Experimental Modelling and Amplitude-Frequency Response Analysis of a Piecewise Linear Vibration System

The amplitude-frequency response of a nonlinear vibration system with the coexistence of stiffness and viscous damping piecewise linearities are analysed by means of analytical, numerical and experimental investigations. First, a mechanical model of the piecewise linear system under simple harmonic base excitation is established, and the amplitude-frequency response equation is obtained by the averaging method. Second, an experimental device is built to realize the piecewise linear system. The stiffness and damping coefficients are identified by the least square method. Third, case studies are conducted to illustrate the effect of the clearance and base excitation amplitude on the amplitude-frequency response. The experimental results show that the introduction of the piecewise linear stiffness and damping significantly decreases the response amplitude at the primary resonance. The piecewise linear stiffness, damping coefficients, primary resonance frequency and frequency range of the bi-stable state depend on the clearance and excitation amplitude. The experimental results are consistent with the theoretical predictions and numerical simulation results of the method of backward differentiation formulas. This research provides instructive ideas to the design of the nonlinear isolator in practical engineering.

INDEX TERMS Piecewise linearity, the averaging method, stability, amplitude-frequency response, parameter identification. frequency of the base excitation (Hz) τ dimensionless time x dimensionless relative displacement of the proof mass l dimensionless clearance ω 1 primary frequency characteristic εg dimensionless base excitation characteristic εξ 1 , εξ 2 dimensionless damping characteristic εη dimensionless stiffness characteristic εf 1 dimensionless piecewise restoring force εf 2 dimensionless piecewise damping force ε small dimensionless parameter a dimensionless relative displacement amplitude β phase of the dimensionless relative displacement ϕ angle of the dimensionless relative displacement ( ) d( )/dτ

I. INTRODUCTION
Piecewise linear systems are systems where the stiffness or damping coefficients remain constant over a range of amplitude and dramatically change to another set of constant values once a threshold is reached [1]. In general, piecewise linear systems can be classified into two categories. The vector field of the first category is discontinuous due to the rigid constraint or dry friction. The vector field of the second category is nonsmooth but continuous, and the nonsmoothness may be caused by a clearance or elastic constraint. Piecewise linear systems have been used to represent switching circuit and resistors [2], [3], mechanical system with Coulomb friction [4], [5], gene regulatory networks [6], [7], and so on. Owing to the practical significance and wide application, a great deal of effort has been devoted to the study of piecewise linear systems over the years.
The study on piecewise linear systems dates back to the early 1930s. Den Hartog and Mikina [8] were the first scholars to find an approximate solution for periodic motion of a system with bilinear stiffness. Shaw and Holmes [9] analysed the harmonic, subharmonic and chaotic motions of a periodically forced SDOF system with a piecewise linear restoring force. Based on Shaw-Pierre nonlinear modes and Rauscher's approach, Uspensky et al. [10] presented a method to calculate the forced vibration of the piecewise linear systems near superharmonic resonances. Yu [11] and Ma et al. [12] presented Bozzak-Newmark and Bozzak-Newmark-LCP numerical schemes, respectively, to determine the responses of MDOF piecewise linear systems. Xu et al. [13] computed the periodic solution of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by the incremental harmonic balance (HIB) method. Zou and Nagarajaiah [14] were the first to study a piecewise linear system with negative and positive stiffness by a modified Lindstedt-Poincaré method. For the forced vibration of an oscillator with piecewise linear asymmetrical damping, Silveira et al. [15] obtained the exact analytical solutions by joining the solutions for the compression and expansion phases and the approximate solutions by the HIB method. Wang et al. [16] studied the effect of the system parameters on the dynamical behaviours of a piecewise linear SDOF oscillator with fractional-order derivative by the averaging method.
The application of the piecewise linearity in vibration control is well developed. Deshpande et al. [1] proposed the optimum parameters of the primary suspension and jump-avoidance conditions of the secondary suspension for a piecewise linear vibration isolation system. Zhong and Chen [17] established the relationship between the system parameters and the topological bifurcation solution of a piecewise linear vehicle suspension. Joglekar and Mitra [18] presented piecewise linear SDOF/MDOF oscillators to represent cracked beams and employed a wavelet-based method to analyse the vibration behaviours. Based on Rauscher method and calculations of the autonomous system nonlinear modes, Uspensky and Avramov [19], [20] studied nonlinear modes of the free and forced torsional vibrations for a piecewise linear system. Shui and Wang [21] and Mustaffer et al. [22] presented a dynamic vibration absorber with an adjustable piecewise linear stiffness and experimentally analysed the characteristic and performance of the absorber, respectively. Yao et al. [23] attached a nonlinear energy sink (NES) with piecewise linear stiffness to suppress the vibration of a forced primary vibration system, and the effectiveness of the NES was proven by experiments. In design of energy harvesters, Tien and D'souza [24] proposed a new vibration harvester composed of a piecewise linear oscillator and an adjustable gap, which has an optimal vibration performance over a broad frequency range and the best performance at resonance. Zhang et al. [25] experimentally designed a vibration harvester with piecewise linear stiffness characteristic, which may work in a broadband and low-frequency range. Using the methods of Floquet theory, Filippov method and finite different method, El Aroudi et al. [26] studied the stability and bifurcation behaviour of a piecewise linear springmass-damper system for vibration-based energy harvesting applications. Shi et al. [27] investigated the vibration transmission characteristic of the SDOF oscillator and coupled 2DOF oscillators with bilinear stiffness and bilinear damping by the harmonic balance method and numerical integrations. Dai et al. [28] studied the dynamic behaviour, vibration transmission and power flow of impact oscillators with linear and quasi-zero-stiffness (QZS) nonlinear constraints. Dai et al. [29] further revealed the effects of the design parameters and locations of the nonlinear constrains on the response and vibration transmission of impact oscillators with nonlinear motion constrains created by the diamond-shaped linkage mechanism. Narimani et al. [30] focused on the availability of the averaging method to find the closed-form solution for the frequency response of a piecewise linear isolator with a hard nonlinearity. The analytical result of the averaging method agreed well with both experimental results and numerical simulation.
From the above literature review, many studies have examined piecewise linear systems in terms of theoretical prediction and numerical simulation, including the analysis of stability, bifurcation, chaos motion, and amplitude-frequency characteristic. However, limited work has been done in experimental studies. In this paper, a mechanical structure with stiffness and viscous damping piecewise linearities is designed and assembled, which has the advantages of simple structure, easy implementation and low cost. Different from the research focus of Ref. [30], the objective of this paper is twofold. The first objective is to identify the piecewise linear coefficients of the built experiment model and verify the correctness of the identification results. The other objective is to discuss the effect of the clearance and base excitation amplitude on the amplitude-frequency response of the piecewise linear system by experimental verification combined with theoretical analysis and numerical calculation.
The paper is organized as follows. In Section II, the mechanical model of a SDOF piecewise linear system is presented. The amplitude-frequency response of the system is obtained by the averaging method. In Section III, the stability analysis of the steady-state response is completed through the eigenvalue analysis method. Section IV is devoted to explore the amplitude-frequency characteristic experimentally. Conclusions are given in Section V.

II. MECHANICAL MODEL AND PERTURBATION ANALYSIS
The mechanical model of the piecewise linear system under investigation is presented in Fig. 1(a). x 1 and u are the absolute displacements of proof mass m 1 and the base, respectively. x 2 = x 1 -u is the relative displacement of proof mass m 1 with respect to the base. F 1 (x 2 ) and F 2 (dx 2 /dt) are the piecewise linear restoring force and damping force, respectively. The stiffness and damping coefficients are illustrated in Fig. 1(b) and (c). is the clearance. The stiffness and damping coefficients are k 1 and c 1 for x 2 ≤ . The stiffness and damping coefficients increase to k 2 and c 2 for x 2 > . According to Fig. 1, the governing differential equation of the system is Assuming u = u 0 sin (ωt) and substituting x 1 = x 2 + u into (1), we obtain where To analyse the primary resonance response of the system, small damping, weak nonlinearity, and soft excitation are assumed. Letting x 0 = 0.01 m, the dimensionless transform parameters are where ε is a small dimensionless parameter. Setting x = dx dτ , the dimensionless form of (2) is By the averaging method [30], the approximate solution of (3) can be written as where a andϕare the functions of τ . The velocity is required to have the same form as the case when ε = 0, that is, Equations (6) and (7) imply that a sin ϕ + aβ cos ϕ = 0 Differentiating (7) with respect to τ , we have Substituting for x and x in (3) yields a r cos ϕ − ar(r + β ) sin ϕ + a sin ϕ + εξ 1 ar cos ϕ + εf 1 (a sin ϕ) + εf 2 (ar cos ϕ) = εg sin (rτ ) (10) Solving (8) and (10) fora andβ , we obtain aβ r = −ar 2 + a − εg cos β sin 2 ϕ Equations (11) and (12) indicate that a and β slowly vary with τ for small ε and primary resonance. In other words, a and β hardly change during the period of oscillation 2π, which enables us to average out the variations in ϕ in (11) and (12). Averaging these equations over the period of 2π and considering a and β to be constant while performing the averaging, the following equations describing the slow variations in a and β are obtained: in which ϕ 0 = arc sin l a . For the stationary solutions of (13) and (14), that is, a = β = 0, bysin 2 β + cos 2 β = 1, we have

III. STABILITY ANALYSIS
The perturbation of the stationary solutions is assumed to be a = a 0 + a and β = β 0 + β, where a 0 and β 0 are the stationary solutions of (15) and (16), and a and β are small perturbations. Substituting a = a 0 + a and β = β 0 + β into (13) and (14) and applying Taylor series, the linear equations of a and β can be obtained: Thus, the characteristic equation is where λ is the characteristic root, Based on Routh-Hurwitz criterion, the stationary solutions are stable for P > 0 and Q > 0; otherwise, the solutions are unstable. We emphasize that a saddle-node-type bifurcation occurs when a real eigenvalue of (19) changes sign, which results in the jumping phenomena. When ω 1 = 20π rad/s, εξ 1 = 0.01, εξ 2 = 0.015, εg = 0.01, εη = 0.5, and εδ = 0.1, the frequency-amplitude response of the piecewise linear system compared to a linear response for primary resonance is shown in Fig. 2. From Fig. 2, the primary resonance peak is significantly reduced, and the multi-steady state arises due to the piecewise linear stiffness and damping. Additionally, the theoretical solution is consistent with the numerical solution obtained by the method of backward differentiation formulas (BDF). Thus, the averaging method can handle the presented model.

A. EXPERIMENTAL SETUP
To validate the above analysis, an experimental structure is designed and assembled as shown in Fig. 3. The proof mass 1 is attached with base 2 via five steel sheets 3 and 4. Two limiting stoppers 5 are fixed on a perforated aluminium plate 6, which are symmetrical about the middle steel sheet 4. Acceleration sensors 7 and 8 are used to measure the absolute acceleration signals of the proof mass and base, respectively.      (1) is used to describe the vibration of the experimental structure. Some notes on the physical parameter of (1) are as follows.

The dynamical model in
(a) During the experiments, the base is harmonically excited by a shaker. The amplitude u 1 and frequency ω of the base acceleration can be adjusted by the M + p vibration controller.
(b) The value of is determined by the installation location of the two limiting stoppers.
(c) k 2 -k 1 and c 2 -c 1 are used to denote the deformation and energy loss during the collision, respectively. Taking (15) as the fitting model, the value of k 2 and c 2 can be determined to fit the experimental data by the least square method [31].
(d) As a preliminary, the values of m 1 , k 1 and c 1 are identified in the case of no collision. Appling sine acceleration excitation with an amplitude of 0.294 m/s 2 to the base, the measured frequency-amplitude response of the linear system is recorded as the red dots shown in Fig. 4, where = ω/(2π). By the least square method, the parameters VOLUME 9, 2021   For = 10.25 Hz, the measured steady-state acceleration time histories of the system at different initial conditions are plotted in Fig. 6. Fig. 7 demonstrates the FFT plot of time histories shown in Fig. 6. In Figs. 6-7, the red and green lines correspond to the large and small initial displacement x 1 (0) and velocity dx 1 (0)/dt of the proof mass, respectively. From Fig. 6, with the constant acceleration amplitude of 0.49 m/s 2 of the base, the acceleration amplitude of the proof mass increases from 2.81 m/s 2 to 31.75 m/s 2 when the initial displacement and velocity of the proof mass increase.
By the signal processing, the experimental results of the relative displacement amplitude of the proof mass versus excitation frequency are shown in Fig. 8. Taking (15) as the fitting model to fit the experimental data shown in Fig. 8, the values of k 2 and c 2 are identified as k 2 = 3747.44 N/m and c 2 = 0.560 N · s/m by the least square method. Fig. 9 shows the experimental amplitude-frequency response compared to the theoretical and numerical responses. The green solid and dotted lines represent the stable and unstable steady-state response given by the averaging  method. The mark ''+'' represents the experimental results, and '' '' represents the numerical results obtained by numerically solving (2). The black solid line represents the theoretical result of the corresponding linear system (that is, = ∞). Obviously, the experimental result is consistent with the theoretical and numerical results, which proves the correctness of the identification result. Thus, the averaging method is valid to predict the dynamical behaviour of the experimental model. From Fig. 9, the maximal relative displacement amplitude of the proof mass is reduced by 25.8% when the piecewise linearity is introduced.   of the proof mass versus excitation frequency. In Fig. 13, the maximum relative displacement is 11.48 mm.
Based on the experimental data in Fig. 13 and (15), k 2 and c 2 are calculated as k 2 = 3630.01 N/m and c 2 = 0.766 N · s/m. Fig. 14 displays the comparison of the experimental, theoretical and numerical results of the amplitude-frequency response. These three results are basically consistent. Compared to that of the linear system, the resonance peak of the piecewise linear system is reduced by 41.89%. Fig. 15 shows the experiment-based relative displacement time history of the proof mass for = 10 Hz, which is validated by the numerical simulation shown in Fig. 16.   = 10Hz (red and green lines correspond to large and small initial displacement x 2 (0) and velocity dx 2 (0)/dt of the proof mass, respectively.). Using the experimental data in Fig. 18 and (15), k 2 and c 2 are determined to be k 2 = 3759.20 N/m and c 2 = 0.795 N · s/m. Fig. 19 shows the comparison of the experimental, theoretical and numerical results of the amplitude-frequency   4.637 mm (The marks ''+'' correspond to the large initial displacement x 2 (0) and initial velocity dx 2 (0)/dt of the proof mass; the marks '' ''correspond to the small initial displacement x 2 (0) and initial velocity dx 2 (0)/dt of the proof mass.).  the experiment-based and numerical relative displacement time histories of the proof mass for = 11 Hz, respectively. The experimental result is clearly consistent with the numerical result. Table 1 shows the effect of u 1 on the experimental amplitude-frequency response of the piecewise linear system for = 4.637 mm, where = x 2 max | linear − x 2 max | piecewiselinear x 2 max | linear × 100%. As indicated in Table 1, some characteristics are summarized.
1) The primary resonance peak of the piecewise linear system is lower than that of the corresponding linear system.  = 5 mm (The marks ''+'' correspond to the large initial displacement x 2 (0) and initial velocity dx 2 (0)/dt of the proof mass; the marks '' ''correspond to the small initial displacement x 2 (0) and initial velocity dx 2 (0)/dt of the proof mass.).
In terms of engineering applications, it is beneficial for the vibration isolation.
2) Frequency range of the bi-stable state is widened, and the primary resonance frequency increases when u 1 increases.
3) k 2 remains basically unchanged, and c 2 increases when u 1 increases. Thus, the value of k 2 depends on .  The experimental result of the amplitude-frequency response is plotted in Fig. 24 in comparison with the numerical and theoretical responses. From Fig. 24, the primary resonance peak of the piecewise linear system is 47.33% lower than that of the linear system. The experiment-based relative displacement time history of the proof mass for = 10.5 Hz   =4.32 mm (The marks ''+'' correspond to the large initial displacement x 2 (0) and initial velocity dx 2 (0)/dt of the proof mass; the marks '' ''correspond to the small initial displacement x 2 (0) and initial velocity dx 2 (0)/dt of the proof mass.).
is drawn in Fig. 25, which is constant with the numerical simulation result in Fig. 26.  . Experiment-based time history for = 11.5 Hz (red and green lines correspond to large and small initial displacement x 2 (0) and velocity dx 2 (0)/dt of the proof mass, respectively.).
Case 5 u 1 = 1.47 m/s 2 and = 4.32 mm Based the above four groups of experiments and corresponding identification results, k 2 and c 2 in Case 5 are predicted to be higher than those in Case 4. For u 1 = 1.47 m/s 2 and = 4.32 mm, the measured absolute acceleration and relative displacement response versus excitation frequency are plotted in Figs. 27 and 28, respectively. The observed frequency range of the bi-stable state is [10 Hz,12.125 Hz]. From Fig. 28, the primary resonance peak of the proof mass is 13.05 mm. k 2 and c 2 are identified as k 2 = 4400 N/m and c 2 = 1.356 N · s/m, which are consistent with the prediction. Fig. 29 shows the experimental amplitude-frequency response in comparison with the numerical and theoretical responses. From Figure 29, the maximum amplitudefrequency response in the case of piecewise linearity is reduced by 67.37% compared to that in the case of linearity.
Figs. 30 and 31 present the experiment-based and numerical relative displacement time histories of the proof mass for = 11.5 Hz, respectively. The two results are clearly corresponded.

V. CONCLUSION
In this paper, the amplitude-frequency response of a piecewise linear system subjected to the base harmonic excitation is investigated. Based on the theoretical analysis, numerical calculation and experimental verification, the following conclusions are drawn.
(1) The analytical results given by the averaging method is consistent with the numerical and experimental results. Therefore, the averaging method is applicable to predict the dynamic behaviour of such piecewise linear system.
(2) The primary resonance response peak of the piecewise linear system is obviously less than that of the linear system. In particular, for u 1 = 1.47 m/s 2 , the primary resonance response peak at = 4.32 mm is reduced by 67.37% compared to that at = ∞. From the viewpoint of engineering application, it may provide new ideas for the design of nonlinear isolators.
(3) The piecewise linear stiffness and damping coefficients depend on the installation location of the limiting stoppers and amplitude of the base acceleration excitation. The primary resonance frequency, primary resonance response peak and frequency range of the bi-stable state change accordingly.
(4) With the fixed installation location of the limiting stoppers (that is, = const), k 2 almost remains constant, and c 2 increases with the increase in excitation amplitude. With the fixed excitation amplitude, k 2 and c 2 decrease with the increase in .