# Stochastic Responses for the Vibro-Impact System under the Broadband Noise

Authors: S. W. Guan; Q. Ling; M. Xu
Journal: Shock and Vibration
 (2019)

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## Abstract

The stochastic averaging is applied to derive the random responses of the SDOF vibro-impact system with one-side wall under additive and multiplicative broadband noise, where the impact is described by the modified Hertzian contact model. Compared with the popular simplified “classical model” (i.e.,x˙−=rx˙+), the whole impact process is captured by the modified Hertzian contact model without any assumptions. One example is given to illustrate the proposed technique. The comparison between the analytical results and those from Monte Carlo simulations manifests the accuracy of the proposed technique. It should be pointed out that the technique proposed in this paper is powerful for the cases of weak excitation and big enough bandwidth of the broadband noise. Finally, the influences of various parameters are also discussed in detail.


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## Body

## 1. Introduction

A vibro-impact system is a significant nonsmooth system and can be frequently found in the mechanical engineering. The main distinguish between the vibro-impact system and ordinary vibration system is the impact. The impact always introduces the complicate nonlinear behaviors, such as homoclinic bifurcations [1]. In view of the wide engineering demands, the impact has been investigated by many engineers and physicists. Dimentberg reviewed the past work on the random dynamics of the vibro-impact system, where the impact is described by the “classical impact model” (i.e., x˙−=rx˙+) and the Hertzian contact model [2]. Iourtchenko investigated the probability density function (PDF) of the stochastic system with inelastic impact [3]. Dimentberg used the path integration method and obtained the numerical PDF of the vibro-impact system [4]. Gu and Zhu developed a novel stochastic averaging to analyze the vibro-impact system under Gaussian white noise excitation [5]. Li et al. studied the stochastic response of the vibro-impact system by the generalized cell mapping method [6]. Recently, Zhu studied the stochastic responses of the vibro-impact system under the different excitations [7, 8].Nearly all of the literatures on the vibro-impact systems presented the “classical model” [2]. The impact reflects under the constraint condition of the velocity jump, which can be dealt by the Zhuravlev transformation to eliminate the discontinuities [9]. In spite of the successful application in vibro-impact system, two shortages should still be pointed out. The comprehensive impact process is ignored by “classical model”, and the restitution coefficient, which is considered as a constant for a random oscillator, is not always reasonable. Recently, one impact model named modified Hertzian contact model, which overcomes the shortages mentioned above, has caught the author’s eyes [10]. The impact on the vibrating oscillator is an inelastic force instead of the discontinuous velocity jump. By introducing the modified Hertzian contact model, the stochastic response of the vibro-impact system under Gaussian white noise has been studied by the stochastic averaging [10].The broadband noises widely existed in the ambient environment and play an important role in dynamic behaviors of mechanical and structural engineering. The broadband noises can always degrade performance of the dynamic system. Therefore, lots of researchers pay lots of attentions to evaluate the system responses, reliability, and security under broadband noise excitations. Ling et al. studied the response and stability of the SDOF viscoelastic system subjected to the wideband noise excitations by using the stochastic averaging [11]. Janevski obtained the Lyapunov exponent and moment Lyapunov exponent of Hill’s equation with frequency and damping coefficients fluctuated by correlated wideband random processes by the stochastic averaging, both first order and second order [12]. Hu and Zhu proposed the stochastic optimal bounded control strategy for the MDOF nonlinear system under combined harmonic and wideband noise excitations [13]. Deng et al. analyzed the stochastic moment stability and almost sure stability by the adopting method of higher-order stochastic averaging [14]. Li et al. studied the first passage failure problem of the time-delayed feedback control system under wideband random excitation to evaluate the system safety [15]. Under the broadband noise, the responses of the random vibration with inelastic impact represented by the modified Hertzian contact model are still open.In an effort to derive the stationary PDF, a stochastic averaging for the inelastic vibro-impact system with one-side wall is proposed. The vibro-impact system is firstly equivalent to a nonlinear system by the energy dissipation balance technique. It is the most crucial step and paves the way to apply the stochastic averaging. Secondly, one coordination transformation is introduced and the motion equation is converted to two one-order ordinary differential equations with reference to slowly varying variables. Then, the system energy’s Itô equation is given, and the stationary PDF of the system energy can be carried out by the FPK equation derived from the system energy’s Itô equation. The marginal PDF of the displacement and velocity can be calculated from the stationary PDF of the system energy directly. Last, one example is given to manifest the effectiveness. The limitations of the proposed technique are also discussed.


## 2. Vibro-Impact System

One linear oscillator is discussed in the present paper, and the schematic diagram of the oscillator can be found in Figure1. A wall is on the right side of the oscillator and cannot be seen as a rigid body. Once collision occurs, the wall will experience local permanent indentation and the oscillator can crash into the wall. It means the PDF of the vibro-impact system behind the wall will not be equal to zero, which is significantly different from “classical model”. The dimensionless motion equation of the linear oscillator can be given as(1)X¨+2ζω0X˙+ω02X+fIX,X˙=ξ1t+X˙ξ2t,where the linear damping coefficient and the natural frequency are ζ and ω0, respectively. The distance between oscillator and wall is denoted as δr. The impact fIX,X˙ is not a constant and depends on the system state when the oscillator collides with the wall. Here, fIX,X˙ is depicted by the modified Hertzian contact model, shown in Figure 2 [10, 16]. It is obvious that the loading process and unloading process are quite different. Moreover, the unloading path is under the loading path, which reflects the collision is a dissipative impact process. The mathematical formulas of the impact force and the relative displacement can be given as(2a)fI+=k3x−x03/2+k2x0−δr3/2,x∈x0,a1k2x−δr3/2,x∈δr,x00,x∈−a2,δrforx˙>0loading phase,(2b)fI−=k2x−δr−Δr3/2,x∈δr+Δr,a10,x∈−a2,δr+Δrforx˙<0unloading phase.where parameters k2 and k3 are the additional stiffness and x0 is the elastic limitation. a1 and a2 are the maximum right-side and left-side displacements of the oscillator, respectively. In general, a1 is smaller than a2. Parameter Δr is the residue deformation caused by the local permanent indentation and governed by the following equation:(3)k2a1−δr−Δr3/2=k3a1−x03/2+k2x0−δr3/2.Figure 1
Schematic of the inelastic vibro-impact system.Figure 2
The constitutive relationship of the modified Hertzian contact model.However, when the collision is not serious (i.e., the displacement of the mass is smaller than the elastic limitationx0), the deformation of the wall is completely elastic, and no residue deformation is left. In this situation, the impact fIX,X˙ does not satisfy (2) and obeys the law as follows:(4)fI=k2x−δr3/2,x∈δr,a1,0,x∈−a2,δr.So, the loading line and unloading line overlap in this case, and the collision can be seen as an elastic impact process.The right side of the vibro-impact system (1) is the independent random noises ξit,i=1,2. The random noise here is modeled as the broadband excitation. The spectral density of random noise ξit is assumed to be in the follow form:(5)Φiω=Diπ1ω2+αi2,i=1,2.The bandwidth of the broadband noise is determined by the parameterαi. The greater the parameter αi, the bigger the bandwidth of the broadband noise. The random noise ξit can also be regarded as the response of the first-order filter subjected to a white noise with the excitation intensity 2Di [17].


## 3. Solving Procedure

The linear oscillator in system (1) incorporates the impact. The exact solution of this system cannot be derived directly. One compromise way is to use the asymptotical technique to derive the approximate results. Among the approximate technique, the stochastic averaging is one of the most efficient methods and has the advantages of the dimensionality reduction and good accuracy. Observing the impact formula in (2), the impact process simultaneously includes the energy dissipation and the elastic deformation. There still are some difficulties to use the stochastic averaging straightforward. A good choice is to use the energy dissipation balance technique, which is a powerful technique and can deal with the complex interaction. As a preparation for the stochastic averaging, the first procedure is to separate the impact into the stiffness and damping term.


### 3.1. Equivalent Nonlinear System

The oscillator vibrating in a period experiences one circle of loading and unloading. The effects from the impact represented by the modified Hertzian contact model reflected on the linear oscillator system are considered to be the additional stiffness and damping terms.The equivalent stiffness can be expressed as the derivative of the potential energy. The potential energy can be evaluated by the work done along the unloading path. Because the loading phase and unloading phase are significantly different, the potential energy will be given, respectively. For the loading case, the system potential energy is(6)U+=ω02x22+25k2y−δr−Δr5/2,x∈x0,a1,ω02x22+25k2x−δr5/2,x∈δr,x0,ω02x22,x∈−a2,δr,x˙>0,while in the unloading phase, it can be given as(7)U−=ω02x22+25k2x−δr−Δr5/2,x∈δr+Δr,a1,ω02x22,x∈−a2,δr+Δr,x˙<0.The intermediate variabley in loading potential energy U+ can be expressed as(8)k2y−δr−Δr3/2=k3x−x03/2+k2x0−δr3/2.The equivalent damping is determined by the criterion that the dissipation energy of the impact in one circle is equal to the equivalent linear damping with the coefficient depending on the system energy instead of a constant. Considering the dissipation in each period of the impact process is different and depends on the system state, the equivalent damping is assumed to be only the system energy dependent. Thus, the equivalent damping is easy to derive, and the associated mathematical formulation is [18](9)∮2ζ1Hx˙dx=Ar,where Ar denotes the dissipation energy induced by the inelastic impact and can be expressed as(10)Ar=∫δra1g+xdx+∫a1δr+Δrg−xdx.From (6) and (7), the equivalent linear damping coefficient can be derived as(11)2ζ1H=Ar∫−a2a12H−2U+xdx−∫a1−a22H−2U−xdx.When the impact is elastic, only the stiffness effect is considered and the modified Hertzian contact model degrades to the Hertzian contact model. Obviously, when the impact equals zero, system (1) is a linear motion equation of the oscillator.


### 3.2. Stochastic Averaging Procedure

The inelastic impact is replaced by the equivalent damping and stiffness, and system (1) is reduced to the following system:(12)X¨+2ζω0+2ζHX˙+∂U∂X=ξ1t+X˙ξ2t.To solve (12), one coordinate transformation is introduced [19]:(13a)sgnXUX=Hcosθ,(13b)X˙=−2Hsinθ,0≤θ<2π.The phaseθ can be separated into fast-varying term and slow-varying term, and the expression is as follows [17]:(14a)θ=∫0tωsds+φ,(14b)ωs=∂UX/∂X2Hcosθ.By using (13) and (14), system (1) is transformed to two-differential equations:(15a)H˙=−2H2ζ+2ζHsin2θ−2Hsinθ⋅ξ1t+2Hsin2θ⋅ξ2t,(15b)φ˙=−2ζ+2ζHsinθcosθ−12Hcosθ⋅ξ1t+cosθ⋅sinθ⋅ξ2t.In (15a) and (15b), if the damping is small and the excitation is weak enough, the stochastic process H and φ are both the slowly varying process. The stochastic averaging can be applied, and the system energy H can be approximated to a Markovian process [20]. By the time averaging, (15a) can be reduced to the stochastic Itô equation without the slowly varying the variable φ:(16)dH=mHdt+σHdBt,(17a)mH=−2H2ζ+2ζHsin2θ+∫−∞02Hsinθt+τ∂∂H2Hsinθt+cosθ2Ht+τ∂∂θ2HsinθtR1τdτ+∫−∞02Hsin2θt+τ∂∂H2Hsin2θt+cosθsinθt+τ∂∂θ2Hsin2θtR2τdτt,(17b)σ2H=∫−∞+∞2Hsinθt+τ2HsinθtR1τdτ+∫−∞+∞2Hsin2θt+τ2Hsin2θtR2τdτt,where Bt represents the unit wiener process.In (17a) and (17b), t denotes time averaging in one period T, and Riτ,i=1,2 is the self-correlation function of the stationary broadband noise. The relationship of the spectral density Φiω,i=1,2 defined in (5) and the self-correlation Riτ is given as follows:(18)Φiω=1π∫−∞0Riτcosωτdτ.In (17a) and (17b), the drift coefficient mH and the diffusion coefficient σ2H both contain the variable θt. To perform the time averaging, two approximate relations are needed [17]:(19a)θt+τ≈θt+ω¯Hτ,(19b)Xt+τ≈Xt+2Hω¯Hcosθt+ω¯Hτ−cosθt,where ω¯H represents the average frequency and can be obtained by ω¯H=1/T∫0Tωsdt=2π/TH, the quasi-period TH is satisfied as TH=∮dx/2H−2Ux.Substituting (19) into (17), the drift coefficient is given as(20)mH=−2ζω0+2ζ1HX˙2t+πΦ1ω¯+πHΦ20+πHΦ22ω¯=−2ζω0+2ζ1HT∮X˙ⅆX+πΦ1ω¯+πHΦ20+πHΦ22ω¯,and the diffusion coefficient is,(21)σ2H=2πX˙2tΦ1ω¯+πH22Φ20+Φ22ω¯=2πT∮X˙dX⋅Φ1ω¯+πH22Φ20+Φ22ω¯.The governing equation of the stationary PDF with reference to the system energyH is(22)−∂mHpH∂H+12∂2σ2HpH∂H2=0.The exact analytical solution of the stationary PDF can be written as(23)pH=Cexp∫0H2mx/σ2xdxσ2H,where C denotes a constant to satisfy the condition of normalization.From (23), it is possible to write the joint stationary PDF of system displacement x and velocity x˙ as follows:(24)px,x˙=pHTHH=Hx,x˙.Once the joint stationary PDF of system displacement and velocity is obtained, the marginal PDF of system displacement and velocity can be calculated by(25a)px=∫−∞∞px,x˙dx˙,(25b)px˙=∫−∞∞px,x˙dx.The mean square displacement can be represented as(26)EX2=∫−∞∞∫−∞∞x2px,x˙dx˙dx.


## 3.1. Equivalent Nonlinear System

The oscillator vibrating in a period experiences one circle of loading and unloading. The effects from the impact represented by the modified Hertzian contact model reflected on the linear oscillator system are considered to be the additional stiffness and damping terms.The equivalent stiffness can be expressed as the derivative of the potential energy. The potential energy can be evaluated by the work done along the unloading path. Because the loading phase and unloading phase are significantly different, the potential energy will be given, respectively. For the loading case, the system potential energy is(6)U+=ω02x22+25k2y−δr−Δr5/2,x∈x0,a1,ω02x22+25k2x−δr5/2,x∈δr,x0,ω02x22,x∈−a2,δr,x˙>0,while in the unloading phase, it can be given as(7)U−=ω02x22+25k2x−δr−Δr5/2,x∈δr+Δr,a1,ω02x22,x∈−a2,δr+Δr,x˙<0.The intermediate variabley in loading potential energy U+ can be expressed as(8)k2y−δr−Δr3/2=k3x−x03/2+k2x0−δr3/2.The equivalent damping is determined by the criterion that the dissipation energy of the impact in one circle is equal to the equivalent linear damping with the coefficient depending on the system energy instead of a constant. Considering the dissipation in each period of the impact process is different and depends on the system state, the equivalent damping is assumed to be only the system energy dependent. Thus, the equivalent damping is easy to derive, and the associated mathematical formulation is [18](9)∮2ζ1Hx˙dx=Ar,where Ar denotes the dissipation energy induced by the inelastic impact and can be expressed as(10)Ar=∫δra1g+xdx+∫a1δr+Δrg−xdx.From (6) and (7), the equivalent linear damping coefficient can be derived as(11)2ζ1H=Ar∫−a2a12H−2U+xdx−∫a1−a22H−2U−xdx.When the impact is elastic, only the stiffness effect is considered and the modified Hertzian contact model degrades to the Hertzian contact model. Obviously, when the impact equals zero, system (1) is a linear motion equation of the oscillator.


## 3.2. Stochastic Averaging Procedure

The inelastic impact is replaced by the equivalent damping and stiffness, and system (1) is reduced to the following system:(12)X¨+2ζω0+2ζHX˙+∂U∂X=ξ1t+X˙ξ2t.To solve (12), one coordinate transformation is introduced [19]:(13a)sgnXUX=Hcosθ,(13b)X˙=−2Hsinθ,0≤θ<2π.The phaseθ can be separated into fast-varying term and slow-varying term, and the expression is as follows [17]:(14a)θ=∫0tωsds+φ,(14b)ωs=∂UX/∂X2Hcosθ.By using (13) and (14), system (1) is transformed to two-differential equations:(15a)H˙=−2H2ζ+2ζHsin2θ−2Hsinθ⋅ξ1t+2Hsin2θ⋅ξ2t,(15b)φ˙=−2ζ+2ζHsinθcosθ−12Hcosθ⋅ξ1t+cosθ⋅sinθ⋅ξ2t.In (15a) and (15b), if the damping is small and the excitation is weak enough, the stochastic process H and φ are both the slowly varying process. The stochastic averaging can be applied, and the system energy H can be approximated to a Markovian process [20]. By the time averaging, (15a) can be reduced to the stochastic Itô equation without the slowly varying the variable φ:(16)dH=mHdt+σHdBt,(17a)mH=−2H2ζ+2ζHsin2θ+∫−∞02Hsinθt+τ∂∂H2Hsinθt+cosθ2Ht+τ∂∂θ2HsinθtR1τdτ+∫−∞02Hsin2θt+τ∂∂H2Hsin2θt+cosθsinθt+τ∂∂θ2Hsin2θtR2τdτt,(17b)σ2H=∫−∞+∞2Hsinθt+τ2HsinθtR1τdτ+∫−∞+∞2Hsin2θt+τ2Hsin2θtR2τdτt,where Bt represents the unit wiener process.In (17a) and (17b), t denotes time averaging in one period T, and Riτ,i=1,2 is the self-correlation function of the stationary broadband noise. The relationship of the spectral density Φiω,i=1,2 defined in (5) and the self-correlation Riτ is given as follows:(18)Φiω=1π∫−∞0Riτcosωτdτ.In (17a) and (17b), the drift coefficient mH and the diffusion coefficient σ2H both contain the variable θt. To perform the time averaging, two approximate relations are needed [17]:(19a)θt+τ≈θt+ω¯Hτ,(19b)Xt+τ≈Xt+2Hω¯Hcosθt+ω¯Hτ−cosθt,where ω¯H represents the average frequency and can be obtained by ω¯H=1/T∫0Tωsdt=2π/TH, the quasi-period TH is satisfied as TH=∮dx/2H−2Ux.Substituting (19) into (17), the drift coefficient is given as(20)mH=−2ζω0+2ζ1HX˙2t+πΦ1ω¯+πHΦ20+πHΦ22ω¯=−2ζω0+2ζ1HT∮X˙ⅆX+πΦ1ω¯+πHΦ20+πHΦ22ω¯,and the diffusion coefficient is,(21)σ2H=2πX˙2tΦ1ω¯+πH22Φ20+Φ22ω¯=2πT∮X˙dX⋅Φ1ω¯+πH22Φ20+Φ22ω¯.The governing equation of the stationary PDF with reference to the system energyH is(22)−∂mHpH∂H+12∂2σ2HpH∂H2=0.The exact analytical solution of the stationary PDF can be written as(23)pH=Cexp∫0H2mx/σ2xdxσ2H,where C denotes a constant to satisfy the condition of normalization.From (23), it is possible to write the joint stationary PDF of system displacement x and velocity x˙ as follows:(24)px,x˙=pHTHH=Hx,x˙.Once the joint stationary PDF of system displacement and velocity is obtained, the marginal PDF of system displacement and velocity can be calculated by(25a)px=∫−∞∞px,x˙dx˙,(25b)px˙=∫−∞∞px,x˙dx.The mean square displacement can be represented as(26)EX2=∫−∞∞∫−∞∞x2px,x˙dx˙dx.


## 4. Example and Discussion

One vibration system with impact under additive random noise is given to manifest the efficiency and precision of the proposed stochastic averaging for the inelastic vibro-impact system with one-side wall. The results from the Monte Carlo simulation are seemed as the exact solution, and the analytical solutions from the stochastic averaging are compared with them. The dimensionless parameters of the vibro-impact system are selected as,ω0=2, k2=20, k3=14, x0=0.3, δr=0.1, ζ=0.01, D1=0.2, D2=0, α1=1.5, and α2=2.0. Firstly, the stationary PDF of the system displacement and velocity are plotted in Figure 3. It can be seen that the results from stochastic averaging (solid line) are close to the results from the Monte Carlo simulation (discrete symbol). In other words, the precision of the proposed technique is good. From Figure 3(a), the stationary PDF of the system displacement is not symmetric due to the existence of the right-side wall, while the PDF of the system velocity is still symmetric in Figure 3(b). It is worth pointing out that the stationary PDF of the system displacement exists behind the wall which is quite different from the “classical model”. Then, the influences of the parameters in the broadband noise excitation are discussed. The relation of the mean square displacement EX2 under different parameter D1 is plotted in Figure 4. As the parameter D1 increases, the mean square displacement increases, and the deviation between the analytical results and the Monte Carlo simulation results also increases. It means that the precision is reducing as the parameter D1 increases. Despite all this, the applicable scope of the parameter D1 is still large. The influence of the parameter α, which represents the bandwidth, is discussed in Figure 5. The PDF of the system displacement (α=0.5) is depicted in Figure 6. Figure 5 shows that the larger the α, the higher accurate the results. As α decreases, the results from stochastic averaging will significantly underestimate the system response (it can be verified at α=0.5 in Figure 5). From Figure 6, it can also be seen that the PDF of the system displacement (α=0.5) does not agree with the Monte Carlo simulation results, which is consistent with Figure 5. Thus, in order to apply the proposed technique and retain high accuracy, the parameter D1 of the broadband noise excitation should be small enough and the bandwidth should be large enough. Large D1 and small α will both bring a large deviation. Last, the elastic limitation x0 is discussed in Figure 7. When the elastic limit x0 increases, the mean square displacement also increases and the precision of the results is good. At last, the influences of the additive and multiplicative random noises on the stationary PDF of system (1) are discussed in Figure 8. Figure 8(a) gives the stationary PDF of the displacement, while Figure 8(b) plots the PDF of the velocity of system (1). As shown in Figure 8(a), the multiplicative random noise will enlarge the system displacement response, but the influences are not so significant. In the limitation of the weak excitation intensities, the system response mainly depends on the additive excitation due to the small system velocity. The stationary PDF of the system velocity is given in Figure 8(b). The stationary PDF of the system velocity is symmetric, though the right-side barriers make the stationary PDF of the system displacement nonsymmetric.Figure 3
The marginal probability densities of system responses. (a) The probability density of displacementX; (b) the probability density of velocity X˙.
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(b)Figure 4
The mean square displacementEX2 vs. excitation parameter D (solid line: the analytical results; discrete symbols: Monte Carlo simulation results).Figure 5
The mean square displacementEX2 vs. excitation parameter α (solid line: the analytical results; discrete symbols: Monte Carlo simulation results).Figure 6
The marginal probability densities of system responses (α=0.5) (solid line: the proposed results; discrete symbols: Monte Carlo simulation results).Figure 7
The relations of mean square responsesEX2 to the limit of elastic deformation x0 (solid line: the proposed technique; discrete symbols: Monte Carlo simulation results).Figure 8
The marginal probability densities of system responses (solid lines: the proposed results; discrete symbols: Monte Carlo simulation results). (a) The probability density of displacementX; (b) the probability density of velocity X˙.
(a)
(b)


## 5. Conclusions

In this paper, a stochastic averaging is developed to deal with the inelastic vibro-impact system with one-side wall under the additive and multiplicative broadband noise. The vibro-impact system is firstly transformed into an equivalent nonlinear system without inelastic impact. Secondly, by introducing a transformation, two one-order stochastic differential equations associated with the system energy and residue phase are obtained. Then, the system energy is approximated by a Markov process. The PDF of the system energy can be derived from the FPK equation. The PDF of the system displacement and velocity and the mean square displacement can all be derived from the PDF of the system energy. One example is performed to illustrate the proposed stochastic averaging. The comparison between the results from the proposed technique and the Monte Carlo simulation reveals that the proposed technique is efficient in case of the large bandwidth and small excitation intensity. Especially, the accuracy will dramatically drop for cases of small bandwidth.



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