subharmonic response of single-degree-of-freedom linear vibroimpact system to narrow-band random...

Appl. Math. Mech. -Engl. Ed., 32(9), 1159–1168 (2011)DOI 10.1007/s10483-011-1489-xc©Shanghai University and Springer-Verlag

Berlin Heidelberg 2011

Applied Mathematicsand Mechanics(English Edition)

Subharmonic response of single-degree-of-freedom linear vibroimpactsystem to narrow-band random excitation∗

Hai-wu RONG (��)1, Xiang-dong WANG (��)1, Qi-zhi LUO (��)1,Wei XU (� �)2, Tong FANG (� �)2

(1. Department of Mathematics, Foshan University, Foshan 528000,

Guangdong Province, P. R. China;

2. Department of Applied Mathematics, Northwestern Polytechnical University,

Xi’an 710072, P. R. China)

Abstract The subharmonic response of a single-degree-of-freedom linear vibroimpact

oscillator with a one-sided barrier to the narrow-band random excitation is investigated.

The analysis is based on a special Zhuravlev transformation, which reduces the system

to the one without impacts or velocity jumps, and thereby permits the applications of

asymptotic averaging over the period for slowly varying the inphase and quadrature re-

sponses. The averaged stochastic equations are exactly solved by the method of moments

for the mean square response amplitude for the case of zero offset. A perturbation-based

moment closure scheme is proposed for the case of nonzero offset. The effects of damping,

detuning, and bandwidth and magnitudes of the random excitations are analyzed. The

theoretical analyses are verified by the numerical results. The theoretical analyses and

numerical simulations show that the peak amplitudes can be strongly reduced at the large

detunings.

Key words single-degree-of-freedom linear vibroimpact system, subharmonic response,

Zhuravlev transformation method, random averaging method

Chinese Library Classification O324

2010 Mathematics Subject Classification 34A36

1 Introduction

An impact oscillator, often named the vibroimpact system, is the term used to represent asystem that is driven in some way and also undergoes intermittent or a continuous sequence ofcontacts with motion limiting constraints[1]. The analyses of the impact systems are importantin various engineering applications. Certain useful applications of vibration are known, wherethe impacts are involved, such as vibratory pile drivers and tie placers. The analyses of theimpact motion are important in the proper design of the corresponding machines and devices[2].However, it is very difficult to investigate those systems. The main difficulty is that such systems

∗ Received Nov. 12, 2010 / Revised Jun. 16, 2011Project supported by the National Natural Science Foundation of China (Nos. 10772046 and50978058) and the Natural Science Foundation of Guangdong Province of China (Nos. 7010407 and05300566)Corresponding author Hai-wu RONG, Professor, Ph.D., E-mail: [email protected]

1160 Hai-wu RONG, Xiang-dong WANG, Qi-zhi LUO, Wei XU, and Tong FANG

are not continuous but rather of the intermittent type. In practice, engineering structures areoften subjected to the time dependent loadings of the stochastic nature, such as the naturalphenomena due to wind gusts, earthquakes, ocean waves, and random disturbance or noise thatalways exists in a physical system. The influence of the random excitation on the dynamicalbehavior of an impact dynamical system has caught the attention of many researchers. Someanalysis methods, e.g., the linearization method[3], the quasistatic approach method[4–5], theMarkov process method[6–7], the stochastic averaging method[8–9], the variable transformationmethod[10–11], the energy balance method[12], the mean impact Poincare map method[13], andthe numerical simulation method[14] have been developed. In Ref. [2], the authors tried to reviewand summarize the existing methods, results, and literatures available for solving the problemsof stochastic vibroimpact systems. Most studies focused on the responses of the linear impactoscillator (here, “linear” means that the differential equation of motion between the impactsis linear) under wild-band random excitations. Few focused on the responses of the impactoscillator under narrow-band random excitations[15]. In Ref. [15], Dimentberg et al. discussedthe response of a linear impact system with a rigid one-sided barrier under a special kindof narrow-band random excitations— the sinusoidal force with the disorder or random phasemodulation in detail. However, as pointed by Dimentberg et al., for certain applications, themodel of the filtered Gaussian white noise may be more appropriate for the basic narrow-bandrandom excitation rather than the one used in Ref. [15].

In this paper, the subharmonic response of a single-degree-of-freedom linear vibroimpactoscillator with a one-sided barrier, which slightly offsets from the equilibrium position of thesystem, is investigated under the general narrow-band random excitation— the filtered Gaus-sian white noise. The impact considered here is an instantaneous impact with the restitutionfactor e. The paper is organized as follows. In Section 2, the Zhuravlev transformation and thestochastic averaging method are used to obtain the mean square amplitude of the response. InSection 3, the direct numerical simulations verify the analytical results. The conclusions arepresented in Section 4.

2 System description and theoretical analyses

Consider the single-degree-of-freedom linear vibroimpact oscillator to the random excitations

{y + 2βy + Ω2y = hξ(t), y < Δ,

y+ = −ey−, y = Δ,(1)

where the dot indicates the differentiation with respect to the time t, β and Ω are the dampingcoefficient and the natural frequency, respectively, Δ represents the distance from the staticequilibrium position of the system to the single rigid barrier, 0 < e � 1 is the restitution factorto be a known parameter of the impact losses, and the subscripts “+” and “−” refer to thevalues of the response velocity just before and after the instantaneous impact. Thus, y+ andy− are the actually rebound and impact velocities of the mass, respectively. They have thesame magnitude whenever e = 1. Therefore, this is a special case of the elastic impact, whereasin the case of e < 1, some impact losses are observed. h denotes the intensity of the randomexcitation. ξ(t) is chosen to be a zero-mean Gaussian narrow-band random excitation. It couldbe obtained by filtering the white noise through a linear filter, i.e.,

ξ + γξ + Ω21ξ =

√γΩ1W, (2)

where Ω1 is the center frequency of ξ(t), and γ is the bandwidth of the filter. W (t) is the unitwhite noise with the autocorrelation function RW (τ) = 2πδ(τ), and δ is the Dirac delt function.The response of the system (1) is discussed in detail by Dimentberg et al.[15] when ξ(t) is a

Subharmonic response of single-degree-of-freedom linear vibroimpact system 1161

sinusoidal force with the disorder as follows:

ξ(t) = sin ϕ(t), ϕ(t) = Ω1 + γW (t). (3)

However, as pointed by Dimentberg et al., for certain applications, the model of the filteredGaussian white noise governed by Eq. (2) may be more appropriate for the basic narrow-bandrandom excitation rather than the one used in Ref. [15]. From Eq. (2), ξ(t) can be rewrittenas[16]

ξ(t) = ξ1(t) sin(Ω1t) + ξ2(t) cos(Ω1t), (4)

where ξ1(t) and ξ2(t) are the slowly varying random functions of the time. In fact, substitutingEq. (4) into Eq. (2) and performing the deterministic and stochastic averaging of the equationsdescribing the modulations of ξ1(t) and ξ2(t) with the time, one obtains

ξ1 +γ

2ξ1 =

√γ

2W1, ξ2 +

γ

2ξ2 =

√γ

2W2. (5)

The unit white noise components W1 and W2 are independent. The autocorrelation functionsof ξ1(t) and ξ2(t) are

Rξ1(τ) = Rξ2(τ) = πe−γ|τ|2 .

The correlation time of ξ1(t) and ξ2(t) is O( 1γ ). This means that for the sufficiently small values

of γ, ξ1(t) and ξ2(t) are the slowly varying random functions of the time. From Eq. (4), ξ(t)can be rewritten as

ξ(t) =√

ξ21 + ξ2

2 sin(Ω1t + ϕ(t)), ϕ(t) = arctanξ2

ξ1. (6)

Following Zhuravlev[17], the nonsmooth transformation of the state variables is introducedas follows:

y = |x| + Δ, y = xsgnx, (7)

where sgnx is the signum function such that sgn x = 1 for x > 0 and sgn x = −1 for x < 0.Obviously, this transformation makes the transformed velocity x be continuous at the impactinstants (i.e., x = 0) in the special case of the elastic impact (i.e., e = 1). Thereby, theproblem is reduced to the one without velocity jumps. However, this is not the case of a generalvibroimpact system with impact losses. The jump of the transformed velocity x becomes to beproportional to 1− e instead of 1 + e for the jump of the original velocity y. This jump can beincluded in the transformed differential equation of motion by using the Dirac delta functionδ(x). Since x(t∗) = 0 at the impact instant t∗ and δ(t − t∗) = |x| δ(x), the impulsive term canbe obtained as

(x+ − x−)δ(t − t∗) = (1 − e)x |x| δ(x).

The transformed equation of motion can be written by substituting Eqs. (6) and (7) into Eq. (1)as follows:

x + Ω2x = −2βx − ΔΩ2sgn x − (1 − e)x |x| δ(x) + h√

ξ21 + ξ2

2 sgnx sin(Ω1t + ϕ(t)). (8)

Thus, the original impact system (1) is reduced to the “common” vibration system (8)without impacts. The term (1−e)x |x| δ(x) on the right-hand side of Eq. (8) describes the impact


losses of the system, which can be regarded as an impulsive damping term. The transformedequation (8) permits a rigorous analytical study by the asymptotic method of averaging overthe period, as long as the coefficients β, Δ, h, and 1 − e are all small and proportional to asmall parameter. Moreover, only the subharmonic resonant responses will be considered, i.e.,the frequency Ω1 of the random excitation is near the subharmonic resonant frequency 2nΩ,and Ω1 ≈ 2nΩ, where n is an arbitrary positive integer. The detuning parameter μ is definedas μ = Ω1 − 2nΩ. μ is assumed to be small and proportional to a small parameter. Then, theresponse of Eq. (8) can approximately be represented as

x = A(t) sin Φ(t), x = ΩA(t) cos Φ(t). (9)

Introduce a new slowly varying phase shift θ(t) = Ω1t + ϕ(t) − 2nΦ(t). Then, Eq. (8) can betransformed to the following pair of first-order equations:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A =cosΦ

Ω(−2βΩA cosΦ − ΔΩ2sgn(sin Φ) − (1 − e)Ω2A cos Φ |A cos φ| δ(A sin Φ)

+h√

ξ21 + ξ2

2 sgn(sin Φ) sin(θ + 2nΦ)),

θ = μ +2n sinΦ

ΩA

(−2βΩA cosΦ − ΔΩ2sgn(sin Φ)−(1 − e)Ω2A cosΦ |A cosφ| δ(A sin Φ)

+h√

ξ21 + ξ2

2 sgn(sin Φ) sin(θ + 2nΦ) +√

γ

2W2ξ1 − W1ξ2

ξ21 + ξ2

2

).

(10)

Under the foregoing assumption that the damping, impact losses, and excitation terms aresmall, the right-hand sides of both equations of Eq. (10) are proportional to a small parameter.A, θ, ξ1, and ξ2 are the slowly varying random processes with respect to the time t, and Φ, W1,and W2 are the fast varying random processes. By averaging over the fast state variables Φ, W1,and W2, the following shortened equations can be obtained[15,18]:⎧⎪⎪⎪⎨

⎪⎪⎪⎩A = −αA + qζ cos θ, θ = μ − q

Aζ sin θ +

δ

A,

α = β +1 − e

πΩ, q =

4nh

(4n2 − 1)πΩ, δ =

4nΩΔπ

, ζ =√

ξ21 + ξ2

2 .

(11)

Equation (11) shows that the difference between the elastic impact (e = 1) and the inelasticimpact (e < 1) is that the inelastic impact increases the damping of the system from β toα = β + 1−e

π Ω. Introduce another new pair of state variables

u = A cos θ, v = A sin θ. (12)

Equation (11) can be transformed to

⎧⎪⎪⎨⎪⎪⎩

u = −αu − μv − δv√u2 + v2

+ qζ,

v = −αv + μu +δu√

u2 + v2.

(13)

It should be noted that an exact analytical study to the system (13) seems to be impossibledue to the nonlinear nature. Thus, the approximate solutions of the second-order moments ofthe subharmonic response are proposed. Equation (13) seems to be linear if Δ = 0 (δ = 0), and


thus is amenable to an exact analysis by the method of moments[19]. For the case of Δ = 0,Eq. (13) can be written as

u = −αu − μv + qζ, v = −αv + μu. (14)

The response moments of any order can easily be predicted. Only the mean square amplitudeEA2 = E(u2 + v2) will be considered here, where E denotes the mathematics expectation. Forthe steady state response, one has

dEu2

dt=

dEv2

dt=

dEuv

dt=

dEuζ

dt=

dEvζ

dt= 0. (15)

From Eqs. (15), (14), and (5), applying the expectation operator and equating the time deriva-tive to zero for a steady state solution, one can obtain

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−αEu2 − μEuv + qEuζ = 0, μEuv − αEv2 = 0,

μEu2 − 2αEuv − μEv2 + qEvζ = 0,

(α +

γ

2

)Euζ + μEvζ = qEζ2, μEuζ −

(α +

γ

2

)Evζ = 0.

(16)

Form Eq. (5), one has Eζ2 = E(ξ21 + ξ2

2) = 1. Substituting this expression into Eq. (16), onecan obtain the solution of Eq. (16) as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Eu2 =q2(α3 + α2γ

2 + μ2γ4 )

α(α2 + μ2)((α + γ2 )2 + μ2)

,

Euv =q2μ(α + γ

4 )(α2 + μ2)((α + γ

2 )2 + μ2),

Ev2 =q2μ2(α + γ

4 )α(α2 + μ2)((α + γ

2 )2 + μ2),

Euζ =q(α + γ

2 )(α + γ

2 )2 + μ2, Evζ =

qμ

(α + γ2 )2 + μ2

.

(17)

Then, the mean square amplitude can be obtained as

EA2 = Eu2 + Ev2 =q2(α + γ

2 )α((α + γ

2 )2 + μ2). (18)

It can be seen from the formula (18) that the mean square amplitude-frequency response curvesare symmetric with respect to the corresponding resonant frequency of any given order n, i.e.,they are the same for the positive and negative detunings.

In some applications, the first-order moments may be required and can easily be obtainedby the method of moments from Eq. (14) as follows:

Eu =αq

α2 + μ2Eζ, Ev =

μq

α2 + μ2Eζ.

Next, we discuss the mean square amplitude of the system (13) in the case of Δ �= 0.Equation (13) is nonlinear for the nonlinear Δ, which requires the use of some closure schemes.


Denote A2∗ = EA2 = E(u2 + v2), and substitute all

√u2 + v2 terms in the right-hand side of

Eq. (13) by A∗. Equation (13) can be transformed to the following linear equations:

u = −αu −(μ +

δ

A∗

)v + qζ, v = −αv +

(μ +

δ

A∗

)u. (19)

Equation (19) is similar to Eq. (14). From the formula (18), one obtains the equation of A∗ as

A2∗ =

q2(α + γ2 )

α((α + γ2 )2 + (μ + δ

A∗)2)

. (20)

Equation (20) has the following solution:

A∗ =−αμδ ± √

α2μ2δ2 + α((α + γ2 )2 + μ2)(q2(α + γ

2 ) − αδ2)α((α + γ

2 )2 + μ2). (21)

It can clearly be seen that, with Δ → 0 (δ → 0), the squared quantity A2∗ approaches the

“exact” mean square amplitude EA2 governed by Eq. (18).

3 Numerical simulation

In this section, the analytical results can be shown and compared with the direct numericalresults. All the direct numerical simulations by the Monte-Carlo method are based on the origi-nal system dominated by Eq. (1), which give the powerful validation with the analytical results.For the method of numerical simulations, readers can refer to Zhu[19] and Shinozuka[20–21]. Inthis paper, the power spectrum of ξ (t) is taken as

S(ω) =

⎧⎪⎨⎪⎩

12π

, 0 < ω � 2Ω,

0, ω > 2Ω.

For the numerical simulation, it is more convenient to use the pseudorandom signal given by[19]

ξ (t) =

√2ΩNπ

N∑k=1

cos( Ω

N(2k − 1)t + ϕk

),

where ϕk are independent and uniformly distributed in (0, 2π], and N is a larger integer.The Monte-Carlo simulations are focused on the first-order subharmonics (n = 1 and Ω1 ≈

2Ω), though the higher-order subharmonic (Ω1 ≈ 2nΩ with n = 2, 3, 4, · · · ) simulations shouldbe of the same importance. In the numerical simulation, the parameters in the system (1) arechosen as

h = 2.0, Δ = 0.1, Ω = 1, n = 1.

The governing equation (1) is numerically integrated by the fourth-order Runge-Kutta algorithmbetween the impacts, which is valid until the first encounter with the barriers, that is, until theequality y = Δ is satisfied. The impact condition y+ = −ey− is then imposed by using thenumerical solution y−. This results in the rebound velocity y+. Thereby, the initial values areprovided for the next numerical calculation. The numerical results are shown in Figs. 1–3.

We first consider the effect of the damping coefficient β and the center frequency Ω1 onthe response amplitude A∗ of the system. The variations of the steady-state response A∗ with


Ω1 are shown in Fig. 1 in the case of γ = 0.1 and e = 0.9 when β = 0.15 and β = 0.1.For comparison, the theoretical results given by Eq. (21) are also shown in Fig. 1. The meansquare response amplitude is calculated as A2

∗ = 2E( xΩ)2 in the numerical simulation, where the

angular brackets denote the common time averaging for the response sample. Figure 1 showsthat the response amplitude predicted by the averaging method is in good agreement with thatobtained by the numerical results.

Fig. 1 Frequency response of system (1) (γ = 0.1 and e = 0.9)

Fig. 2 Response of system (1) (Ω1 = 2.08 and β = 0.1)

It can be seen from Fig. 1 that the response amplitude decreases when the damping βincreases, which is in accordance with the physical intuition. The peak response amplitudebecomes large when the frequency Ω1 is near the resonant frequency Ω1 = 2, and decreasesstrongly when Ω1 departs from the resonant frequency. In comparison with the numericalsolution, the accuracy of the analytical solution is reduced a little in the case of the largedetuning. This can be partly caused by the inaccuracy of the averaging method at the largedetuning.

Next, we consider the effect of the restitution factor e and the bandwidth γ on the responseamplitude A∗ of the system. The variations of the steady-state response A∗ with γ are shownin Fig. 2 in the case of Ω1 = 2.08 and β = 0.1 when e = 1.0 and e = 0.9. For comparison, thetheoretical results given by Eq. (21) are also shown in Fig. 2.


It can be seen from Fig. 2 that the response amplitude A∗ decreases when γ increases or edecreases. There are some impact losses when e < 1, and the impact losses increase when edecrease, effectively leading to a large viscous damping factor, which is in accordance with theforegoing theoretical analyses.

Now, we consider the effect of Δ on the response amplitude A∗ of the system. The variationsof the steady-state response A∗ with Δ are shown in Fig. 3 in the case of Ω1 = 2.08, β = 0.1,e = 0.9, h = 2.0, and γ = 0.1. For comparison, the theoretical results given by Eq. (21) are alsoshown in Fig. 3.

It can be seen from Fig. 3 that the response amplitude A∗ decreases when Δ increases.The response time history of the system (1) is shown in Fig. 4 in the case of Ω1 = 2.08, β =

0.1, e = 1.0, and γ = 0.1, where z(t) = y(t) denotes the velocity of the mass.

Fig. 3 Response of system (1) (Ω1 = 2.08, β = 0.1, e = 0.9, h = 2.0, and γ = 0.1)

Fig. 4 Numerical results of Eq. (1) (γ = 0.1, β = 0.1, Ω1 = 2.08, and e = 1.0)

4 Conclusions and discussion

In this paper, the methods of the Zhuravlev transformation and the stochastic averaging areused to analyze the response of a linear impact system under the disordered periodic excitation.So far, the exact solutions of the nonlinear impact system under the random excitation areonly available for a very limited number of problems. Thus, approximate methods have beendeveloped and used to treat many of these problems. They include the method of equivalentor stochastic linearization, perturbation methods, stochastic averaging, series expansions, etc.


In fact, it is difficult or impossible to solve exactly even in the nonlinear determinationsystem with the single-degree-of-freedom. Hence, the approximate method is widely used inthe analysis of the nonlinear determination system. They include the small parameter method,the method of coordinate transformation, the multiple scale method, the method of slowlyvarying parameter, the Krylov-Bogoliubov-Mitropolsky (KBM) method, the method of equiv-alent linearization, the method of harmonic balance, etc. The approximate methods in thedetermination system can be extended to the random system. For example, the harmonic bal-ance method[22] and the polynomial approximation method[23] have also been extended to therandom systems.

The theoretical analyses and numerical simulations show that the peak response amplitudebecomes large when the frequency Ω1 is near the resonant frequency Ω1 = 2, and decreasesstrongly when Ω1 departs from the resonant frequency. The peak response amplitude decreaseswhen γ and β increase, or e decreases.

Small parameters are often required in many averaging and perturbation methods. However,the numerical simulation shows that the approach presented in this paper is also appropriatefor small h or large h (h = 2.0).

The analysis used in this paper seems to be helpful for other types of nonlinearities, whichcan be suggested for the continuation of this research.

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