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 . 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 .
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 . 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
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.
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 ,L (2r+1)/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:
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 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:
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 z and x is due to the sign convention of thekinematic coupling.
Noll et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)
Figure 2. Computational model of frame containing 20nodes and elements.
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
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 . 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
Noll et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)
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
Figure 7. Flow chart describing the methodology toidentify methodology the properties of elastomeric
bushing, where is the modal damping ratio, 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 propertie