Dhupia_JVC_5-08

Experimental Identification of the NonlinearParameters of an Industrial Translational Guidefor Machine Performance Evaluation�

JASPREET S. DHUPIAA. GALIP ULSOYREUVEN KATZNSF Engineering Research Center for Reconf igurable Manufacturing. System Universityof Michigan, Ann Arbor, 2250 GGBL, 2350 Hayward Street, Ann Arbor, MI 48109, U.S.A.([email protected])

BARTOSZ POWALKATechnical University of Szczecin, Piastow 19, 70-310 Szczecin, Poland

(Received 8 August 2006� accepted 30 March 2007)

Abstract: Prediction of machine dynamics at the design stage is a challenge due to lack of adequate methodsfor identifying and handling the nonlinearities in the machine joints, which appear as the nonlinear restoringforce function of relative displacement and relative velocity across the joint. This paper discusses iden-tification of such a nonlinear restoring force function for an industrial translational guide for use with theNonlinear Receptance Coupling Approach (NLRCA) to evaluate machine dynamic characteristics. Transla-tional guides are among the most commonly used joints in machine tools. Both parametric and nonparametrictechniques have been employed to identify the nonlinearities. A novel parametric model based on Hertziancontact mechanics has been derived for the translational guide. A nonparametric method based on two-dimensional Chebyshev polynomials is also used. The models derived from the two techniques, i.e., para-metric and nonparametric, are fitted to the experimental data derived from static and dynamic tests to get therestoring force as a function of relative displacement and relative velocity across the joint. The nonlinear rep-resentation obtained from both techniques is later converted into the describing function representation whichis needed for evaluation of machine dynamic characteristics using the NLRCA. The describing function rep-resentations obtained from the two approaches are compared. The design of experiments for evaluating thenonlinearities in such industrial machine tool joints is a challenge, requiring careful alignment and calibra-tion, because they are typically very stiff. This constrains the dynamic experiments to be carried out at highfrequencies (e.g. 2000–7000 Hz) where the experimental readings are very sensitive to errors in geometryand calibration.

Keywords: machine dynamics, nonlinear joints, receptance coupling, frequency response functions.

� This paper was contributed by Professor N. G. Chalhoub

Journal of Vibration and Control, 14(5): 645–668, 2008 DOI: 10.1177/1077546307081325

��2008 SAGE Publications Los Angeles, London, New Delhi, Singapore

Figures 3–12, 14 appears in color online: http://jvc.sagepub.com

646 J. S. DHUPIA ET AL.

1. INTRODUCTION

Accurate prediction of machine dynamics at the design stage, before the machine is con-structed, is an important goal, which has become even more important with introductionof high-speed and high-performance machines. Research has been done on automaticallygenerating the possible machine configurations for a given kinematic task (Moon and Kota,2002).However, the related work (Moon, 2000) on finding the dynamic performance of eachmachine configuration is preliminary and is based on the linear receptance coupling theory(Bishop and Johnson, 1960). The nonlinearities present in the machine structure make itdifficult to predict machine dynamics. Most current techniques are limited by linear modelassumptions, and are not suitable for handling nonlinearities such as those arising in joints.Research has been done to formulate the effects of nonlinearities on machine dynamics inthe frequency domain (Ferreira and Ewins, 1996� Yigit and Ulsoy, 2002). However, iden-tification of the nonlinear parameters in machine tool joints in a manner suitable for usewith frequency domain techniques can be viewed as a major barrier that prevents successfulindustrial application of such techniques.

The load experienced at the machine tool joints appears as the restoring force developeddue to joint deflections. The nonlinearity in the joint manifests itself as a nonlinear restoringforce function of relative displacement and velocity at the joint. Much work has been done tomodel the stiffness and damping components of this restoring force. The stiffness has beenmodeled based on material properties along with Hertzian mechanics (Johnson, 1982) or theasperity model (Greenwood and Williamson, 1966). The energy dissipation, or damping,can be modeled in many ways. While sliding motion in a translational slide may need tobe modeled through the static or dynamic friction models (Dahl (Dahl, 1976), LuGre (deWit et al., 1995) and Lueven (Swevers et al., 2000)), our work is focused on the dynamicproperties in the normal direction. The damping is quite small and a viscous damping modelwas experimentally found to be sufficient, as will be described later.

System identification techniques can be broadly classified into parametric or nonpara-metric methods. While parametric methods seek to determine the values of the parameter inan assumed model structure for the system to be identified, nonparametric methods seek todetermine the functional representation as well as parameters of the system to be identified.Most of the parametric and nonparametric identification methods employ the least squaresapproach, in which the square of the errors between the measured response and that of theidentified model is minimized, to provide the best estimate. The most commonly used non-parametric methods employ the Volterra series (Lee, 1997) and two-dimensional Chebyshevpolynomials to represent the restoring force surface (Masri et al., 1982a,b�Worden and Tom-linson, 2001). Most of the parametric identification methods are time based (Kapania andPark, 1996� Mohammad et al., 1992� Yasuda and Kamiya, 1999). Time-domain techniqueshave the advantage of requiring less time and effort for data acquisition than the sine-dwelltechniques used for frequency domain identification, and can be used for the identificationof strongly nonlinear systems. Potential drawbacks of such time-domain approaches includeproblems of differentiating noisy signals and inability to accurately estimate the coefficientof terms which are small (Malatkar and Nayfeh, 2003).Frequency domain techniques includeapproaches based on the backbone (or skeleton) curve and limit envelope (Benhafsi et al.,1995� Fahey and Nayfeh, 1998) and harmonic balance method (Yasuda et al., 1997).

MACHINE PERFORMANCE EVALUATION 647

This paper considers identification of the nonlinear restoring force function parametersfor an industrial translational guide and its subsequent conversion to describing functionrepresentation (e.g., Atherton, 1975) for use with Nonlinear Receptance Coupling Approach(NLRCA) (Ferreira and Ewins, 1996� Yigit and Ulsoy, 2002) to evaluate the dynamic charac-teristics of a machine structure. The paper considers both a parametric and a nonparametricapproach to identify the joint parameters. In the parametric approach, a nonlinear stiffnessrelationship has been derived for translational guides with preloaded ball bearings based onHertzian mechanics. The parameters for the model are experimentally obtained from staticand dynamic tests on the translational guide. The results from this approach are comparedwith a generalized nonparametric approach using a two-dimensional Chebyshev polynomialdeveloped by Masri, Sasri and Caughey (Masri et al., 1982a,b). The paper emphasizes thedesign of experiments for estimating the nonlinearities in the translational guide. This is achallenge because the machine tool joints are typically very stiff. Most literature on estima-tion of nonlinearities in the joints considers systems with low natural frequency (less than50 Hz), while the isolated machine tool joints (e.g. translation guide) have high natural fre-quencies (e.g., 2000-7000 Hz). Thus, the dynamic experiments are constrained to be carriedout at high frequencies, where only small displacements may be obtained and therefore thesystem is very sensitive to alignment.

Subsequently, the paper discusses conversion of the experimentally obtained nonlinearrestoring force function in the time domain to a describing function representation in thefrequency domain. The parametric approach for nonlinear joint identification will typicallyyield a system of equations describing the relationship between the restoring force and rela-tive displacement and relative velocity across a joint. As is the case here, it may not alwaysbe possible to analytically convert such a representation into a describing function. Thus,one may compute the describing function numerically for use with the NLRCA for evaluat-ing system dynamics. However, a nonparametric approach, employing the two-dimensionalChebyshev polynomials for nonlinear joint identification, yields a polynomial representationfor the nonlinear restoring force function. This paper presents a derivation for obtaining theclosed form describing function for such a nonlinear restoring force function. Finally, theevaluated describing functions are compared in the frequency domain within the frequencyrange of interest. The effect of these nonlinearities in the joints on overall machine perfor-mance is beyond the scope of this paper, but is described separately in Dhupia et al. (2007).

2. TRANSLATIONAL GUIDE MODELING FOR PARAMETRIC

ANALYSIS

2.1. Geometrical Description of Translational Guide

Translational guides were chosen for experimental evaluation as they are among the mostcommon joints in machine tools. The translational guide chosen for the experiment was aBosch-Rexroth linear guide system: R1621, size 30. The cross-section of the translationalguide used for the experiment is shown in Figure 1. The joint consists of two components,the rail and the runner block and the contact is made through the preloaded ball bearings. Amodel for joint stiffness under normal load P is developed using Hertzian mechanics in this


Figure 1. (a) Cross-section of the translational guide, (b) Detailed view of ball bearing contact.

section. The model accounts for joint stiffness as a property based on material properties,geometry and preload, but can be directly estimated from experiments.

2.2. Hertzian Contact Model for Translational Guide

The model is derived based on the assumption that the runner block and the rail are rigidand all compliance in the system may be attributed to the ball bearings. The ball bearingsare located in four tracks as shown in Figure 1. The lower track bearings are denoted withsubscript L and the upper track by subscript U. The objective of the model is to describe therelationship between external load P and relative displacement �z, taking into account thestiffness relationship at the ball bearings:

P � f ��z�

or ,

�z � f �1�P� (1)

where,�z � z1 � z2�


The contact deformation of the single ball within two grooves can be found by Hertziananalysis as dependent on the normal load Pball as (Rivin, 1999).

dball � 2� 1�41n�3

�P2

ball

2R1 � R2

R1 R2

�1� �2

1

E1� 1� �2

2

E2

�2

(2)

where,

dball � Deformation of a single ballPball � Force acting on a single ballR1 � Radius of the ballR2 � Radius of the grooveE1� �1 � Young’s modulus, Poisson ratio of ball materialE2� �2 � Young’s modulus, Poisson ratio of groove materialn� � Parameter that depends on the ratio R1 and R2. If R2 � � (i.e. ball in contactwith flat surface), then n�� 1

Or, the deformation-load relationship in equation (2) may be defined simply by the model

dball � �P2�3ball (3)

where, � depends upon material and geometrical properties of the joint.Since the bearings are preloaded, a nominal deformation d0 is present even when no

external load P is applied. When load P is applied, the lower set of bearings experiencedeformation dL and the above set of bearings experience deformation dU . When, the balls inall tracks are in contact with the grooves, the amount of compression the balls in the lowertracks experience must be equal to the amount of decompression the balls in upper tracksexperience (see Figure 1) and they can be related to the relative displacement�z through thefollowing equation:

�dL � d0� sin � �d0 � dU � sin � �z� (4)

Let PL be the normal load experienced by the balls in the lower track and PU be theload experienced by the balls in the upper track. Then, the relationship between the jointdeflection and the normal loads developed at the ball bearing may be obtained by usingequation (3) and equation (4):

�z ��P2�3

L � �P2�30

�sin �

��P2�3

0 � �P2�3U

�sin� (5)

Finally, to relate normal loads at the ball bearing (PL and PU ) to external load P, weconsider the free body diagram of the runner block and equate the forces in the verticaldirection to get the relationship:

P � 2PU sin � 2PL sin� (6)


Using equation (6) to substitute for PU in equation (5), we obtain the relationship be-tween load at the lower bearings, PL and the external load, P. This equation may be used toevaluate PL �

2PL sin � P

2 sin

�2�3

� 2P2�30 � P2�3

L � (7)

Equation (7) is solved numerically for PL as a function of the external load P and preloadP0. Putting the resulting PL into equation (5), the relationship between external load P andjoint deflection �z is obtained which is valid when both upper and lower tracks balls are incontact.

Similarly if PL is substituted in equation (5), using equation (6)

P2�3U � 2P2�3

0 ��

2PU sin � P

2 sin

�2�3

� (8)

Equating, PU � 0, we get

P � 4

2P0 sin� (9)

If P exceeds this value, the balls in the upper track will cease to be in compressionand the joint stiffness is provided by the balls in the lower track only, i.e., the relationshipbetween P and PL becomes P � 2PL sin. Putting this result into equation (5) gives therelationship between external load and joint deflection which is valid when only balls in thelower track are in contact.

Our parametric analysis assumes the model structure described by equations (5)–(9).The difference between the experimental and analytical results is attributed to any errorsassociated with geometrical and material properties as well as the change in preload fromthe factory settings, which is expected from regular wear and tear of the slide. Therefore, thevalues of � and P0 are estimated from experimental data for�z vs. P.

3. NONPARAMETRIC ANALYSIS USING TWO-DIMENSIONAL

CHEBYSHEV POLYNOMIALS

A procedure described by Masri et al. (1982a,b) has been used for estimating the governingnonlinear joint relationship when the joint dynamics can be represented by a single degree-of-freedom system. It was assumed that only the translational motion is present betweenthe slide and the runner block of the translational guide. Thus, there is no relative rotationbetween the slide and the runner block and the joint can be modeled as a single degree-of-freedom system as shown in Figure 2(a). The restoring force which is a function ofrelative displacement and velocity may be computed using equation (10), where z2 is directlymeasured by accelerometers mounted on the rail.

f ��z��z� � �m2 z2� (10)


Figure 2. (a) Mass-spring-damper representation for the relative motion of the translational guide, (b)Mass-spring-damper representation of translational guide with steel block.

The translational guide in itself has a very high natural frequency for normal translationalmode (e.g. 6000–7000Hz). However, as the dynamic experiments are carried out at higherfrequencies, the displacements observed reduce significantly. This is because the energyrequired for same displacement is proportional to 2 where the dynamic experiments arecarried at the input excitation frequency . However, high displacement range is desiredfor nonlinear parameter estimation to capture the nonlinear trend effectively. Therefore, thenatural frequency of the system is lowered by attaching a steel block (see Figure 2(b) andFigure 8) of mass m3 to the rail. Due to the high frequency nature of the experiment, the railand the steel block cannot be assumed to act as a single rigid body. Thus, the experimentalevaluation of the restoring force function is modified as:

f ��z��z� � �m2 z2 � m3 z3 (11)

where, z3 is the measured acceleration of the steel block (Figure 2(b)). This equation may beused to calculate the restoring force value at every relative displacement and velocity valuefound through experiment. Once the surface is obtained, the restoring force may be expressedas a nonlinear function expanded in the form of two-dimensional Chebyshev polynomials:

f ��z��z� �Nx�

i�0

Ny�j�0

Ci j Ti ��z� Tj ��z� (12)

Ti and Tj being the ith and jth order Chebyshev polynomials, the Ci j being the associatedcoefficient in Chebyshev expansion of the function.


The property of orthogonality may be used to calculate the Ci j coefficients, which giveresults similar to the minimax polynomial fit, where the largest deviation in error is madesmallest (Worden and Tomlinson, 2001). However in this research we use least square esti-mates using the Chebyshev polynomial basis. The advantage of the Chebyshev basis func-tion is that we can avoid the ill-conditioned matrices that are obtained when fitting regularpolynomial basis with higher order.

4. EXPERIMENTAL PROCEDURE AND RESULTS

4.1. Hertzian Model Fit to Static Experiments

While static experiments were done to determine the parametric model, dynamic experimentsare needed to determine damping properties and for the nonparametric analysis. It was foundexperimentally that the net energy loss due to damping per cycle was less than 2% of themaximum potential energy, and thus damping had an insignificant role in the overall systemdynamics. This low damping is because it is of mostly material nature rather than frictional.Frictional damping is commonly observed in bolted or riveted structures and is much larger.Also, for these experiments the translational guides did not have any lubricant, which mayincrease damping. Therefore, the viscous damping model was assumed sufficient to modelthe joint damping when using Hertzian analysis. Static experiments allow large input forcesto be transmitted to the joint, and allow an understanding of the overall nonlinear relationshipbetween the restoring force and displacement. The restoring force function, f ��z��z�, canbe related to the force exerted during the static experiments by:

f ��z��z� � P ��z�� c��z (13)

where, P is the restoring force developed due to static deflection or due to stiffness alone andc��z represents the contribution due to the viscous damping.

The translational guide was clamped inside a two-piece vise (Figure 3). A Brinell cali-brator was used to measure the amount of force transmitted to the joint. Two displacementsensors were aligned symmetrically to measure displacement on either side of the runnerblock to ensure that only normal translation was observed when external load is applied.Care was taken to minimize any rotation effects in the joint and to ensure a uniform distrib-ution of force on the joint surface.

The experimental results and the estimated nonlinear stiffness curve are shown in Fig-ure 4. These results were obtained by sweeping the preload values and finding � using aleast square estimate. The Root Mean Square (RMS) error for each fit is calculated and thefit giving the minimum RMS error is chosen (Figure 5). The parameters found for the givenresults were P0 � 630 N, � � 4�8403� 10�7 m/N2�3 with a RMS value of 40.1 N.

4.2. Modal Simulation and Experiments

The formulation for both parametric and nonparametric methods assumes that the rail-runnerblock system responds only in the translational direction when an external load P is applied


Figure 3. Top view of the experimental setup for determining static stiffness.

Figure 4. Comparison of analytically derived Hertzian model for joint with the static experimental data.


Figure 5. Variation in the RMS error as preload is varied for the analytically derived Hertzian.

Figure 6. (a) Three degrees-of-freedom model for translational guide, (b) Translational mode alongZ -axis at 4800 Hz, (c) Rotational mode around Y -axis at 2882 Hz, and (d) Rotational model alongX -axis at 2656 Hz.

to it. This is correct, when the system is symmetrical and free from any imperfections andthe load P is applied through the center of mass of the rail-runner block system. However,in reality the load P can cause rotational motion around the X and Y-axes apart from thetranslational motion along the Z-axis (Figure 6). Thus, modeling these three degrees-of-freedom, i.e., relative translation about Z-axis and relative rotation around X and Y-axesbetween the rail and the runner block is needed. Therefore, a corresponding six degrees-of-freedom model, which included the translation displacement and the two rotational motions


for both the rail and the runner block was developed for simulation (for the purpose of thissimulation, the steel block was assumed to be rigidly connected to the rail) and later verifiedexperimentally to determine the natural mode corresponding to the translational response ofthe system. The combined system will have three rigid body modes and three relative motionmodes. One of the relative motion modes corresponds to the relative translation between therail and the runner block. The dynamic experiments for estimating the nonlinearities of thejoint may be done around this natural frequency as the single degree-of-freedom assumptionfor parametric and nonparametric model can be well justified near this natural frequency.

The various motions that need to be modeled are the translational displacements z1 andz2 of the runner block and rail respectively and the angular displacements � x 1 and � x 2 aroundthe X-axis and � y1 and � y2 around the Y-axis respectively. The mass of the runner block, therail and the steel block are denoted by m1, m2 and m3. These were measured and found tobe m1 � 0�942 kg and m2 � m3 � 2�2 kg. The corresponding moment of inertia aroundthe X-axis was evaluated as Ix1 � 5�9927� 10�4 kg-m2 and Ix2 � Ix3 � 7�6785� 10�4 kg-m2. Similarly, around the Y-axis they are evaluated as Iy1 � 9�1761 � 10�4 kg-m2 andIy2 � Iy3 � 66�867� 10�4 kg-m2. The effective stiffness due to the ball bearings is k, and isassumed to be distributed symmetrically at the four points as shown in Figure 6. The lengthbetween equivalent springs along the X-axis is lx � 12�5 mm and along the Y-axis is ly � 21mm. After preliminary investigation of the joint, k was chosen to be 600 N/�m. The systemdynamics can be represented by the mass-spring system equation as

Mx�Kx � f (14)

where,

x � �z1 � x1 � y1 z2 � x2 � y2

�T

is the state vector,

M � diag�

m1 Ix1 Iy1 m2 �m3 Ix2 � Ix3 Iy2 � Iy3

�is the mass matrix, and,

K �

�

k 0 0 �k 0 0

0 l2x k 0 0 �l2

x k 0

0 0 l2yk 0 0 �l2

yk

�k�4 0 0 k 0 0

0 �l2x k 0 0 l2

x k 0

0 0 �l2yk 0 0 l2

yk

� �is the stiffness matrix. The input force vector for this system is

f � �0 0 0 P 0 0

�T�


Figure 7. The frequency response magnitude of the translational guide from sweep sine tests.

It may be noted that the ideal system is decoupled in translation and for each rotationaldirection. However, as mentioned earlier, the imperfections in the joint and excitation ofthe system causes response in the rotational directions also. Modeling them is necessary toverify that these do not lie close to the natural frequency corresponding to the translationalmotion, in which case the single degree-of-freedom approximation will no longer be valideven near the natural frequency corresponding to the translational mode.

The system has six eigenvalues, three of which correspond to the rigid body motions andare zero. The remaining eigenvalues correspond to rotation around the X-axis at 2656 Hz,rotation around the Y-axis at 2882 Hz, and translational motion along the Z-axis at 4800 Hz.The natural frequency for translation was experimentally verified by doing a sine sweeptest and the frequency response magnitude is shown in Figure 7. The natural frequencycorresponding to the translational mode along Z-axis is found to be 4819 Hz from the sinesweep test.

4.3. Hertzian Model Fit to Dynamic Experiments

The Hertzian model derived from the static test, along with the viscous damping coefficientobtained from the data is used to obtain the restoring force function. Since the energy loss percycle is less than 2% of the maximum potential energy, damping has little role in the overalldynamics of the joint and can be modeled adequately using a viscous damping assumption.The setup for the dynamic test is shown in Figure 8. An electromagnetic shaker is used toexcite the translational guide at 4800 Hz which is close to the natural frequency of normal


Figure 8. Experimental setup for dynamic tests.

translational mode. Since displacement and velocity are orthogonal, the viscous damping cmay be calculated as:

c ��

nf ��z��z� ��z�

n��z2

� 503N �s�m (15)

where n is the number of data points. Figure 9 shows the evaluated restoring force surfacefrom the Hertzian model vs. the experimental data from the dynamic test used to derive theviscous parameter for parametric analysis.

4.4. Polynomial Fit to Dynamic Experiments using Chebyshev Polynomials

The experimental setup for nonparametric analysis of the translational joint was same asthat for determining the viscous damping coefficient in parametric analysis (see Figure 8).The experimental results described here were carried out at 4800Hz, the natural frequencyattributed to the normal translational motion. The amplitude of the input excitation wasmanually changed continuously during the experiment. This is done to distribute the datapoints over the entire velocity and displacement range and hence improve the quality ofestimation of nonlinear restoring force function. A third order polynomial fit in displacementand first order polynomial fit in velocity was chosen as it achieved most of the improvementin the Root Mean Square (RMS) of residues. The evaluated nonlinear restoring force functionis as follows:


Figure 9. Experimental data and evaluated nonlinear restoring force surface from parametric analysis.

f ��z��z� � �70�6689� 654�1640�z � 28�5820�z2 � 1�7083�z3

� 463�0620��z � 22�3744�z��z � 19�8415�z2��z � 44�3015�z3��z (16)

where �z is in �m and ��z is in mm/sec. The RMS error was evaluated by calculating thesquare root of the mean of the sum of squares of the residues at each data point. The RMSerror of the polynomial fit is 166N, which is less than 3% of the maximum of restoring forcemeasured. The experimental data used for determining the restoring force function and theevaluated restoring force function are shown in Figure 10.

5. NONLINEAR RECEPTANCE COUPLING APPROACH USING

DESCRIBING FUNCTIONS

After determining the nonlinearities in the translational guide these characteristics are thenused to evaluate the dynamic characteristics of a machine structure. The relevant dynamiccharacteristics of a machine structure can be obtained using the Frequency Response Func-tions (FRFs) and the Stability Lobe diagrams (SLDs) of the machine structure. However,the SLD itself is found analytically by using the FRFs along with the information about thecutting process. Thus, the evaluation of the FRF for the machine structure, including thenonlinear joint, is discussed here. The displacement-force FRF is also referred to as recep-tance. The receptance for the machine structure can be found by the Nonlinear ReceptanceCoupling Approach (NLRCA) described by Ferreira and Ewins (1996).


Figure 10. Experimental data for polynomial fitted nonlinear restoring force surface and evaluatednonlinear restoring force surface.

The NLRCA represents the response of a “post-coupled system” to various input excita-tions based on relations determining response of the “pre-coupled system” to various inputexcitations. The post-coupled system contains the pre-coupled system along with all the“connections”. The local nonlinearities that may be contained within the connections areapproximated by describing functions to represent them in the frequency domain.

Two coordinate sets are defined for this approach: an internal coordinate set and a con-nection coordinate set. While those coordinates related to “connections” are in the connec-tion coordinate set (referred by subscript n or N), the coordinates not related to “connections”are in the internal coordinate set. The connection coordinates are always a union of a pairof coordinates (p j � q j ), which represent the locations across a connection. Index i refers tothe ith connection among a total of M. In the following equations lowercase indices are usedto represent the pre-coupled configuration, while uppercase indices are used to represent thepost-coupled configuration. The pre-coupled response of the system is defined through thereceptance matrices of the system and given by:��

xn

xp

xq

��

�Hnn Hnp Hnq

Hpn Hpp Hpq

Hqn Hqp Hqq

� ��

fn

fp

fq

�� (17)

where Hmn is the receptance matrix between the coordinate sets m and n. In the post-coupledsystem, the response can be written as


��xN

xP

xQ

��

�HN N HN P HN Q

HP N HP P HP Q

HQN HQ P HQQ

� ��

fN

fP

fQ

�� (18)

Ferreira derived the relationship for the post-coupled relationship (equation 18) in termsof the pre-coupled receptance matrices as��

xN

xP

xQ

��

��

Hnn Hnp Hnq

Hpn Hpp Hpq

Hqn Hqp Hqq

� ��Hnp �Hnq�

�Hpp �Hpq�

�Hqp �Hqq�

� �

� [B]�1

��Hnp �Hnq�

�Hpp �Hpq�

�Hqp �Hqq�

� �T��

��fN

fP

fQ

�� (19)

where

B � Hpp �Hqq �Hpq �Hqp � � with � � diag�1�Gi� (20)

Gi is the describing function (e.g., Atherton, 1975) representation for the local nonlinearitiesin the connections. Under the assumption that the output of the nonlinearity to a harmonicinput is primarily dominated by the first harmonic, describing functions are a good approx-imation for the nonlinearity in the frequency domain. The describing function gain is thefundamental component of the Fourier series representation of the periodic output obtainedfrom the nonlinear function to a sinusoidal input of frequency . Let fi��zi ��zi� representsthe ith nonlinear restoring force function for a joint in the machine structure, where �zi isthe relative displacement for the joint along the direction of the restoring force. Thus,

gi�t� � fi�A sint� A cost� (21)

is the output to the sinusoidal input at the joint and can be expanded in a Fourier series. Thedescribing function is then evaluated as:

Gi �� Ai� �

Ai

� 2

0gi �t� �sint � j cost� dt� (22)

In the subsequent sections, the describing function evaluation from the nonlinear parametersobtained from the two different approaches, i.e., parametric and nonparametric approach isconsidered. It will be shown, that the even though the describing functions obtained from thetwo approaches are different because the translational guide is modeled differently, however,they yield similar results when evaluating a machine’s response. The detailed procedure


to evaluate a machine’s dynamic response in terms of Frequency Response Functions andStability Lobe Diagrams is described in Dhupia et al. (2006, 2007).

6. DESCRIBING FUNCTION FROM PARAMETRIC ANALYSIS

RESULTS

The NLRCA approach for determining machine dynamic characteristics involve represent-ing all modules by their pre-coupled receptance matrices H and have the governing nonlinearrestoring force relationships, fi��zi ��zi�, of the interface between them. Equation 19 con-verts the nonlinear differential equations into a nonlinear algebraic approximation using thedescribing function representation of nonlinearities at the joint. This nonlinear algebraicsystem of equations needs to be solved using some numerical technique, e.g. the Newton-Raphson method. Use of the NLRCA to determine the receptance matrix of the structurefor different nonlinearities is described in (Ferreira and Ewins, 1996� Yigit and Ulsoy, 2002�Dhupia et al., 2006). All these have an assumed and simple analytical expression for non-linear restoring force expression. However, as in this case, it may not be straightforward toderive a simple analytical form from the evaluated nonlinearities for the joint, especially inthe parametric approach. This may happen because the parametric description of the non-linearity cannot always be reduced to a simple nonlinear restoring force function descriptiondescribed by one explicit equation. Thus, at every Newton-Raphson iteration step, the de-scribing function has to be numerically evaluated for the chosen amplitude, which may beobtained from the response variables �xn� xp� xq� at that iteration and the chosen frequencyfor the iteration. Figure 11, shows the describing function plot as the amplitude and fre-quencies are changed. At different amplitude levels of relative vibration amplitude, A, thenonlinear stiffness of the joint leads to different DC gains. The roll-off frequency is deter-mined by the stiffness-damping ratio. Since the damping is assumed to be viscous and thechange in stiffness is very small, the roll-off frequency, c, remains almost a constant atabout 1.2 MHz.

Besides the numerical computation of the describing functions, the Newton-Raphsontechnique to find a solution of nonlinear equations requires the computation of the Jacobianof the system of equations describing the system (equation (19)). Because of the lack ofclosed form representation of describing function this Jacobian has to be numerically com-puted by perturbing every variable in the connection coordinate set around the nominal valueof that iteration. The solution to this nonlinear equation is prone to the ill conditioning ofthe Jacobian matrix and may require tuning of several parameters before the desired solu-tion may be obtained. In the next section, the NLRCA approach using the nonparametricanalysis is discussed. Because of the representation of the nonlinearity in the form of a twodimensional polynomial function, a closed form describing function can be obtained, whichavoids several of the above mentioned difficulties in applying the results obtained to theNLRCA.


Figure 11. Describing function response for parametric analysis, c � 1�2 MHz.

7. DESCRIBING FUNCTION FOR NONPARAMETRIC

ANALYSIS RESULTS

The two-dimensional Chebyshev polynomial obtained from the nonparametric analysis canbe expanded as a two dimensional polynomial representation, as in equation (16), of thenonlinear restoring force function as:

f ��z��z� �Nx�

i�0

N y�j�0

Cxyi j �zi��z j (23)

where Cxyi j represents the coefficient of the term containing ith power of relative displacement

and jth power of relative velocity across the joint. Substituting �z � A sint and ��z �A cost

g�A sint� A cost� �N x�i�0

N y�j�0

Cxyi j

j Ai� j sini �t� cos j �t� � (24)

Thus, the describing function representation is given by:

G �� A� �

Ai

� 2

0�sint � j cost�

N x�i�0

N y�j�0

Cxyi j

j Ai� j sini �t� cos j �t� dt� (25)


Substituting � � t

G �� A� �Nx�

i�0

Ny�j�0

1

Cxy

i j j Ai� j�1

�� 2

0sini�1 � cos j �d� �

� 2

0sini � cos j�1 �d�

�� (26)

To evaluate the above describing function consider the integral:

I �� 2

0sinm � cosn �d�� (27)

Substituting, � � sin2 � ,

I �� 1

0�

m�12 �1� � � n�1

2 d� � (28)

The beta function (Kreyszig, 1999), B�x� y�, is defined as:

B �x� y� �� 1

0� x�1 �1� � �y�1 d� and satisfies the property:

B �x� y� � � �x� � �y�

� �x � y�� B �y� x� (29)

where, the gamma function (Kreyszig, 1999), � �x� � ��0 �x�1e��d� and satisfies the prop-

erties:

� �x � 1� � x� �x�

� �1� � 1, and

� �1�2� � � (30)

Thus, the integral I � � �20 sinm � cosn �d� , can be evaluated using gamma function proper-

ties for integers m and n, and may be written as:

I �� 2

0sinm � cosn �d� � B

�m � 1

2�

n � 1

2

��

�m�1

2

��

n�12

2��

�m�n�2

2

� (31)

The describing function integral [equation (26)] can thus be evaluated using the result ob-tained in equation (31) as:


Figure 12. Describing function response from nonparametric analysis, c � 1�2 MHz.

G �� A� � 2

!0��N x�i�odd

0��N y�j�e�en

Cxyi j

j Ai� j�1 B

�i

2� 1�

j � 1

2

�

� j0��N x�

i�e�en

0��N y�j�odd

Cxyi j

j Ai� j�1 B

�i � 1

2�

j

2� 1

�"� (32)

Thus, the describing function representation for the evaluated nonlinear restoring functionsusing nonparametric analysis (equation (16)) is:

G �� A� � 654�1640� 1�7083� 3A2

4� j��10�6

�463�0620� 19�8415

A2

4

�(33)

The describing function response from the nonparametric approach is shown in Figure 12.In the nonparametric approach using Chebyshev functions, both the stiffness as well as

damping is modeled. While damping is quite small and leads to energy dissipation of lessthan 2% of the maximum potential energy in the translational guide per cycle, the nonlinearrepresentation indicates that the damping function is itself quite nonlinear. This results invarying roll-off frequencies for each of the describing function plots for different vibrationamplitudes. However, the small nonlinearity in stiffness, which affects the machine dynam-


Figure 13. Single degree-of-freedom system used for comparison of evaluated joint description.

ics significantly, affects the DC gain of the describing function response. The describingfunction response plot from both approaches show similar trend till the roll-off frequency.

8. DISCUSSION

The nonlinear restoring force function has been evaluated using two different approaches,i.e., parametric and nonparametric approach. Any joint may be evaluated by either of thesetwo techniques. The parametric approach requires considerable modeling effort and it maybe difficult to derive a physically relevant model for each joint. However, it is easier toidentify experimental errors from the parametric approach. Modeling of the joint also givesconsiderable insight for the design of experiments. The nonparametric approach does notrequire such extensive modeling and can be extended to different joints. But problems withexperiments may go unnoticed and yield unexpected or incorrect results later on. The mod-eling of the translational guide during the parametric approach is beneficial for the designof experiment. However, with a good design of the experiment, the nonparametric approachalso gives good results. The objective of the nonlinear estimation of machine joints is thatthe evaluated nonlinear parameters should be used to evaluate the machine dynamics at thedesign stage. We are currently using a nonlinear receptance coupling approach to evaluatethe machine performance at the design stage. This approach uses the describing functionsto represent the joint nonlinearities. It is possible to obtain good closed-form describingfunction representation for the polynomial fitted nonparametric approach presented in thispaper. This is an advantage because this approach may be used to solve for the dynamicperformance of machine tools efficiently and several problems regarding ill-conditioning ofJacobian matrices and evaluation of each iteration step for the nonlinear algebraic equationsolver can be avoided.

The describing function evaluated for the translational guide using the two approacheshas a slightly different representation because of the different approaches that have been em-ployed. In the Hertzian, or the parametric approach, the nonlinearity has been assumed onlyin stiffness and damping has been assumed to be viscous. In the Chebyshev approach, orthe nonparametric approach, a two-dimensional polynomial function models nonlinearity in


Figure 14. (a) SDOF response with parametric describing function, (b) SDOF response with non-parametric describing function.


both stiffness and damping. Thus, while the describing function response at lower frequen-cies is quite similar, the higher frequencies response, i.e. after the roll-off frequency is quitedifferent. However this roll-off frequency is very high (c � 1�2 MHz) since the joint isvery stiff and the damping is quite small. To compare the effect of this describing functionon the machine structure a single degree of freedom system with the evaluated parametersof the translational guide and a mass of 60 kg as shown in Figure 13 is considered. Thefrequency response function for the SDOF system for parametric vs nonparateric approachwith constant displacement amplitudes vibration is shown in Figure 14(a) and (b). Despitedifferences in the describing function response plot the FRFs of the SDOF system are verysimilar. Even though the natural frequencies shift significantly for a small amplitude changefrom 1�m to 5 �m due to the nonlinearity, the difference between the describing functionsat high frequencies does not affect the SDOF system whose frequency range of interest isaround its natural frequency and has the range up to kHz.

It can be observed from the static experiment results (Figure 4) as well as subsequentdiscussion of results in the frequency domain, that the relative displacement vs. normal loadfollows almost a linear trend. The dominating linear terms can also be seen in the restoringforce function representation in equation 16. Thus the joint is only weakly nonlinear. How-ever, it should be noted that even such small nonlinear terms as observed in this joint cancause significant variations in the machine dynamic performance. This important point isdiscussed in detail in (Dhupia et al., 2006) and is also evident from Figure 14.

9. SUMMARY AND CONCLUSIONS

The nonlinear parameters for restoring force function of an industrial translational guidehave been evaluated and an approach to use these parameters for measuring the machinedynamic performance in the design stage has been described. A translational guide has beenselected because it is among the most common joints found in machine structures. Thejoint was found to have a weak nonlinear stiffness term. Two approaches: a parametricapproach based on Hertzian mechanics and a nonparametric approach based on Chebyshevpolynomials has been used. The parameters are later converted into their describing functionrepresentation for use with the Nonlinear Receptance Coupling Approach to determine themachine dynamic characteristics in the design stage. An algorithm to find the closed-formdescribing function from the nonparametric approach using polynomial representation is alsodescribed.

While the modeling of the translational guide via the parametric approach is essential forproper design of experiments, the generalized nonparametric approach is desirable becauseof the possibility to extend it to different joints and the nonlinear parameter results from thepolynomial fitted restoring force function yield closed-form describing function which maybe used to efficiently evaluate the dynamic performance of machine tools at design stage.

Acknowledgments. The authors are pleased to acknowledge the f inancial support of the NSF Engineering ResearchCenter for Reconf igurable Manufacturing Systems (NSF grant # eec-9529125) and the Foundation for Polish Science.


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