Vibration Power Transmission Through Multi-Dimensional Isolation Paths over High Frequencies

Transcription

1 Vibration ower ransmission hrough Multi-Dimensional Isolation aths over High Frequencies Copright 001 Societ of Automotive Engineers, Inc. Seungbo im and Rajendra Singh he Ohio State Universit ABSRAC In man vibration isolation problems, translational motion has been regarded as a major contributor to the energ transmitted from a source to a receiver. However, the rotational components of isolation paths must be incorporated as the frequenc range of interest increases. his article focuses on the fleural motion of an elastomeric isolator but the longitudinal motion is also considered. In this stud, the isolator is modeled using the imoshenko beam theor (fleural motion) and the wave equation (longitudinal motion), and linear, timeinvariant sstem assumption is made throughout this stud. wo different frequenc response characteristics of an elastomeric isolator are predicted b the imoshenko beam theor and are compared with its subsets. A rigid bod is emploed for the source and the receiver is modeled using two alternate formulations: an infinite beam and then a finite beam. ower transmission efficienc concept is emploed to quantif the isolation achieved. Further, vibration power components are also eamined. he roles of isolator parameters such as the static stiffness ratios, shape factors and material properties are investigated. INRODUCION Vibration isolators are often characterized as discrete elastic elements, with or without viscous or hsteritic damping [1-4]. he compressional stiffness term is tpicall used to develop isolation sstem models [-4] though the transverse (shear) and rotational components are also sometimes specified or included [5-7]. Additionall, at higher frequencies, inertial or standing wave effects occur within the isolator [8-9]. Nonetheless, the isolators are still modeled b man researchers in terms of spectrall-invariant discrete stiffness elements without an cross-ais coupling terms [5-7]. Such descriptions are clearl inadequate at higher frequencies. Consequentl, one must adopt the distributed parameter approach. It is the main focus of this article. Onl a few articles have eamined the elastomeric devices using the continuous vibration sstem theories [10-15]. For eample, the longitudinal stiffness of an isolator has been described b the wave equation to characterize the material propert of an isolator [10-13]. Also, rubber-like material has been modeled using the wave closure relationship [14]. he Euler beam theor has been adopted to describe the fleural motions of a mount for an active vibration control sstem [10, 11] and to characterize an isolator [1]. However, no prior article has eamined the shear deformation and rotar inertia effects of an isolator. Further, the influence of component parameters on the behavior of an isolation sstem has been investigated using a standing wave description in the longitudinal direction and a static stiffness term in the fleural direction [14]. Yet, the frequenc-dependent characteristics of an isolator have not been considered in previous isolation studies. his article proposes to overcome this particular void in the literature. Vibration transmitted via multi-dimensional motions of an isolator is conceptuall shown in Figure 1 in the contet of source, path (isolator) and receiver. Figure 1. Vibration transmission via multi-dimensional motions of an isolator. (a) Sstem configuration with a beam receiver; (b) a clindrical isolator with static stiffness components used for parametric studies. Here, is the aial (longitudinal or compressional) stiffness, S L Isolator path Beam receiver z is the lateral (shear) stiffness and S S rotational stiffness. d (a) z (b) S Rigid mass source S S is the

2 rimar objectives of this stud are as follows. 1. Develop the frequenc response characteristics (mobilit or stiffness) of an isolator based on the continuous sstem theor that includes imoshenko beam (in the transverse direction) and the longitudinal () wave equation formulation. In particular, criticall eamine two tpes of solutions to the imoshenko beam for a clindrical rubber material.. Investigate the role of isolator parameters such as the static stiffness ratios, shape factors and material properties on isolation measures over a broad range of frequencies. he effects of shear deformation and rotar inertia on the characteristic mobilities for a semi-infinite rubber beam are eamined and shown in Figure. MOBILIIES OF IMOSHENO BEAM he classical imoshenko beam theor that describes the effects of shear deformation and rotar inertia has been well studied b man researchers [16-19]. Literature shows that there are two tpes of solution and two modal functions eist at high frequencies [17]. However, the high frequenc solution has been ignored b man since this phenomenon has been believed to occur onl at etremel high frequencies [16]. Onl a few studies have been conducted to eamine the dispersion and spectrum relations of the imoshenko beam structure at high frequencies [18, 19]. In our stud, we eamine this issue and the characteristic mobilities are obtained for a semi-infinite elastomeric beam. Detailed mathematical treatment is given in the recent journal article we wrote [0]. It is assumed that the shear modulus (G) is ver low that is true for a rubber-like material. he eample case considers a rubber beam with circular shape. he spectrall-invariant material properties and dimensions of the beam that is considered as an isolator are shown in able 1. able 1. Material properties and dimensions of source, isolator and receiver sstem. ropert or dimension Source (Cubic rigid bod) Isolator: baseline (Circular beam) Receiver (Rectangular beam) m (kg) E (Ma) X 10 4 G (Ma) η ρ (kg/m 3 ) Dimensions in mm L = 50 L = 30 r = 1 L = 670 (finite beam) b = 100 and t = 10 (finite or infinite beam) Figure. Characteristic mobilities of a semi-infinite beam. (a) Force mobilit v f ; (b) coupling mobilit v q ; (c) moment mobilit w q. e:, imoshenko beam; o, Euler beam with shear deformation;, Euler beam with rotar inertia;, classical Euler beam. Here, ω is 1700 Hz.

3 Note that the frequenc response characteristics of a imoshenko beam changes at the transition frequenc ( ω ). Details are given in reference [0]. It is observed from Figure that the inclusion of shear deformation increases the magnitudes of force and moment mobilities. Conversel, the rotar inertia decreases the magnitudes of force and moment mobilities. Further, the shear deformation does not affect the coupling mobilit of Figure (b) that is frequenc-invariant for the Euler beam model and the one with shear deformation onl. For this circular rubber beam, ω is approimatel 1.7 khz and the nature of spectra changes beond this transition. Beond ω, the characteristic force and moment mobilities of imoshenko beam model remain frequenc-invariant as the frequenc increases, unlike the Euler beam model. Further, the coupling mobilit decreases b the rotar inertia effect and the one of the imoshenko beam solution decreases more rapidl beond ω. Also, the coupling mobilities of the Euler beam with or without shear deformation are the same as shown in Figure (b). RANSMIED VIBRAION OWER FROM SOURCE O RECEIVER AND EFFICIENCY he time-averaged vibrational power ( Π ) components transmitted to a beam receiver are obtained b using harmonic responses and interfacial forces (F and V) and moment (M). Define Π, Π and Π as the lateral (), aial () and rotational ( ) power components respectivel: = 1 1 Re[ F( ω) v * 1 * ( ω)] = Re[ v ( ω) F ( ω)], (1a) 1 * ( ω) ] = Re[ v ( ω) V ( ω) ], (1b) * 1 * ( ω)] = Re[ w( ω) M ( ω)].(1c) [ ( ) * Π ( ω) = Re V ω v 1 = [ ( ω) Re M w Here, v, v and w are the aial (shear direction for isolator), vertical (aial direction for isolator) and rotational ( ) velocit amplitudes of the receiver beam respectivel. Finall, the total vibration power transmitted to a receiver beam is = + +. otal () Additionall, define vibration power transmission efficienc as otal Γ ( ω) =. IN (3) where Π is the harmonic power supplied to a rigid IN bod source. VIBRAION OWER RANSMIED O AN INFINIE BEAM RECEIVER SYSEM CONFIGURAION: he vibrational behavior is eamined for an isolation sstem (Figure 1a) with an infinite beam receiver. Harmonic ecitation is applied at the mass center of a cubic rigid bod. Circular isolator is shown in Figure 1(b) along with vibration components transmitted through the path. he isolator is modeled using the imoshenko beam theor (fleural motion) and the wave equation (longitudinal motion). 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. Material properties of the isolator are listed in able 1. he Young s modulus E for a rubber material is found from the relation E = 3G (1 + Q ) where Q is an empirical constant and is the shape factor. For a circular rubber clinder, Q is and is equal to r /(4L ) where r and L are the radius and length of the isolator beam respectivel [16]. Also, a frequenc-invariant loss factor η is included in the calculation with the comple-valued Young s modulus as E = E (1 + jη ) to incorporate hsteretic damping within the isolator. Material properties of the receiver beam are, as well the dimensions of source, also shown in able 1. A loss factor η of R is used to represent a lightl damped structure and included in the comple-valued E R. Given the sstem properties, the effects of isolator material and geometric properties on the vibration power transmitted to receiver are eamined. EFFEC OF ISOLAOR ROERIES ON VIBRAION OWER RANSMISSION: In order to understand the effect of isolator properties, it is useful to eamine static stiffnesses ( ) of isolator. It should be noted that S fleural stiffnesses have to be dealt with in a matri form since there eist coupling terms between lateral (shear ) and rotational ( ) stiffnesses. Further, note that G (or E) is common to all stiffness terms. Highl damped material with a loss factor of 0.3 is used for this isolator so that overall frequenc-dependent characteristics are observed without the influence of isolator resonances. Results for Γ are shown in Figure 3(a) for a variation in G values for an isolator when a harmonic moment is applied to the mass center of a rigid bod source. It is observed in Figure 3(a) that Γ rises due to an increase in G as the frequenc increases. Additionall, the Γ spectra of Figure 6(a) decrease as the frequenc increases. he power efficienc is also shown in Figure 3(b) when a harmonic force ( f ) 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, Γ grows with G as the frequenc increases. Also, note that the Γ spectra of Figure 3(b) for the aial power transmission are close to unit at low frequencies unlike the one of Figure 3(a) 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 4 for the shear modulus variation.

4 Figure 3. Effect of shear modulus G of an isolator on efficienc ( Γ ) with an infinite beam receiver. (a) For a imoshenko beam isolator model given moment ecitation; (b) given force ecitation f. e:, 0.5G;, G;, G. 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 4(b) that the lateral component dominates. Commonl, designers specif mounts in terms of lumped stiffness elements rather than continuous sstem properties. herefore, the following static stiffness ratios ( α ) are defined. Here, each ratio is normalized with respect to the aial component: S S S α =, α c =, α =. (4a-c) S S S where α, α and α are the ratios of shear, coupling c and rotational stiffness components to the aial stiffness respectivel. For a clindrical isolator, ke parameters in the static fleural stiffnesses include the slenderness ratio ( S / L), material properties (G and ν ) and. In S this case, the fleural stiffnesses change when L is varied proportionall to S, unlike the value. S Figure 4. Normalized vibration power transmitted to an infinite beam given moment ecitation. (a) 0.5G; (b) G; (c) G. e:, aial ();, rotational ( );, lateral (). Note that this behavior is also true for the Euler beam case. Further, α decreases but both α c and α increase as L or S value is increased, while holding S / L, G and ν. he effects of α on efficienc ( Γ ) are eamined in Figure 5 for the case when L is proportionall varing with S.

5 Figure 5. Effect of α (ratio of shear to aial components of static stiffness) of an isolator on efficienc ( Γ ) with an infinite beam receiver. (a) For a imoshenko beam isolator model given moment ecitation; (b) given force ecitation f. e:, 0.7 α ;, α ;, 1.3 α. Figure 5(a) shows that Γ increases as α is increased at higher frequencies when a harmonic moment is applied at mass center of a source. Note that the minimum value of α produces the best vibration isolation (hence the lowest Γ ) for a sstem with a imoshenko beam isolator as shown in Figure 5(a). he Γ spectra are shown in Figure 5(b) when a harmonic force ( f ) is applied to a source. Note that is the S onl component that affects the power transmission and is unchanged as α S is varied in this case. As epected, Γ remains unchanged for the α variations at lower frequencies. However, it is observed in Figure 5(b) that Γ increases as α increases at higher frequencies. Similar to the previous case, vibration power components are calculated in Figure 6. he dominance of lateral and rotational power components changes at certain frequenc for the lowest α value as shown in Figures 6(a). Figure 6. Normalized vibration power transmitted to an infinite beam given moment ecitation. (a) 0.7 α ; (b) α ; (c) 1.3 α. e:, aial ();, rotational ( );, lateral (). Observe for the lowest α value case that the rotational power component is dominant at lower frequencies and continues to dominate up to the mid-frequenc regime where the lateral component is important. However, the lateral component becomes more significant when α is increased and the rotational component is negligible for the highest α case. Net, the effects of the isolator

6 shape on isolation measures are eamined. he shape factor ( β ) of an isolator is defined as β = L / d. Note that an increase in β reduces the static fleural and aial stiffnesses. Results are shown in Figure 7. Figure 7. Effect of isolator shape factor β on efficienc ( Γ ) with an infinite beam receiver. (a) For a imoshenko beam isolator model given moment ecitation; (b) given force ecitation f. e:, 0.5β ;, β ;, β. he Γ value decreases at higher frequencies as β is increased when a moment is applied. Similar to the moment ecitation case, Γ decreases at higher frequencies as β is increased for a force ( f ) ecitation case as shown in Figure 7(b). Normalized vibration power components are also shown in Figure 8. Like the previous cases, the lateral component is larger than the rotational component over all frequencies. However, the rotational component becomes more important at lower and higher frequencies as β is increased. VIBRAION OWER RANSMIED O A FINIE BEAM RECEIVER Figure 8. Normalized vibration power transmitted to an infinite beam given moment ecitation. (a) 0.5 β ; (b) β ; (c) β. e:, aial ();, rotational ( );, lateral (). A finite beam receiver (with clamped ends) is emploed to eamine the vibration transmission through the isolator for a sstem of Figure 1(a). Similar to the sstem with an infinite beam receiver, the imoshenko beam model represent fleural motion of an isolator along with the wave equation for longitudinal motion. An isolator connected to a rigid bod source at one end is assumed to be located off-center ( L = L / 4 ) of the receiver RI 3 R

7 beam in order to incorporate the effect of coupling mobilit of the receiver beam. Note that such a coupling mobilit does not eist for a centrall driven beam (with both ends clamped) and for an infinite beam. he effects of G of a imoshenko beam isolator on Γ are shown Figure 9(a) when a moment is applied at a source. Net, the effect of α is eamined while holding the slenderness and material properties of the mount. Results are shown in Figure 10 for a moment ecitation case. Figure 9. Effect of isolator G on efficienc ( Γ ) with a finite beam receiver. (a) For a imoshenko beam isolator model given moment ecitation; (b) given force ecitation f. e:, 0.5G;, G;, G. Like the infinite beam receiver case, Γ rises especiall at higher frequencies as G of an isolator is increased. However, the deviation from the aforementioned behavior is observed at certain frequencies (around khz) due to the coupling mobilit and resonances of the receiver beam. Like the sstem with an infinite beam receiver, Γ of the imoshenko beam isolator shows the relativel flat spectra over the frequencies as shown in Figure 9(a). When a force f is applied to a source, Γ increases as G is increased as shown in Figure 9(b). Unlike the case of an infinite beam receiver, Γ is not close to unit at low frequencies. Figure 10. Effect of isolator α on efficienc ( Γ ) with a finite beam receiver. (a) For a imoshenko beam isolator model given moment ecitation; (b) given force ecitation f. e:, 0.3 α ;, α ;, 1.3 α. Similar to the infinite beam receiver case, Γ increases at higher frequencies as α of the imoshenko beam isolator is increased as shown in Figure 10(a). And, Γ is shown in Figure 10(b) when a force f is applied and observe that Γ increases at higher frequencies as α is increased like the moment ecitation case even though the static aial stiffness ( ) plas a major role in this S force ecitation case; is kept unchanged for all α S variations. Note that Γ remains unchanged up to a certain frequenc (around 800 Hz in this case) as α is varied, as shown in Figure 10(b). Finall, the effects of β on Γ are eamined in Figure 11. Similar to the infinite

8 beam receiver case, an increase in β decreases Γ for both moment and force ecitation cases. found to be more pronounced than the role of the rotar inertia at higher frequencies. he behavior of a tpical vibration isolation sstem has been eamined using the power transmitted to an infinite beam or a finite beam receiver, when ecited b a harmonic moment or force at the source. arametric stud of isolator parameters on the transmission measures has been conducted using the imoshenko beam isolator model and an infinite beam receiver. Material and geometric parameters of an isolator have been eamined along with the static stiffness ratios (between, and S S S components). he vibration power efficienc for an isolation sstem with an infinite beam structure increase with frequenc as the isolator shear modulus is increased. Resulting characteristics for a sstem with a finite beam receiver confirm the trends. Future work is required to quantif the vibration source in terms of power transmission. roper 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. ACNOWLEDGMENS he General Motors Corporation (Noise and Vibration Center) and the Goodear ire and Rubber Compan (ransportation Molded roducts) are gratefull acknowledged for supporting this research. REFERENCES Figure 11. Effect of isolator shape factor β on efficienc ( Γ ) with a finite beam receiver. (a) For a imoshenko beam isolator model given moment ecitation; (b) given force ecitation f. e:, 0.5β ;, β ;, β. CONCLUSION Chief 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.wo different frequenc response characteristics of an elastomeric isolator are predicted b the imoshenko beam theor. he second tpe solution, that has been previousl believed to occur at etremel high frequencies (sa around 80 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. he continuous sstem analsis clearl shows that the shear deformation and rotar inertia must be considered in order to properl describe the spectral behavior of mount in fleural motions at higher frequencies. In particular, the shear deformation effect is 1.. JEONG and R. SINGH 000 Journal of Sound and Vibration, Inclusion of Measured Frequenc- and Amplitude-Dependent Mount roperties in Vehicle or Machiner Models (in press).. H. G. D. GOYDER and R. G. WHIE 1980c Journal of Sound and Vibration, 68(1), Vibrational ower Flow from Machines into Built-Up Structures, art III: ower Flow hrough Isolation Sstems. 3. J. I. SOLIMAN and M. G. HALLAM 1968 Journal of Sound and Vibration, 8(), Vibration Isolation between Non-Rigid Machines and Non- Rigid Foundations. 4. J. C. SNOWDON 1968 Vibration and Shock in Damped Mechanical Sstems. New York: John Wile & Sons, Inc. 5. M. A. SANDERSON 1996 Journal of Sound and Vibration, 198(), Vibration Isolation: Moments and Rotations Included. 6. W. L. LI and. LAVRICH 1999 Journal of Sound and Vibration, 4(4), rediction of ower Flows hrough Machine Vibration Isolators. 7. J. AN, J. AN and C. H. HANSEN 199 Journal of the Acoustical Societ of America, 9(), otal ower Flow from a Vibrating Rigid Bod to a hin anel hrough Multiple Elastic Mounts. 8. E. UNGAR and C. W. DIERICH 1966 Journal of Sound and Vibration, 4(), High-Frequenc Vibration Isolation.

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