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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 35 (2013) 324–344

0888-32

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n Corr

E-m

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Envelope and order domain analyses of a nonlinear torsional systemdecelerating under multiple order frictional torque

Osman Taha Sen, Jason T. Dreyer, Rajendra Singh n

Acoustics and Dynamics Laboratory, Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, Ohio 43210, USA

a r t i c l e i n f o

Article history:

Received 24 February 2012

Received in revised form

17 July 2012

Accepted 7 September 2012Available online 28 September 2012

Keywords:

Friction-induced machinery vibration

Signal processing for speed-dependent

process

Analytical methods

Nonlinear dynamics

Harmonic balance method

Hilbert transform

70/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ymssp.2012.09.008

esponding author. Tel.: þ1 614 292 9044.

ail address: [email protected] (R. Singh).

a b s t r a c t

The broader goal of this article is to re-examine the classical machinery shut down vibration

problem in the context of a two degree of freedom nonlinear torsional system that

essentially describes a braking system example. In particular, resonant amplifications

during deceleration, as excited by a multi-order rotor surface distortion and pad friction

regime, are investigated using a nonlinear model, and the order domain predictions are

successfully compared with an experiment. Then a quasi-linear model at higher speeds is

proposed and analytically solved to obtain closed form expressions for speed-dependent

torque as well as its envelope curve. The Hilbert transform is also utilized to successfully

calculate the envelope curves of both quasi-linear and nonlinear systems. Finally, the multi-

term harmonic balance method is applied to construct semi-analytical solutions of the

nonlinear torsional model, and the order domain results are successfully compared with

measurements. New analytical solutions provide more insight to the speed-dependent

characteristics given instantaneous frequency excitation.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Resonant build-up in rotating machinery during start up or shut down operations is a classical vibration problem [1–5].Lewis [1] introduced this problem in the context of a single degree of freedom linear torsional system that acceleratedfrom rest with a constant angular acceleration and approximated the response envelopes for different acceleration ratesand damping values. Newland [2] claimed that only numerical or approximate analytical methods can be used for suchproblems. The single degree of freedom linear system problem has been numerically solved by Newland [2] for bothacceleration and deceleration cases, and the response amplitudes (with a varying frequency of excitation) have beencompared to those observed under steady state operation (under a constant frequency excitation). In particular, the peakresponse amplitude location (with an instantaneous frequency excitation) depends on whether the system accelerates ordecelerates. Similar observations regarding the peak amplitude levels and the speeds at which they occur have been madefor both linear [3,4] and nonlinear [5] mechanical systems.

The dynamic amplification issue is of vital importance in the vehicle braking systems where the deceleration iscontrolled by the frictional torque, and related vehicle sub-systems are excited by one or more resonances [6–9]. Jacobsson[6,7] has specifically related the dynamic response amplification to the brake judder phenomenon. Jacobsson [6,7] hasemployed a single degree of freedom model under a single order frictional torque (Tf (t)) excitation and then extended thismodel to a simplified vehicle model. In a recent study, Sen et al. [8] have experimentally investigated brake judder and

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www.elsevier.com/locate/ymssp

dx.doi.org/10.1016/j.ymssp.2012.09.008

dx.doi.org/10.1016/j.ymssp.2012.09.008

dx.doi.org/10.1016/j.ymssp.2012.09.008

mailto:[email protected]

dx.doi.org/10.1016/j.ymssp.2012.09.008

O.T. Sen et al. / Mechanical Systems and Signal Processing 35 (2013) 324–344 325

then developed a two degree of freedom model to describe a laboratory experiment. Duan and Singh [9] have approachedthe judder problem using a source-path-receiver network concept where the brake rotor, suspension system, and steeringwheel constitute the essential elements; the resonant growth is calculated in terms of the angular motions of the steeringwheel. Accordingly, the broader goal of this article is to re-examine the Lewis’s [1] and Newland’s problem [2] in thecontext of a two degree of freedom nonlinear torsional system that describes a braking system problem. In particular,exact and approximate analytical solutions to the speed-dependent envelope curves and order domain analyses will besought given a multi-order frictional excitation; predictions will be compared with experimental studies.

2. Problem formulation

Lewis [1] defined the instantaneous frequency excitation to a single degree of freedom linear system asf ðtÞ ¼ Fcosð0:5Lt2þf0Þ where F, f0, and L denote the force magnitude, phase, and acceleration of the rotating system,respectively. The excitation f (t), a sinusoidal or periodic function with a linearly increasing speed or frequency, can arisedue to many sources including mass imbalance. Lewis [1] converted the problem from the time (t) domain to the orderdomain by defining 0.5Lt2

¼2pr, where r corresponds to the number of revolutions from t¼0 to current time and utilizedthe contour integration procedure to determine the envelope curves in the vicinity of critical speed. The current articleextends Lewis’s classical problem by investigating the speed-dependent shaft torque T(t) behavior of a two degree offreedom nonlinear torsional model that is displayed in Fig. 1. Here, a brake rotor (I1) is connected to a flywheel (I2)with nonlinear spring (K12) and damping (C12) elements. The system is decelerated from a certain speed (O1) under theapplication of motion-dependent frictional torques.

The chief external excitation is the multiple order frictional torque Tf (t), and it is generated at the interfaces betweenthe brake rotor and pads on both finger (superscript f) and piston (superscript p) sides of I1, as denoted in Fig. 1. Theseinterfaces are described with a simple point contact model, where kp and cp represent the lumped, linear contact stiffnessand viscous damping terms, respectively. Motion excitations, specifically displacement (x(y1)) and velocity ð _xðy1, _y1ÞÞ, inthe contact models arise due to the rotor surface distortion profiles on both the finger and piston sides. Accordingly themain objectives of this study are as follows: (1) Develop a nonlinear torsional model of Fig. 1 and validate numericalpredictions using experiments; (2) Obtain a closed form analytical expression of the shaft torque T(t) for a simplified quasi-linear model of the example case of Fig. 1 and verify the analytical solution using numerical integration and numericalconvolution methods; (3) Propose a method to calculate the envelope function of T(t) for both linear and nonlinear modelsusing the Hilbert transform concept; and (4) Apply the multi-term harmonic balance method to construct the enveloperesponses of a nonlinear torsional system with focus on the order domain.

3. Nonlinear model

The governing equations for the torsional model of Fig. 1 are derived as follows where y1 is the angular displacement ofI1, y2 is the angular displacement of I2, and d12¼y1�y2 is the relative angular displacement:

I1€y1þC12gðd12Þ

_d12þK12hðd12Þd12 ¼�Tf ðy1, _y1Þ, ð1Þ

I2€y2�C12gðd12Þ

_d12�K12hðd12Þd12 ¼�Td: ð2Þ

Fig. 1. Braking system example: two degree of freedom nonlinear torsional model.

O.T. Sen et al. / Mechanical Systems and Signal Processing 35 (2013) 324–344326

The nonlinear elastic (h(d12)) and dissipative (g(d12)) functions describe a clearance nonlinearity in the shaft and/ortorsional coupling elements. They are expressed in a piecewise manner as follows where 7b represents the amount ofbacklash:

gðd12Þ ¼

1 d124b

0 �brd12rb

1 d12o�b

,

8><>: ð3Þ

hðd12Þ ¼

1�b=d12 d124b

0 �brd12rb

1þb=d12 d12o�b

:

8><>: ð4Þ

The external drag torque (Td) in Eq. (2) is assumed to be time invariant; it combines the dissipative effects from thefriction at bearings, off-brake drag torque at the pad/rotor interface and structural damping in the experiment. However,experiments performed (with and without the brakes) show that the contribution of Td is minimal compared to Tf ðy1, _y1Þ,and the deceleration rate for brakes-on case is an order of magnitude higher than the brakes-off case. The frictional torqueTf ðy1, _y1Þ in Eq. (1), an explicit function of y1(t) and _y1ðtÞ, is divided into rotational Trf (t) and alternating Taf(t) componentssimilar to prior work [6–9]. The rotational part Trf (t) is related to the actuation (hydraulic) pressure (pr(t)), and in the caseof constant pr, Trf becomes time-invariant. The alternating part Taf (t) arises due to distortions on the rotor surface. Hence,the total friction torque ðTf ðy1, _y1ÞÞ is calculated as

Tf ðy1, _y1Þ ¼ Trf þTaf ðy1, _y1Þ, ð5aÞ

Trf ¼ mRðkfpx

fmþkp

pxpmÞ, ð5bÞ

Taf ðy1, _y1Þ ¼ mRðcfp_x

fðy1, _y1Þþkf

pxfðy1Þþcp

p_x

pðy1, _y1Þþkp

pxpðy1ÞÞ, ð5cÞ

where m and R represent the friction coefficient and effective radius respectively. Here, it should be noted that meandeformations (xm) are due to pr, which is constant during the braking event; these mean deformations on both sides arecalculated as xf

m ¼ prA=kfp and xp

m ¼ prA=kpp, where A is the area of the master cylinder. In another word, Eq. (5b) is related to

pr, and the terms inside the parenthesis of Eq. (5c) generate the alternating part of total pressure. The validity of thisstatement has recently been demonstrated by the same authors [8], since the same trends are observed in measuredpressure and surface distortion profiles.

The multiple order distortions in the rotor surface are simultaneously considered in the current paper, unlike some prioranalyses [6,7,9]. These distortions are measured with non-contact displacement sensors and split into multiple orders. They aredescribed by xn(t)¼Xn(t)sin(ny1(t)) and _xnðtÞ ¼ _XnðtÞsinðny1ðtÞÞþn _y1XnðtÞcosðny1ðtÞÞ, where n and Xn represent the order indexand the surface distortion amplitude of the nth order, respectively. Expand these using the Fourier series as

xfðy1Þ ¼

XNo

n ¼ 1

Xfnsinðny1þf

fnÞ, ð6Þ

xpðy1Þ ¼

XNo

n ¼ 1

Xpnsinðny1þf

pnÞ, ð7Þ

where No and fn represent the total number of orders used for the Fourier expansion and the phase at order n, respectively.Here, it should be noted that Trf(t) and Xn(t) are slowly varying functions of t with respect to y1(t). When thermal and mechanicaldeformations are negligible, Xn also becomes time-invariant.

4. Experimental observations and validation of nonlinear model

An analogous laboratory experiment, as depicted in Fig. 2, is designed based on the kinetic energy scaling concept. A singlebrake corner assembly, excluding the suspension system and wheel-tire assembly, is used. Like the mathematical model, theexperiment consists of a flywheel (I2) and a brake rotor (I1) which are connected through a shaft and torsional couplingelements. The kinetic energy of the experiment is about 1/5 of a vehicle and the flywheel inertia is determined accordingly. Sincethe total kinetic energy in this experiment is less than that of a vehicle, lower actuation pressures pr(t) are used. Consequentlythe maximum temperature on the brake rotor during a typical stop does not exceed 50 1C, which minimizes the thermo-elasticeffects, and thus, associated changes in Xn(t) are minimal. In addition, the experiment is performed with constant actuationpressures, thus pr(t)¼pr. A clearance nonlinearity, as defined by Eqs. (3) and (4), is located at the constant velocity jointconnection. Instrumentation includes a strain-gage based torque patch for T(t), a pressure transducer for pr(t), non-contactdisplacement probes for xf(t) and xp(t), and a quadrature encoder on the shaft for y1(t).

Fig. 3(a) shows measured TðO1Þ during a braking event with respect to the measured angular velocity (O1), where theshaft torque is normalized as: TðO1Þ ¼ TðO1Þ=Trf . First, observe several key events in different speed regions, such as

Fig. 2. Schematic of the laboratory experiment and instrumentation.

Fig. 3. Measured shaft torque during the deceleration process. (a) Speed-dependent torque TðO1Þ vs. time-varying speed O1(t); letters A to F denote key

speed regions; (b) Short-time Fourier transform of TðtÞ; shaded areas represent amplitudes in dB re 1.0.



a region with nearly constant oscillatory amplitude (OBoO1oOA, OEoO1oOD), resonant growth at multiple speeds,which implies the existence of a system resonance and multiple orders (ODoO1oOC, OFoO1oOE), and then anasymmetric torque behavior that indicates the presence of a clearance (OCoO1oOB). Second, the lower bound of TðO1Þ

follows the zero torque line over the OCoO1oOB region; this seems to be related to the clearance nonlinearity as well. Togain more insight, discrete short-time Fourier transform of the TðO1Þ record is calculated, and the correspondingspectrogram is given in Fig. 3(b). Initially, several order lines are visible that prove the existence of multiple orders in therotor profile. Moreover, the system resonant frequency (o1) is experimentally found around 22 Hz, and the super-harmonics of the excitation frequency evolve as the orders cross o1.

The order lines are projected onto the speed (O1) axis in Fig. 4. These order lines are extracted from Fig. 3(b) with aconstant bandwidth filter, whose center frequency is an integer multiple of the excitation frequency. For a detailedexplanation of this order extraction process, refer to a recent paper by the authors [8]. It is seen that the first order does notcross the system resonance over the experimental speed limits. In addition, the first and second orders contribute most tothe total response at higher speeds. When the system slows down to medium speeds, second and third orders excite thesystem resonance at O1¼o1/2 and O1¼o1/3 respectively. Hence the speed range between adjacent orders is given withthe general expression (o1/(n�1))�(o1/n)¼o1(1/(n(n�1))), where n should be the higher order number between theadjacent orders. Note that this expression is simply o1(n�2)!/n!. This expression shows that the speed ranges betweenadjacent orders reduce as n increases, as clearly seen in Fig. 4. Note that peaks are very close to each other in the lowerspeed region, and thus all of the orders seem to contribute to the total response almost on the same basis. Similar toFig. 3(b), super-harmonics of the excitation frequency are again apparent as small peaks on the n¼3 and 4 order lines areseen in Fig. 4 at O1¼650 rpm. These super-harmonics arise due the excitation of the system resonance by the 2nd ordersurface distortion profile at O1¼650 rpm. For further discussion of the experimental observations, refer to [8].

The nonlinear model of Fig. 1, as given by Eqs. (1) and (2), is solved numerically using the Runge–Kutta integrationscheme. Predictions are compared with measurements in Table 1 in terms of the normalized peak-to-peak torqueamplitudes ðTppðtÞÞ. Since a good match for TppðtÞ is obtained over several speed ranges, this nonlinear model is used forfurther analytical investigations.

Fig. 4. Measured speed-dependent torque order amplitudes ðTðO1ÞÞ for n¼1, 2, y, 8. Key: , n¼1; , n¼2; , n¼3; , n¼4;

, n¼5; , n¼6; , n¼7; , n¼8.

Table 1

Experimental validation of the nonlinear torsional model in terms of shaft torque TppðtÞ

amplitudes (peak-to-peak). Refer to Fig. 3 for speed regions.

Speed region Measured TppðtÞ Predicted TppðtÞ

(numerical method)

OBoO1oOA 1.37 1.32

OCoO1oOB 4.87 4.53

ODoO1oOC 15.92 15.60

OEoO1oOD 2.33 2.18

OFoO1oOE 5.40 4.41

O1oOF 1.99 1.50


5. Analytical solution of the quasi-linear model for damped case

5.1. Mathematical formulation of the quasi-linear model

The two degree of freedom model is simplified by ignoring the clearance nonlinearity, and the resulting quasi-linearmodel, as given below by Eqs. (8) and (9), is analytically solved.

I1€y1þC12ð

_y1�_y2ÞþK12ðy1�y2Þ ¼ �Tf ðy1, _y1Þ, ð8Þ

I2€y2þC12ð

_y2�_y1ÞþK12ðy2�y1Þ ¼ �Td: ð9Þ

For the analytical solution, identical elastic and dissipative properties are assumed on finger and piston sides. Therefore,the rotor surface distortion inputs merge, and the combined amplitude and phase at the nth order are defined as

Xn ¼ 9Xfnexpðiff

nÞþXpnexpðifp

nÞ9, ð10Þ

fn ¼ tan�1 ImðXfnexpðiff

nÞþXpnexpðifp

nÞÞ

ReðXfnexpðiff

nÞþXpnexpðifp

nÞÞ

!: ð11Þ

Substituting Eqs. (6), (7), (10) and (11) into Eqs. (5a)–(5c), the resulting expression for Tf ðy1, _y1Þ is

Tf ðy1, _y1Þ ¼ mRprAþmRcp

XNo

n ¼ 1

n _y1Xncosðny1þfnÞþmRkp

XNo

n ¼ 1

Xnsinðny1þfnÞ: ð12Þ

In addition, it is assumed that y1 and y2 are composed of mean and alternating parts, i.e. y1(t)¼yr1(t)þya1(t),y2(t)¼yr2(t)þya2(t). Here, the mean parts refer to the pure rotational (spinning) motion without any oscillations, and ya2(t)is assumed to be negligible due to I2b I1. Since the experiments show that the deceleration of the system is almostconstant, €yr2 ¼�L, _yr2ðtÞ ¼�LtþO0, and yr2(t)¼�0.5Lt2

þO0tþC0, where O0 and C0 represent the initial angularvelocity and position of I2, respectively. After importing these expressions to Eqs. (8) and (9), yr1(t) and ya1(t) are expressedas follows:

yr1ðtÞ ¼�L2

t2þO0tþF�LI1�mRprA

K12expðzo1tÞ

� �cos

gI1

t

� �þ

C12

2g singI1

t

� �� , ð13Þ

ya1ðtÞ ¼

Z t

0Xðt�uÞYðuÞdu, ð14Þ

where F¼C0þLI1=K12�mRprA=K12, g¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI1K12�ðC12=2Þ2

q, o1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK12=I1

p, and z¼ C12=2

ffiffiffiffiffiffiffiffiffiffiffiffiI1K12

p. Observe that Eq. (14) is

the convolution of two functions, say X(t) and Y(t), in time domain which are defined as follows:

XðtÞ ¼expð�o1ztÞ

g singI1

t

� �, ð15Þ

YðtÞ ¼ �mRcp

XNo

n ¼ 1

n _y1ðtÞXncosðny1ðtÞþfnÞ�mRkp

XNo

n ¼ 1

Xnsinðny1ðtÞþfnÞ: ð16Þ

5.2. Analytical solution of the alternating part of rotor displacement

Eq. (16) suggests that y1(t) must be known before an explicit expression of Y(t) can be found. However, only the meancomponent of y1(t) is known as given by Eq. (13). Thus, Y(t) is approximated by ignoring the contribution of ya1(t) toTf ðy1, _y1Þ. However, Eq. (14) still has no analytical solution due to the oscillating component in Eq. (13). In order to obtain aclosed form solution, this oscillatory component must be ignored, and Eq. (13) is thus simplified by yr1(t)ffi�0.5Lt2

þ

O0tþF. This is a reasonable assumption since the amplitude of the oscillating component (LI1/K12�mRprA/K12) is around0.008 rad at the beginning of the stop and decreases with time due to the exp(zo1t) term in the denominator. On the otherhand, the dissipation term in Eq. (16) has a minimal effect on Tf ðy1, _y1Þ due to small cp and decreasing _yr1ðtÞ with time.Thus, this term is dropped in Eq. (16) for the sake of simplicity. After these assumptions, importing Eqs. (15) and (16) intoEq. (14) yield:

ya1ðtÞ ¼�XNo

n ¼ 1

mRkpXn

g

Z t

0expð�o1zðt�uÞÞsin

gI1ðt�uÞ

� �sin �

nL2

u2þnO0uþFn

� �du, ð17Þ


where Fn ¼ nFþjn. Using Euler’s identity, Eq. (17) is written in the form of

ya1ðtÞ ¼ �Im

(XNo

n ¼ 1

mRkpXn

2ig

Z t

0expð�o1zðt�uÞÞ exp

igI1ðt�uÞ

� ��exp �

igI1ðt�uÞ

� �� exp �

inL2

u2þ inO0uþ iFn

� �du

)

ð18Þ

Combine the exponential terms, one gets,

ya1ðtÞ ¼ ReXNo

n ¼ 1

mRkpXn

2g

exp �o1ztþ igtI1þ iFn

� � R t0 exp � inL

2 u2þ o1z� igI1þ inO0

� �u

� �du

�exp �o1zt� igtI1þ iFn

� � R t0 exp � inL

2 u2þ o1zþ igI1þ inO0

� �u

� �du

0B@

1CA

8><>:

9>=>;: ð19Þ

Rewrite Eq. (19) using the complex variable transformation u¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=inL

pand du¼ dv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=inL

p:

ya1ðtÞ ¼ ReXNo

n ¼ 1

mRkpXn

gffiffiffiffiffiffiffiffiffiffiffi2inLp


� � R tffiffiffiffiffiffiffiffiffiffiinL=2p

0 expð�v2þ2w1vÞdv


� � R tffiffiffiffiffiffiffiffiffiffiinL=2p

0 expð�v2þ2w2vÞdv

0BB@

1CCA

8>><>>:

9>>=>>;, ð20Þ

where

w1 ¼o1z�ig=I1þ inO0ffiffiffiffiffiffiffiffiffiffiffi

2inLp , w2 ¼

o1zþ ig=I1þ inO0ffiffiffiffiffiffiffiffiffiffiffi2inLp : ð21a;bÞ

Further manipulation of Eq. (20) provides a more recognizable expression:

ya1ðtÞ ¼ ReXNo

n ¼ 1

mRkpXn



� �expðw2

1ÞR t

ffiffiffiffiffiffiffiffiffiffiinL=2p

0 expð�ðv�w1Þ2Þdv


� �expðw2

2ÞR t


0 expð�ðv�w2Þ2Þdv

0BB@

1CCA

8>><>>:

9>>=>>;: ð22Þ

Integrals given in Eq. (22) could be solved by using the error function [10]:

erfðzÞ ¼2ffiffiffiffipp

Z z

0expð�t2Þdt: ð23Þ

Since the forms of integrals of Eq. (22) and (23) differ, define new variable transformations for each integral, i.e.z1¼v�w1, dz1¼dv, z2¼v�w2 and dz2¼dv. Eq. (22) is rewritten as:

ya1ðtÞ ¼ ReXNo

n ¼ 1

mRkpXn



� �expðw2

1ÞR t


�w1

�w1expð�z2

1Þdz1


� �expðw2

2ÞR t


�w2

�w2expð�z2

2Þdz2

0BB@

1CCA

8>><>>:

9>>=>>;: ð24Þ

Note that the lower limits of the integrals in Eq. (24) are non-zero values. By considering this, the solution is found as

ya1ðtÞ ¼ ReXNo

n ¼ 1

mRkpXnffiffiffiffipp

2gffiffiffiffiffiffiffiffiffiffiffi2inLp


� �expðw2

1Þ erf tffiffiffiffiffiffiinL

2

q�w1

� ��erfð�w1Þ

� ��exp �o1zt� igt

I1þ iFn

� �expðw2

2Þ erf tffiffiffiffiffiffiinL

2

q�w2


� �0B@

1CA

8><>:

9>=>;: ð25Þ

Next, the dynamic torque in the shaft is calculated with TðtÞ ¼ C12½_y2ðtÞ� _y1ðtÞ�þK12½y2ðtÞ�y1ðtÞ�, which requires the

knowledge of _yr1ðtÞ and _ya1ðtÞ. Note that _ya2ðtÞ was assumed to be zero, and thus _yr2ðtÞ ¼ �LtþO0. Finally, _yr1ðtÞ and _ya1ðtÞ

are obtained by taking the derivatives of Eqs. (13) and (25) with respect to t.

_yr1ðtÞ ¼�LtþO0�LI1�mRprA

K12expðzo1tÞ

� �zo1þ

C12

2I1

� �cos

gI1

t

� �þ

C12zo1

2g �gI1

� �sin

gI1

t

� �� , ð26Þ

_ya1ðtÞ ¼ ReXNo

n ¼ 1



�o1zþ igI1

� �exp �o1ztþ igt

I1þ iFnþw2

1

� �erf t

ffiffiffiffiffiffiinL

2

q�w1


� �þexp �o1ztþ igt

I1þ iFnþw2

1

� � ffiffiffiffiffiffiffiffi2inLp

qexp � t


2

q�w1

� �2� �

� �o1z� igI1

� �exp �o1zt� igt

I1þ iFnþw2

2

� �erf t


2

q�w2


� ��exp �o1zt� igt

I1þ iFnþw2

2

� � ffiffiffiffiffiffiffiffi2inLp

qexp � t


2

q�w2

� �2� �

0BBBBBBBBBBB@

1CCCCCCCCCCCA

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;: ð27Þ


5.3. Calculation of envelope curves from the analytical solution

In order to calculate the envelope curve of T(t), rewrite Eq. (19) by combining the exponential terms and separating thereal and imaginary parts.

yalðtÞ ¼ ReXNo

n ¼ 1

mRkpXn

2g

R t0 exp �o1ztþo1zuþ i gt

I1þFn�

nL2 u2�

gI1

uþnO0u� ��

du

�R t

0 exp �o1ztþo1zuþ i � gtI1þFn�

nL2 u2þ

gI1

uþnO0u� ��

du

0B@

1CA

8><>:

9>=>;: ð28Þ

Imaginary exponents in Eq. (28) can be further expanded in sine and cosine terms by using Euler’s identity; thus,these terms are the periodic functions in the above equation. However, for the envelope curve, such periodic functionscould be eliminated by taking an absolute value of the expression. Therefore the envelope function of ya1(t) is defined asfollows:

ya1ðtÞ ¼XNo

n ¼ 1



exp �o1ztþ O0o1zL �

go1zI1Ln

� �erf t


2

q�w1


� ��exp �o1ztþ O0o1z

L þgo1zI1Ln

� �erf t


2

q�w2


� �264

375

��: ð29Þ

It should be noted that the second terms in Eq. (29) contribute little to the total envelope curve. To explain this, writethe arguments of error functions as

t

ffiffiffiffiffiffiffiffiinL

2

r�w1 ¼

�o1zþo1

ffiffiffiffiffiffiffiffiffiffiffiffi1�z2

q�n _y2

2ffiffiffiffiffiffiffinLp

0@

1Aþ i

o1zþo1


q�n _y2


0@

1A, ð30Þ

t


2

r�w2 ¼

�o1z�o1


q�n _y2


0@

1Aþ i

o1z�o1


q�n _y2


0@

1A: ð31Þ

Due to lower z values, the o1


qterm is almost equal to the resonant speed of the system. Thus, the o1


q�n _y2

term is negative in Eq. (30) before the nth order excites this resonance. In addition, the real part of Eq. (30) will always beslightly lower than the imaginary part due to the o1z term. However, the real and imaginary parts of Eq. (31) are alwaysnegative. Accordingly Eqs. (30) and (31) yield the following conditions:

Re t


2

r�w1

!o Im t


2

r�w1

!o0 for n _y24o1, ð32aÞ

0oRe t


2

r�w1

!o Im t


2

r�w1

!for n _y2oo1, ð32bÞ

Re t


2

r�w2

!o Im t


2

r�w2

!o0 for _y240: ð32cÞ

The error function of complex arguments that satisfy the first and third conditions in Eq. (32) is approximately equal to�1. However, the error function generates significantly large numbers for the second condition. Hence, the second term inEq. (29) is dropped as long as the direction of the (mean) rotor rotation stays the same. Similarly, the envelope function ofya1(t) is

_ya1ðtÞ ¼XNo

n ¼ 1



�o1zþ igI1

� �exp �o1ztþ O0o1z

L �go1zI1Ln

� �erf t


2

q�w1


� �þ

ffiffiffiffiffiffiffiffi2inLp

qexp �o1ztþ O0o1z

L �go1zI1Ln�

�o1zþo1

ffiffiffiffiffiffiffiffi1�z2p

�n _y2

2nL

� �2 !

26664

37775

��

��: ð33Þ

6. Verification of analytical solutions for quasi-linear model

Besides the analytical solution, T(t) is also obtained by solving Eqs. (8) and (9) by using a numerical Runge–Kuttaintegration method in the state space form to calculate both angular displacements and velocities. Further, Eq. (14) issolved using the numerical convolution technique; the result of Eq. (14) is numerically differentiated, and then_ya1ðtÞ is calculated. The results for all three solutions are displayed in Fig. 5, and a summary of peak-to-peak torqueamplitudes ðTppðtÞÞ, at certain speeds is listed in Table 2. The speed range used for simulations (Fig. 5) is wider thanthe experimental limits (Fig. 3(a)). In order to be consistent with Fig. 3(a), speeds labeled with A–F designationsremain unchanged, and two new event labels (G, H) are designated in the higher speed region to highlight the role ofn¼1 term.

Fig. 5. Predicted shaft torque TðtÞ for the quasi-linear model; letters A to G denote key speed regions. (a) Analytical solution; (b) numerical solution of

Eqs. (8) and (9); and (c) solution of Eq. (14) using numerical convolution. Refer to Table 2 for a quantitative comparison of methods.


Table 2Numerical verification of the analytical solution for the quasi-linear model in terms of

TppðtÞ amplitudes (peak-to-peak). Refer to Fig. 5 for speed regions.

Speed region Analytical

solution

Numerical

integration

Convolution

integral

OGoO1 0.76 0.66 0.65

OHoO1oOG 13.67 13.66 13.71

OBoO1oOH 0.90 0.86 0.86

OCoO1oOB 22.68 22.63 22.84

OEoO1oOD 1.46 1.47 1.47

OFoO1oOE 3.51 3.57 3.50

O1oOF 2.37 2.44 2.51

Table 3

Comparison of calculated shaft torque TppðtÞ and rotor

surface distortion X amplitudes for first six orders. All

values are normalized with respect to the first order.

Order index n Tpp,nðtÞ=Tpp,1ðtÞ Xn/X1

1 1.00 1.00

2 1.66 1.85

3 0.26 0.24

4 0.17 0.17

5 0.12 0.09

6 0.15 0.16

Fig. 6. Analytical prediction of TðtÞ and its envelope for the quasi-linear model. Key: , TðtÞ vs. O1; , envelope of TðtÞ.


All three methods are in good agreement in Fig. 5, and they show the same trends: (i) almost constant TðtÞ amplitudesfor OGoO1, OBoO1oOH and OEoO1oOD, and (ii) resonant growth for OHoO1oOG, OCoO1oOB and OFoO1oOE. Inaddition, other dynamic amplifications occur in the O1oOF region; significant amplifications are observed for n¼1 andn¼2. However, resonant amplifications are not as significant for nZ3, especially at the lower speeds. In order to explainorder-dependent amplification levels, values of Xn and TppðtÞ are compared in Table 3, where the nth order amplitude isnormalized by the corresponding value at n¼1. Results show an almost linear relationship between TppðtÞ and Xn. Noamplifications are observed in T(t) for n46, since the Xn values for n46 are much smaller than X1.

Envelope curves of T(t), as calculated with Eqs. (29) and (33), are compared in Fig. 6. The closed form analytical solutionsuccessfully predicts the envelope of T(t). As seen in Fig. 6, the envelope curve tracks the peak amplitude values and thus itmay be used as a metric in the product design process.


7. Envelope analysis using the Hilbert transform

The envelope function of ya1(t) is obtained by using the Hilbert transform which is defined as: ~yðtÞ ¼ yðtÞnðptÞ�1¼

ð1=pÞðR1�1

yðtÞ=ðt�tÞdtÞ, where ~yðtÞ represents the Hilbert transform of y(t), and * is the convolution product; essentially~yðtÞ shifts the phase of the spectral content of y(t) by �p/2 while keeping the magnitude intact [11]. Since the Hilberttransform definition has a singularity at t¼t, this integral should be considered as ‘Cauchy’s principal value’ [11].The Hilbert transform of the integral in Eq. (25) is as follows.

~ya1ðtÞ ¼ Re

XNo

n ¼ 1

mRkpXn

2gffiffiffiffiffiffiffiffiffiffiffiffiffiffi2inLpp expðiFnþw2

1Þ

Z 1�1

expð�o1ztþðigt=I1ÞÞerfðtffiffiffiffiffiffiffiffiffiffiffiffiffiffiinL=2

p�w1Þ

t�t dt !

�XNo

n ¼ 1

mRkpXn


1Þerfð�w1Þ

Z 1�1

expð�o1ztþðigt=I1ÞÞ

t�tdt

� �

�XNo

n ¼ 1

mRkpXn


2Þ

Z 1�1

expð�o1ztþðigt=I1ÞÞerfðtffiffiffiffiffiffiffiffiffiffiffiffiffiffiinL=2

p�w2Þ

t�tdt

!

�XNo

n ¼ 1

mRkpXn


2Þerfð�w2Þ

Z 1�1

expð�o1ztþðigt=I1ÞÞ

t�t dt� �

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

: ð34Þ

Since there is no known analytical solution of Eq. (34), ~ya1ðtÞ is calculated in the frequency domain by calculating thespectral content of ya1(t) in terms of magnitude and phase and then regenerating ~ya1ðtÞ by shifting the phase content ofya1(t) by �p/2. Fig. 7 shows the analytical TðtÞ and its envelope for single and multiple orders. The Hilbert transformpredicts the envelope of TðtÞ reasonably well for both single (n¼1 or 2) and multiple order cases. In addition, the sameapproach is employed for the nonlinear model, given by Eqs. (1) and (2), and the calculated envelope functions aredepicted in Fig. 8, again for n¼1 or 2 and multiple orders. The Hilbert transform predictions are reasonable, though furtherprocessing of data [11] could yield better results; it is however beyond the scope of this article.

8. Construction of semi-analytical solution for the nonlinear model for order domain analysis

8.1. Nonlinear single degree of freedom system formulation

The two degree of freedom, semi-definite, nonlinear model of Fig. 1 is simplified to a single degree of freedom system asshown in Fig. 9. The resulting governing equation, with d12 as the new response variable, is

Ic€d12þC12g1ðd12Þ

_d12þK12h1ðd12Þd12 ¼�Tf cðy1, _y1ÞþTdc , ð35Þ

where Ic¼ I1I2/(I1þ I2), Tf cðy1, _y1Þ ¼ Tf ðy1, _y1ÞIc=I1, and Tdc¼TdIc/I2. The discontinuous functions g(d12) and h(d12) areapproximated as

g1ðd12Þffi1þtanhðsðd12�bÞÞ

2�

tanhðsðd12þbÞÞ

2, ð36Þ

h1ðd12Þffi1þðd12�bÞtanhðsðd12�bÞÞ

2d12�ðd12þbÞtanhðsðd12þbÞÞ

2d12: ð37Þ

The regularizing factor (s) in Eqs. (36) and (37) dictates the approximation; indeed prior research [12] suggests that itshould be at least 100. In Eq. (37), it is obvious that h1(d12)�41 when d12�40; thus, no singularity would exist.

As explained in Section 3, Tf ðy1, _y1Þ is an explicit function of y1 and _y1; thus, by defining y1¼O1t with y1 2 ½0,2pÞ, thederivatives in Eq. (35) are replaced with _d12 ¼O1d

0

12 and €d12 ¼O21d00

12 where ð0Þ ¼ dðÞ=dy1. The governing equation becomes:

IcO21d00

12þC12g1ðd12ÞO1d0

12þK12h1ðd12Þd12 ¼�Tf cðy1, _y1ÞþTdc: ð38Þ

8.2. Application of multi-term harmonic balance method

The harmonic balance method has been primarily utilized to calculate the steady-state responses of nonlinear systemswhen excited by sinusoidal or periodic excitations [13–16]. In the current article, this method is applied to the singledegree of freedom nonlinear model of Fig. 9 given the instantaneous frequency excitation to construct the speed-dependent response envelope. Since the multi-term harmonic balance method is essentially a residue minimizationprocess for constructing periodic responses, periodic functions of the entire solution space are used as a basis. Accordingly,the periodic response in the order domain (y1) is written as

d12ðy1Þ ¼ a0þXNh

m ¼ 1

a2m�1sinðmy1Þþa2mcosðmy1Þ, ð39Þ

Fig. 7. Predicted TðtÞ and calculated envelopes (using the Hilbert transform concept) for the linear system for various orders; (a) n¼1; (b) n¼2; and

(c) n¼1, 2, y, 8. Key: , TðtÞ vs. O1; , envelope of TðtÞ.


Fig. 8. Shaft torque TðtÞ and calculated envelopes (using the Hilbert transform) for the nonlinear model for various orders; (a) n¼1; (b) n¼2; and

(c) combined n¼1, 2, y, 8. Key: , TðtÞ vs. O1; , envelope of TðtÞ.


Fig. 9. Simplified nonlinear single degree of freedom system model for the application of multi-term harmonic balance method.


where Nh is the number of harmonics retained in the Fourier series expansion. Note that Eq. (39) does not include any sub-harmonics of the excitation frequency since they are not observed in measurements. Eq. (39) is discretized as d12¼Cawhere a and C are the vectors of Fourier coefficients and discrete Fourier transform matrix, respectively. The discreteFourier transform matrix for the order domain (y1) is defined as

C¼

1 sinðy1,0Þ cosðy1,0Þ . . . sinðNhy1,0Þ cosðNhy1,0Þ

1 sinðy1,1Þ cosðy1,1Þ . . . sinðNhy1,1Þ cosðNhy1,1Þ

^ ^ ^ & ^ ^

1 sinðy1,N�1Þ cosðy1,N�1Þ . . . sinðNhy1,N�1Þ cosðNhy1,N�1Þ

266664

377775, ð40Þ

where N is the number of the data points in the discretized signal. Similarly, g1(d12) and h1(d12) are expanded using thetruncated Fourier series as g1(d12)¼Cb and h1(d12)¼Cc where b and c are the Fourier coefficient vectors of nonlinearfunctions.

Derivatives of d12 with respect to y1 are obtained as d012 ¼CDa and d0012 ¼CD2a, where D is the differential operatormatrix defined as

D¼

0

0 �1

1 0

0 �2

2 0

&

0 �Nh

Nh 0

2666666666666664

3777777777777775: ð41Þ

By using the aforementioned Fourier series expansions, derivatives of d12, and substituting Eqs. (39)–(41) in to Eq. (38),the discretized equations in time domain are constructed as follows:

O21CD2aþ

C12

IcCbO1CDaþ

K12

IcCcCa¼CQ , ð42Þ

where Q is the excitation vector. The first term in the Q vector corresponds to the constant part of the total excitation andthe rest of the terms describe the periodic sub-components of Tf ðy1, _y1Þ with No orders. Since the Fourier expansions arecarried out with sine and cosine terms, there are two terms at each order. In summary, the Q vector has the followingform:

Q ¼ q0 qs1 qc

1 . . . qsNo

qcNo

0 . . .h iT

, ð43Þ

where the superscript T indicates the transpose, and q0, qsn and qc

n represent the constant (dc), sine, and cosine parts of thenth order, respectively. From Eqs. (5–7), these terms are found as follows:

q0 ¼�mRðkf

pxfmþkp

pxpmÞ

I1þ

Td

I2, ð44Þ


qsn ¼�

mR9kfpX

fnexpðiff

nÞþkppX

pnexpðifp

nÞ9

I1, ð45Þ

qcn ¼�

mR9cfpnoXf

nexpðiffnÞþcp

pnoXpnexpðifp

nÞ9I1

: ð46Þ

As seen in Eq. (43), the number of orders in the Q vector must be equal to the number of orders (No) retained for therotor surface profile.

Eq. (42) must be transferred to the order domain to obtain a. However, this equation cannot be directly transferred tothe order domain due to nonlinear g1(d12) and h1(d12) functions. Thus, the nonlinear dissipative and elastic functions arecombined with d012 and d12 terms, respectively, and the combined expressions are represented as:

G1ðd12,d012Þ ¼ g1ðd12Þd0

12, ð47Þ

H1ðd12Þ ¼ h1ðd12Þd12: ð48Þ

Like the previous expansions, it is assumed that G1ðd12,d012Þ and H1(d12) can be represented in a truncated Fourier seriesform as G1ðd12,d012Þ ¼Cd and H1(d12)¼Ce. With this assumption, Eq. (42) turns into:

O21CD2aþ

C12

IcO1Cdþ

K12

IcCe¼CQ , ð49Þ

which can be transferred to the order domain by pre-multiplying with the pseudo-inverse of C as Cþ¼(CTC)�1CT. Theequation must be balanced to obtain the correct a vector, and the residue function R (as defined below) should be close tozero for the correct solution.

R¼O21D2aþ

C12

IcO1dþ

K12

Ice�Q : ð50Þ

In Eq. (50), d and e are not known and need to be replaced with functions G1ðd12,d012Þ and H1(d12). The residue isminimized with the Newton–Raphson method, which requires the calculation of the Jacobian matrix (J) of R, defined as

J¼@R

@a¼O2

1D2þ

C12

IcO1C

þ @G1ðd12,d012Þ

@aþ

K12

IcCþ

@H1ðd12Þ

@a: ð51Þ

The partial derivatives of G1ðd12,d012Þ and H1(d12) with respect to a in Eq. (51) are calculated by using the chain rule as

@G1ðd12,d012Þ

@a¼ d012

@g1ðd12Þ

@aþg1ðd12Þ

@d012

@a¼ d012

@g1ðd12Þ

@d12

@d12

@a|ffl{zffl}C

þg1ðd12Þ@d012

@a|ffl{zffl}CD

, ð52Þ

@H1ðd12Þ

@a¼ d12

@h1ðd12Þ

@aþh1ðd12Þ

@d12

@a¼ d12

@h1ðd12Þ

@d12

@d12

@a|ffl{zffl}C

þh1ðd12Þ@d12

@a|ffl{zffl}C

: ð53Þ

By using Eqs. (52) and (53) in Eq. (51), J becomes:

J¼O21D2þ

C12

IcO1C

þ CDa@g1ðd12Þ

@d12Cþg1ðd12ÞCD

� �þ

K12

IcCþ Ca

@h1ðd12Þ

@d12Cþh1ðd12ÞC

� �: ð54Þ

Since the Newton–Raphson method is an iterative procedure, a is updated at each step (say kþ1) with the followingequation:

akþ1 ¼ ak�J�1k Rk, ð55Þ

where the subscript k represents the iteration index. The iterations continue until a converges to the solution where theconvergence criterion is defined as :R:oe where : � : denotes the Euclidian norm operator, and e is a very small number.

8.3. Arc-length continuation and stability of solution

In order to obtain the envelope curve of the nonlinear system, a is calculated over a given speed range. However, unlikethe prior envelope curves of Sections 6 and 7, multiple solutions at any speed are expected for the nonlinear system. Oneway of obtaining the envelope curve is to simply sweep O1 continuously by either decreasing or increasing the speed.However, this approach fails at the turning points where multiple solutions exist, and thus, this procedure may becomecumbersome when calculating the unstable branches. The angular velocity sweep is instead performed with a predictor–corrector method where the predictor is the arc-length continuation and corrector is the Newton–Raphson method. Thearc-length continuation method [15–17] treats the sweeping parameter (such as O1) as an unknown and adds the arc-length equation to a set of algebraic equations such that the number of unknowns is equal to the number of equations.Predictor step calculates the tangent vector of previous O1 data and estimates an initial guess for the current O1 by


multiplying the tangent vector with a variable arc-length. At the corrector step, the initial guess is iteratively corrected asgiven by Eq. (55). These steps are performed by an orthogonal-triangular decomposition, and the calculated tangential andnormal sub-spaces are used at the predictor and corrector steps respectively. Since O1 is an unknown like a, thedecomposition is carried out on the augmented Jacobian matrix, which is obtained by adding a column @R=@O1 to J.

Since the predictor–corrector method does not provide any information about the stability of the solution, a methodthat is based on the Hill’s method [15,16] is used to examine the stability of the solution. This method determines thebehavior of the periodic solutions under a small perturbation as: d12ðy1Þ ¼ dn

12ðy1ÞþDðy1Þexpðly1Þ. The term dn

12ðy1Þ in theprevious expression is the periodic solution and D(y1) and exp(ly1) represent the periodic and exponential decay terms ofthe perturbation. Taking the derivatives of the perturbed solution with respect to y1 and importing them to Eq. (38), onegets:

IcO21d00n

12þC12O1G1ðdn

12,d0n12ÞþK12H1ðdn

12ÞþTf cðy1, _y1Þ�Tdc

þ

ðIcO21D0Þl2þ 2IcO2

1D1þC12O1

@G1ðd12 ,d012Þ

@d012

��d0n12

D0

� �l1

þ IcO21D2þC12O1

@G1ðd12 ,d012Þ

@d12

��dn

12

D0þC12O1

@G1ðd12 ,d012Þ

@d012

��d0n12

D1þK12

@H1ðd12Þ

@d12

��dn

12

D0

� �l0

0BBB@

1CCCADexpðly1Þ ¼ 0: ð56Þ

Here, it should be noted that D emerges due to the derivatives of D(y1) with respect to y1 and G1ðd12,d012Þ and H1(d12) arelinearized using the Taylor series expansion about the previous solution. The first line that includes dn

12 and its derivativesin Eq. (56) is already zero, since it represents the residue at the solution, and the rest is a polynomial equation in terms of lmultiplied with the perturbation term. Since D(y1)exp(ly1)¼0 yields the trivial solution, it is dropped. Hence, theremaining part of Eq. (56) forms a polynomial problem, and its solution provides 2Nhþ1 eigenvalues (li) as complexconjugate pairs. Finally, the solution is deemed unstable when Re(li)40.

9. Nonlinear order domain responses

The response envelopes are obtained by using the method explained in the previous section, with parameters N¼80,Nh¼20, s¼1e6 and e¼10�6. First, the envelopes of the multiple (combined) order case are displayed in Fig. 10. The resultsare compared for both linear (b¼0) and nonlinear systems; note that the displacement is normalized with d12 ¼ d12=b. Thehorizontal lines indicate the upper (d12¼b) and lower (d12¼�b) boundaries of the clearance nonlinearity, i.e. whend12o�b, the flywheel drives the brake rotor; conversely when d124b, the brake disc drives the flywheel. Within thebacklash region (�brd12rb), the flywheel and brake rotor rotate as independent rigid bodies with no interaction.Fig. 10(a) utilizes the measured Xn and jn values. In contrast, Fig. 10(b) represents a hypothetical case with same Xn valuesbut zero jn for all orders. Comparing Fig. 10(a) and Fig. 10(b), observe that the hypothetical case provides a moreinteresting behavior in the lower speed region as the adjacent orders in Fig. 10(b) have a significant interaction due tohigher Xn values and resonant peaks shift up. However the Xn values at higher orders in Fig. 10(a) are not sufficiently highto excite the nonlinearity, thus only the linear system response is seen in the lower speed region. In addition, resonantpeaks narrow down as n increases for both cases.

Fig. 11 represents the case where only the n¼1 order is considered. Comparing the linear and nonlinear systemresponses, it is seen that the nonlinearity introduces higher mean values and a branch in the response. The nonlinearsystem behaves like a linear system for d12o�b. However, when d12Z�b, the response curve bends towards the lowerspeeds until d12¼b. This occurs due to a decrease in the effective stiffness as a result of single-sided impacts. When d124b,double-sided impacts begin, and hence, the effective stiffness increases and the curve bends towards the higher speeds.

To validate the semi-analytical solutions, measured and predicted torques are compared in Fig. 12 for n¼2, 3, and 4cases; these figures are enlarged for a better visualization of the resonant regime(s). Fig. 12(a) shows the same branchingbehavior as in Fig. 11, though the response is slightly different for the n¼3 and 4 cases due to lower X3 and X4. The effectof single-sided impacts is significant for n¼3, but the system barely exhibits any double-sided impacts. The curve bendsslightly towards lower speeds for n¼4 due to very small single-sided impacts. Overall, the multi-term harmonic balancemethod yields good comparisons with measurements. Here, it should be noted that the multi-term harmonic balancemethod provides multiple stable and unstable solutions, although no unstable solutions are experimentally observed. Inaddition, the measurements are obtained by extracting the order lines from the T(t) spectrogram, and therefore they do notrepresent the true behavior of a single order (pure) rotor.

Speed-dependent torques of the nonlinear system for higher orders (n44) are shown in Fig. 13. Since the Xn

amplitudes at these orders are lower than 0.2X1, the external forces do not seem to excite the nonlinearity. The analyticald12 is superimposed in Fig. 14 on the numerical solution. In addition, predicted TðtÞ is compared with measuredspeed-dependent torque in Fig. 15. The multi-term harmonic balance method is able to predict the peak amplitudeseven for the single-sided impact region (�brd12rb). However, when the double-sided impacts occur, the multi-term harmonic balance method solution cannot seem to predict the amplitudes that are found with the numericalintegration.

Fig. 11. Predicted d12 for linear (b¼0) and nonlinear (b¼0.41) torsional systems for n¼1. Horizontal lines represent the upper and lower boundaries of

the clearance (71). Key: , stable solution; , unstable solution - - - - � , linear system response.

Fig. 10. Predicted d12 for linear (b¼0) and nonlinear (b¼0.41) single degree of freedom torsional systems. Horizontal lines represent the upper and lower

boundaries of the clearance (71). Key: , nonlinear system response; - - - - � , linear system response.


Fig. 12. Measured and predicted shaft torques TðtÞ for linear (b¼0) and nonlinear (b¼0.41) torsional systems for lower orders; (a) n¼2; (b) n¼3; (c)

n¼4. Key: , stable solution; , unstable solution; - - - - � , linear system response; , measured data.


Fig. 13. Measured and predicted shaft torques TðtÞ for linear (b¼0) and nonlinear (b¼0.41) torsional systems for higher orders; (a) n¼5; (b) n¼6. Key:

, stable solution - - - - � , linear system response; , measured data.

Fig. 14. Predicted speed-dependent d12 using multi-term harmonic balance and numerical integration methods. Horizontal lines represent the upper and

lower boundaries of the clearance (71). Key: , multi-term harmonic balance solution; , numerical integration.


Fig. 15. Measured and predicted shaft torques TðtÞ. Horizontal line represents the zero torque line. Key: , prediction using the multi-term harmonic

balance; , measurement.


10. Conclusion

This article has re-examined the classical instantaneous frequency excitation problem [1,2] and proposed newanalytical solutions and insights. Specific contributions of this article include the following. First, a nonlinear torsionalmodel of the brake system example (with rotor surface profile distortions) is successfully developed and experimentallyvalidated using order domain results. Second, the previous analytical solution [8] is enhanced by including the viscousdamping term, and the new analytical solution compares well with numerical convolution or integration method results.Third, a Hilbert transform based envelope function prediction method is developed for linear and nonlinear systems withsingle and multiple order cases, and predicted envelopes virtually match the analytical and numerical solutions. Finally,the multi-term harmonic balance method is successfully implemented to examine the transient problem with focus on theenvelope responses at individual orders. Some multiple stable or unstable solutions, which are not obtained by eitherexperimental studies or with numerical integration, are calculated with this method, and these lead to an improvedunderstanding of the dynamic responses. The main limitation of this article is the assumption of a simple point contactmodel that is to estimate Tf ðy1, _y1Þ, and thus detailed dynamic models of the pad-rotor friction contact regime and thecaliper would be necessary to better understand dynamic interactions.

Acknowledgment

The authors gratefully acknowledge Honda R&D Americas, Inc. for supporting this research. The following individualsare thanked for their contributions: W. Post, B. Nutwell, S. Ebert, F. Howse and P. Bray. The authors also appreciateT.E. Rook’s assistance with the literature.

References

[1] F.M. Lewis, Vibration during acceleration through a critical speed, Trans. Am. Soc. Mech. Eng. 54 (1932) 253–261.[2] D.E. Newland, Mechanical Vibration Analysis and Computation, Dover, New York, 2006.[3] R. Gasch, Acceleration of unbalanced flexible rotors through the critical speed, J. Sound Vib. 63 (3) (1979) 393–409.[4] B.S. Prabhu, R.B. Bhat, T.S. Sankar, Analysis of deceleration phenomenon of high speed rotor systems, Mech. Syst. Sig. Process. 1 (3) (1987) 293–299.[5] A. Carrella, M.I. Friswell, A. Zotov, D.J. Ewins, A. Tichonov, Using nonlinear springs to reduce the whirling of a shaft, Mech. Syst. Sig. Process. 23

(2009) 2228–2235.[6] H. Jacobsson, Disc brake judder considering instantaneous disc thickness and spatial friction variation, Proc. IMechE Part D: J. Automob. Eng. 217

(2003) 325–341.[7] H. Jacobsson, Wheel suspension related disc brake judder, ASME, in: Proceedings of the Design Engineering Technical Conferences, Sacramento, CA,

USA, September 14–17, 1997.[8] O.T. Sen, J.T. Dreyer, R. Singh, Order domain analysis of speed-dependent friction-induced torque in a brake experiment, J. Sound Vib. 331 (2012)

5040–5053.[9] C. Duan, R. Singh, Analysis of the vehicle brake judder problem by employing a simplified source-path-receiver model, Proc. IMechE Part D:

J. Automob. Eng. 225 (2010) 141–149.[10] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, 55,

National Bureau of Standards, Washington DC, 1972.[11] M. Feldman, Hilbert transform in vibration analysis, Mech. Syst. Sig. Process. 25 (2011) 735–802.


[12] T.C. Kim, T.E. Rook, R. Singh, Effect of smoothening functions on the frequency response of an oscillator with clearance non-linearity, J .Sound Vib.263 (2003) 665–678.

[13] C. Duan, R. Singh, Super-harmonics in a torsional system with dry friction path subject to harmonic excitation under a mean torque, J .Sound Vib.285 (2005) 803–834.

[14] T.C. Kim, T.E. Rook, R. Singh, Effect of nonlinear impact damping on the frequency response of a torsional system with clearance, J .Sound Vib. 281(2005) 995–1021.

[15] T.C. Kim, T.E. Rook, R. Singh, Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonicbalance method, J .Sound Vib. 281 (2005) 965–993.

[16] G. von Groll, D.J. Ewins, The harmonic balance method with arc-length continuation in rotor/stator contact problems, J .Sound Vib. 241 (2001)223–233.

[17] M. Peeters, R. Viguie, G. Serandour, G. Kerschen, J.-C. Golinval, Nonlinear normal modes, Part II: Toward a practical computation using numericalcontinuation techniques, Mech. Syst. Sig. Process. 23 (2009) 195–216.

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