Static Torsional Stiffness From Dynamic Measurements UsingImpedance Modeling Technique

Hasan G. Pasha, Randall J. Allemang, David L. Brown and Allyn W. Phillips

University of Cincinnati – Structural Dynamics Research Lab (UC-SDRL), Cincinnati, Ohio, USA

Abstract. Static torsional stiffness of a structure is the measure of the resistance it offers to twisting. It is animportant design parameter for structures subjected to torsion loading. Estimating the static torsional and bending

stiffness values of a structure from test rigs is resource intensive. In order to obtain reasonable estimates of static

torsional stiffness at a minimal cost and less time, diagnostic procedures that are efficient and accurate are needed.A method that utilizes the impedance modeling technique to extract the static stiffness from free-free modal test is

discussed in this paper. The results of applying this technique on a rectangular plate structure are compared with

analytical estimates for static torsional stiffness. The merits and limitations of this technique are also discussed.

Keywords. Impedance Modeling, Dynamic Stiffness method, Compliance method, Hybrid Impedance method, Static

Torsional Stiffness

Notation

Symbol Description[·]−1 Inverse of matrix [·][·]+1 Pseudo-Inverse of matrix [·]

∆ Vertical deflection (in)θ Angular deflection (rad)ω Frequency (rad/s)F Applied force (lbf)

[H(ω)] Free boundary FRFs (XF )

[Hmod] Modified compliance/FRF matrix[∆Kdyn] Modification dynamic stiffness matrix

[Kmod] Modified dynamic stiffness matrixKSS Driving-point stiffnessKT Static torsional stiffnessLf Distance between front DOFs (in)Lr Distance between rear DOFs (in)T Applied Torque (lbf in)

DOF Degree of freedomDOF 1 Left front DOF, z–directionDOF 2 Right front DOF, z–directionDOF 3 Left rear DOF, z–directionDOF 4 Right rear DOF, z–direction

1. Introduction

The performance of an automobile with respect to vehicle dynamics and passenger ride comfort is characterizedby the auto-body’s global static and dynamic stiffnesses [2]. Currently, special test rigs are used to estimate staticstiffnesses. A method based utilizing dynamic measurements (usually taken from a free-free modal test) togetherwith modeling techniques to estimate the static stiffnesses has been developed. Both existing approaches takeconsiderable resources to arrive at a reasonable estimate of the global torsion and bending stiffnesses. In order toobtain reasonable estimates of static global stiffnesses at a minimal cost and less time, diagnostic procedures that

are efficient and accurate are needed. A method based on impedance modeling technique to estimate the staticglobal stiffnesses involving dynamic testing tools is described here. Experimental validation of the method using arectangular plate structure is presented.

2. Impedance Modeling Method

To solve engineering problems such as machine chatter and flutter, it is often desired to predict the effect of structuralmodification on the dynamics of a system. This modification may be additive or subtractive, that is two subsystemsmay be combined together to form a larger system, or a large system might be broken down into its componentparts. Several different modeling methods have been developed to solve this problem: finite element analysis, modelmodeling, and impedance modeling to name a few. Unlike modal modeling, which operates on the modal vectors ofa system, impedance modeling can be viewed as taking one step back and operating on the raw measured FRFs. Asa result, the amount of data required to yield similar results to the modal model is much greater, as the informationcaptured by the data has not yet been distilled down. This was a problem historically, as computing power andstorage were at a premium. As a result, the impedance modeling technique has been limited to the investigationof the effect of structural modification on a subset of individual FRFs. The technique can be expanded, however,to include the full set of FRFs needed for extraction of the modal parameters using standard parameter estimationtechniques.

2.1. Theory. This modeling technique attempts to describe the dynamic behavior of an assemblage of subsystems.All subsystems are considered have known properties from which to expand the model. In an analytical case, input-output relationships at each DOF can be calculated. In an experimental case, measurements must be taken at eachDOF. Graphically stated, these systems can be represented by shown in Fig. 1.

Figure 1: Dynamics Stiffness Method: Substructures

The points s1 and s2, at which the systems are to be connected, are considered the driving points of the system.The driving point FRFs must be known for each body in order to make modifications or connections at these points.The points r and t represent points of interest in the combined system. To accurately predict the modified FRFsafter attaching the additional component to the main structure, cross FRFs must also be known for each body atthese points. We may utilize one of three methods to develop an impedance model to predict the response of thealtered system:

(1) Dynamic Stiffness Method(2) Compliance Method(3) Hybrid Method

2.2. Dynamic Stiffness Method. The dynamic stiffness method begins with the input-output relationship of theconnected systems in the form

(1) {F} = [K]{X}with {F} representing the forcing vector, {X} representing the response vector, and [K] representing the dynamicstiffness matrix. The equation of motion for each component of the system is described theoretically by:

(2) {f} =[−ω2[M ] + [K] + jω[C]

]{x}

or experimentally for component 1 and 2 by Eq. 3 and Eq. 4 respectively,

(3)

{{Fr}{Fs1}

}=

[[Krr] [Krs1 ][Ks1r] [Ks1s1 ]

]{{Xr}{Xs1}

}

(4)

{{Ft}{Fs2}

}=

[[Ktt] [Kts2 ][Ks2t] [Ks2s2 ]

]{{Xt}{Xs2}

}Rigidly connecting the components at points s1 and s2, setting the internal forces equal to zero, and assuming noexternal forces yields the combined expression:

(5)

{Fr}{0}{Ft}

=

[Krr] [Krs1 ] [0][Ks1r] [Ks1s1 ] + [Ks2s2 ] [Ks2t][0] [Kts2 ] [Ktt]

{Xr}{Xs1}{Xt}

While overlapping the matrices in this fashion models the connection as perfectly rigid, it is also possible to connectthese points using springs. In the latter case, the off diagonal terms of the matrix corresponding to the connectedpoints would hold a negative value. This should generate almost the same answer, though there will be a slightnumerical difference. The stiffer the connection, the more closely they will agree. However, we must be careful whenconnecting components using very high stiffness values, as this may cause the solution to be numerically unstable.To solve for the impedance FRFs, we simply loop through each frequency to build the matrix.

2.3. Compliance Method. This method is the most commonly used method for impedance modeling when dealingwith experimental data. It is mathematically identical to the dynamic stiffness method, but FRFs are easier togenerate from an arbitrary input force than it is to precisely control motions experimentally. The compliancemethod begins with the input-output relationship of the system in the form

(6) {X} = [H]{F}with [H] representing the FRF matrix, which is the inverse of the dynamic stiffness matrix, [K].Consider systems A and B, as shown in Fig. 2,

Figure 2: Compliance Method: Substructures

with Xa1 and Xa2 representing displacements at points of interest, FaCP and FbCP representing the internal forcesat the connection points, and Fe representing an external force. The equations relating the inputs (Fe and FCP ) andoutputs (Xa1, Xa2, and XCP ) for each of the disconnected subsystems are as shown in Eq. 7 and Eq. 8,

(7)

{X1}{X2}{XCP }

A

=

[H1,CP ] [H1,e][H2,CP ] [H2,e]

[HCP,CP ] [HCP,e]

A{{FCP }{Fe}

}A

and

(8){{XCP }

}B=[HCP,CP

]B {FCP

}Bwith the elements of the [H] matrices corresponding to either the driving-point or cross-point FRF measurements(experimental) or calculations (analytical) of the separate subsystems. When the subsystems are connected, themotion at the connection point on each object will be equal, and the forces acting on each body will be equal andopposite. This is equivalent to writing the constraint equations shown in Eq. 9 and Eq. 10,

FACP + FB

CP = 0(9)

XACP −XB

CP = 0(10)

These constraint equations, along with the equations relating the inputs and output of the individual subsystems andthe value of the added external force are then arranged into a matrix form allowing for their simultaneous solution.To simplify the calculations in the following examples, we assume here that Fe = 1. This arrangement isolates all ofthe unknowns (the force and displacement variables) into a single vector as shown in Eq. 11,

(11)

1 0 0 −HA1,CP −HA

1,e 0 00 1 0 −HA

2,CP −HA2,e 0 0

0 0 1 −HACP,CP −HA

CP,e 0 00 0 0 0 0 1 −HB

CP,CP

0 0 1 0 0 −1 00 0 0 1 0 0 10 0 0 0 1 0 0

XA1

XA2

XACP

FACP

Fe

XBCP

FBCP

=

0000001

The left-most matrix is known, as the FRFs of each system are known, and the right hand side only contains thevalue of the external force applied. Solving for the vector of unknowns gives the solution to the combined system fora given external force.By setting the external force to unity, the responses in the solution vector above will correspond to the FRF ratiobetween the external force input and the corresponding responses. This system can be solved several times, each timemoving the location of the external force to solve for the entire FRF matrix of the combined system as a function offrequency.

2.4. Hybrid Impedance Modeling. The theoretical formulations for impedance modeling shown in the previoussections were the standard formulations used in the 1970’s and 1980’s. These formulations reduced the size of theproblem by algebraically reducing the size of the solution to a minimum of measurements such that the FRF matrixcould be inverted and that the problem was small enough to run in the existing computer systems. With the adventof much more powerful computers and linear algebra programing languages, such as Matlab, it became possible toformulate the problem so that a singular FRF matrix and/or a Dynamic Stiffness matrix could be estimated, whichcaptured all of the information of interest in the modeling process.The response of a mechanical system excited by external force can be estimated directly from its FRF matrix by thefollowing relationship,

(12) {xn}mx1 = [Hn]mxm {fn}mx1

where [Hn] is the Compliance and/or the FRF matrix for component n, it includes all the points of interest on anygiven component n.The system matrix [Hs] is generated simply by constructing a matrix of all the components. The component matricesare simply put into a system matrix whose size is equal to the summation of the sizes of the component matrices.The components are arranged along the diagonal of the system matrix similar to what was done in the lumped massmodeling,

(13) [Hs] =

[H1] 0 0 . . . 0

0 [H2] 0 . . . 0

0...

.... . . 0

0 0 0 . . . [Hn]

The Dynamic Stiffness Matrix is the inverse of the Frequency Response Function matrix as mentioned in lumpedmass modeling or,

(14) [Ks]mxm = [Hs]−1mxm ≈ [Hs]

+1mxm

The components FRF matrix [Hn] may be partially measured and as a result the inverse of the system matrix[Hs] may not exist. In fact, [Hs] is normally ill conditioned and, in general, a pseudo inverse solution can be used.The pseudo inverse will generate a system dynamic stiffness matrix that can be used predict the response of themodified system only at the measured points of interest. The measured FRFs require driving-point measurement atall connection points and at points where external forces might act on the system model. Cross measurement arerequired between the connection/external force points and response points of interest.The primary reason for choosing this matrix formulation is that it is very simple to develop the modification matrixwhere the measurement DOF corresponds to the row or column space of the Dynamic Stiffness matrix. Thismodification matrix is also the same as the matrix used in the Lumped Modeling and Modal Modeling processes.The modification dynamic stiffness matrix [∆Kdyn] in terms of [M ], [C] and [K] matrices is,

(15) [∆Kdyn] =[−ω2[∆M ] + jω[∆C] + [∆K]

]where ω is the frequency measured in rad/s.The modified dynamic stiffness matrix [Ksmod

] is equal to system dynamic stiffness matrix plus the modificationdynamic stiffness matrix or:

(16) [Ksmod] = [∆Ks] + [∆Kdyn]

and the modified Compliance/Frequency Response Function matrix [Hmod] is the inverse or the pseudo inverse ofthe Modified Dynamic Stiffness Matrix.

(17) [Hmod] = [Ksmod]−1 ≈ [Ksmod

]+1

This is important, because the modification matrix will be common between the Impedance and Modal Modelingcases, which means that it is simple to program systems with both analytical and experimentally based components.The system matrices [H] will always be square and the system matrix will be generated by placing the componentmatrices along the diagonal. As a result, the matrix size of the system matrix will be equal to the sum of all thecomponent matrices sizes. This tremendously reduces the book-keeping for the keeping track of the modifications.Hence it is relatively easy using indexing to keep track of information. The same indexing can be used in theImpedance, Model Modeling and analytical modeling scripts.This method is simple to program and on analytical datasets will estimate the modifications of the system to thenumerical accuracy of the computer. For example, in Matlab simulations, the accuracy of solution is to 15 decimalplaces.

3. Experimental Validation

Impedance modeling is a valuable technique for predicting the result of a structural modification to the system.The proposed method to evaluate the static torsional stiffness involves the use of the dynamic stiffness method.The technique is demonstrated to estimate the static torsional stiffness of a rectangular steel plate structure. Theaddition of stiffness constraints to the measured FRF data for three perturbed mass configurations will be utilizedto numerically model the fixed-free configurations utilized in the static testing test-rigs.

3.1. Model Calibration and Validation. An FE model of the rectangular steel plate structure of dimensions34”x22.5”x.25” was developed. A cold rolled steel rectangular plate structure was fabricated (E = 2.05x1011 Pa,ν = 0.29 and ρ = 7850 kg/m3), with 160 points marked on a 2”x2” grid. Each of these 160 points were impactedand FRFs were measured at 21 reference locations using uniaxial accelerometers.The FE model was calibrated to the match measured modal frequencies and modal vectors. The FE model wasvalidated by comparing analytical FRFs from the model with the measured FRFs. In addition, the results obtainedfor two perturbed mass configurations with unconstrained boundaries were compared with the predictions from theupdated model to check its robustness. The predicted values were within 1.5% relative error for modal frequencies,which satisfied the validation criteria. The comparison of modal frequencies for the no added mass case is shown inTable 1.

Table 1. Comparison of Modal Frequencies

Mode Mode descriptionModal Frequency (Hz) Rel error (%)

FE FE w/ spring support Experimental Vs FE Vs FE w/ springs

1 I Torsion 41.3 41.7 42.3 2.19 1.272 I X Bending 43.7 43.8 44.2 1.11 0.953 II Torsion 95 95.4 95.7 0.71 0.364 I Y Bending 103.1 103.2 104.7 1.51 1.445 II X Bending 118.2 118.2 119.2 0.81 0.776 Antisymm X Bending 137.4 137.7 138.1 0.54 0.327 III Torsion 175.5 175.9 176.5 0.53 0.348 X and Y Bending 202.9 203.4 204 0.55 0.279 III X Bending 244.2 244.8 245.2 0.4 0.17

3.2. Estimation of static torsional stiffness analytically. The static torsional stiffness of the rectangular platecan be estimated analytically. Four points on the calibrated plate model are chosen such that they are symmetricabout the centerline. Points 23 and 135 represent the front DOF and points 28 and 140 represent the rear DOFs asshown in Fig. 3. The distance between the front and rear DOF is 10 in and the distance between points 23–135 andpoints 28–140 is Lf , Lr = 14 in. The rear DOFs are constrained in the vertical direction and a couple is applied atthe front DOFs. The static torsional stiffness is estimated using Eq. 18,

Figure 3: Rectangular plate: Front and rear DOF

(18) KT =T

θ=FLf

θwhere KT is the static torsional stiffness, T is the torque, F is the applied couple, Lf is the moment arm and θ is the

angular deflection. θ = 2∆Lf

rad, when the rotation angle meets small angle criteria, where ∆ is the vertical deflection.

A static analysis was performed on the plate. The static torsional stiffness was estimated to be 3607.15 lbf in/deg forthe no added mass configuration.

3.3. Stiffness estimation from Impedance Modeling. The modal modeling technique can be applied to estimatethe static torsional stiffness. However, due to modal truncation, to get reasonable estimates of the static stiffness, thefrequency range for modal testing should contain all the influential modes. Typically, the first 17− 30 deformationmodes are required for the static stiffness estimates that are in the right range. Impedance modeling method lendswell for this purpose as the measured FRF data contains the contribution of the out-of-band modes in addition tothe modes in the frequency range of interest.In order to estimate the static torsional stiffness of the plate using the dynamic stiffness method, a reduced datasetcontaining driving-point and cross-point measured frequency response functions corresponding to the front and rearDOFs needs to be assembled. Stiffness constraints need to be applied to the dynamic stiffness matrix (inverse of theFRF matrix) at locations corresponding the rear DOFs. Subsequently, the torsion loading points (front DOF driving-points) dynamic stiffness functions should be plotted and an appropriate parameter estimation method should beused to fit the stiffness in the low frequency region. This fit yields the torsion loading point stiffness KSS , which inturn is related to the static torsional stiffness by Eq. 19. A visual illustration of this relationship for a built-up beamstructure is shown in Fig. 4.

(19) KT =KSS ∆ Lf

θ=KSS L2

f

2

(a) (b)

Figure 4: Relation between driving-point stiffness and static torsional stiffness

3.3.1. Rigid body modes. Since the vertical motion was only measured using the uniaxial accelerometers, modalparameter estimation would yield three rigid body modes, namely the pitch, bounce and roll modes, fairly close to0 Hz. By applying constraints using high stiffness at the rear DOFs, it is anticipated that the rigid body modes geteliminated and the asymptote of the first deformation mode would yield the driving-point stiffness value KSS . Whilethe bounce and roll modes were pushed high up in frequency, the pitch mode was still observable at 9.1 Hz as shownin Fig. 5(a).In order to estimate the static torsional stiffness, this mode has to be eliminated. The torsional modes contributesignificantly to the static torsional stiffness. These modes have nodal lines along the centerline of the structure. Athird constraint point could be incorporated at any point along the centerline to eliminate the 9.1 Hz pitch mode.The CMIF plot of the structure with three constraints is shown in Fig. 5(b).

(a) With two rear constraints (b) With two rear constraints and one centerline constraint

Figure 5: Torsion loading point FRFs

3.3.2. Measurement noise issue. Data acquisition during modal testing is subjected to measurement noise issues.Signal processing techniques can be used to reduce measurement noise, however it is difficult to eliminate noiseissues completely. When the DOFs are constrained, the useful portion of the signal that corresponds to motion

of the structure is discarded, but the measurement noise is not eliminated. This can limit the applicability of theimpedance modeling method for estimation of static torsional stiffness in the presence of high measurement noise.

3.3.3. Parameter estimation. Using a zeroth order polynomial to fit the dynamic stiffness in the low frequency range(10−20 Hz) yielded a value of 2.08x103 lbf/in, which corresponds to a static torsional stiffness of 3557.68 lbf in/deg. Acomparison of the impedance modeled data with the analytical constrained data is shown using a CMIF plot in Fig.6(a). The frequencies and the overall level of the modeled and analytical data are in close proximity. The dynamicstiffness of the two driving-points obtained from the constrained measurements and the driving-point stiffness fit areshown in Fig. 6(b) and Fig. 6(c) respectively.

(a) Complex Mode Indicator Function (b) Driving-point Dynamic Stiffness

(c) Stiffness fit

Figure 6: Parameter Estimation

4. Conclusions

A method based on impedance modeling technique to estimate the static torsional stiffness of a structure waspresented. In contrast to the resource intensive methods based on static testing and modal modeling, this methodis simple to implement and realize. This method requires FRF driving-point and cross-point measurements at thetwo front and rear locations. With the rear locations are constrained, the torsion loading point stiffness values canbe utilized to estimate the static torsional stiffness. When low frequency rigid body modes exist, it is necessary toconstrain an additional point along the centerline (nodal line for torsion modes) to simplify the parameter estimationprocess. This method did was not as successful as it was hope. Currently work is being done on other methods toovercome some of the shortcomings of this method.

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