# A Study on Vibration Characteristics and Stability of the Ambulance Nonlinear Damping System

Authors: Meng Yang; Xinxi Xu; Chen Su
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Considering the impact of the nonlinear stiffness, a 2-DOF vibration model with cubic terms was established according to the structural feature and nonlinear behavior. Ignoring the impact of nonlinear terms, the system was linearly analyzed. In the case of primary resonance and 1 : 1 internal resonance, a multiscale method was used to obtain a first-order approximate solution. Taking the parameters of one tracked ambulance for instance, the approximate solution was corroborated and the influence of the parameters on damping effect was investigated. Finally, motion stability of the damping system was analyzed with singularity theory. The theoretical bases for improving efficiency of the damping system were provided.


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## Body

## 1. Introduction

The tracked ambulance can go through a variety of complex terrain and perform first aid on the sick and wounded. To achieve safe transfer and implement first aid on the way, it is often necessary to demand good mobility of tracked ambulance and meet the special needs of the sick and wounded. For the tracked ambulance that is refitted by crawler chassis, the installation of the vehicle damping system becomes the main way to improve the ride comfort of the sick and wounded.The tracked ambulance damping system is comprised of the carriage, the stretcher base, the chassis, and the nonlinear shock absorber. Therefore, it can be easily converted into a multidegree-of-freedom nonlinear vibration system. The use of the nonlinear vibration system presents numerous advantages, such as better performance in inhibiting broadband vibration, especially low-frequency vibration. However, complex mechanical properties usually exist in a nonlinear vibration system such as chaos and bifurcation, which makes it difficult to be analytically calculated and analyzed; therefore approximate analytical algorithm are widely used. Lee and Perkins [1] investigatesuspended, elastic cables driven by planar excitation with nearly commensurable natural frequencies in a 2 : 1 ratio. The first-order analysis shows that there are saturation and jump phenomena, and the cubic nonlinearity disrupts saturation. Li et al. [2] apply multiple scales method and Runge-Kutta to study the nonlinear vibration characteristics of the axial movement, multilayered cylindrical shells made from composites. The results show some nonlinear properties of the system, such as the phenomenon of internal resonance, and point out that the excitation amplitude, damping, and velocity can affect the response amplitude, range of internal resonance, and soft feature of the system. Xin et al. [3] use the average method to analyze piecewise nonlinear characteristics of the viscoelastic shocker absorber and the relationship between amplitude-frequency characteristics and system parameters. Xinye et al. [4] use the average method to study the possibility of delay feedback control over the gyroscope system under force. Inoue and Ishida [5] investigate vibration phenomena of the one-degree-of-freedom magnetically levitated system considering the effect of the nonlinearity of the electromagnet, using a shooting method. Moon and Kang [6] put forward an analytical method for evaluation of the steady-state periodic behavior of a nonlinear system. The method is based on the substructure synthesis formulation and a multiple-scale procedure.In this paper, the differential equations of the 2-DOF tracked ambulance nonlinear damping system, including the cubic nonlinear spring, were presented. In the case of primary resonance and 1 : 1 internal resonance, multiple-scale method was used to obtain a first-order approximate solution of the differential equations. Taking the parameters of the tracked ambulance for instance, the accuracy of the approximate solution was established compared with numerical results. The impact of the parameters on damping effect and motion stability was also investigated. Furthermore, the theoretical bases for improving efficiency of the damping system were put forward.


## 2. Damping System Physical Model

The tracked ambulance damping system is shown in Figure1. The damping system is mainly constituted of rubber damping shock absorber and zero stiffness damper. The linear model is used to describe the stiffness and damping of the rubber damping shock absorber. For zero stiffness damper, the damping is described by a linear model, and stiffness is described by the positive and negative stiffness parallel model [7], shown in Figure 2.Figure 1
Ambulance damping system:(1) carriage, (2) stretcher base, (3) zero stiffness damper, (4) rubber damping shock absorber, and (5) coach chassis.Figure 2
Positive and negative stiffness parallel model.The stiffness, the original length, and the initial deformation of horizontal spring, in Figure1, are defined as k, L, and λ. k0 stands for the stiffness of the vertical spring. The vertical elastic restoring force of the model can be expressed in the form
(1)f(x)=k0x-k[x-L-λ1-(x/L)2xL].By using the Taylor series, a second-order expansion is sought in the form(2)11-(x/L)2=1+12(xL)2+1324(xL)4+⋯.Substituting the first two terms into (1) results in
(3)f(x)=(k0-kλL)x+L-λL3k2x3⋯.Therefore, the restoring force of zero stiffness damper can be expressed in the form(4)f(z)=Ksx+βKsx3⋯,
where Ks=(k0-(kλ/L)), βKs=((L-λ)/L3)(k/2), and β is a small parameter.According to the occupant of the vehicle ride (lying) comfort evaluation standards, occupant comfort is mainly affected by the vertical vibration acceleration. Neglecting the other two directions of vibration, a 2-DOF model of the tracked ambulance damping system is shown in Figure3, including the following:M1: the quality of stretcher and decubital body;M2: the quality of carriage;Ks: the stiffness of zero stiffness damper;C1: the damping of zero stiffness damper;K2: the stiffness of rubber damping shock absorber;C2: the damping of rubber damping shock absorber;x1, x2, and x3: stretcher base displacement, carriage displacement, chassis displacement.Figure 3
The 2-DOF damping system.The differential equations describing the motion of the damping system are(5)M1x¨1+C1(x˙1-x˙2)+Ks(x1-x2)+βKs(x1-x2)3=0,M2x¨2-C1(x˙1-x˙2)-Ks(x1-x2)-βKs(x1-x2)3+C2(x˙2-x˙3)+K2(x2-x3)=0.We rewrite (5) as
(6)x¨1+ω12x1=l1x2-2u1x˙1+2u1x˙2-b1(x1-x2)3,x¨2+ω22x2=fcosΩt+2u2x˙1-2u3x˙2+l2x1+b2(x1-x2)3,
where
(7)ω12=KsM1,l1=KsM1,2u1=C1M1,b1=βKsM1,2u2=C1M2,ω22=K2+KsM2,fcosΩt=K2x3+C2x˙3,2u3=C1+C2M2,l2=KsM2,b2=βKsM2.


## 3. Linear Analysis

Ignoring the impact of nonlinear terms, the nonlinear equations (5) are changed into linear equations as follows:
(8)M1x¨1+C1(x˙1-x˙2)+Ks(x1-x2)=0,M2x¨2-C1(x˙1-x˙2)-Ks(x1-x2)+C2(x˙2-x˙3)+K2(x2-x3)=0.By making the substitution,(9)x3=eiωt,x1(t)=H(ω)x1-x3eiωt,x˙1(t)=iωH(ω)x1-x3eiωt,x¨1(t)=-ω2H(ω)x1-x3eiωt,x˙3=iωeiωt,x2(t)=H(ω)x2-x3eiωt,x˙2(t)=iωH(ω)x2-x3eiωt,x¨2(t)=-ω2H(ω)x2-x3eiωt.Equations (8) become
(10)(-ω2M1+iωC1+Ks)H(ω)x1-x3=(iωC1+Ks)H(ω)x2-x3,(-ω2M2+iωC1+Ks+K2+iωC2)H(ω)x2-x3=(iωC1+Ks)H(ω)x1-x3+K2+iωC2.SettingB1=iωC1+Ks, B2=-ω2M1+iωC1+Ks, and B3=-ω2M2+iωC1+Ks+K2+iC2ω, we obtain
(11)H(ω)x1-x3=B1(K2+iωC2)B3B2-B12,H(ω)x2-x3=B2(K2+iωC2)B3B2-B12.The steady-state amplitude from (11) is
(12)|H(ω)|x1-x3=γ[(1-4λ2ξ1ξ21/γu)2+4λ2(ξ21/γu+λξ1)2Δ]1/2,|H(ω)|x2-x3=γ[{12}((1-λ2-4λ2ξ1ξ21/γu)2+4λ2(-λ2ξ21/γu+ξ21/γu+ξ1)2)×(Δ)-1{12}]1/2,
where
(13)Δ=[(-λ2+1)(1+γ-λ2u)-4λ2ξ1ξ2γu-1]2+4λ2ξ12[γ-(1+1u)λ2+ξ2ξ1γu(1-λ2)]2,ξ1=C12M1Ks,ξ2=C22M2K2,u=M1M2,λ=ωω1,ω1=KsM1,γ=K2Ks,ω2=K2M2.The amplitude-frequency curves are plotted with the parameters of a tracked ambulance. The parameters of the tracked ambulance areM1=180 kg,  M2=2000 kg,  Ks=217582 N/m, C1=4200 N·s/m,  f=1500 N,  β=0.1, K2=2200000 N/m,  and C2=19540 N·s/m. The amplitude-frequency curves are shown in Figure 4. Although the model is 2-DOF, there is only one resonance peak. This is because ω12≈ω22.Amplitude-frequency curves.
(a)
(b)


## 4. Perturbation Analysis

Utilizing multiscale method, the response of damping system is solved. The new independent time scales(14)Tn=εnt,n=0,1,…
are introduced, where ε represents a small positive parameter and Tn, n=0,1,… are “slow” time scales which capture the response due to the nonlinearities, damping, and external excitation. And we note that
(15)ddt=D0+εD1+⋯,d2dt2=D02+2εD0D1+ε2(D12+2D0D2)+⋯,
where Dn=∂/∂Tn, n=0,1,…. We expand the time-dependent variables x1 and x2 in powers of ε as
(16)x1=x11(T0,T1,…)+εx12(T0,T1,…)+⋯,x2=x21(T0,T1,…)+εx22(T0,T1,…)+⋯.Then we substitute (15)-(16) into (6) and obtain the following equations.Order (ε0):
(17)D02x11+ω12x11=0,D02x21+ω22x21=0.Order (ε1):
(18)D02x12+ω12x12=-2D0(D1x11+u1x11-u1x21)+l1x21-b1x113+3b1x112x21-3b1x11x212+b1x213,D02x22+ω22x22=-2D0(D1x21-u2x11+u3x21)+l2x11+b2x113-3b2x112x21+3b2x11x212-b2x213+fcos(ΩT0).The solution of (17) can be expressed as
(19)x11=A1(T1)exp(iω1T0)+cc,x21=A2(T1)exp(iω2T0)+cc.To express 1 : 1 internal resonance and the nearness of the excitation frequency to the first-order natural frequency, we introduce two detuning parametersσ1and  σ2defined by ω2=ω1+εσ1, Ω=ω1+εσ2. Substitution of (19) and ω2=ω1+εσ1, Ω=ω1+εσ2 into (18) leads to secular terms. By eliminating these secular terms, the state equations are obtained as follows:
(20)-2A1′iω1-2u1A1iω1+2u1iω2A2exp(iσT0)+l1A2exp(iσT0)-3b1A12A-1+3b1A22A-2exp(iσT0)+3b1A12A-2exp(-iσT0)+6b1A1A-1A2exp(iσT0)-6b1A1A-2A2-3b1A-1A22exp(2iσT0)=0,-2A2′iω2-2u3A2iω2+2u2A1iω1exp(-iσT0)+l2A1exp(-iσT0)+3b2A12A-1exp(-iσT0)-3b2A22A-2-3b2A12A-2exp(-2iσT0)-6b2A1A-1A2+6b2A1A-2A2exp(-iσT0)+3b2A-1A22exp(iσT0)+12fexp(iσ2T1-iσ1T1)=0,
where An′=D1An, σ=εσ1. Introducing the polar form
(21)An=12anexp(iθn),n=1,2
into (20) and separating the equation into real and imaginary parts result in the following four state equations:
(22)-a1′ω1-u1a1ω1+u1ω2a2cosγ+12l1a2sinγ+38b1a23sinγ+38b1a12a2sinγ-38b1a1a22sin2γ=0,a1θ1′ω1-u1ω2a2sinγ+12l1a2cosγ-38b1a13+38b1a23cosγ+98b1a12a2cosγ-34b1a1a22-38b1a1a22cos2γ=0,-a2′ω2-u3a2ω2+u2a1ω1cosγ-12l2a1sinγ-38b2a13sinγ+38b2a12a2sin2γ-38b2a1a22sinγ+12fsinφ=0,a2ω2θ2′+u2a1ω1sinγ+12l2a1cosγ+38b2a13cosγ-34b2a12a2-38b2a12a2cos2γ+98b2a1a22cosγ-38b2a23+12fcosφ=0,
where γ=σT0+θ2-θ1, φ=σ2T1-σ1T1-θ2. At steady state, an′=θn′=0 and the average equations (22) become
(23)-u1a1ω1+u1ω2a2cosγ-38b1a1a22sin2γ+38b1a12a2sinγ+38b1a23sinγ+12l1a2sinγ=0,(24)a1σ2ω1-u1ω2a2sinγ+12l1a2cosγ-38b1a13+38b1a23cosγ+98b1a12a2cosγ-34b1a1a22-38b1a1a22cos2γ=0,(25)-u3a2ω2+u2a1ω1cosγ-38b2a13sinγ+38b2a12a2sin2γ-12l2a1sinγ-38b2a1a22sinγ+12fsinφ=0,(26)a2ω2(σ2-σ1)+u2a1ω1sinγ+12l2a1cosγ+38b2a13cosγ-38b2a23-38b2a12a2cos2γ-34b2a12a2+98b2a1a22cosγ+12fcosφ=0
which provide the steady-state amplitudes and phases.


## 5. Simulation Analysis

The numerical simulation is to analyze the relationship between the vibration amplitudes of the stretcher base and carriage and the parameters of tracked ambulance. To establish the accuracy of the average equations, simulation analysis is performed for the parameters shown previously [8]. For ω1=ω2=34.8 rad/s, the 1 : 1 resonance may occur. In Figure 5, we show the comparison between numerical solutions, obtained by the Runge-Kutta method, and perturbation solutions.Amplitude-frequency curve.
(a)
(b)In Figure5, the trend and resonance position of perturbation solutions and numerical solutions are the same, but the amplitudes are different. This is because we only use the first-order approximation, and it does not affect our qualitative analysis of the dynamic behavior of the system. The damping system amplitude-frequency curve is similar to the linear results, where jump phenomenon does not occur. Figures 6–10 show the impact of the tracked ambulance’s parameters on the damping effect, where a1 and a2 represent the vibrating amplitudes of the carriage and the stretcher base.The influence of the exciting force.
(a)
(b)The influence of zero stiffness damper’s damping.
(a)
(b)The influence of rubber damping shock absorber’s damping.
(a)
(b)The influence of zero stiffness damper’s stiffness.
(a)
(b)The influence of rubber damping shock absorber’s stiffness.
(a)
(b)As can be seen from Figure6, when the amplitude of the exciting force becomes bigger, the vibration amplitudes of stretcher base and carriage all become bigger.Figures7 and 8 clearly show that the damping of zero stiffness damper has a great impact on the vibration amplitude of stretcher base, but little on the vibration amplitude of carriage and the damping of rubber damping shock absorber has a great impact on the vibration amplitudes of stretcher base and carriage. The damping is greater and the amplitude is smaller. Hence, increasing the damping of rubber damping shock absorber, to some degree, is more effective reducing vibration.Figures9 and 10 show that the stiffness of zero stiffness damper only has a major impact on the vibration amplitude of stretcher base, and the stiffness of rubber damping shock absorber only has a major impact on the vibration amplitude of carriage on the premise of meeting 1 : 1 internal resonance roughly. However, both stiffness of zero stiffness damper and rubber damping shock absorber affect the resonance frequency. Increasing the stiffness of zero stiffness damper or decreasing the stiffness of rubber damping shock absorber can increase the resonance frequency, which is helpful in decreasing vehicle vibration [9]. With comprehensive comparison of Figures 7–10, damping has a great influence on the amplitude of vibration, and stiffness has a great impact on the resonance frequency.


## 6. Stability Analysis

To analyze the stability of the system in the primary resonance, we need to convert the average equations in polar form into a rectangular form by introducing [10–14] p1=a1cos(γ+φ), q1=a1sin(γ+φ), p2=a2cosφ,   and  q2=a2sinφ resulting in
(27)p˙1=-u1p1+u1p2-l12ω1q2-38ω1b1(p22+q22)q2-σ2q1-38ω1b1(p12+q12)q2-38ω1b1(q1p22-q22q1-2p1p2q2)-34ω1b1(p1p2+q1q2)q1+34ω1b1(p22+q22)q1+38ω1b1(p12+q12)q1,p˙2=-u3p2+u2p1-l22ω2q1-38ω2b2(p12+q12)q1-(σ2-σ1)q2+38ω2b2(2p1q1p2-p12q2+q12q2)-38ω2b2(p22+q22)q1+38ω2b2(p22+q22)q2+34ω2b2(p12+q12)q2-34ω2b2(p1p2+q1q2)q2,q˙1=-u1q1+u1q2+l12ω1p2+38ω1b1(p22+q22)p2+σ2p1+38ω1b1(p12+q12)p2-38ω1b1(p1p22-q22p1+2q1p2q2)+34ω1b1(p1p2+q1q2)p1-34ω1b1(p22+q22)p1-38ω1b1(p12+q12)p1,q˙2=-u3q2+u2q1+l22ω2p1+38ω2b2(p12+q12)p1+f2ω2-38ω2b2(p12q1-q13+2p1q1q2)+38ω2b2(p22+q22)p1+(σ2-σ1)p2-38ω2b2(p22+q22)p2-34ω2b2(p12+q12)p2+34ω2b2(p1p2+q1q2)p2,
where the average equations become more complex and the exact analytical solution cannot be obtained. At steady state, p˙1=0, p˙2=0, q˙1=0,  and q˙2=0, and we use the Newton Method to calculate the value of the equilibrium point of the average equations (22) by repeatedly changing the initial value of the equilibrium point. There are three sets of equilibrium points(28)ϕ1={-3.9041,-1.2622,2.0696,1.8797},ϕ2={12.1124,7.7186,-13.7048,-10.4106},ϕ3={-10.9811,-6.7073,11.9741,8.9831}.The stability of the system at the equilibrium point is governed by the eigenvalue of the Jacobian matrix of (27) based on the singularity theory. The eigenvalues are obtained as follows:
(29)λ1={-13.5988+22.7426i,-13.5988-22.7426i,-4.0362+5.5237i,-4.0362-5.5237i},λ2={-31.75+120.17i,30.18,-31.75-120.17i,-1.95},λ3={-28.95+103.88i,-1.70,-28.95-103.88i,24.33}.Equation (23) is the Jacobian matrix of (22) at equilibrium point, where the expressions of  nij(i=1,…,4,j=1,…,4) are given in the appendix:
(30)A=[n11n12n13n14n21n22n23n24n31n32n33n34n41n42n43n44].After singularity analysis, the system is only stable in the first equilibrium point. Since there is only one stable equilibrium point, the jump phenomenon does not occur. The Runge-Kutta method is used to validate the singularity analysis. Figure11 presents the final stable position of (22) at three equilibrium points.System stability location.
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(c)Figure11 clearly illustrates that the system is only stable in the first equilibrium point and diverges to infinity (Figure 11(b)) or converges to the stable equilibrium point (Figure 11(c)) at an unstable equilibrium point. Therefore, the system is impossible to get a stable state in the unstable equilibrium point.


## 7. Conclusion

This paper established the dynamic model of a tracked ambulance damping system containing cubic nonlinear terms. We utilized a multiscale method to investigate the dynamic model and obtained the average equations. Average equations were corroborated with the actual parameters. The influence of damping system parameters on the damping effect as well as the stability of the damping system was analyzed. The result explained the reasons why there is no jump phenomenon. This analysis method is suitable for multidegree-of-freedom bearing motion system, particularly suitable for vehicles. The research results are valuable for the vehicle damping system design as well as forecast the damping system dynamic behavior.



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