Vibrational Power Flow Considerations Arising From Multi-Dimensional Isolators. Abstract
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1 Vibrational Power Flow Considerations Arising From Multi-Dimensional Isolators Rajendra Singh and Seungbo Kim The Ohio State Universit Columbus, OH , USA Abstract Much of the vibration isolation research has focused on uni-directional behavior of the sstem. For man real-life problems, the role of rotational and shear stiffness components must be understood and a multi-dimensional formulation needs to be developed, especiall at higher frequencies. Consequentl, characterization and modeling of vibration isolators is increasingl becoming more important in mid and high frequenc regimes where ver few methods are known to eist. This paper presents a new eperimental identification method that ields frequenc-dependent multi-dimensional dnamic stiffnesses of an isolator. The proposed identification method is applied to one practical rubber isolator, and eperimental results are successfull compared with data measured on commercial equipment for aial motions. The effects of multi-dimensional isolator on vibration power transmitted to the receiver structure are analticall investigated. Rigid bod and Timoshenko beam models are emploed to describe source and path respectivel. Infinite beam is used to represent a compliant foundation. Also, linear, time-invariant sstem assumption is made. 1. Introduction Vibration isolators are often characterized as discrete elastic elements, with or without viscous or hsteritic damping [1]. The compressional stiffness term is tpicall used to develop isolation sstem models though the transverse (shear) and rotational components could be a significant contributor to the vibration transmission [2]. Additionall, at higher frequencies, inertial or standing wave effects occur within the isolator. Nonetheless, the isolators are still modeled b man researchers in terms of spectrall-invariant discrete stiffness elements without an cross-ais coupling terms [2]. Such descriptions are clearl inadequate at higher frequencies. Consequentl, one must adopt the distributed parameter approach. Onl a few articles have eamined the elastomeric devices using the continuous vibration sstem theories [3-4]. Eperimental methods must be adopted to dnamicall characterize stiffnesses of rubber, hdraulic, air and metallic isolators since the invariabl ehibit frequenc-dependent properties and are sensitive to mean loads and dnamic ecitation levels. Historicall, characterization methods have focused on aial or compressional stiffness. Also, in most
2 studies onl the lower frequenc range has been considered [1-2], and consequentl the direct measurement of dnamic stiffness on commercial machines is tpicall limited to lower frequencies. Several approimate identification methods for transfer stiffness of resilient elements have been proposed at higher frequencies and have been refined for lower frequencies [5]. See references [4, 6] for a detailed literature review and a list of relevant articles. Overall, an appropriate characterization method for the measurement of multidimensional stiffnesses of an isolator has et to be proposed. The underling measurement and estimation issues are even more difficult as the frequenc increases [1, 5]. Also, no prior article has eamined the shear deformation and rotar inertia effects of an isolator. In this article, we propose a new dnamic characterization method that should be valid over low, mid and high frequenc regimes, based on a new isolator stiffness matri formulation. Further, we eamine the influence of isolator parameters on the behavior of an isolation sstem using a continuous sstem model. Problem formulation is conceptuall shown in Figure 1 in the contet of source, path (isolator) and receiver. The scope is limited to a linear time-invariant (LTI) sstem, and the effects of preload, etc. are not considered. θ θ θ θ θ ISOLATOR (P) θ SOURCE REGIME (S) RECEIVER OR TERMINATION (R) Isolator path Beam receiver (c) Rigid mass source V S1 V S2 = V P2 V P3 = V R3 V R4 d 1 SOURCE 2 PATH 3 RECEIVER 4 (S) (P) (R) L ~ θ ~ F S1 F S2 = -F P2 (b) F P3 = -F R3 F R4 z z (d) ~ Figure 1. Problem formulation. Isolator is depicted as a multi-dimensional path for an practical problem; (b) source-path-receiver sstem and their mobilit matrices, and. Here F and V are vectors; (c) Vibration transmission via multi-dimensional motions of an isolator to a beam receiver; (d) a clindrical isolator with static stiffness components. 2. identification of mobilit matri of an isolator Multi-dimensional mobilit matri of a vibration isolator can be identified using a mobilit snthesis formulation. Figure 2 shows a schematic that is used for eperimental work. Multi-degree of freedom connections at both ends of an isolator are modeled since the information at both ends of a sub-sstem is needed. Two masses are attached to an isolator as shown in Figure 2 and the snthesized mobilit for the overall sstem is formulated first.
3 Then the mobilit matri of an isolator is reformulated given the snthesized formulation. Finall, the mobilit matri of an isolator can be obtained b substituting the snthesized mobilit matri with measured mobilit matri for the combined sstem. It is possible to measure all elements of the multi-dimensional mobilit matri from the fact that an application of force with an offset from the reference point results in both force and moment simultaneousl. Mobilit model for one rubber isolator is identified using the proposed procedure. Two masses are attached to the ends of each isolator and the combined sstem of Figure 2 is suspended to simulate free boundaries. The reciprocit principle has been applied throughout the snthesis procedure since small inconsistencies or noise in frequenc response function measurements can significantl contaminate results via the numerical inversion process that is essential to the entire procedure. Tpical results for transfer stiffnesses are shown in Figures 3 and 4 for an isolator eample of Figure 2(b). Note that the eperimental validations are conducted using the MTS machine (up to 1 Hz). The MTS method emplos the blocked end boundar and onl the transfer stiffness is measured. First, stiffness modulus and loss angle in aial direction are shown in Figures 12. Static stiffness in aial motion is 5 N/mm for the isolator. Identified results from the proposed eperiment are given from to 2 Hz. Predictions for zero preload are compared with the MTS test results that are obtained under three different preloads, up to 1 khz. Ecellent agreements are observed between results based on the proposed identification method and eperimental data ield b the MTS machine. Similar results are seen for two additional mounts [4] mm Rigid Mass Isolator Rigid Mass 19. mm Aluminum Plate Neoprene Aluminum Plate Figure 2. Proposed identification method. Simplified scheme to identif mobilities of an isolator (b) practical rubber isolator used for eperimental studies K ~ 1 (N/mm) Figure 3. Comparison of aial dnamic stiffnesses for isolator. Dnamic stiffness modulus; (b) loss angle. Ke:, Mobilit model;, Measured (MTS): mean = 3 N;, Measured (MTS): mean = 12 N;, Measured (MTS): mean = 23 N. (degrees) φ K (b)
4 6 3 Mag: N/mm 4 2 Mag: N/rad 2 1 Mag: N Mag: Nm/rad Figure 4. Transfer stiffnesses in fleural motion for isolator, as etracted using the identification scheme. Lateral stiffness modulus; (b) coupling stiffness modulus; (c) coupling stiffness modulus; (d) rotational stiffness. Ke:, Eperimental result;, curve fit. 3. Vibration power transmitted to an infinite beam receiver The vibrational behavior is eamined for an isolation sstem (Figure 1c) with an infinite beam receiver. Harmonic ecitation is applied at the mass center of a cubic rigid bod with a mass of 1 kg and a length of 5 mm. Circular isolator is shown in Figure 1(d) along with vibration components transmitted through the path. The isolator is modeled using the Timoshenko beam theor (fleural motion) and the wave equation (longitudinal motion). Detailed mathematical treatment is given in the recent journal article we wrote [6]. Note that the coupling mobilit does not eist for an infinite beam receiver. However, a coupling arises because the motion (or force) in shear direction of an isolator is coupled with the longitudinal direction of a receiver beam. The Young s modulus, shear modulus and mass densit of a rubber isolator of for this stud are 1.62 MPa, 5 MPa and 1 kg/m 3 respectivel. Also, the circular clindrical isolator has the length of 3 mm and the radius of 12 mm. Material properties of a receiver beam having a thickness of 1 mm and a width of 1 mm are MPa, 2723 kg/m 3 and.1 for Young s modulus, mass densit and loss factor respectivel. In order to understand the effect of isolator properties, it is useful to eamine the static stiffnesses ( ) of an isolator. It should be noted that fleural stiffnesses have to be dealt with in a matri form since there eist coupling terms between lateral (shear ) and rotational ( θ ) stiffnesses. Further, note that (or ) is common to all stiffness terms. Highl damped material with a loss factor of.3 is used for this isolator so that overall frequencdependent characteristics are observed without the influence of isolator resonances. Results for are shown in Figure 5 for a variation in values for an isolator when a harmonic moment is applied to the mass center of a rigid bod source. It is observed in Figure 5 that rises due to an increase in as the frequenc increases. Additionall, the spectra of
5 Figure 5 decrease as the frequenc increases. The power efficienc is also shown in Figure 5(b) when a harmonic force ( ) is applied to the mass center of a source. In this case, onl aial stiffness of the mount affects vibration power transmission. Like the moment application case, grows with as the frequenc increases. Also, note that the spectra of Figure 5(b) for the aial power transmission are close to unit at low frequencies unlike the one of Figure 5 for the fleural power transmission. Normalized power components with respect to the total actual power transmitted to the receiver beam are also shown in Figures 5(c-d) for the shear modulus variation. As discussed previousl, aial and coupling power components do not eist in this case and therefore the sum of the normalized lateral (shear direction of mount) and rotational power components is equal to unit. Overall, the lateral power component is larger than the rotational component. It is shown in Figure 5(c-d) that the lateral component dominates Γ 1 2 Π / Π Total Γ (b) Π / Π Total (c) Figure 5. Effect of shear modulus of an isolator on efficienc ( ) with an infinite beam receiver. For a Timoshenko beam isolator model given moment ecitation; (b) given force ecitation. Ke:,.5;, ;, 2. (c) normalized vibration power transmission given moment ecitation.5g; (d) normalized vibration power transmission given moment ecitation 2G. Ke:, aial ();, rotational ( θ );, lateral ().
6 Conclusion A new characterization method has been proposed for the identification of multidimensional frequenc-dependent transfer stiffnesses of an isolator. Our method uses a phsical sstem that consists of two inertial elements and an isolator. Further, refined multidimensional mobilit snthesis and decomposition procedures have been formulated. Results of the proposed scheme compare well with test data for one practical isolator in aial motions on a commercial machine up to 1 khz. Another main contribution of this paper is the application of continuous sstem theor to an elastomeric isolator and the eamination of associated fleural and longitudinal motions of the source-path-receiver sstem. Two different frequenc response characteristics of an elastomeric isolator are predicted b the Timoshenko beam theor. The second tpe solution, that has been previousl believed to occur at etremel high frequencies (sa around 8 khz) for metallic structures and therefore not of interest in structural dnamics, takes place at relativel low frequencies (sa around 1.5 khz) for a rubberlike material. The behavior of a tpical vibration isolation sstem has been eamined using the power transmitted to an infinite beam receiver, when ecited b a harmonic moment or force at the source. Parametric stud of isolator properties on the transmission measures has been conducted using the Timoshenko beam isolator model and an infinite beam receiver. The vibration power efficienc for an isolation sstem with an infinite beam structure increase with frequenc as the isolator shear modulus is increased. Future work is required to quantif the vibration source in terms of power transmission. Proper interpretation of various vibration isolation measures for a multi-dimensional sstem, such as power efficienc and effectiveness, must also be sought over a broad range of frequencies. Finall, nonlinear effects of an isolator should be eamined. Acknowledgements The General Motors Corporation (Noise and Vibration Center) and the Goodear Tire and Rubber Compan (Transportation Molded Products) are gratefull acknowledged for supporting this research. References 1. T. JEONG and R. SINGH 2, Inclusion of Measured Frequenc- and Amplitude-Dependent Mount Properties in Vehicle or Machiner Models (in press). 2. M. A. SANDERSON 1996, (2), Vibration Isolation: Moments and Rotations Included. 3. P. GARDONIO, S. J. ELLIOTT and R. J. PINNINGTON 1997, (1), Active Isolation of Structural Vibration on a Multiple-Degree-of-Freedom Sstem, Part I: The Dnamics of the Sstem. 4. S. KIM and R. SINGH 2, Multi-Dimensional Characterization of Vibration Isolators over a Wide Range of Frequencies (in press). 5. D. J. THOMPSON, W. J. VAN VLIET and J. W. VERHEIJ 1998, (1), Development of the Indirect Method for Measuring the High Frequenc Dnamic Stiffness of Resilient Isolator. 6. S. KIM and R. SINGH 21, Vibration Transmission Through an Isolator Modeled b Continuous Sstem Theor.
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