INTRODUCTIONElastomeric components are used as engine mounts,

suspension bushings, torque struts, shock and strut mounts,jounce bumpers, cradle mounts, body frame mounts, andexhaust hangers. Their geometries are shaped to providefavorable properties in certain directions based upon thediagonal terms [1]. Nonetheless, non-diagonal (coupling)terms are often unknown though they are intrinsic to thedesign of complex automotive assemblies [2, 3]. Elastomericjoints present many modeling challenges such as amplitudeand frequency dependent properties. Their small amplitudedynamic stiffness and damping characteristics influencevehicle noise, vibration, and harshness performance; theirlarge-amplitude static force-deflection characteristics arecrucial for vehicle ride and handling performance; on theother hand, when a vehicle is driven on a rough road,elastomeric components experience large-amplitude dynamicloads [4].

To complicate the matter further, the dynamic propertiesof an elastomeric joint embedded within an assembly maydiffer from those of components due to preload and boundaryconditions [2]. Real world automotive frames, engine cradles,and suspension subframes are complicated structures;

consequently, there is a need for a laboratory frame designpermitting combinations of bushings and frames forexperimental and analytical characterization. In view of theabove-mentioned challenges, this article attempts to fill thisvoid by designing and constructing a simple, yet high fidelitylaboratory frame and investigate the influence of non-diagonal elastomeric joint properties on the modal propertiesthrough computational modeling and experimental validation.

PROBLEM FORMULATIONThe scope of this investigation is limited to the frequency

range of the first nine modes which includes six rigid bodyand three elastic deformation modes (say up to 300 Hz). Onlynatural frequencies and mode shapes will be examined. Sinceonly lower frequencies are examined, the elastomeric mountswill be modeled as massless with lumped linear stiffness.Thick beam (Timoshenko) theory will be used for beamflexure and the joints are selected with low damping lossfactors (say less than 5% in the small amplitude regime), thusthe assembled structure is assumed to have normal modes.The specific objectives of this article are as follows: 1. designand construct a laboratory ladder beam frame with cornermounted (four) elastomeric mounts in the context of

2013-01-1904Published 05/13/2013

Copyright © 2013 SAE Internationaldoi:10.4271/2013-01-1904saepcmech.saejournals.org

Effect of Local Stiffness Coupling on the Modes of a Subframe-Bushing System

Scott Noll, Jason Dreyer and Rajendra SinghThe Ohio State University

ABSTRACTThe elastomeric joints (bushings or mounts) in vehicle structural frames are usually described as uncoupled springs

(only with diagonal terms) in large scale system models. The off-diagonal terms of an elastomeric joint have beenpreviously ignored as they are often unknown since their properties cannot be measured in a uniaxial elastomer testsystem. This paper overcomes this deficiency via a scientific study of a laboratory frame that is designed to maintain ahigh fidelity with real-world vehicle body subframes in terms of natural modes under free boundaries. The steel beamconstruction of the laboratory frame, with four elastomeric mounts at the corners, permits the development of a highlyaccurate, yet simple, beam finite element model. This allows for a correlation study between the experiment and model thathelps shed light upon the underlying physical phenomenon. In particular, the effect of local stiffness coupling ofelastomeric bushings or mounts is demonstrated through computational modeling and experimental validation. It is seenthat the joint stiffness matrices strongly influence the modal properties of a laboratory subframe-mount system. Forinstance, a strong correlation between the rigid body modes (r = 1 to 6) and the first three elastic modes (r = 7 to 9) is onlypossible when the coupling (non-diagonal) terms are included in the bushing model.

CITATION: Noll, S., Dreyer, J. and Singh, R., "Effect of Local Stiffness Coupling on the Modes of a Subframe-BushingSystem," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1904.

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automotive subframes (engine cradle or rear suspension), 2.conduct modal measurements in freely suspended andconstrained boundary conditions, 3. develop a computationalmodel of the frame and frame-bushing system experiment, 4.examine the influence of off-diagonal stiffness terms on theeigensolution.

DESIGN OF A LABORATORYFRAME

A ladder frame design was selected due to its simplicity inconstruction, modeling, and robustness in capturing themodal properties of real-world frames. The laboratory ladderframe (Figure 1) is comprised of two symmetrical rail beamsthat connect together via two symmetrical cross-memberbeams. The dimensions of the laboratory frame were selectedusing modal similitude with that of real-world frames for thefirst three elastic deformation modes that are brieflydescribed in Table 1. A simple design equation is used toestimate the natural frequency of the first rail beam bendingmode which can be approximated with the free-free conditionbeam flexure formula:

(1)where β,L ≈ (2r+1)π/2, E is young's modulus, I is the areamoment of inertia, A is the cross-sectional area and L is thelength of the rail beam. For beam models with the the samegeometry, the natural frequency ratio can be scaled fordifferent materials (from material 1 to 2) as:

(2)

Figure 1. Schematic of the laboratory frame. Bushingsupports are at the four corners.

Table 1. Comparison of measured normalized naturalfrequencies ( ) of elastic deformation modes betweenvehicle and laboratory frames under freely suspendedconditions. Natural frequencies normalized by r = 7 of

the vehicle frame.

COMPUTATIONAL MODEL OFFRAME

The computational model is comprised of 20 beam finiteelements shown in Figure 2. 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[5] contains a complete derivation of the finite elementstiffness and mass matrices for a Timoshenko beam. Thefinite element model of the laboratory frame is implementedinto a self-written script. The bushings are modeled withlumped dynamic stiffness matrices at nodes 1, 7, 10, and 16.The axi-symmetric bushings are assumed to have a stiffnessmatrix of the form:

(3)

when one end is attached to ground. Note that the vertical andtorsional stiffnesses, k22 and k55, are uncoupled (lackingterms in the same row or column) and that the x and ztranslational and rotational stiffness terms are assumedidentical as are the coupling terms. The negative signs for k61that couple the δz and δθx is due to the sign convention of thekinematic coupling.

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Figure 2. Computational model of frame containing 20nodes and elements.

SUBFRAME-BUSHING MODALEXPERIMENT

Four identical elastomeric mounts are connected to thelaboratory frame, one to each of the four corners as shown inFigure 1. Driving point measurements were taken in threedirections at points corresponding to nodes 6 and 7. Theaccelerance plots are shown in Figure 3 where the first ninevibration modes, including six rigid body modes and the firstthree elastic deformation modes are observable. Note thatmodes 2 and 4 prominently appear as pairs for excitations inthe y- and z-directions. This experimental results suggeststhat strong modal coupling exists between modes 2 and 4(rigid body modes).

Figure 3. Driving point accelerance measurements takenon the restrained laboratory frame at (a) Point 7 in the x-direction; (b) Point 6 in the y-direction; (c) Point 6 in the

z-direction.

Figure 4. Modal mapping of the laboratory frame fromfreely suspended conditions to restrained by four

elastomeric bushings (as shown in Fig. 1).

Figure 5. Coupled modes of restrained laboratory frameexperiment. (a) r = 2 mode; (b) r = 4 mode.

The modal parameters are estimated for this work using apolyreference least-squares complex frequency-domainmethod. This implementation estimates the naturalfrequencies, damping parameters, and mode shapes in aglobal sense [6]. The mapping of the modes is illustrated inFigure 4. where the natural frequencies are normalized by thefirst elastic deformation mode in the freely suspendedcondition. The mode shapes 2 and 4 visually exhibit thegreatest coupling as shown in Figure 5. The coupled modes

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include motion in the z-direction coupled to rotation about thex-axis. When animated, mode shapes 2 and 4 have a glidingaction where mode shape 2 has a pivot point below the frameand mode shape 4 has a pivot point above the frame.

IDENTIFICATION OF MULTI-AXISBUSHING PROPERTIES

Figure 6. Resonant beam experiment used to identifymulti-axis stiffness properties of the elastomeric

bushings.

Figure 7. Flow chart describing the methodology toidentify methodology the properties of elastomeric

bushing, where ζ is the modal damping ratio, ψ is themode shape vector, M is the mass matrix, C is the viscous

damping matrix, K is the stiffness matrix.

A simplified resonant experiment of a beam connected bytwo bushings to ground (Figure 6) was used to identifylimited multidimensional bushing dynamic properties. Theflow-chart of the identification method from the resonantbeam tests is depicted in Figure 7. The methodologycombines experimental and modeling information in order toextract the joint properties and is based upon the work of

Noll, et al. [7] and the flow-chart replaces the mathematicaldetails and conceptually states the procedure from anapplications viewpoint.

MODELING OFSUBFRAMEBUSHING EXPERIMENT

The in-plane properties of two bushings are identified andaveraged for use in the subframe-bushing experiment. Theremaining multidimensional joint properties are estimatedbased upon the axisymmetric geometry of the bushings. Allfour bushing properties are assumed to be identical, althoughcomponent variations have been observed. The nominalbushing properties are determined to be:

(4)

Only the k55 term is unmeasured in the simplifiedresonant beam experiment. This parameter is identifiedthrough in an iterative manner. The identified bushingproperties are incorporated into the computational framemodel as elastic boundary conditions. The comparison of thetheory with the experiment can be accomplished through boththe natural frequencies and mode shapes. Table 2 contains themeasured and predicted natural frequencies of the first sixrigid body modes and three elastic modes (total of ninemodes). The predicted and measured natural frequencies arewithin +/− 3.0%. with the exception of mode 4 (a highlycoupled mode) which deviates from the experiment by 21%.The modal assurance criterion (MAC) is used to correlate themode shapes between theory and experiment. A modalassurance criterion value of 1.00 indicates perfect correlationand it was found that the correlation contained only strongdiagonal terms (high correlation shown in Figure 8) with eachgreater than 0.93 with the exception of mode 4 (highlycoupled) which exhibited a MAC value of 0.88, still a strongcorrelation. The non-diagonal MAC values fall in the rangeof 0.00 to 0.10, with the largest values appearing betweenmodes 2 and 4 as well as 3 and 5 (highly coupled) as depictedin Figure 9.

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Table 2. Comparison of natural frequencies betweenlaboratory frame and computational model when the

frame is restrained by four bushings.

A separate computational analysis is conducted to assessthe need of the non-diagonal stiffness terms as given in Eqn.(4). Here, the non-diagonal terms are set to zero and thenatural frequencies and mode shapes are recalculated. Whendone so, the natural frequencies of the coupled rigid bodymodes (modes 2, 3, 4 and 5) are most affected. The naturalfrequencies for the first rigid body mode and the elasticmodes (modes 1, 7, 8, and 9) for this bushing-frame systemare unaffected. This is due to the orthogonal placement of thebushings relative to the frame, in that the vertical bushingstiffness (k22) is uncoupled with the other directions due tothe component symmetry. The most drastic effect is noticedin the MAC values (Figure 8). Here, the MAC values formodes 2 and 4 drop from 0.88-0.93 to 0.29-0.38. Coupledmode sets 2-4 and 3-5 exhibit a similar type of motion. Thedifference between the two coupled mode sets is the motionof the mode set 2-4 occurs with rotation about the x-axis withinfluence of the width (w) of the frame, whereas, the coupledmode set 3-5 exhibits rotation about the z-axis with influenceof the length (L) of the frame. Thus, the most sensitivecoupled mode set is 2 and 4 since the width dimension is one-third the length. The deviation in predicted natural frequencyof mode 4 is attributed to component variation within the fourbushings, since only limited multidimensional properties oftwo bushing are identified and average properties attributedto each of the four bushings in the computational model.

Figure 8. Diagonal terms of the modal assurancecriterion (MAC) between theory and experiment for thebushing-frame subsystem. Values of 1.0 indicate perfect

correlation. Key: Theory with coupling terms; Theory without coupling terms.

Figure 9. All terms of the modal assurance criterion(MAC) between theory and experiment for the bushing-

frame subsystem.

CONCLUSIONThe off-diagonal terms of elastomeric joints have been

previously ignored as they are often unknown. This paperovercomes this deficiency via a scientific study of laboratoryframe. The frame is designed to maintain a high fidelity withreal-world vehicle body subframes in terms of natural modesunder free boundaries. A correlation study between theexperiment and model helps shed light upon the underlyingphysical phenomenon. In particular, the effect of localstiffness coupling of elastomeric bushings or mounts isdemonstrated and experimentally validated. It is seen thatstiffness matrices strongly influence the modal properties of alaboratory subframe-mount system. A strong correlationbetween the rigid body modes (r = 1 to 6) is only possiblewhen the coupling (non-diagonal) terms are included in thebushing model. The methods in this paper were limited inscope to the linearized properties of the bushings in thefrequency domain. Additionally, the relationship between

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bushing damping properties and the bushing-frame systemmodal damping is not considered. Despite these limitations,the proposed approach builds a clear foundation allowing fora greater understanding of the dynamic interactions betweenframes and elastomeric bushings.

Although superior modeling results are achieved utilizingthe non-diagonal terms, it comes at the expense of having anaccurate knowledge of these properties. Conventionaluniaxial elastomer test machines cannot directly measurethese properties, and thus indirect methods, such as the recenttechnique proposed by Noll, et al. [7], must be used toidentify the stiffness and damping properties.

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. Kim, S. and Singh, R. “Vibration transmission through an isolatormodeled by continuous system theory”. Journal of Sound and Vibration,248:925-953, 2001.

3. Kim, H. J., 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 Vibration,6:215-229, 2010.

4. Kuo, E., “Testing and Characterization of Elastomeric Bushings forLarge Deflection Behavior,” SAE Technical Paper 970099, 1997, doi:10.4271/970099.

5. Friedman, Z., Kosmatka, J. “An improved two-node Timoshenko beamfinite element,” Computers and Structures, 47 (1993) 473-481.

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

7. 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.

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 - length parametersA - cross-sectional areaC - viscous damping matrix

E - Young's modulusF - frequency, HzI - area moment of inertiaK - stiffnessK - joint stiffness matrixK - stiffness matrixL - lengthM - mass matrixW - widthx, y, z - Cartesian coordinatesB - natural frequency parameterΔx - displacement, x-directionΔy - displacement, y-directionΔz - displacement, z-directionδθx - rotational displacement about the x-axis

δθy - rotational displacement about the y-axis

δθz - rotational displacement about the z-axis

Z - damping ratioΘ - rotational coordinateP - densityΨ - mode shape vectorΩ - frequency, rad / sec

Superscript - normalized value

SubscriptR - modal index

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