Journal of Sound and Vibration

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1 Journal of Sound and Vibration 331 (01) Contents lists available at SciVerse ScienceDirect Journal of Sound and Vibration journal homepage: Experimental study of hydraulic engine mounts using multiple inertia tracks and orifices: Narrow and broad band tuning concepts Benjamin Barszcz, Jason T. Dreyer, Rajendra Singh n Smart Vehicle Concepts Center, Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 4310, USA article info Article history: Received 1 July 011 Received in revised form 1 May 01 Accepted 1 July 01 Handling Editor: K. Worden Available online 3 August 01 abstract Hydraulic engine mount tuning concepts with one inertia track and one decoupler are well understood. However, the dynamic response with multiple tracks or orifices is not. To overcome this void in the literature, dynamic tuning concepts of hydraulic engine mounts, with emphasis on multiple (n-)inertia tracks/orifices, are experimentally examined. A new prototype mount concept is designed, built, and experimentally evaluated in a controlled manner. Refined linear time-invariant models of fixed decoupler-type designs are developed to critically assess the dynamic stiffness measurements and to explore a family of alternate designs. Three narrowband devices are investigated for accurately predicting the frequencies corresponding to peak loss angles for the first time, in addition to examining and validating an n ¼ 3 track mount. Two broadband devices are also successfully evaluated by tuning damping introduced by orifice-type tracks. A special broad-tuned design utilizing a controlled leakage path flow area is then suggested, and the role of fluid resistance in achieving the desired performance is clarified. Finally, a production mount with unknown configuration is diagnosed using the proposed models with n tracks. & 01 Elsevier Ltd. All rights reserved. 1. Introduction The dynamics of passive hydraulic engine mounts with one inertia track and one decoupler is well understood [1 13]. However, behavior with multiple tracks, decouplers, or orifices is not. Nevertheless, many inventors [14 1] have proposed mount designs utilizing the multiple inertia track tuning concept to improve motion control or vibration isolation properties. Others have developed one or more tracks that are contained within the mount [ 8]. Scientific investigations on this topic are rather sparse, though a few prior investigations suggest that a hydraulic engine mount with n inertia tracks could be promising for passive, adaptive, and active control [ 4]. For instance, Zhang and Shangguan [] and Shangguan et al. [3] analytically find the peak loss angle frequency o jk for n ¼ and 3 mounts; here o is the circular frequency (rad/s). Their studies indicate that the dynamic stiffness magnitude 9K*(jo)9, loss angle j K* (jo), and the peak loss angle frequency o jk could increase with n ¼ or 3 configurations when compared to the conventional one track mount; here, j is the imaginary unit. Lu and Ari-Gur [4] approximated o jk of a mount with n ¼ tracks, comparable to the studies by Zhang and Shangguan [] and Shangguan et al. [3]. Furthermore, Kim and Singh [5] suggested that a typical hydraulic engine mount with a single inertia track could still be modeled with two independent flow paths by virtue of a secondary leakage path that might exist (between two fluid chambers due to either design or wear). Their K n (jo) prediction matched the experimental results well for a mount with an inherent leakage path. n Corresponding author. Tel.: þ ; fax: þ address: singh.3@osu.edu (R. Singh) X/$ - see front matter & 01 Elsevier Ltd. All rights reserved.

2 510 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) Several deficiencies and unresolved issues on the topic still remain in the literature. For instance, no prior article has experimentally evaluated mounts with a combination of multiple tracks and orifices. Also, no other formal investigation has analytically examined n-track mounts although closed-form K n (jo) models have been developed for n o 3 tracks and for narrow-tuned mounts only. Existing K n (jo) models have not been adequately experimentally validated, especially for n4 tracks. Approximation methods for o jk need to be improved as well. Consequently, there is a need to experimentally and analytically investigate the influence of multiple inertia tracks and orifices on mount dynamics. Furthermore, an examination of both narrow- and broadband tuning issues, including amplitude (in)sensitive behavior, is of interest. This is the main focus of this article, with focus on experimental methods to examine dynamic stiffness spectra up to 50 Hz; analytical formulations are used to critically analyze the measurements. Problem formulation As seen in the schematic of Fig. 1, the mount is excited at the top through its upper rubber #r of mass m r with a displacement x t r ðtþ¼x r þx r ðtþ corresponding to an engine force F t m ðtþ¼f m þf m ðtþ and transmits a force F t T ðtþ¼f T þf T ðtþ to a fixed base or chassis. Here, the superscript t implies total. A bar above the symbols denotes the mean component. The subscript T refers to the force transmitted force 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). Typically Fig. 1. Low frequency models for a passive hydraulic engine mount with two inertia tracks or orifices (G i,1, G i, ). (a) Fluid model and (b) analogous mechanical model for cross-point stiffness. Here x r t (t) is the displacement excitation, and FT t (t) is the force transmitted to a rigid base.

3 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) the mount is tested by a sinusoidal displacement input x t r ðtþ¼x r þx r sinðotþ with X r being the input displacement amplitude of #r and t the time [9]. This type of nonresonant testing assumes m r ¼ 0 and attempts to dynamically characterize the mount while ignoring super- and sub-harmonics [7]. The mount is comprised of upper (#1) and lower (#) fluid-filled chambers which communicate with one another via hydraulic paths (G i,1 and G i, ) on a mid-plate. Each of the fluid control volumes can be described with lumped elements of fluid compliance C, resistance R, and inertance I [1,30]. The hydraulic mount of Fig. 1 is atypical in that although it does not include the decoupler, it has two inertia tracks or orifices (G i,1 and G i, ). It is well-recognized that the hydraulic engine mount is a 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 [5 9]. Nevertheless, linear methods are utilized in this paper from the perspective of designing new mount configurations. Specific research objectives include the following. (1) Design and build a controlled experiment to evaluate a number of track or flow passage combinations (capillary tube, orifice) on the dynamic tuning of a hydraulic engine mount and to validate the linear system theory. () Suggest refinements to the linear time-invariant models for the hydraulic engine mount with n inertia tracks (but without a decoupler) and critically assess the measurements. (3) Diagnose a production mount of unknown configuration using linear models with n inertia tracks. (4) Propose n-track mount designs which could tune a hydraulic engine mount over a narrow- or broadband basis (up to 50 Hz) when compared to the conventional n ¼ 1 track case, and explore the role of fluid resistance 3. Experimental studies on a new prototype mount with multiple configurations A new prototype hydraulic engine mount with three possible external capillary tube-type tracks and three internal orifice-type tracks is built. The mount is created in such a manner that one can change not only n tracks (from the six possible), but also the length l i and diameter d i of each as well. A custom midplate is designed and built to accommodate multiple inertia track configurations. However, the experimental scope is limited to emphasize the effect of n Z and orifice-type tracks on tuning the mount. Table 1 lists six designs that are experimentally evaluated. These are also displayed in Fig.. The external capillary tracks and internal orifice tracks appear as they do in this prototype construction simply for convenience and ease of quick testing of the various track configurations, resulting in the ability to make a substantial number of comparisons to the behavior of a conventional passive hydraulic engine mount with a fixed internal inertia track design in a relatively short period of time. The external flow passages in the cases depicted in Fig. (b) (e) could have been made with different material or geometry such that they would not protrude in the potentially invasive manner that they do in these specific instances. The internal orifice tracks could have fixed lengths and be adjusted by diameter (the most sensitive parameter) by indexing a sealed, nonfloating disc in the midplate to one of several orifice holes of different diameters via an external switching device, instead of having a fixed disc with one orifice present in the midplate. 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 more controlled leakage path of fluid flow between #1 and #. Each configuration is harmonically tested using an elastomer test machine [9] at a preload of F m ¼ 100 N (typical preload due to a four cylinder engine) given excitation amplitude of X r ¼ 0.3 mm and X r ¼ 1.0 mm. Experimental results will be provided in the later sections along with theoretical predictions. 4. Analytical formulation and simplifications for design studies The continuity and momentum equations are utilized to describe the behavior of G i,1 and G i,. The equivalent mechanical analog of Fig. 1(b) can be found as suggested by [1,6,9] and by normalizing the hydraulic elements with the piston area A r. Following the normalization by A r in the momentum equation, the following system parameters are defined: I i ¼ m i /A r, Ri ¼ b i /A r, Ci ¼ A r /ki. Assuming linear models for the fluid resistance (R) of the inertia tracks, the resistance of a capillary tube, R i ¼ (18ml)/(pd 4 ), could be used for long inertia tracks where m is the coefficient of viscosity. The linearized resistance, R i ¼ (rq o )/(c o Ao ), of a sharp edge orifice about an operating point (say flow rate qo )is used for short length inertia tracks where r is the density, A o is the area, and c o is the discharge coefficient. The governing Table 1 Configurations for experimental studies: prototype hydraulic engine mount with multiple inertia track designs. Design n Description (also see Fig. ) Tuning concept D1 1 Baseline configuration (one external capillary-type) Narrow band D 3 Three identical external capillary-type Narrow band D3 3 Two identical external capillary-type, one long external capillary-type Narrow band D4 One external capillary-type, one small internal orifice-type Quasi-broad band D5 One external capillary-type, one large internal orifice-type Broad band D6 One external capillary-type, one internal controlled leakage path Broad band

4 51 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) (a) (b) (c) (d) (e) Fig.. Experimental studies with prototype mount configurations (corresponding to Table 1). (a) Representative cross-section (side view) with internal orifice and external inertia track. Views of prototype mount designs (top view): (b) designs D1 and D6, l t ¼ 1.4 cm, d t ¼ 4.93 mm; (c) design D, l t ¼ 1.4 cm, d t ¼ 4.93 mm; (d) design D3, l t,1 ¼ l t, ¼ 1.4 cm, l t,3 ¼ 4l t,1 ¼ 85.6 cm, d t,1 ¼ d t, ¼ d t,3 ¼ 4.93 mm; (e) designs D4 (l o ¼.00 mm, d o ¼ 1.00 mm) and D5 (l o ¼.00 mm, d o ¼.38 mm), both with l t ¼ 85.6 cm, d t ¼ 4.93 mm. equations for the mechanical system of Fig. 1(b) are as follows where F m (t) is the input and x r (t) is the output: F m ðtþ¼m r x r ðtþþb r _x r ðtþþk r x r ðtþþk 1 ½x r ðtþ x i,1 ðtþ x i, ðtþš (1) k 1 ½x r ðtþ x i,1 ðtþ x i, ðtþš k ½x i,1 ðtþþx i, ðtþš ¼ m i,1 x i,1 ðtþþb i,1 _x i,1 ðtþ () k 1 ½x r ðtþ x i,1 ðtþ x i, ðtþš k ½x i,1 ðtþþx i, ðtþš ¼ m i, x i, ðtþþb i, _x i, ðtþ (3) Taking the Laplace Transform of the system of equations, setting s ¼ jo, and rearranging into matrix form yields the following: G r ðjoþ k 1 k 1 x r ðjoþ >< >= >< F m ðjoþ >= 6 4 k 1 G i,1 ðjoþ k 1 þk 7 5 x i,1 ðjoþ k 1 k 1 þk G i, ðjoþ >: x i, ðjoþ >; ¼ 0 (4) >: >; 0 where G r (jo) ¼ m r (jo) þ b r (jo) þ k r þ k 1, G i,1 (jo) ¼ m i,1 (jo) þ b i,1 (jo) þ k 1 þ k, and G i, (jo) ¼ m i, (jo) þ b i, (jo) þ k 1 þ k. Solving for x r (jo) and simplifying yields the driving-point stiffness K*(jo) model for a hydraulic engine mount with inertia tracks G i,1 and G i, F m ðjoþ x r ðjoþ ¼ Kn 64 ðjoþ¼ G rðjoþ½g i,1 ðjoþg i, ðjoþ ðk 1 þk Þ Š k 1 ½G i,1ðjoþþg i, ðjoþšþk 1 ðk 1 þk Þ ½G i,1 ðjoþg i, ðjoþ ðk 1 þk Þ Š (5) The above K*(jo) model is useful for vehicle system studies. However, for the sake of experimental studies on mounts alone [9], the cross-point dynamic stiffness (K*(jo)) is analytically found next for sinusoidal excitation x r (t)¼x r e jot and response F T (t) ¼ L T (jo) e jot where L T (jo) is the complex amplitude of transmitted force. Assume that m r E 0 and relate

5 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) Fig. 3. Typical K*(jo) predictions for a narrow-tuned fixed decoupler mount using alternate models, given by Eqs. (5) (8). Key:, K n 64 (jo);, K n 54 (jo);, K* 44 (jo);, K* (jo). F m (t) tof T (t) in Eq. (5) to yield the following K*(jo) expressions for a mount with G i,1 and G i, terms: F T ðjoþ x r ðjoþ ¼ Kn 54 ðjoþ¼ G rðjoþ½g i,1 ðjoþg i, ðjoþ ðk 1 þk Þ Š k 1 ½G i,1ðjoþþg i, ðjoþšþk 1 ðk 1 þk Þ ½G i,1 ðjoþg i, ðjoþ ðk 1 þk Þ Š (6) where G r (jo) ¼ b r (jo) þ k r þ k 1. At a low frequency, one may assume that b r E 0, and thus G r (jo) ¼ k r þ k 1. Further, expand the G i,1 and G i, terms to G i,1 (jo) ¼ m i,1 (jo ) þ b i,1 (jo) þ k 1 and G i, (jo) ¼ m i, (jo ) þ b i, (jo) þ k 1, respectively. This yields the following reduced-order K*(jo) model: F T ðjoþ x r ðjoþ ¼ Kn 44 ðjoþ¼ G rðjoþ½g i,1 ðjoþg i, ðjoþ k 1 Š k 1 ½G i,1ðjoþþg i, ðjoþšþk 3 1 ½G i,1 ðjoþg i, ðjoþ k 1 Š (7) Further, define x eq (t) ¼ x i,1 (t) þ x i, (t). The characteristic equation of the equivalent hydraulic element is now represented by a(jo) ¼ m eq (jo) þ b eq (jo) þ k 1. Extend this concept and now define equivalent parameters of n-track mount as R eq ¼ Q n ¼ 1 R i, = P n ¼ 1 R i, ¼ R i,1 R i, =ðr i,1 þr i, Þ, with I eq ¼ Q n ¼ 1 I i, = P n ¼ 1 I i, ¼ I i,1 I i, =ði i,1 þi i, Þ. Likewise, one may define m eq ¼I eq A r, Ieq ¼effective inertance for a parallel combination of I i s; b eq ¼R eq A r. Also, define the following: Req ¼ an effective resistance for a parallel combination of R i s; k eq ¼A r/ceq,andc eq ¼ aneffectivecomplianceforaseriescombinationofc i s. Algebraic manipulation yields the following reduced order (second order in o for both numerator and denominator) representation: h i F T ðjoþ x r ðjoþ ¼ Kn ðjoþ¼ G rðjoþaðjoþ k ðk r þk 1 Þ m eq ðjoþ þb eq ðjoþþ krk1 1 kr þ k 1 ¼ (8) aðjoþ m eq ðjoþ þb eq ðjoþþk 1 when sample parameters (such as k r ¼ 10 N/mm, A r ¼ 95 mm, C 1 ¼.1 mm 5 /kn, R i ¼ 8.81 GPa s/mm 3, I i ¼ 1.98 kg/cm 4, R d ¼ 441 GPa s/mm 3 ) are considered, the K n ðjoþ model of Eq. (8) yields almost the same results in Fig. 3 up to 50 Hz as the higher order models from Eqs. (5) (7). Thus, the K n ðjoþ model is adopted for design studies in the subsequent sections. 5. Narrow band tuning concepts with n identical inertia tracks The K n ðjoþ formulation can be interpreted as a combination of a damped Helmholtz resonator in the denominator (den) and a viscously damped mechanical oscillator in the numerator (num). Their natural frequency and damping ratio expressions are summarized below. From Eq. (8), the numerator has been simplified to a standard representation and interpretation of an LTI lumped parameter second order mechanical system in both the numerator and denominator. Further, both numerator and denominator expressions are normalized by the equivalent p inertia track mass m eq and using the following standard definitions for a mechanical oscillator: z¼b/(mo n )ando n ¼ ffiffiffiffiffiffiffiffiffi k=m. The resulting formulation is given as K n ðjoþ¼ ðk r þk 1 Þ½ðjoÞ þz n,num o n,num ðjoþþo n,num Š ½ðjoÞ þz n,den o n,den ðjoþþo n,den Š (9a) o n,num ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k r I eq ðc 1 k r þa r Þ (9b)

6 514 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) and sffiffiffiffiffiffiffiffiffiffiffi 1 o n,den ¼ I eq C 1 z n,num ¼ R eq z n,den ¼ R eq sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 1 k r þa r Þ I eq k r sffiffiffiffiffiffi C 1 I eq (9c) (9d) (9e) It is well known that a mount can be tuned by changing the geometry (l i, d i ) of the inertia tracks. The parameters of K n ðjoþ suggest that the values of o n,num and o n,den depend on I eq. However, this will also influence z n,num and z n,den. Since the linear R i is very sensitive to d i, it would be necessary to tune both l i and d i simultaneously to obtain the desired performance. Furthermore, the greater the number of inertia tracks n, the more flexible the device is with respect to tunability. The K n ðjoþ model is used to illustrate the benefits of n Z inertia tracks. For the calculation of I eq and R eq, the smallest I i and R i will dominate, respectively, and two limiting cases will bound these equivalent track parameters. When I i,1 and R i,1 are much smaller than I i,, I i,3, y, I i,n and R i,, R i,3, y, R i,n, respectively, I eq E I i,1 and R eq E R i,1. This limiting case is not particularly interesting, since the n-track mount will therefore effectively behave like a single track device. However, the second limiting case is particularly interesting. For a device with n-identical inertia tracks, I i,1 ¼ I i, ¼ I i,3 ¼¼I i,n, R i,1 ¼ R i, ¼ R i,3 ¼ ¼R i,n. It follows that I eq ¼I i,1 /n and R eq ¼R i,1 /n, respectively. This type of device should significantly change the dynamics when compared to a single track case with I eq ¼I i,1 and R eq ¼R i,1. Using this fact, sample dynamic stiffness spectra for 1 r n r 7 are displaced in Fig. 4 using the K n ðjoþ model. In general, not only is a mount with n Z 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 Fig. 4 that as n increases, 9K n ðjoþ9 and j K*(jo) also increase, which is good for motion control of the engine bounce mode. This is expected, as an increase in n will introduce more fluid oscillations to contribute to F T (jo). When comparing the same n ¼ result (shown in Fig. 4) to the largest inertia track case of n ¼ 7 above, it can be seen that ð9k n 9=k r Þ max increased from approximately 4.5 to 6.5. In addition, j K*,max increases from approximately 671 to Asa result, the sensitivity of the mount is increased as well near ð9k n 9=k r Þ max and j 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 o jk. For the n ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi case, o =p ¼ 8 Hz is bounded by o jk d,num=p ¼ 5 Hz and o d,den =p ¼ 10 Hz where o d,num ¼ o n,num 1 z num and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o d,den ¼ o n,den 1 z den. However, for the n ¼ 7 track case, the o d,num =p ¼ 11 Hz and o d,den =p ¼ 0 Hz values bound the o jk =p of 16 Hz. Lastly, it can be seen that an increase in n makes the device more narrowband in nature. Expressions relating the mount parameters o n,num, o n,den, z n,num, and z n,den to change in n are developed below for the n-identical track case (from Eq. (9)) using I eq ¼I i,1 /n and R eq ¼R i,1 /n. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nk o r n,num ¼ I i,1 ðc 1 k r þa r Þ (10a) Fig. 4. Variations in K * (jo) spectra for a narrow band mount with n-identical inertia tracks.

7 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) rffiffiffiffiffiffiffiffiffiffiffi n o n,den ¼ I i,1 C 1 z n,num ¼ R i,1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 1 k r þa r Þ ni i,1 k r sffiffiffiffiffiffiffiffiffi z n,den ¼ R i,1 C 1 ni i,1 (10b) (10c) (10d) The result for o n,den is comparable to analytical approximations for o jk presented by Zhang and Shangguan [], Shangguan et al. [3],and Lu and Ari-Gur [4]. Nevertheless, o n,num and the expressions for z n,num and z n,den in (10c) and (10d) are refinements of the existing theory [1] based on the K n ðjoþ model which can better approximate o j for a K narrow-tuned device. Note that z n,num, and z n,den will decrease with n even though j K* (jo) generally grows. This is because the number of parallel paths between #1 and # increase, and R eq will decrease with n so long as the mount inertia track configuration does not satisfy the first limiting case. Note that the theory described in Section 4 can be conceptually applied to any case of multiple inertia tracks, not just the limiting case of n-identical tracks. These cases exhibit system parameters that will fall somewhere between the n-identical track case and the n¼1 track case. This is true as long as I eq and R eq can be expressed in terms of I i,1 and R i,1, respectively (preferably the track with the smallest values of I and R). Sensitivity analysis of o n,num, o n,den reveals that a good practical limit on the number (n) of inertia tracks is 3 or 4. This takes into consideration the tunability of o n,num, o n,den within this regime, in addition to geometric, design, material and manufacturing, constraints for a passive device. 6. Broad band tuning concepts The order reduction of Section 4 still applies when the mount exhibits a broadband dynamic stiffness response. While there is no typical broadly tuned response, these devices are known to be highly damped. Increases in z num, z den compared to the narrow-tuned case should result in a broad-tuned mount. Shown in Fig. 5 are the same order reductions performed for the narrowband device in Fig. 3 simply by increasing the previous R eq by a factor of 5; all other mount parameters remain unchanged. For a broad-tuned device, the dynamic stiffness spectrum remains almost unchanged as the order of K n ðjoþ is reduced. As a result of increasing R eq by 5, z num and z den have increased by the same factor. Some of the same issues near o/p ¼ 0 Hz occur once again due to numerical calculations, as expected. One very important note to make is the inability to approximate o for broadband responses. The o jk n,num/p, o n,den /p of the mount will remain the same as those values in narrowband cases (6.44 Hz and 11.0 Hz, respectively), since R eq was the only parameter changed. However, o jk =p is seen to decrease significantly from 8 Hz to around Hz, since the damping ratio increases. The system now becomes overdamped in a broadband device. As a result, f d - 0 asz - 1, and the o n,num o n,den bound will occur somewhere to the right of o in a broadband device using jk Kn ðjoþ. This illustrates the errors in approximating o j K for broad-tuned devices using the o d,num o d,den bounding method (a reasonable approach for narrow-tuned devices). Furthermore, since the broad-tuned behavior is very dependent on z 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 Fig. 5. Typical K*(jo) predictions for a broadband fixed decoupler mount using alternate models, given by Eqs. (5) (8). Key:, K n 64 (jo);, K n 54 (jo);, K 44 * (jo);, K * (jo).

8 516 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) narrowband behavior, since z decreases with n. Therefore, although it is difficult to approximate o jk for a broad-tuned device, it is possible to tune broadband behavior with z by introducing highly damped inertia tracks to the system. Note that since all published results [1 13] focus on narrow-tuned devices, it is difficult to validate the accuracy of broad-tuned K n ðjoþ spectra with prior formulations. Nonetheless the Kn ðjoþ expression is developed in the same manner as the linear models formulated by Singh et al. [1] for n ¼ 1. By normalizing the K n ðjoþ model here appropriately to behave like an n ¼ 1 track mount and comparing it to the model of Singh et al. [1], the accuracy of K n ðjoþ is illustrated as virtually the same results are obtained (similar to Fig. 5). 7. Diagnosis of a production mount of unknown configuration 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 ¼ 4 6 mm and l i ¼ 10 5 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, and thus further predictions will focus on such devices. Some production mounts with an n ¼ 1 capillary tube-type track seem to exhibit broad band responses, primarily based on a leakage path (of area A between #1 and #). For this reason, the linearized fluid resistance (R o ) concept is also used to describe the leakage orifice (with subscript o), about an operating point such as flow q o where c o is the coefficient of discharge [30]. ddp dq ¼ R o ¼ r q o (11) q ¼ qo Using the K n ðjoþ model, broadband dynamic stiffness predictions are made via specification of k r and k 1, fluid properties in #1 and #, and l i and d i. This is done by best-fitting to a set of measured broadband dynamic stiffness data from a sample black box production mount. Note that the internal dimensions of this mount or its configuration are not available. Nevertheless, the K n ðjoþ model utilizing different combinations of linear capillary tube and orifice-type tracks in n ¼, 3, and 4 track configurations are attempted in order to match the measured dynamic stiffness of an unknown track configuration at both X r ¼ 1.00 mm and X r ¼ 0.5 mm excitations. Models assume the following values to ensure that adequate j K*,max can be represented near low frequency resonance [1]: r ¼ 1059 kg/m 3, m ¼ 1.00 mpa s for a percent ethylene glycol water antifreeze solution by volume [9], steady volumetric flow rate q o ¼ 5 cm 3 /s and c o ¼ 0.61 for a sharp-edged orifice. Fig. 6 summarizes the results of broadband response approximations using K n ðjoþ to simulate the measured data. It is apparent that designs utilizing one capillary tube-type track and n 1 orifice-type tracks describe a broad-tuned device particularly well. The total volumetric flow within a given orifice-type track is approximated using their appropriate areas as given by q o ¼ A o = P n ¼ 1 A i, qeq. This assumes the flow velocity in each of the tracks is equal, i.e. _x i,1 ðtþ¼_x i, ðtþ¼ ¼ _x i,n ðtþ. The results of Fig. 6 show a reasonable match between measurements and assumed models, especially for the shapes and absolute values of 9K n ðjoþ9=k r. However, the linear theory is rather limited in predicting j K*,max, especially at low excitation amplitude. When X r ¼ 1.00 mm, K n ðjoþ does a reasonable job at matching o j K and j K*,max ; however, o d,num o d,den does a poor job of bounding o jk. This approximation error is the result of the large amounts of damping c o A o Fig. 6. Diagnosis of a production broadband mount using predicted and measured dynamic stiffness spectra. (a) For X r ¼1.00 mm; (b) for X r ¼0.30 mm. Key:, measured stiffness of a production mount with unknown configuration;, K n (jo) prediction with n¼ tracks (one capillary tube, one sharpedged orifice);, K n (jo) prediction with n¼3 tracks (one capillary tube, two identical sharp-edged orifices);, K n (jo) prediction with n ¼ 4 tracks (one capillary tube, three identical sharp-edged orifices).

9 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) from z num and z den in broad-tuned devices. Furthermore, for X r ¼ 0.5 mm, the shape of the j K* (jo) could not be matched well at all within the o=p ¼ 150 Hz tuning range. Note that there is no distinct o jk present for this particular device. Interestingly, this production mount resembles the response yielded by a Voigt-type model of a purely rubber mount (with fluid drained) [1]. This suggests that the hydraulic path is nearly negligible at this excitation amplitude. Consequently, inertia tracks have a higher R eq at this smaller excitation amplitude, as evidenced by the large z num and z den needed to reproduce 9K n ðjoþ9=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 amplitude-dependent properties; this issue must be considered in future studies. 8. Experimental results for narrow-tuned mounts Mount designs utilizing only capillary tube-type orifice tracks can be categorized as narrow-tuned devices. This corresponds to designs D1 through D3 as listed in Table 1. Geometries for l i, d i of designs D and D3 are chosen to study the effect of tuning a device using n ¼ 3 tracks when compared to a baseline configuration (design D1). Namely, introducing more tracks will change the system parameters o n,num, o n,den, z num, and z den as discussed earlier. Design D is representative of the limiting case of n-identical tracks and is expected to display the biggest change in system parameters compared to design D1. Furthermore, the system parameters of design D3 are expected to lie somewhere between those for designs D1 and D. Predictions using K n ðjoþ and measurements are summarized in Figs. 7 9 for both the X r ¼ 0.3 mm Fig. 7. Dynamic stiffness comparisons for design D1. (a) Without empirical coefficients; (b) with empirical coefficients. Key: X r ¼ 1.0 mm;, K n (jo) prediction with X r ¼ 1.0 mm;, measured with X r ¼ 0.3 mm;, K n (jo) prediction with X r ¼ 0.3 mm., measured with Fig. 8. Dynamic stiffness comparisons for design D. (a) Without empirical coefficients; (b) with empirical coefficients. Key: X r ¼ 1.0 mm;, K * (jo) prediction with X r ¼ 1.0 mm;, measured with X r ¼ 0.3 mm;, K n (jo) prediction with X r ¼ 0.3 mm., measured with

10 518 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) Fig. 9. Dynamic stiffness comparisons for design D3. (a) Without empirical coefficients; (b) with empirical coefficients. Key: X r ¼1.0 mm;, K n (jo) prediction with X r ¼1.0 mm;, measured with X r ¼0.3 mm;, K n (jo) prediction, with X r ¼ 0.3 mm., measured with and X r ¼ 1.0 mm excitations. Note that the X r values are peak displacement amplitudes [9], and thus all subsequent results are given accordingly. The K n ðjoþ model is in good agreement with measurements at both X r ¼ 0.3 mm and X r ¼ 1.0 mm. In specific, K n ðjoþ is seen to match the shape of 9K n ðjoþ9=k r and j K* (jo) over the frequency range of interest, especially near o. Note jk that K n ðjoþ 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 R i. This is attributed to the simplifications needed for this fluid resistance model. The capillary tube model assumes steady flow for a very long tube of l i without the end effects. This is a gross approximation for this mount, as flow here is oscillatory, l i is finite and will have some end effects due to area expansion at #1 and #, and the tubes themselves are curvilinear. These additional effects (including turbulence) are expected to add more damping to the mount. For this reason, each K n ðjoþ prediction is also made using an empirical coefficient (g) to better approximate the actual system in Figs. 7b, 8b, and 9b. Introduce an empirical coefficient g to enhance the capillary tube resistance R i as follows: R i ¼ g 18ml i (1) pd 4 i p Here, g is chosen to be 14 ffiffiffiffiffiffiffiffi 4=3. This enhancement seems to improve the damping via improvements in ð9k n 9=k r Þ max in addition to j K*,max near o. To account for end effects, l jk eq ¼ (4/3)l i can be substituted for the geometric length in I i to obtain finer tuning of o jk [30]. In general, the mount is seen to display amplitude-dependent response behavior. Upon inspection of k 1 values, k 1 is generally larger for the high amplitude excitation cases when compared to X r ¼ 0.3 mm. This evidences the dominance of the hydraulic path over the parameters of #r when X r is large. The k 1 values from the K n ðjoþ 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 nonlinear due to X r and o, as well as x r from preload F m [1 9]. Furthermore, resonant and anti-resonant peak magnitudes for 9K n ðjoþ9k r are seen to be larger for the low amplitude excitation condition. Upon further inspection, 9K*(jo)9 is generally larger for the X r ¼ 1.0 mm case on an absolute basis. However, the normalization by k r makes this phenomenon unclear. A closer look at the static stiffness of #r reveals that k r is actually larger for X r ¼ 1.0 mm when compared to X r ¼ 0.3 mm. The nonlinear load-deflection curve of the mount with fluid drained is used as an explanation for this k r behavior; such results [7,9] are not displayed here. To study the effect of dynamic tuning of narrowband designs using the n-track concept, consider the system parameters of the mount o n,num, o n,den, z num, and z den. Variations in o n,num, o n,den, z num, and z den as designs and 3 are compared to design D1 are appropriate means of evaluating and predicting the tunability of the mount due to the n-track concept. Namely, approximations for o can be made for these (and similar narrow-tuned) device designs, since o jk n,num, o n,den, z num, and z den are expected to serve as good bounds through o d,num, o d,den. The same K n ðjoþ predictions used to fit the measured results of the prototype for designs 1 3 have now been plotted together at each X r for the sake of tuning comparison purposes in Figs. 10 and 11. Qualitatively speaking, the dynamic stiffnesses of designs D1 through D3 follow the expected trends identified in Section 5 for n-track mounts, as seen in Figs. 10 and 11. When comparing design D1 to design D, R eq and I eq are reduced by a factor of 1/3 due to geometric track relationships. The following equations illustrate the ability to tune the natural frequencies and damping ratios of the mount predicted by K n ðjoþ: p ðo den Þ D - ffiffiffi 3 ðoden Þ D1 (13a)

11 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) Fig. 10. Dynamic stiffness comparison of three narrowband designs with X r ¼ 1.0 mm excitation; (a) without empirical coefficients; (b) with empirical coefficients. Key:, design D1;, design D;, design D3. Fig. 11. Dynamic stiffness comparison of thee narrowband designs (D1, D, and D3) with X r ¼0.3 mm excitation; (a) without empirical coefficients; (b) with empirical coefficients. Key:, design D1;, design D;, design D3. ðz den Þ D - p 1 ffiffiffi ðz den Þ D1 (13b) 3 ðo den Þ D3-3 ðo denþ D1 (13c) ðz den Þ D3-3 ðz denþ D1 (13d) Eqs. (13a) and (13b) compare design D to design D1, and (13c) and (13d) reflect these changes when comparing design D3 D1. It can be seen that a bigger change in o n,den and z den is expected for the limiting case of n-identical tracks. A pattern emerges here when considering these specific designs which is not obvious in the arbitrary limiting case presented before. Namely, when comparing the linear system mount parameters of the n ¼ 1 track mount case to some n 4 1 case, o n,den increases by some direct proportionality constant, while z den 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 should not change when the empirical coefficient g is introduced to the R i model, since this adjustment should be applied to each track equally. The system parameters behave qualitatively as expected, with the parameters of design D3 falling somewhere between designs D1 and D. In particular, the expressions of Eq. (13) are seen to be good approximations when tuning the n ¼ 1

12 50 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) track case of design 1 to some n 4 1 track case, as shown using the system parameters of designs and 3. This is especially true for the o n,num, o n,den, z num, and z den predictions of X r ¼ 0.3 mm. Nonetheless, there are some errors present for the system parameter predictions using the design 1 values when X r ¼ 1.0 mm. This is due to o jk being about 1 Hz less than that of design 1 for the X r ¼ 0.3 mm case. This discrepancy can be seen upon inspection of Fig. 11. As a result, the predicted values for o n,num, o n,den are lower than those for the X r ¼ 0.3 mm case and the values of z num, and z den slightly higher from K n ðjoþ. However, using the o n,num, o n,den, z num, and z den parameters of the X r ¼ 0.3 mm case in place of the X r ¼ 1.0 mm case shows good agreement in predicting o n,num, o n,den, z num, and z den for designs D and D3 when using Eq. (13). For this reason, it is suspected that o for design 1, X jk r ¼ 1.0 mm, may be incorrect on the basis of tuning predictions being otherwise consistent. Furthermore, o n,num, o n,den, z num, and z den are observed to provide good bounds for approximating o with o jk d,num, o d,den in narrow-tuned devices such as these. 9. Experimental results for quasi-broadband and broadband mounts 9.1. Experimental results for quasi-broadband mount The proposed theory does not suggest when a transition from narrowband to broadband behavior might occur. For this reason, design 4 in experimental studies incorporates n ¼ tracks (capillary tube-type and orifice-type track in tandem) with the intent of observing the transition. While the capillary-type and orifice-type tracks are physically dissimilar, they can still be compared qualitatively on the same R and I. Recall the narrowband results where z n,den is expected to increase as the design of the mount deviates from the n-identical track limiting case, while o n,den 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 ¼ 1 track case (which would increase o n,den ), while R eq E R i. When dissimilar R o and R i models are introduced to describe R eq of the tracks G i,1 and G i, (as is the case for design 4), R eq will deviate the furthest from the n-identical capillary-type track case, and R eq E R i. This is especially true when R eq is comprised of one or more R o terms, since R o is typically 3 orders of magnitude larger than R i given typical mount parameters and operating conditions. While R eq remains relatively unchanged compared to the n ¼ 1 track case as dictated by R i, z n,den will increase due to orifice inertance I o. Since V o ) V i, I o ) I i, and as a result, I eq EI o. For this reason, the smaller I o is, the larger z den will be. As a result, z n,den can be controlled by designing A o of the orifice-type inertia track properly. This control of z n,den will be the focus of the broadband device tuning, since o is difficult to bound with o jk d,num, o d,den as I eq decreases and z den increases. The orifice of design 4 is 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 E 1 and is modeled as a short-tube orifice for K n ðjoþ predictions. The discharge coefficient of such an orifice has the following empirical formulation [30] based on the Reynolds number (Re): 8 1=, 1:5þ13:74 lo do Re >< 450 do Re lo c o ¼ 1=, (14) :8þ64 lo do Re >: 450 do Re 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 n ðjoþ predictions match well with the measured results of design 4 in Fig. 1 for X r ¼ 0.3 mm and lo Fig. 1. Bounded dynamic stiffness comparisons for design D4. (a) Without empirical coefficients; (b) with empirical coefficients: X r ¼ 1.0 mm;, K * (jo) prediction with X r ¼ 1.0 mm;, measured with X r ¼ 0.3 mm;, K n (jo) prediction with X r ¼ 0.3 mm., measured with

13 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) X r ¼ 1.0 mm excitations. An empirical parameter e (as shown below) is used to enhance R o to those seen from measurements. It can be thought of as a means to scale c o into an effective discharge coefficient c eq for the orifice model R o ¼ e rq o ¼ e rq o c o A o c eq A o (15a) I o ¼ b rl o A o (15b) Here, b ¼ 0.85, e ¼ 0.004, and c eq ¼ c o =e. TheKn ðjoþ model is seen to match the measured quasi-broadband response of design 4 well. Note that K n ðjoþ predictions without empirical adjustments fit the measurements better than those predictions made with the empirical coefficients, especially near j K*,max. Furthermore, inspection of Fig. 1 reveals that when comparing the quasi-broadband response to the previous narrowband designs, 9K n ðjoþ9=k r is very stiff over a larger range of frequencies. In addition, the j K* (jo) spectra is more distributed over the o/p ¼ 1 50 Hz regime. The shape and angles of j K* (jo) are approximated fairly well here, although it is much more difficult todosoherewhencompared totruenarrowband responses due to the limitations of linear theory. 9.. Experimental results for broadband mount The true broadband device of design D5 in experimental studies is physically similar to design 4 in that it will also have one capillary tube-type and one orifice-type track drilled through the orifice disc. However, the orifice-type track of design 5 has a diameter that is almost.5 times larger than that used in the quasi-broadband device, resulting in an 18 percent increase in the track area. This change results in a smaller I o which is in turn expected to raise z den higher than previously to obtain a true broad-tuning. Using the same c o for a straight tube orifice in Eq. (14), K n ðjoþ predictions for the measurements for design D5 are shown in Fig. 13. Once again, K n ðjoþ predictions match measured results for design 5. This device displays substantial damping via j K* (jo) over a wider range of frequencies when compared to the quasi-broadband device in design 4, in addition to a large magnitude 9K n ðjoþ9=k r with a relatively flat profile. Note that j K* (jo) is seen to be more difficult to predict using linear methods, evidencing a need for analysis of the nonlinear broadband devices. Good data are acquired to only o/p ¼ 35 Hz during testing of this device. This was because it was particularly tough to measure K n ðjoþ within the specified F m and X r tolerances at frequencies higher than this due to the discontinuous nonlinearity in rubber. However, note that 9K n ðjoþ9=k r is still seen to be increasing beyond 35 Hz in the X r ¼ 1.0 mm case. The unadjusted K n ðjoþ predictions are more accurate in approximating 9Kn ðjoþ9=k r and j K* (jo) than the predictions using the empirical coefficients. This was also the case for predictions of design D4, especially when approximating j K* (jo). It is suspected that the large magnitudes of z den are not as influenced by R eq as they are I eq due to the very small I o in such designs. As was seen in the narrowband designs, k 1 values vary based on design and excitation amplitude, evidencing a nonlinear C 1. This is especially true for design D4 when X r ¼ 1.0 mm, where k 1 is seen to spike to almost twice the values used for other K n ðjoþ predictions. These empirical coefficients have no apparent effect on variation in k 1 once again. Finally, it should be mentioned that 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 reasonable to suggest that when z num Z 0.7 and z den Z 0.5, the mount starts to behave in a broad-tuned manner. Fig. 13. Bounded dynamic stiffness comparisons for design D5. (a) Without empirical coefficients; (b) with empirical coefficients: X r ¼ 1.0 mm;, K n (jo) prediction with X r ¼ 1.0 mm;, measured with X r ¼ 0.3 mm;, K n (jo) prediction with X r ¼ 0.3 mm., measured with

14 5 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) Fig. 14. Bounded dynamic stiffness comparisons for design D6. (a) Without empirical coefficients; (b) with empirical coefficients:, measured;, K * (jo) prediction, n ¼ tracks (one capillary tube, one sharp-edged orifice);, K n (jo) prediction, n¼3 tracks (one capillary tube, two sharpedged orifices);, K n (jo) prediction, n¼4 tracks (one capillary tube, three sharp-edged orifices) Experimental results for controlled leakage path design To study leakage in a controlled and simple manner, the gasket and silicone caulking materials used to seal #1 from # around the circumference of the orifice disc in design 1 are removed. In this manner, an essentially constant area, small annular opening now exists for fluid to communicate between #1 and # in tandem to the existing external capillary tubetype track. This new passage introduces a large amount of R and a small amount of I (when compared with design D1). For this reason, modeling this leakage path as a constant area orifice-type track is justified. Design 6 is tested at X r ¼ 1.5 mm to exaggerate the effects of hydraulic force path and illustrate the leakage path distinctly with the prototype. Since flow between #1 and # takes place around the outer edges of the orifice disc, c o ¼ 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 is described on the premise of the n- inertia track concept using orifice-type tracks to broad-tune the device. This has been done in Fig. 14 for 1,, and 3 orificetrack designs. Once again, good agreement is seen between K n ðjoþ predictions and measurements for n ¼, 3, and 4, especially those made without the use of the empirical coefficients. Fig. 14a shows predictions which describe the leakage particularly well, even for j K* (jo), which has been difficult to do in the previous broad-tuned designs. Predictions in Fig. 14b with empirical coefficients (to adjust R i, R o, and I o ) are once again seen to be less accurate in predicting shape and value of 9K n ðjoþ9=k r and j K* (jo) when compared to those without empirical adjustments. 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, 9K n ðjoþ9=k r is seen to be substantially larger than any design yet. This can be attributed to the discontinuous rubber nonlinearity k r, since at this higher X r, the mount dynamics will be much more influenced by the rubber compression. Measurements are made to only o/p ¼ 30 Hz here, since it is particularly difficult to measure K*(jo) within the specified F m and X r tolerances for this design due to the discontinuous-nonlinear characteristic of rubber. 10. Conclusion The chief contribution of this article is the experimental study of a new multiple inertia track prototype mount with the capability of varying the capillary tube and orifice and number n of inertia tracks in a controlled manner. This prototype (with external passages for the sake of experimental convenience) has been used to examine several devices with different n-track configurations for improving performance compared to the conventional n ¼ 1 track case. These examples and those exemplified in the prototype are neither all-inclusive nor comprehensive for how the multiple internal passages the can be implemented in practice. Rather, they serve as a specific subset of such configurations which are possible and how they could be obtained. The multiple inertia track/orifice concept has been successfully used to dynamically tune a fixed decoupler hydraulic mount. Three narrowband devices have been designed and experimentally evaluated to refine and extend the linear system theory for predicting j K* (jo), in addition to validating K*(jo) model predictions for an n ¼ 3 track mount for the first time. Two broadband devices have been designed and evaluated successfully by tuning z den 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.

15 B. Barszcz et al. / Journal of Sound and Vibration 331 (01) The refined extensions of prior linear models are now applicable to both narrow-tuned and/or broad-tuned hydraulic engine mounting devices with n Z. In particular, the role of fluid resistance has been clarified in developing broad band devices. With respect to the existing literature on this topic [ 4], improved K n ðjoþ models have been developed here which are now accurate (for at least three tracks) and can include a combination of orifice-type tracks and capillary tubes. Improvements in analytical approximations of j K* (jo) for narrow-tuned devices have also been made. Future research should include more extensive characterization studies of the prototype mount built in this study, for both narrowband and broadband designs. This could include transient tests with ideal or more realistic inputs [8,9]. Nonlinear investigations of n-track designs (especially with orifices) should be performed as well [5 9]. Lastly, interactions between a free decoupler and multiple inertia tracks could be studied more rigorously in future investigations. Acknowledgments We are grateful to the member organizations (such as Transportation Research Center of Ohio and Honda R&D) of the Smart Vehicle Concepts Center ( and the National Science Foundation Industry/University Cooperative Research Centers program ( for supporting this work. References [1] R. Singh, G. Kim, P.V. Ravindra, Linear analysis of automotive hydro-mechanical mount with emphasis on decoupler characteristics, Journal of Sound and Vibration 158 (199) [] Y.Q. Zhang, W.B. Shangguan, A novel approach for lower frequency performance design of hydraulic engine mounts, Computers and Structures 84 (006) [3] W.B. Shangguan, Z.S. Song, Y.Q. Zhang, K.H. Jiang, C. Xu, Experimental study and simulation analysis of hydraulic engine mounts with multiple inertia tracks, Journal of Vibration Engineering (Zhendong Gongcheng Xuebao) 18.3 (005) [4] M. Lu, J. 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