Journal of Sound and Vibration (1995) 179(1), 13–36

ANALYTICAL SOLUTION FOR MODULATIONSIDEBANDS ASSOCIATED WITH A CLASS OF

MECHANICAL OSCILLATORS

G. W. B

General Motors Corporation, Powertrain Division, 37350 Ecorse Road, Romulus,Michigan 48174-1376, U.S.A.

R. S

Acoustics & Dynamics Laboratory, Department of Mechanical Engineering, The Ohio StateUniversity, Columbus, Ohio 43210-1107, U.S.A.

(Received 12 April 1993, and in final form 10 September 1993)

Sideband structures are a commonly observed phenomenon in measured vibro-acousticsignatures of many types of mechanical systems, and especially in rotating machinery. Suchspectral information is often used for fault diagnostic applications as well as noise andvibration control studies. A new theory is developed that examines in a critical manner thecommonly held belief that simple amplitude and frequency or phase modulation processesare responsible for generating such sidebands, and instead provides a more logicalexplanation. A class of viscosity damped mechanical oscillators is examined havingspatially periodic stiffness and displacement excitation functions that are exponentiallymodulated by the instantaneous vibratory displacement of the inertial element. To this end,a new family of dual domain periodic differential equations is introduced which is shownto be more pertinent to the study of force modulation inherent to many mechanical systems,such as a gear pair. Analytical expressions are then derived to predict sideband amplitudesin terms of key system parameters. The harmonic balance method and direct time domainintegration techniques are employed to study various subsets of the problem.

1. INTRODUCTION

Sideband structures are a commonly observed phenomenon in measured vibro-acousticsignatures of many types of mechanical systems and especially in rotating machinery [1–3].A typical auto-power spectrum of, say, an acoustic pressure or a structural accelerationsignal, may contain many low order and high order harmonic components along withcomplex sideband structures. Such spectral information is often used for noise andvibration control studies [2] as well as fault diagnostic applications [3]. Although thefrequency of each harmonic is ultimately related to the kinematics of a mechanical linkageor mechanism, the amplitude characteristics usually result from complex dynamic acousticprocesses. The intent of this paper is to unravel some of the unknown aspects associatedwith a class of mechanical oscillators that exhibit such spectral characteristics. A newtheory is developed that examines in a critical manner the commonly held belief that simpleamplitude and frequency or phase modulation processes are responsible for generatingsuch sidebands [1, 3].

The genesis of this line of enquiry arises from the study of gear dynamics, in whichsideband structures are especially germane to experimental diagnostic studies [1–3]. Based

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0022–460X/95/010013+24 $8.00/0 7 1995 Academic Press Limited

. . . 14

on the unique problem of a single gear pair system, a simplified mathematical formulationemerges which could be extended to other physical systems. Accordingly, some of theterminology from the context of gear dynamics is retained for the sake of clarity inmathematical and physical interpretation [4].

Perhaps of more fundamental importance is the set of mathematical problems groupedunder the family of ordinary linear differential equations having periodic, time-varyingcoefficients, such as the Hill or Mathieu equations [5]. Historically, these equations havebeen of interest because of their inherent applications to engineering and physical sciences.Most mathematical formulations assume a second order, homogeneous differentialequation with sinusoidal or piecewise linear variations of the coefficients or systemparameters. The independent variable is usually time t and typically only the initial valueproblem is solved, with an emphasis on the dynamic stability and parametric behavior ofthe system [5, 6]. The question naturally arises as to whether these equations are sufficientlywell defined to describe the spectral characteristics discussed earlier. To this end, a newclass of periodic differential equations is introduced, which is later shown to be morepertinent to the study of force modulation inherent to many mechanical systems. A secondorder inhomogeneous differential equation is written, where the coefficients and forcingfunction are assumed to be periodic in a spatial or angular variable u(t) rather than time.This spatial variable represents the instantaneous position of the inertia element or mass,which is assumed to have a uniform velocity under ideal conditions. However, vibratorydeviations of the mass from its ideal uniform motion give rise to dynamic fluctuations inu(t) from its nominal value, thus producing a non-linear feedback effect. This gives riseto rather complex spectral characteristics in the steady state forced response; one of theobjectives here is to demonstrate this.

2. PROBLEM FORMULATION

Consider the single-degree-of-freedom (SDOF) oscillator shown in Figure 1, having aspatially periodic stiffness function k[u(t)]= k[u(t)+2p]. Of particular interest is thesteady state forced response of this system when it is subjected to a time-invariant meanforce F acting on the mass and a spatially periodic displacement excitation functione[u(t)]= e[u(t)+2p] applied in line with the spring as shown. The governing equation ofmotion for this system is given by

mx0+ cx'+ k[u(t)]x(t)=F+ k[u(t)]e[u(t)] (1)

where the prime denotes differentiation with respect to t. The system mass m and viscousdamping coefficient c are both assumed to be time-invariant. In order better to describecertain types of force modulation phenomena, the stiffness and displacement excitation

Figure 1. The physical system: a single-degree-of-freedom oscillator having spatially periodic stiffness k[u(t)]and displacement excitation (error function) e[u(t)] under mean applied load F.

15

functions in equation (1) are intentionally defined in two distinct analysis domains: the time(t) domain and the spatial (u) domain. The angular variable u(t) is defined as

u(t)=Vt+ bx(t), (2)

where V is a time-invariant mean angular velocity and b is a constant determined by thephysical process under study. The second term in equation (2) represents a linear variationin u(t) with x(t). Hence, the stiffness and error functions are exponentially modulated bythe instantaneous vibratory displacement of the mass. Under such circumstances, brepresents the so-called angle modulation depth. For the special case in which b=0,equation (1) is linear with a periodic time-varying stiffness coefficient, and it represents aspecial class of the forced Hill’s equation which has been well studied in variousforms [5, 6]. The governing equation is non-linear when b$ 0. The initial valueu(t=0) is assumed to be zero in the absence of angle modulation without any loss ingenerality.

While the right side (r.h.s.) of equation (1) could be replaced with a more general forcingfunction, it is left in the form given in order to facilitate an investigation of the interactionbetween the mean force, displacement excitation and stiffness variation which has practicalsignificance to many mechanical system problems. Of particular interest are higher orderharmonic components and sideband structures present in the spectra of steady statesolutions of equation (1).

The stiffness function k[u(t)]q 0 is assumed to be positive definite. Furthermore,k[u(t)]= k0 + ka [u(t)] is written as the sum of a time-invariant mean component k0 = �k�uand a spatially periodic alternating component ka [u(t)] with �ka�u =0. A dimensionlesstime variable t=vnt is defined, where vn =zk0/m is the static natural frequency of thesystem. A dimensionless displacement excitation function d[u(t)] is defined such thate[u(t)]=Ed[u(t)]. Here E is some unit displacement amplitude, which is typically of theorder of F/k0. Accordingly, the governing equation (1) can be written in terms of thedimensionless displacement co-ordinate j(t)= x(t)/E

j� +2zj� + {1+ ac[u(t)]}j(t)= f+ {1+ ac[u(t)]}d[u(t)]. (3a)

Here, the dot denotes differentiation with respect to t, z= c/2zk0m is the time-invariantsystem damping ratio, f=F/k0E is the dimensionless mean load andac[u(t)]= ka [u(t)]/k0, where c[u(t)]=c[u(t)+2p] and =c=max =1. Hence, a is a stiffnessmodulation index and furthermore, aQ 1 so that the dimensionless stiffness functionremains positive definite. The angle parameter u(t) is now given by

u(t)=Lt+ bj(t), (3b)

where L=V/vn is a dimensionless frequency ratio and b= bE is the dimensionless anglemodulation depth.

The scope of this paper is the study of steady state forced response of equations (3)for a viscously damped system with b=0 and bW 1. Parameters a and L are chosen suchthat parametric instability issues are not of concern. Consequently, dynamic stability issuesare considered only briefly in section 4.1. Various subsets of equations (3) are solved byusing both analytical (including the harmonic balance method) and computationalapproaches. The main emphasis is on the development of analytical expressions whichrelate sideband amplitudes to key system design parameters, given b=0. Since the scopeof this enquiry is rather vast, the effect of angle modulation (b$ 0) is examinedqualitatively in section 6.

. . . 16

3. FUNCTIONAL STIFFNESS AND ERROR EXPRESSIONS

The forced response of the dynamic system considered here is highly dependentupon the explicit forms of c and d. Prior studies have shown that for the linear case(b=0), the system behavior is more dependent upon the lower harmonic content ofthe periodic coefficient c and less sensitive to its higher harmonics [5]. Furthermore,the simplest possible forms for c and d are desired which are capable of producingforce modulation phenomena which may provide a key to understanding gear dynamicsand gear noise. Of great practical interest is the case in which c(u)=c(Nu) is a highor mesh frequency periodic function of Nu, with Nw 1 being a large positive integerand d(u) being a low or shaft frequency periodic function. For example, in a gearedsystem, c(Nu) arises from variations in the number of gear teeth in contact as afunction of gear rotation angle u, d(u) may represent eccentricity or roundnesserrors associated with a gear wheel and j(t) represents the instantaneous relativedisplacement across the gear mesh interface [4]. Note that c(Nu) has a pumping periodof 2p/N, being commensurate with the fundamental excitation period of d(u) which is ofcourse 2p.

Since the pumping and excitation frequencies are commensurate, the steady stateforced response must be periodic with fundamental frequency L in the absence of anyangle modulation (b=0), provided that the system is asymptotically stable [7]. Whenangle modulation is considered with bW 1, the solution is assumed to remain periodic.In either case, one can readily observe that the steady state forced response will consistof two distinct frequency regimes as shown in Figure 2: (i) The shaft frequency regimeand (ii) the mesh frequency regime, which are well separated provided that N is sufficientlylarge and d is adequately band limited. Accordingly, j(t)= jS (t)+ jM (t) can be writtenas the sum of a shaft frequency component jS and a mesh frequency component jM . Thespectral composition of jS is dictated primarily by that of d and the angle modulation depthb. The spectral composition of jM is considerably more complicated. Even simple harmonicforms of c(Nu) will generate higher order mesh harmonics in jM . Furthermore, theproduct of jS (t) and c(Nu) will generate shaft order sidebands about the respectivemesh harmonic components. For this reason, the jM is written as the sumjM (t)= jN (t)+ jSB (t), where jN contains only the mesh frequency component and itsharmonics, while jSB contains only sideband frequencies. Angle modulation will inducefurther harmonic distortion in jN and also contributes significantly to sideband structuresgiven by jSB .

Figure 2. A typical steady state response spectrum illustrating two distinct (shaft and mesh) frequency regimes.

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In light of the above discussion, only two simple yet very practical harmonic forms forc(Nu) and d(u) are considered:

c(Nu)= cos (Nu), d(u)=DS cos (u+fS ), (4, 5)

where fS is a phase angle and DS is a dimensionless displacement amplitude.

4. SOLUTION STRATEGIES

Defining the dimensionless state vector x(t)= {j� (t) j(t)}T, equations (3) are readilycast into the state space form

ẋ(t)=A[u(t)]x(t)+ f[u(t)], (6a)

where A[u(t)] and f[u(t)] are the time-varying, spatially periodic matrix and forcing vectorgiven, respectively, by

A[u(t)]=$−2z1 −1− ac[u(t)]0 %, f[u(t)]= {f+ {1+ ac[u(t)]}d[u(t)] 0}T. (6b, c)Equation (6) is linear only when b=0. Several methods are available for determining thesteady state forced response analytically for the linear case, provided that the system isasymptotically stable. For instance, Struble [8] presents a variational method in which asolution of the form j(t)=P(t) cos [NLt+Q(t)]+ ej1 + e2j2 + · · · is assumed. HereP(t) and Q(t) are unknown functions that are assumed to vary slowly in time comparedwith frequency NL. Substituting this assumed solution into equations (6) and collectingterms to the first order yields a set of coupled non-linear differential equations which canbe solved for P(t) and Q(t). However, a general solution is not easily obtained. Thismethod has been employed by Miyasar and Barr [9] to determine the stability of a linearoscillator having a parametric stiffness excitation with externally controlled sinusoidalfrequency variation. Another method is to use a harmonic balance type approach byassuming a linear periodic solution and solving for the unknown Fourier coefficientsdirectly. This approach has been widely employed to determine the frequency response ofelectrical N-networks with time-varying or switched parameter values [10–14]. Since theexact solution contains an infinite number of harmonic terms [15], the approximatesolution must be truncated. Hsu and Cheng [7] have also presented an explicit expressionfor the steady state response of equation (6), given b=0 in terms of the fundamentalmatrix of the homogeneous system, which must also be periodic. The existence of a closedform solution in such cases is dictated by the expression for c(u). When c(Nu) is asinusoidal function, the fundamental matrix can be expressed in terms of the well knownMathieu functions [16], but only for specific values of a. Otherwise, the fundamental matrixmust be evaluated numerically or by other approximate methods [5, 7].

When b$ 0, equation (6) is non-linear and no closed form solutions are known to existregardless of the expression for c. In such cases, equations (6) must be solved by usingnumerical integration techniques or other approximate methods [8, 17]. However, forbW 1, an approximate solution for x(t) may be obtained by linearizing equation (6) aboutits static operating point x0 = {0 j0}T, where j0 is the static solution to equation (3) givenL=0. Here, it is assumed that x0 does exist and, furthermore, that the solution x(t) iscontinuous; both assumptions are valid as long as no separation takes place in the physicalsystem. Accordingly, the linearized state equation written in terms of the co-ordinatevector xp (t)= x(t)− x0 is obtained by a Taylor series expansion and is given by

ẋ(t)=A [u*(t)]xp (t)+ f[u*(t)]. (7a)

. . . 18

Here, A [u*(t)] is the linearized coefficient matrix that is periodic in the new spatialparameter u*(t)=Lt+ bj0 and is given by

A [u*(t)]=$−2z1 −1− a[c(u*)+c'(u*)j0]+c(u*)d'(u*)+c'(u*)d(u*)+ d'(u*)0 %,(7b)

where

c'I (u*)=−Nb sin (Nu*), d'(u*)=−bDS sin (u*+fS ). (7c, d)

Hence, the static deflection of the system produces a phase shift of magnitude bj0 in thespatially varying parameters. The angle modulation also gives rise to several additionalparametric terms. The products cd' and c'd produce apparent stiffness terms atsideband frequencies (N2 1)L and d' gives rise to an apparent shaft frequency stiffnessvariation. The number of additional frequencies which must be considered in anyanalytical solution can become overwhelming when the spectral contents of d and/or care expanded to include multiple shaft harmonics and/or additional mesh harmonic com-ponents.

The steady state solution to equations (7) will be periodic with fundamental frequencyL due to their linearity. However, as b is increased, the exact governing equation (6) canbecome strongly non-linear and it may be possible to observe several non-linear phenom-ena such as period doubling bifurcations and aperiodic behavior [17]; such cases are clearlybeyond the scope of this study. No closed form expressions for the fundamental matrixcorresponding to either equations (6) or (7) exist for arbitrary values of a, given theassumed forms of c and d. Furthermore, even though it is possible to compute thefundamental matrix numerically in such cases, this may not provide much physical insight.To this end, a modified harmonic balance approach is employed. Analytical solutions arethen compared with numerical integration results whenever possible.

4.1. Since the fundamental frequency of c(Nu) is NL, unstable parametric resonance can

occur whenever L=2N/k, where k is a positive integer [6]. This phenomenon is illustratedin the Strutt diagram shown in Figure 3, which defines regions of instability for kE 3 asa function of a, z and L/N, given b=0. This diagram was obtained by numerically solvingfor values of a which yield eigenvalues of the monodromy matrix having a modulus equalto unity for given z and L/N. The monodromy matrix M=Z(2p/NL) was determined inthe usual manner [6] by solving the homogeneous matrix equation Z� (t)=A(t)Z(t), giventhe initial condition Z(0)= I2, where I2 is the 2×2 identity matrix. For viscously dampedsystems having ze 0·05 and aE 0·6, the system can only exhibit k=1 instability atfrequencies L/N greater than about 1·72. In Figure 4 are shown the regions of stable k=1parametric resonance as a function of a and z, given L=2/N. The boundaries shown inFigure 4 correspond to lines of constant amplitude for the L=1/N harmonic componentof the normalized steady state forced response jSS(t)/f obtained by the direct time domainintegration of equations (6), given DS =0. The system was considered to be unstable for=j(jL)/f =e 10. The lower bound corresponds to =j(jL)/f ==0·001. The actual zone ofstable parametric resonance is quite narrow and no parametric resonance correspondingto k=1 occurs for ze 0·25 regardless of aQ 1. In real life, the system response in theseunstable regions will be dictated by non-linear effects which are not included in thegoverning equations given here. However, the focus of this study is on the steady state,non-resonant response, in which any such non-linear effects are typically insignificant.

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Figure 3. A Strutt diagram, given b=0 and kE 3. ----, z=0; · · · ·, z=0·05; — · — ·, z=0·1; — · · — · ·,z=0·2.

When b$ 0, the stability of the system can be inferred from equation (7) by examiningthe eigenvalues of the corresponding monodromy matrix M=Z(2p/L), which is deter-mined by solving Z� (t)=A (t)Z(t) given the initial condition Z(0)= I2. Given NbW 1and DS =0, the deviation from the stability regions shown in Figure 3 is insignificant.However, as either b or DS becomes large, the regions of instability can vary from thoseshown in Figure 3. One limiting case occurs when jS is assumed to be zero and u isexponentially modulated by a harmonic shaft frequency function of the formd(t)=DS cos (Lt+fS ). Accordingly, the stiffness function 1+ ac[u(t)] takes on the form1+ a cos [NLt+NbDS cos (Lt+fS )]. This type of problem has been examined in a moregeneral context by Miyasar and Barr [9]. They found that instability can also occur atfrequencies L/N=2N/k(N2 s) where s=0, 1, 2, . . . is an integer shaft order index. The

Figure 4. The region of stable k=1 parametric resonance.

. . . 20

region of instability for a given k becomes an agglomerate of several narrower regions,each associated with one of the above frequencies. The spacing of the individual regionsof instability is dictated by the frequency separation ratios N/(N2 1). However, for largevalues of N, the ratios N/(N2 1) approach unity and the stability of the system approachesthat shown in Figure 3, especially given small values of NbDS .

The primary focus of this study is on the steady state response regions in whichinstability does not occur; such regions are typical of many real-life machines. Accordingly,the remainder of this study considers only values of a and z such that the oscillator isoperating well outside the stability boundaries shown in Figure 4, given L/NQ 1·8 orL/Nw 2. The non-linear oscillator is also assumed to be stable given these conditions;numerical simulation confirms the validity of this assumption.

5. STEADY STATE SOLUTION IN THE ABSENCE OF ANGLE MODULATION (b=0)

In this section the governing equation (3) is solved both analytically and numerically,given b=0. Several parametric studies are conducted and simplified expressions for theamplitude of the first mesh order sideband pair are developed.

5.1. - Before proceeding with other exact solutions, a simpler approximate method is first

employed to solve equation (3) when b=0. This provides physical insight to the role ofmean load and stiffness variation on the generation of higher order mesh harmonics. Thegoverning equation can be written exactly in terms of an equivalent quasi-static loadedexcitation function j( f, u):

j�+2zj�+ {1+ ac[u(t)]}j(t)= {1+ ac[u(t)]}j[ f, u(t)], (8)

where j( f, u) may be computed or measured under quasi-static conditions [4] as L:0 andis given by

j( f, u)= d(u)+ f/[1+ ac(u)]. (9)

The first term of equation (9) represents the original displacement excitation function,while the second term represents an additional displacement arising due to the stiffnessvariation and is directly proportional to the mean load f. A static compliance function isdefined as [1+ ac(Nu)]−1 and is plotted versus u in Figure 5, given c(Nu)= cos (Nu) and

Figure 5. The static compliance function [1+ a cos (Nu)]−1 vs. u, given a=0·6.

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a=0·6. This compliance function is an even function that is periodic in N(u) and cantherefore be expressed in terms of the Fourier series

[1+ a cos (Nu)]−1 = sa

n=0

an cos (nNu), (10a)

where n is the mesh harmonic index and Fourier coefficients an are given by

a0 =1

z1− a2, n=0; an =

2(z1− a2 −1)anz1− a2

, ne 1 (10b, c)

The magnitudes of the first five Fourier coefficients a0–a4 are plotted in Figure 6 as afunction of stiffness variation a. This simple harmonic stiffness function generates higherorder mesh harmonic content in j( f, u), having substantial amplitude for non-zero f wheneven modest values of a are considered. The mean value of the compliance function alsovaries with a. However, no sidebands are present in the spectrum of j( f, u). Hence,displacement sidebands must arise only under dynamic conditions when L$ 0, given b=0.

One obvious simplification of equation (8) is to replace the stiffness term by its spatiallyaveraged mean value to obtain the approximate governing equation

j�+2zj�+ j(t)= j[ f, u(t)], (11)

which is linear with time invariant (LTI) coefficients only if angle modulation is ignored.The effect of stiffness variation on the system natural frequency is completely neglectedin equation (11) and is included implicitly only in j, which is assumed to be available froman independent quasi-static analysis of the oscillator. Equation (11) is particularlyconvenient when modelling physical systems in which j can be measured directly. Giventhe assumed forms of c and d, the steady state solution jss to equation (11) is easily writtenas

jss (t)= f/z1− a2 +DsM1 cos (Lt+8s +81)+ fMN (1−1/z1− a2) cos (NLt+8N )

+ f sa

n=2

anNMN cos (nNLt+8N ), (12a)

Figure 6. The Fourier amplitudes of the static compliance function [1+ a cos (Nu)]−1 vs. the mesh stiffnessvariation a. ----, a0; · · · ·, a1; – – – –, a2; — · — ·, a3; — · · — · ·, a4.

. . . 22

where Mi and 8i are amplitude and phase correction factors respectively, given by

Mi = {[1− (iL)2]2 +4z2(iL)2}−1/2, 8i =tan−1 0 2z(iL)1− (iL)21. (12b, c)A typical forced response spectrum showing the normalized peak to peak amplitude ofjss (t)/f versus mesh frequency ratio L/N is shown in Figure 7, given Ds =0, a=0·4 andz=0·10. Also shown in Figure 7 is the corresponding exact solution obtained by numericalintegration of equation (6). Note that the LTI formulation (11) cannot predict the peakat L/N=2 which arises due to a stable k=1 parametric resonance. Another effect of thestiffness variation is to decrease the apparent natural frequency of the system from its staticvalue of unity. In order to improve the accuracy of the approximate LTI solution, vn isreplaced by an effective value which is related to the static mesh compliance and is givenby vneff =(1− a2)0·25. A corrected forced response curve is also shown in Figure 7, whichagrees more closely with the exact response for L/Ne 0·96, excluding the k=1 parametricresonance at L/N=2.

In the mesh frequency regime LW 1, jS is not affected by a provided that Nw 1/(1− a)such that L remains well below the lower limit of the instantaneous natural frequency1− a. Furthermore, jS is relatively unaffected by the mesh frequency stiffness variationwhen the oscillator is operating in the shaft frequency regime L/Nw 1, provided that Nis sufficiently large. This is attributed to averaging effects which occur when the separationbetween shaft and mesh frequency is large.

The advantage of this approximate LTI solution technique is that it can predict withvery reasonable accuracy jS in the appropriate frequency regimes, as well as thepeak-to-peak amplitude of jN , with minimal effort. However, the method is incapable ofpredicting sidebands in the displacement spectrum and the accuracy of individual meshorder amplitudes of jN is relatively poor, as shown later.

5.2. In order to obtain a more accurate closed form expression for the steady state forced

response which includes the prediction of modulation sidebands, we employ a modified

Figure 7. A comparison of the exact solution jSS /f (----) and the approximate LTI solutions using vn =1·0(— · · — · ·) and vneff =0·9573 (· · · ·), given DS =0, z=0·1 and a=0·4.

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T 1

Frequency mapping of harmonic interactions given b=0

Harmonic Harmonicdescription Frequency indices, m

Mean value 0 0, NShaft order L 1, N2 1Lower sidebands (nN−1)L nN−1, (n2 1)N−1nth mesh order nNL nN, (n2 1)NUpper sidebands (nN+1)L nN+1, (n2 1)N+1

harmonic balance approach [17]. The following expression for jss (t) is assumed:

jss (t)= sa

m=0

Am cos (mLt+ gm ). (13)

Substituting the above expression into equation (3) and collecting even and odd terms oflike frequency produces two independent systems of algebraic equations which yield Amand gm when solved. At this point it is instructive to construct the frequency mapgiven in Table 1, which describes the interactions between the various harmonic termsas excited by the assumed forcing functions given b=0. Observe that in the absence ofangle modulation, the shaft frequency component and sidebands are independent of themean value and mesh harmonic amplitudes. This makes it possible to perform therespective analyses separately. Letting am =Am cos gm and bm =Am sin gm , two independentsystems of linear algebraic equations are obtained for shaft–sideband and mesh frequencyregimes:

K 1− (NL)2 − 12a2 −2z(NL) 12aG

−2z(NL) (NL)2 −1 0 −12aGG 12a 0 1− (2NL)2 −2z(2NL) ·G

−12a −2z(2NL) (2NL)2 −1 · ·G

G 12a 0 · · ·G

−12a · · · ·G· · · · 12aG

G· · · 0G

G · · 1− [(m−1)NL]2G

· −2z(m−1)NLG12aG

GGGGGGk Equation 14 continued overleaf

. . . 24

LHHHHHHHH

−12a H−2z(m−1)NL 12a H

H[(m−1)NL]2 −1 0 −12a H

0 1− (mNL)2 −2zmNL 12a HH

−12a −2zmNL (mNL)2 −1 0 −12a H

12a 0 1− [(m+1)NL]

2 −2z(m+1)NL · HH

−12a −2z(m+1)NL [(m+1)NL]2 −1 · H

12a 0 · H

H−12a · H

· l

aN −faF J F JG H G HbN 0G H G H

a2N 0G H G HG H G Hb2N 0G H G H

· ·G H G HG H G H· ·G H G H

· ·G H G HG H G Ha(m−1)N 0g h g h

b(m−1)N=

0, (14)

G H G HG H G HamN 0G H G H

bmN 0G H G HG H G Ha(m+1)N 0G H G H

b(m+1)N 0G H G HG H G H· ·G H G H

· ·G H G Hf j f j· ·

25

K *1−L2 −2zL 12a 0 12a 0G *

−2zL L2 −1 0 12a 0 −12aG *

12a 0 1− [(N−1)]L]

2 −2z(N−1)L 0 0G *G *0 12a −2z(N−1)L [(N−1)L]2 −1 0 0G *1

2a 0 0 0 1− [(N+1)L]2 −2z(N+1)LG *

0 −12a 0 0 −2z(N+1)L [(N+1)L]2 −1G *

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·G *G *1

2a 0 0 0G *−12a 0 0G *

12a 0G *

G *−12aG *G *G *G *G *G *G *k *

* L* H* H* H* H1

2a* H0 −12a* H0 0 12a* H

* H0 0 0 −12a* H· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·* H

−[(2N−1)L]2 −2z(2N−1)L 0 0 ·* H−2z(2N−1)L [(2N−1)L]2 −1 0 0 · ·* H

* H0 0 1− [(2N+1)L]2 −2z(2N+1)L · · ·* H0 0 −2z(2N+1)L [(2N+1)L]2 −1 · · ·* H1

2a 0 0 0 · · ·* H−12a 0 0 · · ·* H

12a 0 · · ·* H

* H−12a · · ·* H· · ·* l

· ·Equation 15 continued overleaf-·

. . . 26

F J F Ja1 Ds cos fsG H G H

b1 −Ds sin fsG H G HaN−1 12 aDs cos fsG H G H

G H G HbN−1 12 aDs sin fsG H G HaN+1 12 aDs cos fsG H G HbN+1 −12 aDs sin fsG H G H

G H G Ha2N−1 0G H G Hb2N−1

=0

.(15)

g h g ha2N+1 0G H G H

G H G Hb2N+1 0G H G H· ·G H G H· ·G H G H

· ·G H G H· ·G H G H· ·G H G H· ·G H G H· ·f j f j

Each system of equations is governed by a narrow-banded symmetric coefficient matrixand consists of an infinite number of equations which must be truncated before obtaininga solution. The Fourier amplitudes and phase angles are given by Am =za2m + b2m andgm =tan−1 (bm /am ), respectively.

5.3. The mesh harmonic amplitudes AnN of jSS are obtained by solving a truncated system

of equations (14). These mesh harmonic amplitudes are normalized with respect to themean force f and are thus dependent solely upon the values of a and L. In Figures 8(a)and (b) it is shown that AN /f and A2N /f converge rapidly to their exact values as the numberof equations is increased beyond m= n. The exact values of AN /f and A2N /f weredetermined from numerical integration of equation (6), given a=0·4 and z=0·1.Sufficient accuracy is nearly always achieved by letting m= n+1 when solving for AnN .Further analysis reveals that A(n+1)N WAnM for ne 2. Hence, higher order mesh harmonicscorresponding to large n, if desired, may be determined by using successive iterationtechniques. Numerical bounds on truncation error associated with systems of this type canalso be determined [15]. Also shown in Figure 8 are the corresponding harmonicamplitudes obtained from the approximate LTI solution given by equation (11) using vneff .The LTI solution is inadequate for most cases, especially for higher order mesh harmonicswith ne 2. Equation (14) presents an extremely efficient method to compute the exactmesh harmonic response of equation (3) in a fraction of the time required by the directnumerical integration method. The mean value A0 is obtained directly, once AN and gN areknown, by

A0 = f− 12 aAN cos gN . (16)

27

Figure 8. The convergence of the normalized mesh harmonic amplitudes AnN /f obtained by solving equation(15) with m= n (· · · ·) and m= n+1 (— · — ·) to the exact value (----), given DS =0, a=0·4 and z=0·1. (a)n=1; (b) n=2. — · · — · ·, the corresponding LTI solution.

Also of interest is the total harmonic distortion (THD) defined as

THD=Xsan=2 A2nN > sa

n=1

A2nN , (17)

which serves as an indicator to the significance of higher order mesh harmonic termsgiven ne 2. The dependence of THD on a in the mesh frequency regime is shown inFigures 9(a) and (b), given z=0·1 and z=0·5, respectively. Note that the THD increaseswith a, especially in the frequency range L/NQ 1, and is independent of f. THD is notdefined for a=0.

5.4. – The shaft and sideband Fourier coefficients of jSS are similarly obtained by solving a

truncated system of equations (15). Here again, convergence is quite rapid for lower ordersideband amplitudes. Numerical investigation of equation (15) reveals that the mesh orderstiffness variation has no significant effect on the shaft frequency response regardless ofL, and in all cases AnN2 1 WA1 regardless of the mesh order index n. Accordingly, the shaftfrequency response jS can be determined by a straightforward LTI analysis with noappreciable loss in accuracy; i.e., A1 3DSM1 and g1 381, where M1 and 81 are given by

. . . 28

Figure 9. THD versus a in the mesh frequency regime, given DS =0. ----, a=0·2; – – – –, a=0·4; — · — ·,a=0·6; — · · — · · a=0·8. (a) j=0·1; (b) z=0·5.

equations (12a) and (12b), respectively. Further analysis shows that AnN2 1 WAN2 1 forne 2 over the entire frequency range, except in some cases near L/N=1/n when z is verysmall. In accordance with these assumptions, an approximate solution for the first meshorder sideband pair is obtained by solving equation (15), given n=1:

AN2 1 3 12aDSMN2 1M1z1+M−21 −2M−11 cos 81. (18)

The above expression provides a reasonably accurate approximation of AN2 1 overthe entire frequency range. However, when LW 1, M1:1·0 and cos 81 3 1−0·5[2zL/(1−L2)]2. Accordingly, an even simpler approximation is obtained:

AN2 1 3 aDS0 zL1−L21MN2 1, LW 1, (19)which is valid only in the mesh frequency regime, Several observations are made fromequations (18) and (19). First, sideband amplitudes are independent of phase angle fS but,rather, are dictated by the phase correction of the shaft frequency component given by 81.

29

Figure 10. The dependence of normalized sideband amplitudes AnN2 1/DS on z in the mesh frequency regime,given a=0·4 and N=50; AnN+1/DS (----) and AnN−1/DS (– – – –), given z=0·1; AnN+1/DS (— · · — · ·) andAnN−1/DS (— · — ·), given z=0·5. (a) n=1; (b) n=2.

Hence, sideband amplitudes in the mesh frequency regime in which LW 1 are favored bya high damping ratio z which induces a larger phase correction at lower operatingfrequencies; this may be explained as the ‘‘broadening’’ of the resonance around L/N=1.This phenomenon is clearly illustrated in Figure 10(a), which shows AN2 1/DS , given a=0·4and N=50 with z=0·1 and z=0·5 over the mesh range 0EL/NE 1·8. Higher meshorder sideband amplitudes behave in a similar fashion, as shown in Figure 10(b) forA2N2 1/DS . These values were obtained by solving equations (15) given m=5, with nosignificant difference between the corresponding exact solutions obtained by numericalintegration. Note the asymmetric in upper and lower sideband pairs AN2 1 which beginsto occur near L/N=1/n. It also follows from equations (18) and (19) that sidebandamplitudes in the mesh frequency regime will also favor smaller N. This phenomenon isillustrated in Figure 11, which shows the upper mesh order sideband amplitude AN+1/DSversus L/N, given a=0·4 and z=0·1 with N=10, 50 and 100. Similar trends are observedfor higher mesh order sidebands. The accuracy of the approximate solution given by

. . . 30

Figure 11. The dependence of the normalized upper sideband amplitude AN+1/DS on N in the mesh frequencyregime, given a=0·4 and z=0·1 ----, N=10; – – – –, N=50; — · · — · ·, N=100.

equation (19) is compared with the exact solution over the mesh frequency regime0EL/NE 1·8 in Figure 12, which shows AN+1/DS vs. L/N given a=0·4, z=0·1 andN=50. Indeed, equation (19) provides a useful tool to estimate sideband amplitudes.

In the limiting case L=1, equation (18) reduces to

AN2 1 3aDS

2(N2 1)2z1+4z2, L3 1. (20)

Hence, given a large value of N, mesh order displacement sidebands in the shaft frequencyregime are relatively small compared to their corresponding values in the mesh frequencyregime. This is illustrated in Figure 13, which shows AN2 1/DS over the shaft frequencyrange given a=0·4 and N=50 with z=0·1 and z=0·5. Relative maxima at L3 1 occur

Figure 12. A comparison of the approximate upper sideband solution AN+1/DS (– – – –) given by equation (19)with its exact value (----) in the mesh frequency regime, given a=0·4, z=0·1 and N=50.

31

Figure 13. The dependence of the normalized sideband amplitudes AN2 1/DS on z in the shaft frequency regime,given a=0·4 and N=50; AN+1 (----) and AN−1 (– – – –), given z=0·1; AN+1 (— · · — · ·) and AN−1 (— · — ·),given z=0·5.

only for the mesh order sideband pair given small z. Higher mesh order sidebands are ofnegligible amplitude in this frequency range.

5.5. In order to determine whether the amplitude of displacement sidebands is comparable

to that of the mesh harmonics, a sideband amplitude ratio (SBAR) is defined as

SBAR=Xsan=1 (A2nN−1 +A2nN+1) > sa

n=1

A2nN , (21)

which can be normalized with respect to DS /f. SBAR is found to vary linearly with meshfrequency ratio L/N, except for a slight deviation near L/N=1 for small z, as illustratedin Figure 14, which shows SBAR( f/DS ) versus L/N, given a=0·4 and N=50 withz=0·05, 0·1 and 0·5. Furthermore, SBAR is directly proportional to z/N and is completely

Figure 14. SBAR versus L/N, given a=0·4 and N=50. ----, z=0·05; – – – –, z=0·1; — · — ·, z=0·5.

. . . 32

independent of a. An approximate expression for SBAR, valid only for Nw 1 and aq 0,is given by

SBAR3z2zN 0DSf 1(L/N), Nw 1, aq 0. (22)

SBAR is not defined for a=0. Equation (22) is found to deviate from the exact value ofSBAR only near L/N=1 for very small values of z. Typically, 0QDS E 104 and0Q fE 10 for most geared systems. Hence, 0EDS /fE 103 and, indeed, sidebandamplitudes may be of comparable magnitude to the amplitudes of their respective meshharmonic components. Equation (22) is very useful to determine when sideband com-ponents should be included in the calculation of diagnostic and noise perception indices[2].

6. STEADY STATE SOLUTION IN THE PRESENCE OF ANGLE MODULATION (b$ 0)The theory presented in section 5 is capable of predicting only single sideband pairs

about harmonics of the meshing frequency due entirely to mesh order stiffness variationand shaft order displacement. However, higher order sidebands having frequencies(nM2 s)L, where n is a mesh order index and s a shaft order index, are commonlyobserved in measured vibro-acoustic spectra of many mechanical systems [1–3]. Higherorder sidebands sq 1 can be predicted given b=0 by considering d to be a periodicfunction having spectral content to at least order s. Indeed, higher order shaft harmonicsin jS are usually observed in spectra exhibiting multiple sideband structures; especially ingeared systems. However, in many cases the gear wheel typically does not exhibit any lowfrequency runout or index errors of order sq 1. Another limitation to the theory givenb=0 is the inability to predict highly asymmetric sideband structures. It is not uncommonto observe differences between corresponding upper and lower sidebands of varying ordersof magnitude in measured machinery spectra [1, 3]. However, only relatively smallasymmetry can be predicted, given b=0. Hence, an alternative explanation is required andthe theory is extended accordingly. To this end, angle modulation is introduced via theparameter b= bE. For example, in a geared system, b corresponds to the reciprocal ofthe base radius and E is the unit excitation amplitude along the line of action. Typically10−5 E bE 10−3 in a geared system.

6.1. Recall that when b=0, jS (t)+ jSB (t) is independent of jN (t), and it is possible to solve

for the mesh and shaft–sideband response independently. Unfortunately, this is not thecase, given b$ 0, due to strong harmonic interactions which may exist between shaft, meshand sideband terms depending on the values of N, b and DS . However, given NbW 1 andDSbW 1, the governing equation (3) can be linearized to yield the state space form givenby equations (7), which are sufficiently accurate to model the system behavior given theseconditions. The harmonic balance method can be employed to obtain a steady statesolution to equation (7), although the system of equations is considerably morecomplicated. This is evident from Table 2, which illustrates the harmonic interactions givenbW 1.

6.2. DS =0In the case in which DS =0 and NbW 1, the AnN are easily determined by using equations

(14) and substituting aeff , given by

aeff = az1+ (Nbf )2/(1− a2) (23)

33

T 2

Frequency mapping of harmonic interactions given b$ 0

Harmonic Harmonicdescription Frequency indices, m

Mean value 0 0, 1, N, N2 1

Shaft order sL s, s2 1, N2 (s2 1)

Lower sidebands (nN− s)L nN− s, (n2 1)N− s,nN−(s2 1), (n2 1)N−(s2 1)

nth mesh order nNL nN, nN2 1, (n2 1)N, (n2 1)N2 1

Upper sidebands (nN+ s)L nN+ s, (n2 1)N+ s,nN+(s2 1), (n2 1)N+(s2 1)

in lieu of a. Hence the effect of small b on the steady state response, given DS =0, is toincrease the effective mesh stiffness variation. Of course, the phase must also be adjustedaccordingly. Note that aeff is dependent on the mean force f in addition to a and Nb.

6.3. : DS bThe steady state response of equation (6), given DS $ 0 and b$ 0, is considerably more

complex than that corresponding to b=0. No attempt is made here to present anexhaustive study on the effects of angle modulation and a rigorous mathematical treatmentof dual domain problems is let for future investigations. Rather, this study is intended onlyto illustrate the importance of b$ 0 and to determine the feasibility of using this newtheory is a plausible explanation for observed phenomena. Furthermore, only solutionsin the mesh frequency regime are considered. In Figure 15 are shown nine discrete Fourierspectra of jSS (t)/f, given N=50, a=0·4, z=0·1 and L/N=0·8 with DS /f=1, 10 and100 and b=0, 0·0001 and 0·001. The phase angle fS was found not to have any significanteffect on the spectral amplitudes and is therefore set equal to zero. The spectra wereobtained by transforming steady state solutions of equation (6) from direct time-domainintegration. The normalization here is possible only, given NbW 1 and bDS W 1. Anexamination of Figures 15 reveals that the effect of b on the displacement spectra is indeeddramatic, even for the very small values of b considered. Increasing the product bDSproduces higher order shaft harmonics as well as higher order sideband pairs which maybe highly asymmetric. The product bDS has very little effect on lower order mesh harmonicamplitudes provided that nNbDS W 1, where n is the index of the highest mesh harmonicamplitude of interest. The assumption bDS W 1 is violated in Figure 15(i) and non-lineareffects come into play. There is a large increase in both shaft harmonic and sidebandactivity, with sidebands having amplitudes equal to or exceeding their corresponding meshharmonic amplitudes in some cases.

Further numerical investigation reveals that sideband asymmetry is largely a factor ofL and b. Sidebands also exist under quasi static conditions given b$ 0. Most importantly,given nNbDS W 1, jS is largely independent of jM . However, jM is strongly dependent onjS . Hence, there is a unique one-way dynamic coupling which exists between steady statesolutions in the shaft and mesh frequency regimes. This suggests that for certain limitingcases it is possible to solve for jS independently of jM (t) and to then employ jS in a separatedynamic analysis in order to solve for jM . This analysis methodology is being pursued bythe authors in a companion study to determine analytical solutions for selected

. . . 34

Fig

ure

15.

The

effec

tof

bon

the

stea

dyst

ate

resp

onse

spec

tra

ofj

SS/f

,gi

ven

N=

50,a=

0·4,

z=

0·1

and

L=

0·8.

(a–c

)D

S/f

=1;

(d–f

)D

S/f

=10

;(g

–i)

DS/f

=10

0.(a

,d,g

),b

=0;

(b,e

,h),

b=

0·00

01;(c

,f,i

),b

=0·

001.

35

sideband amplitudes for a class of linear oscillators having an exponentially modulatedspatially periodic stiffness term.

7. PRIOR EXPLANATIONS FOR SIDEBANDS BASED ON COMMUNICATION THEORY

Before closing, it is necessary to comment on prior theory offered in an attempt toexplain sideband phenomena such as that shown in Figure 15. Many investigators haveemployed concepts directly from electrical communications theory. In particular, com-bined amplitude modulation (AM) and frequency (FM) for phase modulation (PM)processes have been proposed as viable explanations for sideband phenomenon in gearedsystems [3]. Typically, a spatially periodic mesh frequency carrier function xc [Nu(t)] isassumed in conjunction with two independent harmonic, time-varying, shaft frequencyAM and FM or PM message functions xAM (Lt) and xFM/PM (Lt). These assumedexpressions are combined in the usual fashion [3, 18] to obtain an overall expression forthe steady state response x(t), which generally takes on a form similar to the expressiongiven below for combined AM and PM:

x(t)= [1+ axAM (Lt)]·xc{N[Lt+ bxPM (Lt)]}. (24)

Here, a is an AM modulation index and b is a PM modulation depth. Typically, the AMmessage function is assumed to be free from any phase modulation effects. While suchtheory may provide qualitative understanding of various modulation processes and theirresultant spectral composition, its effective use as an analysis tool to describe the spectraassociated with many types of mechanical systems is limited to certain types of quasi-staticPM phenomenon and certain types of AM and PM phenomenon due to external variationsin L and f [19]. Both L and f are assumed to be time-invariant in the present study.Furthermore, conventional modulation theory is often not capable of predicting sidebandshaving amplitudes comparable to those predicted by using equation (3) when realisticvalues for a and b are employed. While it is usually possible to decompose any measuredspectra containing sidebands into an effective carrier and AM and PM message functions,these functions usually do not carry any physical significance. For these reasons, theauthors have abandoned conventional communication theory in lieu of the new theoryproposed in this study, which is believed to provide a more plausible explanation forsideband phenomena occurring in many types of mechanical systems.

8. CLOSURE

A class of viscously damped mechanical oscillators is examined having spatially periodicstiffness and displacement excitation functions that are exponentially modulated by theinstantaneous vibratory displacement of the inertial element. To this end, a new class ofperiodic differential equations is introduced which is shown to be more pertinent to thestudy of force modulation inherent to many mechanical systems. New analytical ex-pressions are derived to predict sideband amplitudes in terms of key system parametersfor one limiting case when angle modulation is not present (b=0). Under such conditions,sidebands arise only under dynamic conditions and are most prominent in the meshfrequency regime near L/N3 1. When angle modulation is present (b$ 0), the spectralcontents of the steady-state response are considerably more complex. Strong harmonicinteractions can occur between shaft, mesh and sideband components.

Although an attempt is made here to study a generic problem with many potentialapplications to mechanical systems, the reader must realize that only limited information

. . . 36

can be obtained from a single model. Of more importance is the fundamental understand-ing of the various physical mechanisms which generate modulation sidebands. The newtheory challenges the commonly held belief that simple amplitude and frequency or phasemodulation processes are responsible for generating such sidebands and instead providesa more logical explanation for sidebands in terms of fundamental physical parameters.This is believed to be a new contribution to the body of knowledge in the general areasof vibration, acoustics and signal processing.

ACKNOWLEDGMENTS

The authors would like to thank the Powertrain Division of General Motors Corpor-ation for supporting this research.

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Analysis of force coupling and modulation phenomenon in geared rotor systems.