INTRODUCTIONElastomeric joints are widely used in vehicle isolation

systems and their geometries are shaped to provide favorableproperties in certain directions based on the diagonal terms[1, 2]. Nonetheless, non-diagonal (coupling) terms are oftenunknown though they are intrinsic to the design of complexautomotive assemblies [3, 4]. Elastomeric joints presentmany challenges in modeling such as amplitude andfrequency dependent properties. Direct measurement of themultidimensional dynamic properties of practical elastomericjoints is not possible, and these properties are furthercomplicated when an elastomeric component is integratedinto a sub-system assembly. Typically, the dynamicproperties of assemblies may differ from those of componentsdue to preload and boundary condition effects; therefore,there is a need to develop improved experimental methods toexamine these issues [4].

Identification methods that utilize computational andexperimental modal analyses may be integrated into someapplications as long as the structures behave in a linearmanner [5, 6]. For instance, Kim et al. [7] used a modal-based technique to characterize the dynamic stiffness of beamsupports, each modeled by a lumped transverse spring. Thisprior method [7] has recently been extended by Noll et al. [8]

to include the identification of multidimensional matrices.The method uses the physical system matrices developedfrom a discretized model (lumped parameter or finiteelement) without joints, and then measured naturalfrequencies, modal loss factors, and mode shapes are utilizedto extract the joint parameters. This article introduces aresonant identification method [8] and compares the resultswith the conventional dynamic stiffness properties measuredusing a non-resonant test method.

PROBLEM FORMULATIONThe scope of the resonant test is limited to an elastic beam

structure connected to ground through two elastomeric joints,where each joint is comprised of two cylinders. Theassembled system is assumed to be linear time invariant andself-adjoint; the damping of the elastomeric joints isstructural, and the beam is assumed to be proportionallyviscous damped. Further, the joint mass is known a priori,and the dynamic properties of a joint can be represented at apoint by structural damping h and stiffness k matrices. Thespecific objectives of this article are as follows. 1. Design atractable resonant beam experiment that allows a directcomparison with the non-resonant test of an elastomericcomponent. 2. Use a recently extended resonant formulation

2013-01-1925Published 05/13/2013

Copyright © 2013 SAE Internationaldoi:10.4271/2013-01-1925saepcmech.saejournals.org

Comparative Assessment of Multi-Axis Bushing PropertiesUsing Resonant and Non-Resonant Methods

Scott Noll, Jason Dreyer and Rajendra SinghThe Ohio State University

ABSTRACTShaped elastomeric joints such as engine mounts or suspension bushings undergo broadband, multi-axis loading;

however, in practice, the elastomeric joint properties are often measured at stepped single frequencies (non-resonant testmethod). This article helps provide insight into multi-axis properties with new benchmark experiments that are designed topermit direct comparison between system resonant and non-resonant identification methods of the dynamic stiffnessmatrices of elastomeric joints, including multi-axis (non-diagonal) terms. The joints are constructed with combinations ofinclined elastomeric cylinders to control non-diagonal terms in the stiffness matrix. The resonant experiment consists of anelastic metal beam end-supported by elastomeric joints coupling the in-plane transverse and longitudinal beam motion. Thedynamic stiffness and loss factors of the elastomeric cylinders are measured in a non-resonant commercial elastomer testmachine in shear, compression, and inclined configurations and a coordinate transformation is used to estimate thekinematic non-diagonal stiffness terms. Strong agreement is found for both dynamic stiffness and loss factors between theresonant and non-resonant methods at small displacements.

CITATION: Noll, S., Dreyer, J. and Singh, R., "Comparative Assessment of Multi-Axis Bushing Properties Using Resonantand Non-Resonant Methods," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1925.

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of the authors [8] to identify fully populated joint dynamicmatrices given limited modal measurements on the resonantbeam. 3. Compare the dynamic joint properties identifiedwith resonant and non-resonant test methods; havingconsistency between the two different methods would suggesta heuristic validation though these two methods are rarelycompared in the scientific literature.

The experiments are illustrated in Fig. 1: elastomericcylinders (Fig. 1a) are inclined to permit multi-axis loading ofthe components in a non-resonant test method; An elasticbeam system is supported by the same inclined elastomericcylinders at each end, with each elastomeric joint loaded inmultiple directions. Kinematic coupling to the structure isintroduced through inclined attachment of the joints whichcouples the transverse y(x,t) and longitudinal u(x,t) motionsof the beam. The modal parameters of the unconstrained(free-free) beam are selected so that the first elastic flexuralmode is representative of the first elastic mode of real-worldautomotive subframe structures, say between 100 to 200 Hz.Further, the first 3 rigid body and 3 elastic modes (up to 1024Hz) are examined, as both rigid and elastic modes are ofconcern in practical assemblies. Elastomeric joints areintentionally selected even though they exhibit excitationamplitude and frequency dependent properties, as well assample to sample variations (say 10 to 15%). Given theamplitude-dependent properties, comparisons must be madewith a non-resonant method. Both resonant and non-resonantmethods are employed in industry, from system andcomponent perspectives, respectively.

(a). Non-resonant test

(b). Resonant testFig. 1. Schematic of the non-resonant and resonant

tests: (a) elastomeric cylinders loaded in: 1. compression;2. shear; 3. angled; (b) elastic beam is end supported bytwo joints, where each joint is comprised of two inclined

elastomeric cylinders.

JOINT STIFFNESS MATRIX OF ANELASTOMERIC CYLINDER

Analytical Joint Stiffness MatrixEach elastomeric cylinder in the resonant experiment has

a transverse eccentricity ε from the neutral axis of the beamand an inclination angle θ from the beam's nodal coordinatesystem as shown in Fig. 2. Both the eccentricity and theinclination angle introduce a form of kinematic coupling thatmust be considered such that a suitable transformation can bemade from the local coordinate system of the cylinder to thebeam's nodal coordinate system.

Fig. 2. Details of transverse eccentricity, ε, and angle ofinclination, θ.

A uniform circular cylinder when grounded at one endexhibits a stiffness matrix in a local coordinate system of thefollowing form:

(1)

where a is the shear stiffness, b is the compressive stiffness, cis the rotational stiffness, and d is the elastic couplingbetween rotation and shear. Assuming that the point stiffnessat the end of the cylinder follows a rigid connection to thebeam node, the effective stiffness matrix of the elastomericcylinder in the beam nodal coordinate system can becomputed via the following transformations:

(2)

where Tθ and Tε are defined as:

(3a-b)

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Non-Resonant Joint CharacterizationA non-resonant test employs a uniaxial hydraulically

actuated, close loop servo-controlled elastomer test machine.Typically a sinusoidal displacement, x(t), is applied to the topside of the specimen, and the transmitted force, f(t), at thebottom of the specimen is measured at the same frequency.The dynamic stiffness can be defined as:

(4)where δ is the loss angle. The loss factor, γ = tan δ, is definedas:

(5)The elastomeric cylinders are characterized in three

different configurations: (1) compression; (2) shear; and (3)angled as depicted in Fig. 1. The compression configurationconsists of a single cylinder. The shear and angledconfigurations are conducted with a pair of elastomericcylinders to ensure appropriate boundary conditions. Thediameter and height of each cylinder is 25.4 mm. Metalliccaps are bonded to both sides of each cylinder to preventdamage to the elastomer during disassembly. Thenonresonant measurements correspond with specific linearcombinations of the elements of the cylinder stiffness matrixof Eqn. (1) as: 1. shear = 2a, 2. compression = b, and 3.angled = a+b.

The dynamic characterization is conducted at an initialpreset of 0.5 mm from 5 to 200 Hz at 5 Hz increments. Undereach configuration, the peak-to-peak amplitude is specified at0.1 mm, 0.05 mm, or 0.01 mm. The dynamic stiffness ofelastomeric cylinders exhibit considerable amplitude andfrequency dependence in the non-resonant tests as shown inTable 1. The highest sensitivity to frequency dependence is

observed below 50 Hz. The measured increases for eachconfiguration as the amplitude of the excitation decreased bya factor of 2.1, 1.5, and 2.0 for the compression, shear, andthe angled configurations, respectively. The loss factor (Table2) is nearly constant with respect to frequency; however, itdecreases by half when the peak-to-peak amplitude of theexcitation is reduced from 0.1 mm to 0.01.

Validation of Superposition AssumptionSince the elastomer test machine permits only uniaxial

measurements, the specimens are oriented in severalconfigurations to measure relevant diagonal elements of the

dynamic stiffness matrices. Using Eq. (2), is calculatedin Table 3 showing the angled measurement against thecomputed value using the measured data from the shear andcompression measurements shown in Table 1. Thiscomparison demonstrates that superposition holds near theoperating points though the component is inherentlynonlinear. Since there is such a strong agreement, it is

expected that the computed values for the kinematic couplingterms between longitudinal and transverse directions can besimilarly estimated by using Eq. (2).

Table 1. Elastomeric joint characterization using non-resonant test method, dynamic stiffness magnitude (N/

mm), at 100 Hz

Table 2. Elastomeric joint characterization using non-resonant test method, loss factor at 100 Hz

Table 3. Comparison of the dynamic stiffness magnitude(N/mm) of two elastomeric cylinders angled at 45° at 100

Hz

RESONANT BEAM EXPERIMENTProposed Experiment

A rigid body formulation alone will not permit a sufficientnumber of equations to uniquely identify the two differentjoints in the resonant beam experiment; thus an elastic bodyformulation is required. Further, the elastomeric cylinders areselected to have similar stiffness properties to automotivebushings and ratio of joint to structure stiffness. To maintainthe lumped joint stiffness assumption, the ratio of the contactsurface dimension to beam length is kept below 0.05. Withthese guidelines, a steel beam (with Young's modulus, E=207GPa, Poisson's ratio, ν = 0.3, and mass density, ρ =7850 kg m−3) is selected. The beam is 914 mm in length (L) with arectangular cross-section of 25.4 × 50.8 mm where thethickness is 25.4 mm in the y direction for the desired flexuralstiffness in the experiment. Modal measurements are made onthe beam while it is suspended by very compliant springs.

The beam is supported near each end by a combination oftwo angled elastomeric cylinders, specified as Neoprene witha Shore A hardness of 70. To introduce symmetric coupling,angled cylinders with an aspect ratio of 7 are inclined at 45°from above and below the beam. The angled cylinders induce

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local coupling between the transverse and longitudinalmovement of the beam at the joint.

A modal experiment is conducted for the resonant beamas depicted in Fig. 1. The accelerometer is positioned belowpoint 3 (corresponds to node 3 of Fig. 3) and the rovingimpulse hammer technique is employed, with force inputs atpoints 1, 3-9, and 11 in the transverse direction and at points1 and 11 in the longitudinal (x) direction; note that impactsdirectly at the joint location are not possible. Ten impacts ateach location are averaged to minimize random error in the

accelerance measurements. For this work, apolyreference least-squares complex frequency-domainmethod is utilized where this implementation estimates thenatural frequencies, damping parameters, and mode shapes ina global sense [6].

Although this measurement set results in mode shapeestimates at each location, only the results near the jointlocations are utilized in the identification procedure, e.g. atjoint I, the mode shape components at points 1 and 3 areused. The mode shape components are averaged in thelongitudinal (x) direction. After identification, the jointdynamic stiffness matrices are employed with the beammodel to forward predict the natural frequencies andfrequency response functions of the experiment.

Computational ModelThe computational model is comprised of 10 beam finite

elements with lumped dynamic stiffness matrices at nodes 2and 10 as depicted in Fig. 3. A two-node Timoshenko beamelement is superposed with a longitudinal rod element togenerate a two-node six degree of freedom element. Thisformulation assumes that elastic longitudinal and beambending are uncoupled. The work of Friedman and Kosmatka[9] contains a complete derivation of the finite elementstiffness and mass matrices for a Timoshenko beam.

Fig. 3. Computational beam model with 10 elements and11 nodes. Lumped stiffness elements are attached at

nodes 2 and 10.

Each joint is comprised of two cylinders at the left andright hand of the resonant beam test. The cylinder below thebeam is inclined at 45° and has an eccentricity of negative ε;whereas the cylinder above is inclined at negative 135° andhas an eccentricity of positive ε. The effective stiffness is thusthe superposition of both cylinder point stiffnessestransformed to the beam node as using Eqns. (1-3):

(6)

RESONANT JOINT IDENTIFICATIONMETHOD

The flow-chart of the joint identification method from theresonant beam tests is depicted in Fig. 4. The methodologycombines experimental and modeling information in order toextract the joint properties. The flowchart replaces therigorous mathematical details given in the article by Noll etal. [8] and conceptually displays the procedure from anapplications viewpoint.

Fig. 4. Flow-chart of the resonant joint identificationmethod.

EXPERIMENTAL VALIDATIONBeam Modal Properties with FreeBoundaries

The modal testing of the freely suspended elastic beampermits a direct comparison of the theoretical andexperimental natural frequencies and mode shapes as well asthe extraction of C assuming proportional damping of thesteel beam. The measured natural frequencies for the firstthree elastic deformation modes (r = 4 - 6) are listed in Table4. Timoshenko beam theory captures each mode within 1 to 2Hz, whereas the Euler theory is still reasonably accurate(within 10 Hz) at the third elastic mode (r = 6).

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Table 4. Comparison of first three natural frequenciesfor the unsupported elastic beam (under free-free

boundary conditions)

Resonant Beam ExperimentThe end-supported resonant beam experiment exhibits

nearly repeated roots with a separation of 2 Hz over afrequency range of interest of up to 1024 Hz, in which sixmodes of vibration are captured. A statistical analysis of theobservations, curve-fitted values, and residuals is performedto establish a confidence interval for the identified stiffnessmatrix. Fig. 5 shows the plot and good agreement is achievedthrough the full range with a coefficient of multipledetermination of 0.99.

Assuming a normal distribution and a confidence level of95%, the dominant terms of k11,k22,k12, and k21 exhibit arange that is within +/− 3 to 9% with the greatest rangeexisting at joint I for k11 direction with a range of +/− 9%.The range of values within the confidence interval forstiffness terms k13, k31, k32, k23, and k33 pass through zero.These terms are difficult to distinguish from zero;nevertheless, their presence is still required to obtainadequate results. The terms k13, k31, k32, and k23 wereintentionally cancelled with opposing cylinders at each joint;however, misalignments and static preloads can introducestiffness coupling for these terms.

Fig. 5. Observed data against the curve-fitted data; Key:+ joint I; o joint II; linear fit with slope of unity without

a zero offset.

Modeling of Resonant Beam ExperimentThe identified joint dynamic stiffness matrices are

employed to forward predict the modal parameters andfrequency response functions. Table 5 lists the naturalfrequencies and the forward predictions which agree within3% with the highest error occurring at a rigid body mode (r =

2). Forward predictions of the accelerance spectra arecomputed by a direct inversion of the assembled dynamicstiffness matrices:

(7)

Table 5. Comparison of natural frequencies forsupported resonant beam experiment

Fig. 6. Driving point accelerance for the resonant beamexperiment in the transverse direction at point 3. (a)

magnitude; (b) phase. Key: (−) predicted using Eqn. (7);(+) measured.

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The driving point comparison Fig. 6 exhibits good

agreement, though some regions show discrepancies. Thismay suggest a deficiency in a simple (assumed) structuraldamping model. The cross point spectrum between transverseand longitudinal motions in Fig. 7 exhibits influence fromout-of-plane bending vibration at 330 Hz. It is observedduring measurements that impact location errors at the end ofthe beam excite these lightly damped structural modes;whereas the transverse impacts are less sensitive to this issue.The cross-point comparison shows that the forwardprediction and measurement were out of phase near the anti-resonance at 440 Hz, which can be explained by a poor signalto noise ratio in this frequency range.

Fig. 7. Cross point accelerance for the resonant beamexperiment with a longitudinal force input at point 1 andtransverse response at point 3 (a) magnitude; (b) phase.

Key: (−) predicted using Eq. (7); (+) measured.

CORRELATION BETWEENRESONANT AND NON-RESONANT

TESTSA comparison of the resonant and non-resonant dynamic

stiffness is shown in Table 6. Generally, there is goodagreement between the resonant tests and the 0.01 mm peak-to-peak excitation amplitude. Table 7 shows the comparisonbetween the non-resonant loss factor and bounded loss factorfrom the resonant beam experiment. Overall, there isexcellent agreement between the non-resonant and resonanttests at joint II, though the loss factor identified at joint Iexhibits a wider range for the confidence interval.

Table 6. Comparison of the identified stiffness from theresonant beam experiment and non-resonant

measurements from 50 to 200 Hz at 0.01 mm pk-pkamplitude

CONCLUSIONThis article has contributed to the state of the art by

designing new experiments and methods that permit a directcomparison of resonant and non-resonant methods includingnon-diagonal terms in the stiffness matrices of joints. Theproposed methodology is employed to identify the dynamicstiffness properties of joints with dimension 3 in the resonantexperiment consisting of an elastic beam with twoelastomeric supporting elements. A forward modelsuccessfully predicts the measured modal parameters andaccelerance spectra. Excellent agreement is found fordynamic stiffness and loss factors between the resonant andnon-resonant methods.

The methods in this article are not limited to metal beamand elastomeric joint systems, but rather are applicable to amore general class of jointed assemblies. The resonant test inthis article is limited to an examination of the linearized jointproperties; however, the design lends itself well toincorporate different excitation levels to examine theamplitude-dependent properties of elastomers. Further, withtwo opposing joints, preload or displacement can be easilyadded without altering the system. Finally, the work has beenlimited to a single structure connected to ground, whereasmany real-world assemblies contain two or moresubstructures. Despite these limitations, this article providesvaluable insights for interpreting non-resonant componenttest data with resonant system data even when the selectedelastomer exhibits strong amplitude dependent properties.

Table 7. Comparison of the loss factor from the resonantbeam experiment and non-resonant measurements from

50 to 200 Hz at 0.01 mm peak-peak amplitude

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REFERENCES1. Lewitzke, C. and Lee, P. “Application of Elastomeric Components for

Noise and Vibration Isolation in the Automotive Industry” SAETechnical Paper 2001-01-1447 2001, doi:10.4271/2001-01-1447.

2. Oh, H. and Kim, K., “Design of Shape for Visco-Elastic IsolationElement by Topological and Shape Optimization Methods,” SAETechnical Paper 2009-01-2127, 2009, doi:10.4271/2009-01-2127.

3. Kim H.J. and Kim K.J., “Effects of rotational stiffness of bushings onnatural frequency estimation for vehicles by substructure coupling basedon frequency response functions”, Int. J. Vehicle Noise and Vib. 6(2010) 215-229.

4. Kim S. and Singh R., “Multi-dimensional characterization of vibrationisolators over a wide range of frequencies”, Journal of Sound Vibration,245 (2001) 877-913.

5. Cook R.D., Malkus D.S., and Plesha M.E., Concepts and applications offinite element analysis, third edition, John Wiley & Sons, New York,1989.

6. Peeters B., Van der Auweraer H., Guillaume P., and Leuridan J., “ThePolyMAX frequency-domain method: a new standard for modalparameter estimation?” Shock and Vibration, 11 (2004) 395-409.

7. Kim T.R., Ehmann K.F., and Wu S.M., “Identification of joint structuralparameters between substructures,” J. Eng. Ind., 113 (1991) 419-424.

8. Noll, S., Dreyer, J., Singh, R., “Identification of dynamic stiffnessmatrices of elastomeric joints using direct and inverse methods,”Mechanical Systems and Signal Processing, (2013), on-line publication,http://dx.doi.org/10.1016/j.ymssp.2013.02.003i.

9. Friedman Z. and Kosmatka J., “An improved two-node Timoshenkobeam finite element,” Computers and Structures, 47 (1993) 473-481.

CONTACT INFORMATIONProfessor Rajendra SinghAcoustics and Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDept. of Mechanical and Aerospace EngineeringThe Ohio State Universitysingh.3@osu.eduPhone: 614-292-9044www.AutoNVH.org

ACKNOWLEDGMENTSWe acknowledge the member organizations such as

Transportation Research Center Inc., Honda R&D Americas,Inc., YUSA Corporation and F.tech of the Smart VehicleConcepts Center (www.SmartVehicleCenter.org) and theNational Science Foundation Industry/University CooperativeResearch Centers program (www.nsf.gov/eng/iip/iucrc) forsupporting this work.

DEFINITIONSa, b, c, d - arbitrary constantsA - cross-sectional areaA - acceleranceC - viscous damping matrixE - Young's modulusf - forceh - joint structural damping matrix

i - I - area moment of inertia

- dynamic stiffnessk - stiffnessk - joint stiffness matrixK - stiffness matrixL - lengthM - mass matrixT - transformation matrixu - displacement, x-directionw - widthx - Cartesian coordinatesy - displacement, transverse directionz - joint dynamic stiffness matrixZ - dynamic stiffness matrixγ - loss factorδ - loss angleε - transverse eccentricityζ - damping ratioθ - inclination angleν - Poisson's ratioρ - densityψ - mode shape vectorω - frequency, rad / sec

Superscripts- - normalized value

Subscriptsr - modal index

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