between designs 1 and 6 is the absence of gasket and silicone caulk on the orifice disc in design 6 to allow for a controlled leakage path of fluid flow between #1 and #2. 69

86 Table 3.9. Multiple inertia track designs tested using prototype hydraulic engine mount Design Description 1 Baseline - 1 Track (1 External Capillary-Type) 2 3 Tracks (3 Identical External Capillary-Type) 3 3 Tracks (2 Identical External Capillary-Type, 1 Long External Capillary-Type) 4 Quasi-Broadband 2 Tracks (1 External Capillary-Type, 1 Small Internal Orifice-Type) 5 Broadband - 2 Tracks (1 External Capillary-Type, 1 Large Internal Orifice-Type) 6 Controlled Leakage Path - 2 Tracks (1 External Capillary-Type, 1 Internal leakage path) (a) (b) (c) (d) Figure Views of tested prototype mount designs: (a) Designs 1 and 6, l t = 21.4 cm, d t = 4.93 mm ; (b) Design 2, l t = 21.4 cm, d t = 4.93 mm; (c) Design 3, l t,1 = l t,2 = 21.4 cm, l t,3 = 4l t,1 = 85.6 cm, d t,1 = d t,2 = d t,3 = 4.93 mm; (d) Designs 4 (l o = 2.00 mm, d o = 1.00 mm) and 5 (l o = 2.00 mm, d o = 2.38 mm), both with l t = 85.6 cm, d t = 4.93 mm. 70

87 3.3.1 Narrow-Tuned Designs Device designs utilizing only capillary tube-type orifice tracks can be categorized as narrow-tuned devices. This corresponds to designs 1 through 3 listed in Table 3.9. Geometries for l t, d t of designs 2 and 3 were chosen to study the effect of tuning a device using n = 3 tracks when compared to a baseline mount configuration such as design 1. Namely, introducing more tracks will change the system parameters ω n,num, ω n,den, ζ num, and ζ den as discussed earlier. Design 2 is representative of the limiting case of n-identical tracks, and is expected to display the biggest change in system parameters compared to design 1. Furthermore, the system parameters of design 3 are expected to lie somewhere between those for designs 1 and 2. The results of mount characterization using K * 22(jω) predictions to match measured dynamic stiffness spectra are summarized in Figures for both the X r = 0.3 mm and X r = 1.0 mm test conditions, along with the corresponding k 1 values used for these predictions in Tables 3.10 and (a) (b) Figure Bounded dynamic stiffness comparisons for design 1: (a) without empirical coefficients; (b) with empirical coefficients:, measured prototype mount, X r = 1.0 mm;, K * 22(jω) prediction, X r = 1.0 mm;, measured hydraulic mount, X r = 0.3 mm;, K * 22(jω) prediction, X r = 0.3 mm 71

88 (a) (b) Figure Bounded dynamic stiffness comparisons for design 2: (a) without empirical coefficients; (b) with empirical coefficients:, measured prototype mount, X r = 1.0 mm;, K * 22(jω) prediction, X r = 1.0 mm;, measured hydraulic mount, X r = 0.3 mm;, K * 22(jω) prediction, X r = 0.3 mm (a) (b) Figure Bounded dynamic stiffness comparisons for design 3: (a) without empirical coefficients; (b) with empirical coefficients:, measured prototype mount, X r = 1.0 mm;, K * 22(jω) prediction, X r = 1.0 mm;, measured hydraulic mount, X r = 0.3 mm;, K * 22(jω) prediction, X r = 0.3 mm 72

89 Table Mount upper chamber stiffness predicted by K * 22(jω) model for narrowband designs (without empirical coefficients) X r Design k mm x 10 5 N/m 2.75 x 10 5 N/m 3.39 x 10 5 N/m x 10 5 N/m 1.0 mm x 10 5 N/m x 10 5 N/m Table Mount upper chamber stiffness predicted by K * 22(jω) model for narrowband designs (with empirical coefficients) X r Design k mm x 10 5 N/m 2.85 x 10 5 N/m 3.20 x 10 5 N/m x 10 5 N/m 1.0 mm x 10 5 N/m x 10 5 N/m K * 22(jω) is seen to show good agreement to dynamic stiffness response at both X r = 0.3 mm and X r = 1.0 mm. In specific, K * 22(jω) is seen to match the shape of K * (jω) /k r and φ K* (jω) over the frequency range of interest, especially near. Note that K * 22(jω) is seen to have some trouble in predicting the magnitude of damping present in the system near the resonant and anti-resonant peaks using the capillary tube model for 73

90 R i. This is attributed to the simplifications needed for this fluid resistance model. The capillary tube model assumes steady flow for a straight tube of l t without end effects. This is a gross approximation for this mount, as flow here is assumed to be nonsteady (oscillatory), l t is finite and will have some end effects due to area expansion at #1 and #2, and the tubes themselves are curvilinear instead of straight. These additional effects are expected to add viscous damping to the mount. For this reason, each K * 22(jω) prediction was also made using empirical coefficients to better approximate the actual system in Figures 3.12b, 3.13b, and 3.14b. These empirical coefficients are thus a means to adjust mount damping for more non-ideal cases. Namely, the damping of K * 22(jω) predictions can be better matched to measurements by introducing empirical coefficient γ to adjust capillary tube model R t as follows: (3.3.1a) Here, γ is chosen to be to reproduce ζ which reflects values for a typical narrow-tuned device design. This adjustment to R t is seen to improve the damping via improvements in ( K * /k r ) max, in addition to φ K*,max near. To account for end effects, l eq = (4/3)l t can be substituted for the geometric length in I t to obtain finer tuning of [29]. In general, the mount is seen to display amplitude dependent response behavior based on X r. Upon inspection of k 1 values in Tables 3.10 and 3.11, k 1 is generally larger for the high amplitude excitation cases when compared to X r = 0.3 mm. This evidences the dominance of the hydraulic path of force transmission over the parameters of #r when X r is large. k 1 values from the K * 22(jω) are seen to vary based on design and excitation amplitude, showing a nonlinearity in k 1 which is most likely due to C 1. This is expected, as C 1 is a very device-specific parameter which will typically be 74

91 nonlinear due to X r and f, as well as from preload [1-9]. Furthermore, resonant and anti-resonant peak magnitudes for K * (jω) /k r are seen to be larger for the low amplitude excitation condition. Upon further inspection, K * (jω) is generally larger for X r = 1.0 mm case on an absolute basis. It is the normalization by k r that makes this phenomenon unclear; however, a closer look at the static stiffness of #r reveals that k r is actually larger for X r = 1.0 mm when compared to X r = 0.3 mm. The load-deflection curve of the mount with fluid drained can be used as an explanation for this k r behavior. This is shown below in Figure Figure Static stiffness behavior, k r, of #r for prototype hydraulic engine mount The stiffness behavior of #r is nonlinear due to a discontinuous nonlinearity taking place around = 5.3 to 5.5 mm. Hysteresis effects to due unloading #r will cause the static displacement corresponding to this discontinuous nonlinearity to move 75

92 somewhat. Subjecting the mount to the = 1200 N preload typically corresponded to of 4.8 to 4.9 mm during testing. For this reason, perturbing the mount about this operating point will cause the system to approach this discontinuous preload nonlinearity, especially when X r = 1.0 mm. Prior to the discontinuity, k r 250 N/mm, but beyond this it increases by almost a factor of 8 to k r 2000 N/mm. This behavior is characteristic of an elastomer transitioning from its shear mode of deformation into its compressive mode [31]. This will substantially increase the overall dynamic stiffness of the mount, especially when X r = 1.0 mm and x r (t) transitions this discontinuity substantially. For the sake of lumped parameter analysis, k r can be approximated by linearizing about the operating point deflection for each X r case. Accordingly, this will correspond to k r 265 N/mm for X r = 0.3 mm, and k r 335 N/mm for X r = 1.0 mm. These are the representative k r values used for all subsequent K * 22(jω) predictions. K * (jω) is expected to be preload dependent as well, and it is be difficult to control tolerances on measurements beyond = 1200 N and X r = 1.0 mm, especially at the higher frequencies of f = 1 to 50 Hz, as will be seen later in Section To study the effect of dynamic tuning of narrowband designs using the n-track concept, consider the system parameters of the mount ω n,num, ω n,den, ζ num, and ζ den. Variations in ω n,num, ω n,den, ζ num, and ζ den as designs two and three of the prototype mount are compared to the first design are appropriate means of evaluating and predicting the tunability of the mount due to the n-track concept. Namely, approximations for can be made for these (and similar narrow-tuned) device designs, since ω n,num, ω n,den, ζ num, and ζ den are expected to serve as good bounds through ω damped,num, ω damped,den. The same K * 22(jω) predictions used to fit the measured results of the prototype for designs 1 76

93 through 3 have now been plotted together at each X r for the sake of tuning comparison purposes in Figures 3.16 and 3.17 below. (a) (b) Figure Dynamic stiffness comparison of narrowband designs, X r = 1.0 mm: (a) without empirical coefficients; (b) with empirical coefficients:, design 1; ---, design 2; - -, design 3 (a) (b) Figure Dynamic stiffness comparison of narrowband designs, X r = 0.3 mm: (a) without empirical coefficients; (b) with empirical coefficients:, design 1; ---, design 2; - -, design 3 77

94 Qualitatively speaking, the dynamic stiffnesses of designs 1 through 3 follow the expected trends identified in Section 2.4 for n-track mounts, as seen in Figures 3.16 and When comparing design 1 to design 2, R eq and I eq are reduced by a factor of 1/3 due to geometric track relationships. Similarly, when comparing design 1 to design 3, R eq and I eq are reduced by a factor of 4/9. The system parameters of the mount are therefore expected to change accordingly. Equation (3.3.1b) illustrates the ability to tune the natural frequencies and damping ratios of the mount predicted by K * 22(jω) when design 1 is compared to design 2, whereas (3.3.1c) reflects these changes when comparing design 1 to 3. (3.3.1b) (3.3.1c) It can be seen that a bigger change in f n and ζ is expected for the limiting case of n-identical tracks. A pattern emerges here when considering these specific designs which was not obvious in the arbitrary limiting case presented in equations (2.4a-d). Namely, when comparing the linear system mount parameters of the n = 1 track mount case to some n > 1 case, f n increases by some direct proportionality constant, while ζ decreases by the reciprocal of that same constant. This will be true for device designs utilizing capillary tube-type tracks and narrow tuning. The above expressions will not change when the empirical coefficient γ is introduced to the R t model, since this adjustment is applied to each track equally. The values of system parameters for designs 1 through 3 at each X r are listed in Tables 3.12 and 3.13, without and with empirical adjustments, respectively. 78

95 Table Linear system parameters predicted by K * 22(jω) model for narrowband designs (without empirical coefficients) Mount Design, X r = 0.3 mm Mount Configuration, X r = 1.0 mm System Parameter ω n,num /2π 3.12 Hz 5.34 Hz 4.96 Hz 2.64 Hz 5.45 Hz 4.98 Hz ω n,den /2π 4.51 Hz 7.63 Hz 7.41 Hz 3.44 Hz 7.46 Hz 7.17 Hz ζ num ζ den Table Linear system parameters predicted by K * 22(jω) model for narrowband designs (with empirical coefficients) Mount Design, X r = 0.3 mm Mount Configuration, X r = 1.0 mm System Parameter ω n,num /2π 3.37 Hz 5.46 Hz 4.91 Hz 2.73 Hz 5.60 Hz 5.14 Hz ω n,den /2π 4.85 Hz 7.87 Hz 7.30 Hz 3.58 Hz 7.67 Hz 7.22 Hz ζ num ζ den The system parameters behave qualitatively as expected, with the parameters of design 3 falling somewhere between designs 1 and 2. In specific, the expressions of (3.3.1b-c) are seen to be good approximations when tuning the n = 1 track case of design 1 to some n > 1 track case, as shown using the system parameters of designs 2 and 3. This is especially true for the ω n,num, ω n,den, ζ num, and ζ den predictions of X r = 0.3 mm. There are some errors present for the system parameter predictions using the design 1 values when X r = 1.0 mm, however, due to being about 1 Hz less than that of 79

96 design 1 for the X r = 0.3 mm case. This discrepancy can be seen upon inspection of Figure As a result, the predicted values for ω n,num, ω n,den are lower than those for the X r = 0.3 mm case, and the values of ζ num, and ζ den slightly higher from K * 22(jω). However, using the ω n,num, ω n,den, ζ num, and ζ den parameters of the X r = 0.3 mm case in place of the X r = 1.0 mm case shows good agreement in predicting ω n,num, ω n,den, ζ num, and ζ den for designs 2 and 3 when using (3.3.1b-c). For this reason, it is suspected that for design 1, X r = 1.0 mm, may be incorrect on the basis of tuning predictions being otherwise consistent. Furthermore, ω n,num, ω n,den, ζ num, and ζ den are observed to provide good bounds for approximating with ω damped,num, ω damped,den in narrowtuned devices such as these. The remainder of dynamic tuning studies, however, will emphasize tuning a hydraulic engine mount from the perspective of using the multiple inertia track concept to produce a broad-tuned device by simply increasing ζ num and ζ den sufficiently. This will be made clear later in Sections Quasi-Broadband Design As seen earlier in this chapter, using a multiple inertia track design with one capillary tube-type track and at least one orifice-type track should lead to broadband dynamics. However, it is unknown when this transition from narrowband to broadband dynamic stiffness behavior will occur. For this reason, design 4 incorporates n = 2 tracks (capillary tube-type and orifice-type track in tandem), with the intent of observing where this transition behavior will occur based on A o. While the capillary-type and orifice-type tracks are very physically dissimilar, they can still be compared qualitatively on the same R and I basis to evaluate the tuning 80

97 of a device for broadband behavior. Upon observation of the narrowband results, it can be seen that ζ is expected to increase as the design of the mount deviates from the n- identical track limiting case, while f n will decrease. However, using a capillary tube-type inertia track in tandem to an orifice-type track is generally seen to decrease I eq from the n = 1 track case (which would increase f n ), while R eq R t. When dissimilar R o and R t models are introduced to describe R eq of the tracks #i,1 and #i,2 as is the case for design 4, R eq will deviate the furthest from the n-identical capillary-type track case, and R eq R t. This is especially true when R eq is comprised of one or more R o terms, since R o is typically 2 to 3 orders of magnitude larger than R t given typical mount parameters and operating conditions. While R eq remains relatively unchanged compared to the n = 1 track case as dictated by R t, ζ will increase due to orifice inertance I o. Since V o V t, I o I t, and as a result, I eq I o. For this reason, the smaller I o is, the larger ζ will be. As a result, ζ can be controlled by designing A o of the orifice-type inertia track properly. This control of ζ will be the focus of broadband device tuning, since is difficult to bound with ω damped,num, ω damped,den as I eq and ζ. The orifice of design 4 was fabricated by drilling a hole through the orifice disc located within the prototype assembly and de-burring the rough edges. This orifice-type track has an aspect ratio of d o /l o 1, and is modeled as a short-tube orifice for K * 22(jω) predictions. Discharge coefficient of such an orifice has the following empirical formulation based on Re [29]: (3.3.2a) 81

98 Assuming the same q eq as had been used previously and applying the same assumptions for volumetric flow in each track using area percentage, K * 22(jω) predictions are matched to the measured results of design 4 in Figure 3.18 for X r = 0.3 mm and X r = 1.0 mm excitations. (a) (b) Figure Bounded dynamic stiffness comparisons for design 4: (a) without empirical coefficients; (b) with empirical coefficients:, measured prototype mount, X r = 1.0 mm;, K * 22(jω) prediction, X r = 1.0 mm;, measured hydraulic mount, X r = 0.3 mm;, K * 22(jω) prediction, X r = 0.3 mm Similar empirical corrections are applied to R o model as had been done for R t, since flow conditions are non-ideal here for using the linearized orifice model. The empirical constant ε used to adjust R o to those seen from measurements can be thought of physically as a means to scale c o into an effective discharge coefficient c eq for the orifice model in this prototype mount: (3.3.2b) (3.3.2c) 82

99 Here, β = 0.85, ε = 0.004, and. K * 22(jω) is seen to match the measured quasi-broadband response of design 4 well. There is some tradeoff between approximating K * (jω) /k r and φ K* (jω) when comparing the K * 22(jω) predictions with and without the empirical coefficients. Namely, K * 22(jω) predictions without empirical adjustments fit the measurements better than those predictions made with the empirical coefficients, especially near φ K*,max. Furthermore, inspection of Figure 3.18 reveals that when comparing the quasibroadband response to the previous narrowband designs, K * (jω) /k r is very stiff over a larger range of frequencies. Despite X r = 1.0 mm data set being normalized by the larger k r = 335 N/mm here, it is still seen to be larger than the X r = 0.3 mm case. In addition, the φ K* (jω) spectra is more distributed over the f = 1 to 50 Hz regime. These K * (jω) /k r and φ K* (jω) characteristics are more typical of broadband response behavior when compared to narrow-tuned devices which exclusively exhibit large stiffness and damping via distinct peaks ( K * /k r ) max, φ K*,max within small frequency bands, respectively. The shape and angles of φ K* (jω) are approximated fairly well here, although it is much more difficult to do so here when compared to true narrowband responses due to the limitations of LTI theory. The system parameters of the mount ω n,num, ω n,den, ζ num, and ζ den for this design will be discussed with the results of design 5 in the next section for a direct comparison between the quasi-broadband and broadband designs Broadband Design The true broadband device of design 5 is physically similar to design 4 in that it will also have one capillary tube-type and one orifice-type track drilled through the 83

100 orifice disc. However, the orifice-type track of design 5 has a diameter that is almost 2.5 times larger than that used in the quasi-broadband device, resulting in an 18% increase in total track area. This change results in a smaller I o which is in turn expected to drive ζ higher than previously to obtain a true broad-tuning. This increase in d o from the previous design should allow for more fluid to flow between #1 and #2, so q eq = 1.18q eq now. Using the same c o for straight tube orifice in (3.3.2a), K * 22(jω) predictions for the measurements made for design 5 are shown in Figure 3.19 below. Furthermore, the k 1 used in the K * 22(jω) predictions for both the quasi-broadband and broadband designs are compared in Tables 3.14 and Lastly the system parameters ω n,num, ω n,den, ζ num, and ζ den for these two designs are compared in Tables 3.16 and 3.17 based on the K * 22(jω) predictions. (a) (b) Figure Bounded dynamic stiffness comparisons for design 5: (a) without empirical coefficients; (b) with empirical coefficients:, measured prototype mount, X r = 1.0 mm;, K * 22(jω) prediction, X r = 1.0 mm;, measured hydraulic mount, X r = 0.3 mm;, K * 22(jω) prediction, X r = 0.3 mm 84

101 Table Mount upper chamber stiffness predicted by K * 22(jω) model for broadband designs (without empirical coefficients) X r Design k mm x 10 5 N/m 2.41 x 10 5 N/m 1.0 mm x 10 5 N/m x 10 5 N/m Table Mount upper chamber stiffness predicted by K * 22(jω) model for broadband designs (with empirical coefficients) X r Design k mm x 10 5 N/m 2.34 x 10 5 N/m 1.0 mm x 10 5 N/m x 10 5 N/m Table Linear system parameters predicted by K * 22(jω) model for broadband designs (without empirical coefficients) Mount Design, X r = 0.3 mm Mount Configuration, X r = 1.0 mm System Parameter ω n,num /2π 2.68 Hz 5.76 Hz 2.64 Hz 5.10 Hz ω n,den /2π 3.78 Hz 7.95 Hz 4.05 Hz 6.84 Hz ζ num ζ den

102 Table Linear system parameters predicted by K * 22(jω) model for broadband designs (with empirical coefficients) Mount Design, X r = 0.3 mm Mount Configuration, X r = 1.0 mm System Parameter ω n,num /2π 3.06 Hz 3.05 Hz 3.10 Hz 2.83 Hz ω n,den /2π 4.56 Hz 4.18 Hz 4.37 Hz 3.74 Hz ζ num ζ den Once again, K * 22(jω) predictions match measured results for design 5, which is indeed seen to exhibit true broadband behavior. This device displays substantial damping via φ K* (jω) over a wider range of frequencies when compared to the quasi-broadband device in design 4, in addition to a large magnitude K * (jω) /k r with a relatively flat profile. The prototype mount still generally exhibits higher K * (jω) /k r at X r = 1.0 mm on an absolute basis, although normalization by the k r values for each will make X r = 0.3 mm appear larger here once again. φ K* (jω) is seen to be more difficult yet to predict using LTI methods, evidencing a need for some nonlinear modeling for broadband devices. Good data was acquired to only f = 35 Hz during testing of this device. This was because it was particularly tough to measure K * (jω) within the specified and X r tolerances at frequencies higher than this for this design, which is thought to be due to the discontinuous nonlinearity in #r. However, note that K * (jω) /k r is still seen to be increasing beyond 35 Hz in the X r = 1.0 mm case. The un-adjusted K * 22(jω) predictions were more accurate in approximating K * (jω) /k r and φ K* (jω) than the predictions using the empirical coefficients. This was also 86

103 the case for predictions of design 4, especially when approximating φ K* (jω). This suggests that it may be unnecessary to fit the R t model with γ when using the capillary tube-type track in tandem with an orifice-type track. The purpose of γ was to account for the additional damping seen in narrowband design measurements that could not be approximated with K * 22(jω) using R t alone. In actuality, it is suspected that the large magnitudes of ζ will not be so influenced by R eq as they are I eq due to the very small I o in these designs broadband. However, it may also be appropriate to use new empirical coefficients specifically for designs which use a capillary tube-type and orifice-type track in tandem, since they are so physically dissimilar. As was seen in the narrowband designs, k 1 values will vary based on design and excitation amplitude, evidencing a nonlinear C 1. This is especially true for design 4 when X r = 1.0 mm, where k 1 is seen to spike to almost twice the values used for other K * 22(jω) predictions. The empirical coefficients have no apparent effect on variation in k 1 once again. Investigation of the system mount parameters in Table 3.16 reveal that the device is much more damped in the design 5 case than in design 4. For the sake of brevity, the differences between ω n,num, ω n,den, ζ num, and ζ den due to the influence of empirical coefficients will not be discussed, as they seem to be unnecessary and unrepresentative of the mount behavior as discussed above due to γ. The quasi-broadband device is substantially more damped upon observation of ζ when comparing to narrowband designs 1 through 3. However, ζ num, and ζ den of the quasi-broadband design will be nominally 2.5 to 3 times less than those values for the true broadband device of design 5. ζ num, and ζ den are seen to increase with X r in both designs as the mount behavior becomes 87

104 dominated by the hydraulic path of transmission, as expected. Furthermore, ω n,num and ω n,den are seen to increase from design 4 to design 5. This is attributed to the decrease in I o between the two device configurations. It is now clear why tuning for a broadband response with n > 1 should be done on the basis of simply trying to control the amount of ζ num, and ζ den with A o of an orifice-type track rather than trying to approximate with a ω damped,num -ω damped,den or ω n,num - ω n,den bound. It can be seen from Figure 3.19 that as a mount transitions from narrowtuning to broad-tuning, is pushed to smaller and smaller f (a damped natural frequency). In the process, this smashing effect on φ K*,max due to the large amounts of damping is seen to make it more difficult to approximate with ω damped,num - ω damped,den or ω n,num -ω n,den, since it is suggested that I o will cause these values to approach zero or stay well to the right of, respectively, due to I o decreasing I eq and increasing ζ num, and ζ den simultaneously. Since R eq remains unchanged for all intensive purposes for these types of mount designs, I eq is essentially the primary means of controlling ζ num, and ζ den. Furthermore, while the angles of φ K* (jω) are smaller than the φ K*,max values observed in narrowband designs, large ζ num, and ζ den can be thought of as a means to distribute the energy content of φ K* (jω) located at the distinct of a narrow-tuned device over a much larger band of frequency content. For this reason, it is not as important to closely approximate with bounds such as ω damped,num - ω damped,den or ω n,num -ω n,den, since damping will thus be more distributed in broad-tuned devices. No definitive case is identified here for a transition point between broadband and narrowband mount behavior, since this transition is seen to be gradual. However, it is 88

105 reasonable to suggest that when ζ num 0.7 and ζ den 0.5, the mount starts to behave in a broad-tuned manner Controlled Leakage Path Design As briefly discussed earlier, broadband responses in production mounts are suspected to be the cause of some unintentional leakage path due to a constant or loaddependent fluid flow area existing between #1 and #2, which can be described using orifice-type tracks. To study this phenomenon in a controlled and simple manner, the gasket and silicone caulking materials used to seal #1 from #2 around the circumference of the orifice disc in design 1 are removed. In this manner, an essentially constant area, small annular opening will now exist for fluid to communicate between #1 and #2 in tandem to the existing external capillary tube-type track. This new opening will introduce a large amount of R and a small amount of I to designs such as configuration 1 which have only one capillary tube-type track by design. For this reason, modeling this leakage path as a constant area orifice-type track is justified. Design 6 was tested at X r = 1.5 mm to exaggerate the effects of hydraulic force transmission and illustrate the leakage path phenomenon distinctly with the prototype. Since flow between #1 and #2 takes place around the outer edges of the orifice disc, c o = 0.61 for a sharp-edged orifice model here, with the same flow and area assumptions used in all previous predictions utilizing orifice-type tracks in the design. The behavior of the leakage path design can be described on the premise of the n-inertia track concept using orifice-type tracks to broad-tune the device. This has been done below in Figure 3.20 for 1, 2, and 3 orifice-track designs, with the geometries of the n-tracks listed in Table

106 (a) (b) Figure Bounded dynamic stiffness comparisons for design 6: (a) without empirical coefficients; (b) with empirical coefficients:, sample broadband production mount;, K * 22(jω) prediction, n = 2 tracks (1 capillary tube, 1 sharp-edged orifice); ---, K * 22(jω) prediction, n = 3 tracks (1 capillary tube, 2 sharp-edged orifices); - -, K * 22(jω) prediction, n = 4 tracks (1 capillary tube, 3 sharp-edged orifices) Table Orifice-type inertia track geometries predicted by K * 22(jω) model used to explain controlled leakage path device response, #i,1 geometry of l t = 21.4 cm, d t = 4.93 mm K * 22(jω) Prediction n d o Without Empirical Coefficients mm 3.25 mm 2.75 mm mm With Empirical Coefficients mm mm Once again, good agreement is seen between K * 22(jω) predictions and measured results for n = 2, 3, and 4, especially those made without the use of the empirical coefficients. Figure 3.20a shows design predictions which describe the leakage 90

107 particularly well, even for φ K* (jω), which has been difficult to do in the previous broadtuned designs. Predictions in Figure 3.20b using the empirical coefficients to adjust R t, R o, and I o, are once again seen to be less accurate in predicting shape and value of K * (jω) /k r and φ K* (jω) when compared to those without empirical adjustment. It is once suggested that these adjustments will not be necessary based on the earlier justifications regarding γ. As n is increased by adding more orifice-type tracks to the design, d o is seen to decrease. It is suggested that as n is continually increased in this manner, the annulus around the orifice disc can be interpreted as many small orifice-type tracks. Note that at this higher X r, K * (jω) /k r is seen to be substantially larger than any design yet. This can be attributed to the discontinuous nonlinearity of k r, since at this higher X r, the mount dynamics will be influenced by the compressive mode of #r much more. Measurements were made to only f = 30 Hz here, since it was particularly tough to measure K * (jω) within the specified and X r tolerances for this design due to the discontinuousnonlinear k r characteristic of #r. Table 3.19 below lists the system parameters of design 6 found from K * 22(jω) predictions. Looking at the results without the empirical coefficients, it can be seen that the device is predicted to be highly damped. The ω n,num - ω n,den appears to the right of and ω damped,num - ω damped,den 0, as expected for a broad-tuned device such as this based on the results of the previous broadband designs and predictions. In addition, k 1 is very large here, further evidencing the hydraulic path of force transmission dominating at high X r and a possible nonlinearity in C 1. The results for n = 2, 3, and 4 track designs are 91

108 very similar, with some general increases in ω n,num, ω n,den, ζ num, and ζ den as the number of orifice-type tracks in the design is also increased. Table Linear system parameters predicted by K * 22(jω) model for controlled leakage path design Without Empirical Coefficients k 1 = 4.23 x 10 5 N/m With Empirical Coefficients k 1 = 4.14 x 10 5 N/m Parameter n = 2 n = 3 n = 4 n = 2 n = 3 n = 4 ω n,num /2π 6.42 Hz 6.56 Hz 6.79 Hz 4.67 Hz 4.79 Hz 4.71 Hz ω n,den /2π 9.35 Hz 9.55 Hz 9.89 Hz 6.77 Hz 6.93 Hz 6.85 Hz ζ num ζ den Quasi-linear K * 22(jω) Model As discussed for the broadband designs of Section , lumped parameter LTI models such as K * 22(jω) can approximate K * (jω) /k r well; however, it is difficult to approximate device damping via φ K* (jω). The prototype mount device is seen to be and X r dependent, which is suspected to be the result of nonlinear k r, C 1 and R eq. The prototype may also be dependent on f, as C 1 nonlinearities are suspected to depend on frequency as well from the literature [1-13]. Since emphasis is placed on the n-inertia track concept here and φ K* (jω) is particularly tough to predict with K * 22(jω) (especially for broad-tuned devices), a quasi-linear version of the K * 22(jω) model is developed to illustrate the need for nonlinear modeling of n-track devices such as the prototype mount studied here. This quasi-linear K * 22(jω) model assumes all the nonlinearities of the 92

109 mount are due to equivalent track resistance R eq for the sake of mathematical convenience. This is done by assuming lumped parameter R eq is now an amplitude- and frequency-dependent parameter R eq (X r,ω), for an assumed n-linear tracks in parallel. Therefore, all other previous lumped parameter LTI assumptions used in the K * 22(jω) model will still apply, but now with R eq = R eq (X r,ω) being a mount parameter which can vary based on X r and ω of x r (t). It is then possible to use this quasi-linear version of K * 22(jω) with R eq (X r,ω) parameter to match a desired or sample measured K * (jω) mount spectra using complex mathematics. To do this, assume the linear K * 22(jω) model to take the following form in terms of the magnitude and phase information of the sample K * (jω) to be matched: (3.4a) Here, M is just K * (jω) of the sample measured dynamic stiffness data. Assuming that R eq = R eq (X r,ω) = Α + jβ, both the magnitude and phase information contained within the sample K * (jω) can be matched using the linear K * 22(jω) model. Substituting the assumed complex R eq (X r,ω) into (3.4a) and collecting the real and imaginary parts in the numerator and denominator yields the following: (3.4b) Cross multiplication of the above expression and collection of real and imaginary terms on each side results in the following: 93

110 (3.4c) Setting the real and imaginary parts of (3.4c) on each side equal to one another will yield a set of two equations in two unknowns Α, Β. Solving these two equations for Α, Β will produce a closed-form solution for R eq (X r,ω) as so: (3.4d-e) In this manner, R eq (X r,ω) is defined at each excitation amplitude and frequency and so that the linear K * 22(jω) model matches a known sample set of dynamic stiffness data given by M and φ K*. M and φ K* need not be actual response data of a production mount, but it will serve the most practical merit when this is the case. For this reason, consider the sample broad-tuned stiffness data used for the production mount of unknown track configuration in Section 3.2.1, as it is representative of a very broad-tuned design. Shown below in Figure 3.21 are results of the quasi-linear K * 22(jω) model as applied to matching the dynamic stiffness of this sample device, assuming the n = 2 track (1 capillary-type, 1 orifice-type) designs, with k N/mm for X r = 1.0 mm and k N/mm for X r = 0.25 mm as was found for this sample device in Section Note the ability of the quasi-linear K * 22(jω) model to match frequency response data extremely well at both X r = 1.00 mm and X r = 0.25 mm excitations. Vector error norm analysis can be used to demonstrate the accuracy of the numerical method used in obtaining this quasi-linear dynamic stiffness solution. When 94

111 the error norm is of size 2, it will be equivalent to the length of the error vector and can be thought of as an error magnitude for analysis purposes using the following expression: (3.4f) U is the true value of a given quantity, u is the measured, or approximate value of that quantity, and S is the size of error norm v. v is a vector quantity since the values of K * (jω) and φ K* (jω) used here are directional quantities of the hydraulic engine mount. It can be seen in Table 3.20 that this method is accurate to at least an order of For this reason, the R eq (X r,ω) in Figure 3.21b used for this prediction is expected to be accurate as well. Since R eq (X r,ω) can vary with frequency and excitation amplitude, inaccuracies are expected when comparing quasi-linear K * 22(jω) predictions to those of the linear version with lumped R eq. In specific, upon investigation of Figure 3.21 and Table 3.20 for this particular mount example, the quasi-linear K * 22(jω) model predicts the responses very well, with R eq (X r,ω) which tend to be somewhat oscillatory and exhibit no simple relationship vs. f at a given X r. For this reason, R eq (X r,ω) cannot be assumed as lumped parameters R eq. However, the quasi-linear K * 22(jω) model is seen to be capable of showing the relationship between R eq (X r,ω) and f in this manner, which may be very useful for adaptive or active control applications of n-track designs when this relationship is more simple (i.e. linear). For example, geometries l i and d i of one or more of the n- tracks can be adaptively or actively controlled in a manner to yield the same R eq (X r,ω) found from the quasi-linear K * 22(jω) model to match the sample (or desired) dynamic stiffness spectrum. 95

112 (a) (b) Figure Results of quasi-linear K * 22(jω) model for a sample broadband mount response: (a) K * (jω); (b) R eq (X r,ω):, measured production mount, X r = 1.00 mm;, quasi-linear K * 22(jω) prediction, X r = 1.00 mm; +, measured production mount, X r = 0.25 mm;, quasi-linear K * 22(jω) prediction, X r = 0.25 mm Table Error norms of quasi-linear K * 22(jω) model X r 0.25 mm 1.00 mm O ~ N/mm O ~ N/mm O ~ O ~

113 CHAPTER 4: CONCLUSION 4.1 Summary and Contributions The multiple (n-) inertia track concept has been used to dynamically tune a hydraulic engine mount with success, showing promise for passive, adaptive, and active control applications. Progress has been made in overcoming key deficiencies and unresolved issues in the literature on the topic [2-4]. Since there are a wide variety of n- inertia track configurations, K * 22(jω) dynamic stiffness models, with emphasis on a fixed decoupler, have been developed to explain them all on an arbitrary basis. Furthermore, a new multiple inertia track prototype mount concept has been designed, built, and tested in a controlled manner, with the capability of varying the type (capillary tube, orifice) and number n of inertia tracks, in addition to l i and d i of each. This prototype has been used to test several devices with different n-track configurations for improving performance compared to the conventional n = 1 track case. Three narrowband devices have been designed and tested to refine and extend existing theory for predicting, in addition to validate K * (jω) model predictions for an n = 3 track mount for the first time. Two broadband devices have been designed and tested successfully by tuning ζ of the mount with orifice-type tracks for the first time. Several n-track mount designs with orifice-type tracks are also proposed, which successfully describe a special broad-tuned design utilizing a controlled leakage path flow area for the first time. Lastly, a quasilinear K * (jω) model has been developed to study X r - and f-dependent behavior of R eq, 97

114 which should lead to nonlinear models of n-track devices and improved adaptive or active mounts. Contributions of this work are thus experimentally validated extensions of previous K * (jω) lumped parameter LTI models, which are now applicable to both narrow-tuned and/or broad-tuned hydraulic engine mounting devices with n 2. With respect to existing literature on the topic [2-4], improved LTI K * 22(jω) models have been developed here, which are now accurate for at least n = 3 tracks and can now include designs with orifice-type tracks in tandem to capillary tube-type tracks. Improvements in analytical approximations of for narrow-tuned devices have been made using a ω damped,num - ω damped,den bound, with extension to determination of the ω damped,num -ω damped,den bound for an arbitrary n-track case with knowledge of ω damped,num, ω damped,den, and track geometries l i and d i of an n = 1 track mount. Furthermore, a quasi-linear K * 22(jω) now establishes some direction for nonlinear modeling and adaptive and active mount development of n-track mounts, which had not been done previously. 4.2 Recommendations for Future Work A significant body of work still remains on the topic of n-track mounts for dynamic tuning of the response. Future research should include more extensive mount characterization studies of the prototype mount built in this study, for both narrowband and broadband designs. This should include transient testing with ideal or more realistic inputs (such as step-ups or step-downs, pulses, or measured displacement profiles). Nonlinear investigations of n-track designs, especially with respect to modeling R eq (X r,ω) and C 1 (X r, ), should be performed. Nonlinear studies could then be applied to the facilitation of adaptive and active n-track mount designs for improving motion control 98

115 and vibration isolation of the powertrain compared to an n = 1 track mount. Furthermore, the influence of n-track mounts on vehicle system response in 1/4, 1/2, and full vehicle models is of interest, as component modeling alone is not sufficient. System studies could include linear or nonlinear passive, adaptive, or active n-track mount designs. Lastly, interactions between a free decoupler and multiple inertia tracks could be studied more rigorously in future K * (jω) models in order to improve predictions further. 99

116 REFERENCES [1] Singh, R., G. Kim, and P. V. Ravindra. "Linear Analysis of Automotive Hydro- Mechanical Mount with Emphasis on Decoupler Characteristics." Journal of Sound and Vibration (1992): [2] Zhang, Y. Q., and W. B. Shangguan. "A Novel Approach for Lower Frequency Performance Design of Hydraulic Engine Mounts." Computers and Structures 84 (2006): [3] Shangguan, W. B., Z. S. Song, Y. Q. Zhang, K. H. Jiang, and C. Xu. "Experimental Study and Simulation Analysis of Hydraulic Engine Mounts with Multiple Inertia Tracks." Zhendong Gongcheng Xuebao/Journal of Vibration Engineering 18.3 (2005): [4] Lu, M., and J. Ari-Gur. "Study of Hydromount and Hydrobushing with Multiple Inertia Tracks." JSAE Annual Congress Proceedings JSAE Annual Congress, Yokohama, Japan. Vol. 68. JSAE, Ser. 02. [5] Kim, G., and R. Singh. "Nonlinear Analysis of Automotive Hydraulic Engine Mount." Transactions of the ASME 115 (1993): [6] Kim, G., and R. Singh. "A Study of Passive and Adaptive Hydraulic Engine Mount Systems with Emphasis on Non-Linear Characteristics." Journal of Sound and Vibration (1995): [7] Tiwari, M., H. Adiguna, and R. Singh. "Experimental Characterization of a Nonlinear Hydraulic Engine Mount." Noise Control Engineering Journal 51.1 (2003): [8] Adiguna, H., M. Tiwari, R. Singh, H. E. Tseng, and D. Hrovat. "Transient Response of a Hydraulic Engine Mount." Journal of Sound and Vibration 268 (2003): [9] He, S. Development of Nonlinear Hydraulic Engine Mount Models for Transient Responses Given Limited Measurements. Masters Thesis. The Ohio State University,

117 [10] Lee, J. H., and R. Singh. Critical Analysis of Analogous Mechanical Models Used to Describe Hydraulic Engine Mounts. Journal of Sound and Vibration, (2008): [11] Yu, Y, N. G. Naganathan, and R. V. Dukkipatit. "Review of Automotive Vehicle Engine Mounting Systems." International Journal of Vehicle Design 24.4 (2000): [12] Colgate, J. E., C. T. Chang, Y. C. Chiou, W. K. Liu, and L. M. Keer. "Modeling of a Hydraulic Engine Mount Focusing on Response to Sinusoidal and Composition Excitations." Journal of Sound and Vibration (1995): [13] Truong, T. Q., and K. K. Ahn. "A New Type of Semi-Active Hydraulic Engine Mount Using Controllable Area of Inertia Track." Journal of Sound and Vibration (2010): [14] De Fontenay, E. Elastic Vibration Isolation Mounting with Integral Hydraulic Damping and a Rigid Partition with an Adjustable Passage for Conducting Fluid. Caoutchouc Manufacture et Plastiques, Versailles, France, assignee. United States Patent Number 4,909, Mar [15] Hofmann, M., and H. Muller. Two Chamber Engine Mount with Hydraulic Damping. Metzeler Kautschuk GmbH, Munich, Fed. Rep. of Germany, assignee. United States Patent Number 4,676, June [16] Miller, J. W., L. A. Peterson, and C. A. Kingsley. Hydraulic-Elastomeric Mount. General Motors Corporation, Detroit, Michigan, assignee. United States Patent Number 4,765, Aug [17] Quast, J. R. Hydraulic Damping Rubber Engine Mount. Boge GmbH, Eitorf, Fed. Rep. of Germany, assignee. United States Patent Number 4,645, Feb [18] Satori, K., and T. Sakamoto. Liquid Sealed Type Elastic Mount. Yamashita Rubber Kabushiki Kaisha, Saitama, Japan, assignee. United States Patent Number 6,267, July [19] Bodie, M. O., M. W. Long, and S. G. Tewani. Dual Track Variable Orifice Mount. Delphi Technologies, Inc., Troy, MI, assignee. United States Patent Number 6,799, Oct [20] Gennesseaux, A. Hydraulic Antivibration Devices. Hutchinson, Paris, France, assignee. United States Patent Number 5,411, May [21] Bouhours, J. P. Hydraulic Antivibration Devices. Hutchinson, Paris, France, assignee. United States Patent Number 5,123, June

118 [22] Guillemot, T. Hydraulic Anti-Vibration Mount and Manufacturing Process for Same. United States Patent Application US Apr [23] Guillemot, T. Method of Manufacturing a Hydrualic Anti-Vibration Mount. Hutchinson, Paris, France, assignee. United States Patent Number 6,536, Mar [24] Sciortino, G. Hydraulically Damped Engine Mount Having Improved Throttle Ports. Firma Carl Freudenberg, Weinheim/Bergstrl, Fed. Republic of Germany, assignee. United States Patent Number 4,787, Nov [25] Sciortino, G. Engine Mount. Firma Carl Freudenberg, Weinheim an der Bergstrasse, Fed. Republic of Germany, assignee. United States Patent Number 4,796, Jan [26] Tewani, S. G., M. W. Long, M. O. Bodie, R. A. Beer, and J. P. Hamberg. Hydraulic Mount with Reciprocating Secondary Orifice Track-Mass. Delphi Technologies, Inc., Troy, Michigan, assignee. United States Patent Number 7,159, Jan [27] Gugsch, M., and A. Prenning. Hydraulic Damping Two-Chamber Engine Mount. BTR AVS Technical Centre GmbH, Hoer-Grenehausen, DE, Germany, assignee. United States Patent Number 6,357, Mar [28] Sciortino, G. "Hydraulically Damped Engine Mount Having Improved Throttle Ports." Journal of the Acoustical Society of America 89.3 (1991): [29] Doebelin, E. O. System Dynamics Modeling, Analysis, Simulation, Design. New York: Marcel Dekker, [30] ISO/IEC IS 10846:1997: Acoustics and Vibration Laboratory Measurement of Vibro-Acoustic Transfer Properties of Resilient Elements. International Organization for Standardization, Geneva, Switzerland. [31] Beranek, L. L., ed. Noise and Vibration Control. Annapolis: Institute of Noise Control Engineering,

119 APPENDIX A: PHYSICAL DESCRIPTION OF HYDRAULIC ENGINE MOUNT DESIGN A.1 Elastomeric Casing The elastomeric outer casing is typically made of a rubber material and is in the shape of a tube, cylinder, rectangular prism, or cone. It houses brackets on its upper and lower surfaces which are used for securing to the engine and chassis of the automobile, respectively. These brackets consist of several studs to couple each end of the mount accordingly. In order to accommodate the preload introduced by the weight of the engine, the casing is typically reinforced with metallic (often steel) inserts. These inserts create a composite shell with increased shear strength and resistance to bulging due to engine weight and system excitations. With respect to performance, the elastomeric casing of the hydraulic engine mount is primarily responsible for the static path of force and vibration transmission between the powertrain and chassis/frame due to mechanical excitations. Accordingly, the rubber material of the elastomeric casing is often modeled using the Voigt model. Essentially, the Voigt model describes the behavior of a material in terms of mechanical elements of stiffness and damping acting in parallel with one another. In this case, the stiffness k r, and viscous damping, b r, of upper rubber #r will act in parallel to the motion of mass m r. This is shown in Figure 1.1b, where half of the elastomeric casing stiffness and viscous damping is represented symmetrically on each side of the schematic for illustrative purposes. In the traditional manner, the stiffness accounts for storage of 103

120 potential energy as a result of mechanical excitations or initial conditions, while the viscous damping component of the material acts as a dissipater of mechanical energy. Typically, the properties of an elastomer such as that used for the casing are very frequency insensitive (Figure 2.1) and these mechanical properties can be approximated as lumped parameters for the hydraulic engine mount. Furthermore, the viscous damping of the rubber is often small in magnitude for the types of rubber generally used for hydraulic mount applications [1]. These values can be validated using dynamic testing of a hydraulic engine mount with the fluid drained. A.2 Chambers In addition to a static path of force transmission, the majority of dynamic force transmission in the hydraulic engine mount is due to a second path, often called the hydraulic path, contained within the elastomeric casing. Referring to Figure 1.1b, the dynamics of the device can be attributed to upper (#1) and lower (#2) bellow-like fluidfilled chambers, which act in parallel to (same displacement as) the elastomeric casing in which they are encompassed. These chambers are connected to each other and are essentially formed by the inner space of the elastomeric casing. Typically, they contain a relatively temperature-insensitive low viscosity fluid such as water or a water-glycol (antifreeze) mixture (to prevent freezing). The primary function of the chambers is to store fluid energy within fluid compliances C 1 and C 2 by acting as a hydraulic system via the dynamics of fluid transfer from upper to lower chamber and vice-versa. This causes absolute pressures p t 1 (t) and p t 2 (t) to form in each chamber. In large part, the upper chamber stores the pressure of the 104

121 mount, while the lower chamber acts as an accumulator exposed to atmospheric pressure p atm (as represented by the bleed orifice near the base of the mount structure in Figure 1.1b) having a much smaller pressure magnitude than the upper (this will be the case for steady-state behavior) [1]. Compliances C 1 and C 2 due to pressures p t 1 (t) and p t 2 (t) are very nonlinear due to material nonlinearities (geometry, properties), chamber vacuum effects, and engine preload [7-9]. As a result, it is often difficult to explain their behavior. Lumped parameter values can sometimes be approximated via experimentation and linearization of subsequent p-v (pressure vs. volume) curves about an operating pressure or volume using a special test apparatus [5,8]. These values will depend on specific mounting device, however. Some computational methods are used to approximate the chamber compliances (such as finite element methods) [2,3], as well, but these are often supplemented with analytical models and curve fits, also produced from experiment. A.3 Midplate The midplate, or partition, (denoted #α) of a hydraulic engine mount is a structural component of a hydraulic engine mount which physically divides the upper (#1) and lower (#2) chambers. Its primary function is to act as an interface between these upper and lower chambers. It is commonly considered a rigid member (often steel) and is built into the outer casing of the mount itself. Typically, it consists of an upper and lower orifice plate press-fit together. The midplate itself often has negligible influence on the behavior of the mount. However, it houses hydraulic elements, which control the dynamics of the hydraulic path (and the mount itself) to a large extent. These hydraulic 105

122 elements are essentially independent fluid flow paths existing between the upper and lower chambers. The two common hydraulic elements have been identified previously as the decoupler and inertia tracks in Figure 1.1b (#d, #i,1, #i,2). Both of these elements are very nonlinear, i.e. their properties vary with excitation amplitude and frequency. The stiffness and damping properties of the mount are highly dependent on them as well. For this reason, they are very important to understand. A.4 Inertia Track The most common hydraulic element of the midplate (which makes an engine mount an engine mount and not a shock absorber, for example) is the inertia track (also known as an orifice/fluid passage/track/channel). Physically speaking, the inertia track is a long, small cross-section, often spiral, passage starting at the upper surface of the midplate and running through its thickness to its exit at the bottom surface. The inertia track acts like a capillary tube (long, with small diameter) containing a fluid mass which reciprocates within the midplate due to external excitations. An example of a typical inertia track for a hydraulic engine mount is shown in Figure A.1 below. This oscillating fluid acts as an inertia-augmenting mass, which hydraulically magnifies the forces input at the powertrain side of the mount due to area reduction of #1 and #2 [10]. The inertia track can be described in terms of its fluid inertance, I i, and fluid resistance, R i. The fluid capacitance of the track is usually neglected since a minimal amount of fluid volume is actually contained within the small passage at any given time, thus contributing to negligible energy storage capacity (especially when compared to #1 and #2). I i can be approximated using theoretical methods (essentially a mass calculation 106

123 of fluid mass which can be contained within the track). R i has significant nonlinearities associated with it, so linear theory (such as capillary tube or orifice models) can be used in limitation for analytical approximations under controlled conditions. Most of the time, however, R i must be found by experiment using pressure-flow relations found from a specially prepared hydraulic circuit [5]. Similar to C 1 and C 2 approximations, such models can then also be implemented in computational finite element software to simulate the parameter under different operating conditions [2,3]. I i and R i are particularly large due to typical inertia track geometries. For this reason, the inertia track introduces lots of damping to the system. The inertia track also introduces a degree of freedom, since I i (its fluid mass) is significant. This DOF causes the low frequency resonance/mode characteristic of a hydraulic engine mount (usually below f = 50 Hz) [1-13]. Therefore, understanding the inertia track and the ability to accurately tune this natural frequency is especially important in design practice. This resonant frequency is used to approximate the notch frequency, commonly referred to as the peak frequency of the loss angle,, which is simply the frequency of the maximum phase angle of the device s dynamic stiffness K * (jω). The phase response of hydraulic engine mounts is largely a function of damping, and is often the location of maximum phase angle (and thus damping) of the device. Therefore, controlling this frequency (tuning it with the inertia track) is a practical means to introduce the most damping at the low frequency resonance of the device. However, one particular problem introduced by the inertia track is a phenomena commonly referred to as dynamic hardening in the automobile noise, vibration, and harshness community. While the inertia track is a permanent means for the upper and 107

124 lower chambers to communicate hydraulically, it performs very differently at different excitation amplitudes. In specific, the inertia track functions particularly well at high excitation amplitudes of approximately X r = 1.0 mm or greater (which are usually of low frequency content, i.e. below f = 25 Hz) [1]. However, the inertia track causes the upper chamber stiffness to increase with frequency and become very hard, so high frequency excitation of #r typically induces violent pressure fluctuations in #1 that cause significant increases in noise and force transmission to the base. This dynamic hardening is not good for vibration isolation, which requires a compliant transfer path for mitigation of transmissibility. For this reason, the inertia track hydraulic element is often used in tandem to the decoupler in hydraulic engine mounts [11]. (a) (b) (c) Figure A.1. Example of hydraulic engine mount inertia track: (a) Inlet from #1 to #i at upper orifice plate; (b) Entrance to #i from upper orifice plate and corresponding l i, d i along inside of lower orifice plate; (c) #i exit on bottom of lower orifice plate to #2 108

125 A.5 Decoupler The decoupler, sometimes referred to as orifice with decoupling diaphragm or simply as orifice, is another hydraulic mechanism of the midplate. The decoupler regulates hydraulic communication between upper and lower chambers by functioning to allow fluid flow through this path only when excitation amplitudes of the mount are small (nominally around X r = 0.1 mm). It does this by principle of utilizing an orifice-type passageway located between the orifice plates of the midplate. Contained within this passageway is a thin rubber-composite disk of negligible mass which acts like a constricting diaphragm between the upper and lower chambers. This disk floats within the decoupler free travel gap (orifice-type passageway) [1] between stops located on the inside of the upper and lower orifice plates. This is illustrated schematically in Figure 1.1b as #d, and shown physically in Figure A.2 below with an example of a typical hydraulic engine mount. At high excitation amplitudes, the decoupler closes and prevents flow between the upper and lower chambers through this path by virtue of the disk bottoming or topping out on one of the internal stops within the decoupler. When the decoupler is closed in this manner, it is commonly referred to as fixed. However, at low excitation amplitudes, fluid does not move the disk far enough to reach the internal stops of the decoupler, thus allowing fluid to pass. This open condition is also commonly called the free decoupler state. In this manner, the decoupler can be thought of physically as a fluid flow switch between #1 and #2, which changes between its free and fixed states based on the amplitude of mechanical excitation. The decoupler is a very nonlinear device, due to its excitation amplitude and frequency-dependent switching action functionality. However, it mitigates the dynamic 109

126 hardening problem of the inertia track experienced at low amplitude excitations, which is most typical of higher frequency excitation content (above f = 25 Hz). In other words, the decoupler softens the hydraulic path of force transmission of the mount at lower amplitudes, which is especially important at higher frequencies (above f = 25 Hz), where the inertia track cannot. This is done at the expense of some of the low amplitude damping introduced by the inertia track at low frequencies (below f = 25 Hz), however [11]. Just like the inertia track, the primary parameters characterizing the behavior of the decoupler are its fluid resistance R d and inertance I d. The decoupler also stores negligible fluid at any given time, therefore it often has no fluid capacitance associated with it either. The decoupler presents some unique design challenges for hydraulic engine mounts, however. In specific, the dynamics of the decoupler are not well understood, and therefore accurate models of this mechanism are not readily available. This is due to its nonlinear R d and I d, caused by turbulent flows and the switching action. There is currently no effective way to measure R d, so this parameter is often approximated using computational or analytical models and simulation. The degree of freedom introduced by I d is often neglected for the sake of analysis, as it is very small compared to that of the inertia track, especially at low frequencies. In specific, the resonance due to the decoupler typically occurs around f = 150 to 250 Hz (Ω = 9,000 to 15,000 RPM), which is outside the typical operating speeds of an automobile powertrain [1]. 110

127 (a) (b) (c) Figure A.2. Example of hydraulic engine mount decoupler: (a) #d resting in lower orifice plate (lower stop of free travel gap); (b) #d rotated in lower orifice plate (to show thickness); (c) #d resting in upper orifice plate (upper stop of free travel gap) 111