# Dynamic and Stability Analysis of the Machine Hydrostatic Slide with Magnetorheological Damper

Authors: Zhongkui Zhang; Feng Gao; Yan Li; Han Zhang
Journal: Shock and Vibration
 (2019)

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## Abstract

Tangential dynamic behaviors of the machine hydrostatic slide with a magnetorheological (MR) fluid damper are studied, and the effect of the MR damper to control the vibration of the hydrostatic slide is discussed. The dynamic model of the hydrostatic slide with the MR damper is established, and the tangential vibration equation of linear and nonlinear is derived. The multidimensional incremental harmonic balance method (MIHBM) with discrete Fourier transform (DFT) is derived by which the nonlinear response and stability of the system are studied. The resonance response of the Duffing equation under the combined action of harmonic excitation and constant excitation is obtained. In order to investigate the vibration response of the hydrostatic slide with the MR damper in detail, the bifurcation diagram, phase diagram, and Poincaré map are given. Finally, the dynamic response of the machine hydrostatic slide with the MR damper is discussed, and it is verified that the MR damper can suppress the tangential vibration of the hydrostatic slide effectively and the constant controller can control the chaotic behavior of the system well.


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## Body

## 1. Introduction

A hydrostatic slide is widely used in precision machines because of the characteristic of low motion error and high stiffness [1, 2]. The normal vibration characteristics of the joint surface of the hydrostatic slide have been extensively studied by scholars at home and abroad [3–5]. The dynamic characteristics of the machine table is changed instantaneously and abruptly during the start-stop, acceleration to constant speed, and constant speed to deceleration because of the small friction coefficient and tangential low damping of the hydrostatic slide [6–8]. In addition, the system is forced to vibrate under the cutting force, ground vibration, residual vibration, and oil film nonlinear fluid fluctuation force which lead to poor stability of the hydrostatic slide in the tangential direction and will affect the positioning accuracy and work effectiveness. However, there are few research reports on the tangential vibration of the hydrostatic slide.An MR damper and electrorheological (ER) damper are new intelligent vibration suppressors, which have widely been used in industrial and architectural fields in recent years. The magnitude of the force produced by the MR damper is mainly determined by the performance of internal MR fluid. The magnetorheological fluids are intelligent fluids which are very sensitive to magnetic field. When magnetic field is applied to magnetorheological fluids, the viscosity of magnetorheological fluids increases significantly and the MR fluid becomes viscoelastic solid [9]. The output damping force of the MR damper is regulated by input current. In a certain range, the greater the current, the stronger the electromagnetic field and the greater the output damping force. The working principle of the ER damper is similar to that of the MR damper. Kishore et al. [10] developed an electro-magnetorheological damper that can be used to control the process dynamics of the cutting tool and reduced cutting force by 12.9%. Tarng et al. [11] used a piezoelectric inertia actuator which is mounted on the cutting tool to suppress chatter, and it was shown that chatter can be effectively suppressed. Fei et al. [12] experimentally verified the problem of the motion damper to suppress chatter in milling flexible parts. Hosseinabadi and Altintas [13] used active damping to effectively suppress the vibration of the machine tool and increase the bandwidth of the drive. Mori et al. [14, 15] proposed a passive viscoelastic damping system which can be applied to various machine tools to reduce the residual vibration. Aoyama and Inasaki [16] developed an electrorheological fluid damper which improved the stability of machine tool table systems with linear motion rolling element bearings. Chen [17] studied the dynamic characteristics of the linear feed system with the MR damper by virtual prototyping technology and proved the system has shorter positioning time, more stable speed characteristics, and better acceleration and deceleration characteristics. Bayat and Pakar [18–20] studied the nonlinear vibration response of beams using the Hamiltonian approach and variational approach, and it was demonstrated that two methods were very accurate and effective and therefore may find wide applicability in engineering and other sciences. In recent years, artificial neural networks were also developed for parameter identification and vibration prediction of nonlinear models [21, 22]. Most of the above studies improved the machining accuracy by suppressing tool chatter using dampers. However, vibration of the hydrostatic slide is also one of the important factors affecting the machining accuracy, and the influence of vibration of hydrostatic guide on processing accuracy has not been studied in the above literature.In this paper, we studied the dynamic behaviors and stability of the machine hydrostatic slide with the magnetorheological damper. In the next section, the mathematical model of the machine hydrostatic slide with the magnetorheological damper was described and the tangential single-freedom dynamic model equations are derived. In Section3, the MIHBM with DFT was derived and its accuracy is verified. Section 4 analyzed the linear and nonlinear dynamic responses and stability of the machine hydrostatic slide with the magnetorheological damper. Suppression of the vibration of the hydrostatic slide by using the magnetorheological damper is studied in detail in Section 5. Finally, some conclusions are drawn briefly in Section 6.


## 2. Dynamic Model of Hydrostatic Slide with MR Damper

In this section, the dynamic theoretical models of the MR damper and hydrostatic slide system were established, and the single-degree-of-freedom vibration equation of the hydrostatic slide with the MR damper is derived.


### 2.1. Design and Characterization of MR Damper

Figure1 shows the structure of the MR damper, which is used to control the vibration of hydrostatic slide. According to the Bingham model of the MR damper [23, 24], the damping force FMR is equivalent to the parallel connection of viscous damping force and Coulomb damping force, i.e.,(1)FMR=cev+FI,where ce is the damping coefficient of MR fluid, FI is the Coulomb damping force, and v is the relative velocity, which can be deduced by the structure of the magnetorheological damper:(2)ce=12ηAp2lπDDh3,F¯I=3lApDhτsgnv,where l is the length of the piston, D is the inner diameter of the cylinder, Ap is the effective area of the piston, Dh is the diameter of the piston damping hole, τ is the yield shear stress, η is the viscosity coefficient, and sgn () is the symbolic function.Figure 1
Structure of the MR damper.According to Ampere’s loop law and magnetorheological fluid properties, the relationship between yield stress and currentI is that(3)τ=1.43×10−6μIN2h2+4.743×10−3μIN2h−1,where N is the number of coil, I is the input current, and u is the magnetic permeability of the magnetorheological fluid. The substitution of equations (2) and (3) into equation (1) yields(4)FMR=12ηAp2lπDDh3v+3lApDh1.43×10−6μIN2h2+4.743×10−3μIN2h−1sgnv.The hysteresis three-dimensional model of the MR damper is obtained by parameter identification as shown in Figure2.Figure 2
Hysteresis model of the MR damper.Damping force increases with the increase of input current, but it does not increase linearly. When the current reaches 1.5 A, the output force tends to be saturated, which reflects the characteristics of nonlinear saturation of the MR damper. When the speed changes in different directions, the trajectory of the damping force is different, which reflects the hysteresis characteristics of the magnetorheological damper.


### 2.2. Dynamic Model of Hydrostatic Slide with MR Damper

The structure of the hydrostatic slide with the magnetorheological damper is shown in Figure3. The cylinder of the MR damper is fixed. The damper rod is connected with the slide and generates damping force to control vibration of the worktable. The slide moves in the X direction, and the kinematics equation is as follows:(5)mx¨+Fc+Fs+Ff+FL+FMR=Fd,where Fc is the structural damping force, Fs is the elastic force, Ff is the friction, FL is the load, and FMR is the driving force.Figure 3
Structure of the hydrostatic slide system.The structural damping force is affected by the material and internal resistance of the structure, i.e.,(6)Fc=c1x˙,where c1 is the structural damping coefficient. Assume that the friction is linear, i.e.,(7)Fc=c2x˙,where c2 is the linear friction coefficient.In order to analyze the tangential dynamic characteristics of the hydrostatic slide with the magnetorheological damper, the system was simplified to the single-degree-of-freedom mechanical system model as shown in Figure4. Let Fd−FL=F¯sinω¯t, and the vibration equation of the system is expressed as(8)mx¨+cx˙+Fsx=F¯sinω¯t−F¯I,where c=c1+c2+ce.Figure 4
Single-degree-of-freedom mechanical system.


## 2.1. Design and Characterization of MR Damper

Figure1 shows the structure of the MR damper, which is used to control the vibration of hydrostatic slide. According to the Bingham model of the MR damper [23, 24], the damping force FMR is equivalent to the parallel connection of viscous damping force and Coulomb damping force, i.e.,(1)FMR=cev+FI,where ce is the damping coefficient of MR fluid, FI is the Coulomb damping force, and v is the relative velocity, which can be deduced by the structure of the magnetorheological damper:(2)ce=12ηAp2lπDDh3,F¯I=3lApDhτsgnv,where l is the length of the piston, D is the inner diameter of the cylinder, Ap is the effective area of the piston, Dh is the diameter of the piston damping hole, τ is the yield shear stress, η is the viscosity coefficient, and sgn () is the symbolic function.Figure 1
Structure of the MR damper.According to Ampere’s loop law and magnetorheological fluid properties, the relationship between yield stress and currentI is that(3)τ=1.43×10−6μIN2h2+4.743×10−3μIN2h−1,where N is the number of coil, I is the input current, and u is the magnetic permeability of the magnetorheological fluid. The substitution of equations (2) and (3) into equation (1) yields(4)FMR=12ηAp2lπDDh3v+3lApDh1.43×10−6μIN2h2+4.743×10−3μIN2h−1sgnv.The hysteresis three-dimensional model of the MR damper is obtained by parameter identification as shown in Figure2.Figure 2
Hysteresis model of the MR damper.Damping force increases with the increase of input current, but it does not increase linearly. When the current reaches 1.5 A, the output force tends to be saturated, which reflects the characteristics of nonlinear saturation of the MR damper. When the speed changes in different directions, the trajectory of the damping force is different, which reflects the hysteresis characteristics of the magnetorheological damper.


## 2.2. Dynamic Model of Hydrostatic Slide with MR Damper

The structure of the hydrostatic slide with the magnetorheological damper is shown in Figure3. The cylinder of the MR damper is fixed. The damper rod is connected with the slide and generates damping force to control vibration of the worktable. The slide moves in the X direction, and the kinematics equation is as follows:(5)mx¨+Fc+Fs+Ff+FL+FMR=Fd,where Fc is the structural damping force, Fs is the elastic force, Ff is the friction, FL is the load, and FMR is the driving force.Figure 3
Structure of the hydrostatic slide system.The structural damping force is affected by the material and internal resistance of the structure, i.e.,(6)Fc=c1x˙,where c1 is the structural damping coefficient. Assume that the friction is linear, i.e.,(7)Fc=c2x˙,where c2 is the linear friction coefficient.In order to analyze the tangential dynamic characteristics of the hydrostatic slide with the magnetorheological damper, the system was simplified to the single-degree-of-freedom mechanical system model as shown in Figure4. Let Fd−FL=F¯sinω¯t, and the vibration equation of the system is expressed as(8)mx¨+cx˙+Fsx=F¯sinω¯t−F¯I,where c=c1+c2+ce.Figure 4
Single-degree-of-freedom mechanical system.


## 3. MIHBM with DFT Method and Feasibility Analysis

### 3.1. MIHBM with DFT

Wang et al. [25, 26] verified that the dynamic behaviors of the NC table have chaotic characteristics. So the system of the hydrostatic slide with the magnetorheological damper exhibits nonlinear properties. The elastic force [27]: Fs=k1x+k3x3. Let τ=ω¯t,ωn=k/m,ω=ω¯/ωn,ζ=c/2mωn,F=F¯/k1,andFI=F¯I/k1, dimensionless processing, the equation is transformed to(9)ω2x″+2ζωx′+x+εx3=Fsinτ−FI,where ε is the small parameter, ε=k3/k1,0<ε≤1.MIHBM with DFT [28–32] is a new algorithm which replaces the Galerkin procedure [33, 34] with DFT. Equation (9) is a typical Duffing equation under the combined action of harmonic excitation and constant excitation. And the steady-state response of equation (9) is solved by the MIHBM with the DFT algorithm, which improves the computational efficiency.Supposex0, ω0 is the solution of the vibration equation, and increments are introduced as follows:(10)x=x0+Δx,ω=ω0+Δω.By introducing equation (10) into equation (9) and omitting higher-order small quantities, the incremental equations with Δx and Δω are obtained:(11)ω02Δx″+2ζω0Δx′+1+3εx02Δx=Fsinτ+FI−ω02x0″+2ζω0x0′+x0+εx03−2ω0x0″+2ζω0x0′Δω.The excitationPτ, response x0τ, and its increment Δxτ can be expressed as the following Fourier series:(12a)Pτ=Fsinτ−FI=p02+∑n=1Npancosnτ+pbnsinnτ=CSP,(12b)x0τ=a02+∑n=1Nancosnτ+bnsinnτ=CSA0,(12c)Δxτ=Δa2+∑n=1NΔancosnτ+Δbnsinnτ=CSΔA0,where CS=1cosτsinτ⋯cosNτsinNτ, P=FI0F⋯00T, A0=a0/2a1b1⋯aNbN, ΔA=Δa0/2Δa1Δb1⋯ΔaNΔbNT, and N is the number of harmonics.According to reference [35], CS is discretized to a matrix Γ and introducing the differential operator matrix D as follows:(13)Γ=1cosτ0sinτ0…cosNτ0sinτ01cosτ1sinτ1…cosNτ1sinτ1⋮⋮⋮⋮⋮⋮1cosτi−2sinτi−2…cosNτi−2sinτi−21cosτi−1sinτi−1…cosNτi−1sinτi−1,D=001−1002−20⋱0N−N0,where i is the number of the data points in a period. So the incremental equation (11) can be expressed as(14)ω02ΓD2ΔA+2ζω0ΓDΔA+k1+3k3ΓA02Γ=ΓP−ω02ΓD2A0+2ζω0ΓDA0+k1ΓA0+k3ΓA03−2ω02ΓD2A0+2ζΓDA0Δω,When equation (14) is multiplied by transpose matrix ΓT, the following linearized incremental equation is obtained:(15)KTΔA=RmΔω+R,where KT=ω02ΓTΓD2+2ζω0ΓTΓD+ΓTk1+3k3ΓA02Γ, Rm=−2ω0ΓTΓD2A0+2ζΓTΓDA0, R=ΓTΓP−ω02ΓTΓD2A0+2ζω0ΓTΓDA0+ΓTk1ΓA0+k3ΓA03, and KT, Rm, and R are the Jacobian matrix, unbalance force matrix, and corrective matrix, respectively.Adding the constraint equation by the arc length method [31, 35]:(16)ΔATΔA=Δl2,where Δl is the arc length parameter. Figure 5 shows the flow chart of the algorithm.Figure 5
Algorithm of the MIHBM with DFT.


### 3.2. Feasibility Analysis of MIHBM with DFT

In order to verify the exactness and accuracy of the method proposed in the previous section, Figure6 shows a comparisons between the approximate analytical solutions based on the MIHBM with DFT and the numerical results based on the Runge–Kutta methods. The vibration waveform and phase diagram of primary resonance response with F=0.5, FI=0.006, and ω = 1.0 of the hydrostatic slide with the MR damper are shown in Figure 6. NMM represents the numerical results based on the Runge–Kutta method. As demonstrated, the differences between both methods are very slight, so the MIHBM with DFT is feasible to analyze the dynamic behaviors of the hydrostatic slide with the MR damper.Figure 6
Vibration waveforms and phase diagrams.
(a)
(b)


## 3.1. MIHBM with DFT

Wang et al. [25, 26] verified that the dynamic behaviors of the NC table have chaotic characteristics. So the system of the hydrostatic slide with the magnetorheological damper exhibits nonlinear properties. The elastic force [27]: Fs=k1x+k3x3. Let τ=ω¯t,ωn=k/m,ω=ω¯/ωn,ζ=c/2mωn,F=F¯/k1,andFI=F¯I/k1, dimensionless processing, the equation is transformed to(9)ω2x″+2ζωx′+x+εx3=Fsinτ−FI,where ε is the small parameter, ε=k3/k1,0<ε≤1.MIHBM with DFT [28–32] is a new algorithm which replaces the Galerkin procedure [33, 34] with DFT. Equation (9) is a typical Duffing equation under the combined action of harmonic excitation and constant excitation. And the steady-state response of equation (9) is solved by the MIHBM with the DFT algorithm, which improves the computational efficiency.Supposex0, ω0 is the solution of the vibration equation, and increments are introduced as follows:(10)x=x0+Δx,ω=ω0+Δω.By introducing equation (10) into equation (9) and omitting higher-order small quantities, the incremental equations with Δx and Δω are obtained:(11)ω02Δx″+2ζω0Δx′+1+3εx02Δx=Fsinτ+FI−ω02x0″+2ζω0x0′+x0+εx03−2ω0x0″+2ζω0x0′Δω.The excitationPτ, response x0τ, and its increment Δxτ can be expressed as the following Fourier series:(12a)Pτ=Fsinτ−FI=p02+∑n=1Npancosnτ+pbnsinnτ=CSP,(12b)x0τ=a02+∑n=1Nancosnτ+bnsinnτ=CSA0,(12c)Δxτ=Δa2+∑n=1NΔancosnτ+Δbnsinnτ=CSΔA0,where CS=1cosτsinτ⋯cosNτsinNτ, P=FI0F⋯00T, A0=a0/2a1b1⋯aNbN, ΔA=Δa0/2Δa1Δb1⋯ΔaNΔbNT, and N is the number of harmonics.According to reference [35], CS is discretized to a matrix Γ and introducing the differential operator matrix D as follows:(13)Γ=1cosτ0sinτ0…cosNτ0sinτ01cosτ1sinτ1…cosNτ1sinτ1⋮⋮⋮⋮⋮⋮1cosτi−2sinτi−2…cosNτi−2sinτi−21cosτi−1sinτi−1…cosNτi−1sinτi−1,D=001−1002−20⋱0N−N0,where i is the number of the data points in a period. So the incremental equation (11) can be expressed as(14)ω02ΓD2ΔA+2ζω0ΓDΔA+k1+3k3ΓA02Γ=ΓP−ω02ΓD2A0+2ζω0ΓDA0+k1ΓA0+k3ΓA03−2ω02ΓD2A0+2ζΓDA0Δω,When equation (14) is multiplied by transpose matrix ΓT, the following linearized incremental equation is obtained:(15)KTΔA=RmΔω+R,where KT=ω02ΓTΓD2+2ζω0ΓTΓD+ΓTk1+3k3ΓA02Γ, Rm=−2ω0ΓTΓD2A0+2ζΓTΓDA0, R=ΓTΓP−ω02ΓTΓD2A0+2ζω0ΓTΓDA0+ΓTk1ΓA0+k3ΓA03, and KT, Rm, and R are the Jacobian matrix, unbalance force matrix, and corrective matrix, respectively.Adding the constraint equation by the arc length method [31, 35]:(16)ΔATΔA=Δl2,where Δl is the arc length parameter. Figure 5 shows the flow chart of the algorithm.Figure 5
Algorithm of the MIHBM with DFT.


## 3.2. Feasibility Analysis of MIHBM with DFT

In order to verify the exactness and accuracy of the method proposed in the previous section, Figure6 shows a comparisons between the approximate analytical solutions based on the MIHBM with DFT and the numerical results based on the Runge–Kutta methods. The vibration waveform and phase diagram of primary resonance response with F=0.5, FI=0.006, and ω = 1.0 of the hydrostatic slide with the MR damper are shown in Figure 6. NMM represents the numerical results based on the Runge–Kutta method. As demonstrated, the differences between both methods are very slight, so the MIHBM with DFT is feasible to analyze the dynamic behaviors of the hydrostatic slide with the MR damper.Figure 6
Vibration waveforms and phase diagrams.
(a)
(b)


## 4. Dynamic and Stability Analysis of Hydrostatic Slide with MR Damper

In this section, the dynamic behaviors and stability of the hydrostatic slide with the MR damper were studied by MIHBM with DFT.


### 4.1. Linear System Analysis

Assume the stiffness of the hydrostatic slide is much greater so that it can be regarded as a constant, namely,Fs=k1x. Thus, the hydrostatic slide can be regarded as a linear system. Let τ=ω¯t,ωn=k/m,ω=ω¯/ωn,ζ=c/2mωn,F=F¯/k1,andFI=F¯I/k1, dimensionless processing, the equation is transformed to(17)ω2x″+2ζωx′+x=Fsinτ−FI.The steady-state response of equation (17) is(18)xτ=F1−ω22+2ζω2sinτ−φ−FI,φ=arctan2ζω1−ω2.Figure7 shows the vibration response.Figure 7
Vibration displacement atFI = 0, 0.4, and 1.2.The larger the Coulomb damping forceFI, that is, the greater the output force of the magnetorheological damper, the better the vibration suppression effect. When FI = 1.2, the hydrostatic slide reaches a stable state quickly. However, it is known that the force produced by magnetorheological damping tends to be saturated when the current increases to a certain value.The force produced by magnetorheological damping has a maximum value and cannot be increased indefinitely. The amplitude-frequency curve linear system shows that the larger the damping coefficient, the smaller the amplitude of the system, as shown in Figure8. Therefore, the viscous damping force of the MR damper can also reduce the vibrate displacement of the hydrostatic slide.Figure 8
Amplitude-frequency curve.


### 4.2. Effect of MR Damper on Amplitude-Frequency

Combined action of harmonic excitation and constant excitation is a prominent feature of vibration equation (9). In this subsection, the effects of constant excitation Coulomb force on the response of system are investigated using the amplitude-frequency curves. Let ξ = 0.015, ε = 1, and F = 0.05, for FI = 0.2, 0.6, 0.9, 1.1, 1.2, 2.0, respectively, and the amplitude-frequency responses of the system are shown in Figure 9.Figure 9
Amplitude-frequency curve. (a)FI = 0.2, (b) FI = 0.6, (c) FI = 0.9, (d) FI = 1.1, (e) FI = 1.2, and (f) FI = 2.0.
(a)
(b)
(c)
(d)
(e)
(f)It can be seen from Figure9(a) that the amplitude-frequency curve of the vibration system shows complete hardening-type nonlinearity when FI=0.2. With the increase in Coulomb damping force FI to 0.9, the amplitude-frequency curve of the vibration system not only shows hardening-type nonlinearity but also exhibits slight softening-type nonlinearity. However, there is no overlap region between hardening-type nonlinearity and softening-type nonlinearity when FI=0.9. The amplitude-frequency curve of the vibration system show hardening-type nonlinearity and softening-type nonlinearity when FI=1.1, and there is overlap region between hardening-type nonlinearity and softening-type nonlinearity. When FI=1.2, the region of the softening-type nonlinearity covers the region of hardening-type nonlinearity completely. The amplitude-frequency curve of the vibration system shows the softening-type nonlinearity and slight hardening-type nonlinearity.With the increase in Coulomb damping forceF to 2.0, hardening-type nonlinearity of the amplitude-frequency curve of the vibration system disappears completely and the vibration system exhibits softening-type nonlinearity.


### 4.3. Dynamic and Stability Analysis of the Hydrostatic Slide with MRD

The Floquet method [36, 37] is an effective method to determine whether the periodic solution of a nonlinear dynamic system is stable. The periodic solution x0 of the equation is obtained by MIHBM with the DFT method, and by substituting it into equation (11), the following equation is obtained:(19)ω2Δx″+2ζω0Δx′+Δx+3εx02Δx=0.Introduce(20)X=Δx,Δx′T,Qτ=01−1+3εx02ω02−2ζω0.Equation (20) can be rewritten as(21)X′=QτX.According to the Hsu method proposed by Friedmann et al. [38], periodic T is divided into qk equal-length subintervals and interval size is(22)Δk=τk−τk−1=Tqk.The state transition matrix of equation (20) is(23)Φ=∏j=qk1expQkΔk=∏j=qk1I+∑i=1qiΔkQkii!,where I is the unit matrix and qi is the number of the polynomial expanded by the exponential matrix.The eigenvaluesλ1 and λ2 of matrix Φ are the Floquet multipliers of the system. If the modulus of all the Floquet multipliers of the system is less than 1, the solutions are stable and unstable otherwise. Moreover, if a real Floquet multiplier passes through the unit circle along the negative real axis and the rest are located in the unit circle, there is period-doubling bifurcation.In this subsection, the steady-state solution of the hydrostatic slide with the magnetorheological damper system is studied by the Floquet method. Figure10(a) shows the amplitude-frequency curve for FI=1.1, and it exhibits hardening-type nonlinearity and softening-type nonlinearity in different regions. And there is overlap region between hardening-type nonlinearity and softening-type nonlinearity. Each frequency value of the overlap region corresponds to five amplitude solutions, three of which are stable and two of which are unstable, and stable and unstable solutions alternate. At the lower excitation frequency (i.e., ω ∈ [0, 1.5269]), the change of the vibration system steady response is very slight until the limit point at “2” (ω = 1.5269). With the decrease in frequency ω, the amplitude A becomes unstable until the second limit point reaches “7” (ω = 1.4803). By further increasing the frequency ω, the amplitude of the stable response increases obviously and the response amplitude reaches the peak value at point “9” (ω = 1.5237). With the decrease in frequency ω, the vibration system becomes unstable until the point at “5” (ω = 0.5052). Then, the vibration system returns to stable response once again.Figure 10
Nonlinear responses of the vibration system: (a) amplitude-frequency curves and (b) bifurcation diagram.
(a)
(b)With the increase of the excitation frequencyω from point “1” to point “4,” the path of the stable response is 1⟶ 2 ⟶ 3 ⟶ 4. There appears the jumping phenomenon of stable solutions, and the point “2” is the jumped point. With the increase of the excitation frequency from point “4” to point “1”, the path of the stable response is 4 ⟶ 5⟶ 6 ⟶7  ⟶ 8 ⟶ 1. There appears the jumping phenomenon of stable solutions, and the points “5” and “7” are jumped points. The stable solution between points “6” and “9” is achieved by the path 4 ⟶5 ⟶ 6 ⟶ 9 ⟶ 10 ⟶ 4, and the jumped points are “5” and “9.”Figure10(b) presents the bifurcation diagram of the hydrostatic slide with the magnetorheological damper system with the parameter ω. At the lower excitation frequency (i.e., ω ∈ [0, 0.06]), the vibration system exhibits chaos motion. In the range of ω ∈ [0.06, 1.87], the vibration system exhibits various types of motion including periodic-n and chaotic motions, as shown in Figures 11 and 12. With the increasing of excitation frequency, the vibration system exhibits chaos motion again in the range of ω ∈ [1.87, 3], as shown in Figure 13.Figure 11
Dynamic response of the system atω = 0.5. (a) Phase diagram. (b) Poincaré map.
(a)
(b)Figure 12
Dynamic response of the system atω = 1.29. (a) Phase diagram. (b) Poincaré map.
(a)
(b)Figure 13
Dynamic response of the system atω = 2.8. (a) Phase diagram. (b) Poincaré map.
(a)
(b)


## 4.1. Linear System Analysis

Assume the stiffness of the hydrostatic slide is much greater so that it can be regarded as a constant, namely,Fs=k1x. Thus, the hydrostatic slide can be regarded as a linear system. Let τ=ω¯t,ωn=k/m,ω=ω¯/ωn,ζ=c/2mωn,F=F¯/k1,andFI=F¯I/k1, dimensionless processing, the equation is transformed to(17)ω2x″+2ζωx′+x=Fsinτ−FI.The steady-state response of equation (17) is(18)xτ=F1−ω22+2ζω2sinτ−φ−FI,φ=arctan2ζω1−ω2.Figure7 shows the vibration response.Figure 7
Vibration displacement atFI = 0, 0.4, and 1.2.The larger the Coulomb damping forceFI, that is, the greater the output force of the magnetorheological damper, the better the vibration suppression effect. When FI = 1.2, the hydrostatic slide reaches a stable state quickly. However, it is known that the force produced by magnetorheological damping tends to be saturated when the current increases to a certain value.The force produced by magnetorheological damping has a maximum value and cannot be increased indefinitely. The amplitude-frequency curve linear system shows that the larger the damping coefficient, the smaller the amplitude of the system, as shown in Figure8. Therefore, the viscous damping force of the MR damper can also reduce the vibrate displacement of the hydrostatic slide.Figure 8
Amplitude-frequency curve.


## 4.2. Effect of MR Damper on Amplitude-Frequency

Combined action of harmonic excitation and constant excitation is a prominent feature of vibration equation (9). In this subsection, the effects of constant excitation Coulomb force on the response of system are investigated using the amplitude-frequency curves. Let ξ = 0.015, ε = 1, and F = 0.05, for FI = 0.2, 0.6, 0.9, 1.1, 1.2, 2.0, respectively, and the amplitude-frequency responses of the system are shown in Figure 9.Figure 9
Amplitude-frequency curve. (a)FI = 0.2, (b) FI = 0.6, (c) FI = 0.9, (d) FI = 1.1, (e) FI = 1.2, and (f) FI = 2.0.
(a)
(b)
(c)
(d)
(e)
(f)It can be seen from Figure9(a) that the amplitude-frequency curve of the vibration system shows complete hardening-type nonlinearity when FI=0.2. With the increase in Coulomb damping force FI to 0.9, the amplitude-frequency curve of the vibration system not only shows hardening-type nonlinearity but also exhibits slight softening-type nonlinearity. However, there is no overlap region between hardening-type nonlinearity and softening-type nonlinearity when FI=0.9. The amplitude-frequency curve of the vibration system show hardening-type nonlinearity and softening-type nonlinearity when FI=1.1, and there is overlap region between hardening-type nonlinearity and softening-type nonlinearity. When FI=1.2, the region of the softening-type nonlinearity covers the region of hardening-type nonlinearity completely. The amplitude-frequency curve of the vibration system shows the softening-type nonlinearity and slight hardening-type nonlinearity.With the increase in Coulomb damping forceF to 2.0, hardening-type nonlinearity of the amplitude-frequency curve of the vibration system disappears completely and the vibration system exhibits softening-type nonlinearity.


## 4.3. Dynamic and Stability Analysis of the Hydrostatic Slide with MRD

The Floquet method [36, 37] is an effective method to determine whether the periodic solution of a nonlinear dynamic system is stable. The periodic solution x0 of the equation is obtained by MIHBM with the DFT method, and by substituting it into equation (11), the following equation is obtained:(19)ω2Δx″+2ζω0Δx′+Δx+3εx02Δx=0.Introduce(20)X=Δx,Δx′T,Qτ=01−1+3εx02ω02−2ζω0.Equation (20) can be rewritten as(21)X′=QτX.According to the Hsu method proposed by Friedmann et al. [38], periodic T is divided into qk equal-length subintervals and interval size is(22)Δk=τk−τk−1=Tqk.The state transition matrix of equation (20) is(23)Φ=∏j=qk1expQkΔk=∏j=qk1I+∑i=1qiΔkQkii!,where I is the unit matrix and qi is the number of the polynomial expanded by the exponential matrix.The eigenvaluesλ1 and λ2 of matrix Φ are the Floquet multipliers of the system. If the modulus of all the Floquet multipliers of the system is less than 1, the solutions are stable and unstable otherwise. Moreover, if a real Floquet multiplier passes through the unit circle along the negative real axis and the rest are located in the unit circle, there is period-doubling bifurcation.In this subsection, the steady-state solution of the hydrostatic slide with the magnetorheological damper system is studied by the Floquet method. Figure10(a) shows the amplitude-frequency curve for FI=1.1, and it exhibits hardening-type nonlinearity and softening-type nonlinearity in different regions. And there is overlap region between hardening-type nonlinearity and softening-type nonlinearity. Each frequency value of the overlap region corresponds to five amplitude solutions, three of which are stable and two of which are unstable, and stable and unstable solutions alternate. At the lower excitation frequency (i.e., ω ∈ [0, 1.5269]), the change of the vibration system steady response is very slight until the limit point at “2” (ω = 1.5269). With the decrease in frequency ω, the amplitude A becomes unstable until the second limit point reaches “7” (ω = 1.4803). By further increasing the frequency ω, the amplitude of the stable response increases obviously and the response amplitude reaches the peak value at point “9” (ω = 1.5237). With the decrease in frequency ω, the vibration system becomes unstable until the point at “5” (ω = 0.5052). Then, the vibration system returns to stable response once again.Figure 10
Nonlinear responses of the vibration system: (a) amplitude-frequency curves and (b) bifurcation diagram.
(a)
(b)With the increase of the excitation frequencyω from point “1” to point “4,” the path of the stable response is 1⟶ 2 ⟶ 3 ⟶ 4. There appears the jumping phenomenon of stable solutions, and the point “2” is the jumped point. With the increase of the excitation frequency from point “4” to point “1”, the path of the stable response is 4 ⟶ 5⟶ 6 ⟶7  ⟶ 8 ⟶ 1. There appears the jumping phenomenon of stable solutions, and the points “5” and “7” are jumped points. The stable solution between points “6” and “9” is achieved by the path 4 ⟶5 ⟶ 6 ⟶ 9 ⟶ 10 ⟶ 4, and the jumped points are “5” and “9.”Figure10(b) presents the bifurcation diagram of the hydrostatic slide with the magnetorheological damper system with the parameter ω. At the lower excitation frequency (i.e., ω ∈ [0, 0.06]), the vibration system exhibits chaos motion. In the range of ω ∈ [0.06, 1.87], the vibration system exhibits various types of motion including periodic-n and chaotic motions, as shown in Figures 11 and 12. With the increasing of excitation frequency, the vibration system exhibits chaos motion again in the range of ω ∈ [1.87, 3], as shown in Figure 13.Figure 11
Dynamic response of the system atω = 0.5. (a) Phase diagram. (b) Poincaré map.
(a)
(b)Figure 12
Dynamic response of the system atω = 1.29. (a) Phase diagram. (b) Poincaré map.
(a)
(b)Figure 13
Dynamic response of the system atω = 2.8. (a) Phase diagram. (b) Poincaré map.
(a)
(b)


## 5. The Effect of MR Damper to Control Vibration of Hydrostatic Slide

The relationship between amplitudeA and constant excitation Coulomb damping force FI is shown in Figure 14. For ω = 0.2, 1, 1.49, 1.6, 2.0, respectively, the A−FI curve is quite different. At lower Coulomb damping force, there appears the jumping phenomenon of stable solutions. With the increase of Coulomb damping force FI, the amplitude decrease monotonously for the curve ω = 0.2, 1, 1.49, respectively. The fact is that the A−FI curve decreases monotonously as long as ω ≤ 1.49. Moreover, the amplitude A decreases rapidly when Coulomb damping force FI increases to a certain value as shown in Table 1. By further increasing Coulomb damping force FI, the amplitude tends to zero for the curve ω = 0.2, 1, 1.49, respectively. However, there appears the jumping phenomenon of stable solutions for the curve of ω = 1.6 and 2. It can be seen from the above analysis that the MR damper can reduce the vibration amplitude of the hydrostatic slide effectively.Figure 14
Amplitude-force curves of the system.Table 1
Comparison of amplitude corresponding toFI = 0.1, 0.2, 0.5, 1, 2, 3.5.
Case
Vibration amplitude of system
F
I = 0.1
F
I = 0.2
F
I = 0.5
F
I = 1
F
I = 2
F
I = 3.5
ω = 0.2
0.051
0.046
0.055
0.019
0.014
0.009
ω = 1
0.370
0.309
0.121
0.037
0.018
0.010
ω = 1.49
1.012
1.002
0.953
0.077
0.023
0.013
ω = 1.6
1.527
1.458
1.445
1.355
0.036
0.016
ω = 2
2.681
2.113
2.049
1.975
1.904
0.017Figure15(a) presents the bifurcation diagram of the hydrostatic slide with the magnetorheological damper system with the parameter FI. The system exhibits chaotic motion after a very short period-doubling bifurcation. With an increase in FI, the vibration system displays period-doubling bifurcation again in the interval [0.76, 1.09]. When FI > 1.47, the system present period-doubling bifurcation stably. Figures 15(b) and 15(c) present the phase diagrams and Poincaré diagrams, respectively, when FI = 1.7. It can be seen that the constant controller can control the chaotic behavior of the system well, which is consistent with that conclusion [39]. So the effect on the magnetorheological damper to control the vibration of the hydrostatic slide is excellent.Figure 15
Dynamic response of the system. (a) Bifurcation diagram. (b) Phase diagram. (c) Poincaré map.
(a)
(b)
(c)


## 6. Conclusions

In this paper, the tangential dynamic characteristics and stability of the machine hydrostatic slide with the MR damper are studied by using a single-freedom dynamic model. The nonlinear responses the vibration equation are calculated using the IHBM with DFT, and the stability of the solutions is checked using the Floquet method. Finally, the effect of the MR damper to control the vibration of the hydrostatic slide is discussed, and it is verified that the constant controller can control the chaotic behavior of the system well.(1)
The MIHBM with DFT is derived which is proven to be feasible to analyze nonlinear dynamic behavior.(2)
For the machine hydrostatic slide with the MR damper, the hardening- and softening-type nonlinearities are observed in amplitude-frequency curves. There is an overlap region between hardening-type nonlinearity and softening-type nonlinearity whenF = 1.1 and each frequency value of the overlap region corresponds to five amplitude solutions. With the increase in frequency, hardening-type nonlinearity disappears completely and the vibration system exhibits softening-type nonlinearity. In addition, the maximum amplitude of the machine hydrostatic slide with the MR damper decreases obviously.(3)
The bifurcation diagram, phase diagram, and Poincaré map for the vibration system under different excitation frequency are illustrated. With the increasing excitation frequency, the vibration system exhibits more complex dynamic behaviors including wide variety of period motion, chaotic behavior motion, and jumping discontinuous phenomena.(4)
The appropriate damping force provided by using the MR damper can reduce the tangential vibration of the hydrostatic slide effectively.(5)
Constant controller can control the chaotic behavior of the Duffing system well.Overall, the researches in this paper can not only understand the dynamic response of the machine hydrostatic slide with the MR damper but also provide some reference to suppress vibration and design and optimize the vibration system.



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