Vibration reduction of vehicle-bridge-coupled system using smooth-and-discontinuous absorber

Abstract

Three-degree-of-freedom coupled system of vehicle, beam, and smooth-and-discontinuous absorber is constructed to detect the application of smooth-and-discontinuous absorber in vibration reduction of beam subjected to an infinite series of moving vehicles. Smooth-and-discontinuous absorber can be linear absorber or nonlinear one depending on the value of the smoothness parameter. The moving vehicles are simplified as an infinite series of moving spring masses. The amplitude–frequency response functions of the system are obtained using a series of elliptic integrals of the first and second kind. Furthermore, power flow approach is used to detect the effectiveness of smooth-and-discontinuous absorber with parameter variation. The results show that smooth-and-discontinuous absorber can adapt itself to effectively reduce the vibration energy of beam subjected to an infinite series of moving spring masses, which broaden the application of smooth-and-discontinuous absorber in engineering.

Introduction

It is well known that the vibrations of beam bridge caused by moving loads will decrease the strength and stability of bridge structure. ¹ However, the safety of vehicles will be affected by the excessive vibration of beam bridge. In fact, the dynamics of the vehicle–bridge system become more and more complex with the ever-increasing flow of high-powered heavy vehicles traveling at higher and higher traffic speeds. Therefore, the investigations on vibration and control of beam bridges subjected to moving loads have been of great significance in engineering.^2–11 The effects of interaction between the vehicle and the bridge pavement were investigated in Chatterjee et al. ⁶ An analytical approach was presented to solve the problem of vehicle–bridge dynamic interaction taking into account both flexural and torsional mode shapes in Marchesiello et al. ⁷ The effects of separation between the moving vehicle and bridge were investigated in Cheng and Au. ⁸ The dynamic response and the resonance curve of the bridge were obtained using the mode analyzing method and Runge–Kutta method in Chen et al. ⁹ The vibration of a beam excited by an elastic body with conformal contact was studied in Ouyang and Mottershead. ¹⁰ Using the numerical–analytic approach, the equation of motion of the beam and the moving body were established separately and a reattachment condition of the moving oscillator to the beam after separation was proposed in Stancioiu et al. ¹¹ However, these results were limited in vibration of the beams subjected to a moving load.

In fact, there are very few papers on the vibration of beams subjected to an infinite series of moving loads.^12,13 Traditionally, theoretical investigations of beam vibration due to moving loads were focused on dynamic response of beam subjected to a moving load. But, the vibration of a beam subjected to an infinite series of moving loads is composed of not only linear superposition of dynamic response caused by a moving load but also the nonlinear dynamic characteristics. Hence, bifurcation and chaos should be detected. The problem of repetitive moving loads was classified as a linear beam subjected to an infinite series of moving loads with constant velocity, which repeated at regular time intervals. ¹² Moreover, nonlinear stiffness of dynamic vibration absorber can lead to complex dynamics such as quasi-periodic, chaotic, and sub-harmonic responses. The mid-span deflection of the nonlinear beam under traveling vehicles through the bridge successively had rich dynamic behaviors, which may lead to the potential hazard of the bridge. ¹³ Therefore, how to eliminate bridge vibrations under an infinite series of moving loads becomes a challenging task.

In engineering, dynamic vibration absorbers were applied in order to control vibration. A semi-active vibration absorber with piece-wise linear elastic components was studied in Qian and Hu. ¹⁴ The energy pumping with a strong nonlinear attachment was detected and its advantages were analyzed in comparison with classical tuned mass damper (TMD) in Gourdon et al. ¹⁵ The optimization of parameters for four semi-active DVAs was presented in Shen et al. ¹⁶ In practice, all kinds of moving loads can lead to the different external excitation frequency. Frequency often shifts from one mode to another in the beam subjected to moving loads. Hence, we are encouraged to find a versatile dynamic damper which leads itself to linear dynamic damper or nonlinear one in order to meet vibration control demands. It happened to be coincided with the characteristics of smooth-and-discontinuous (SD) oscillator whose natural frequency can be changed depended on the value of the smoothness parameter.^17–19 Based on SD oscillator, we constructed SD absorber and used it to reduce the vibration of the 2-degree-of-freedom coupled system of vehicle and bridge in Tian et al. ²⁰ In this article, we focus on the 3-degree-of-freedom coupled system in which moving vehicles are simplified as moving spring masses and the effectiveness of SD absorber with parameter variation in this case.

This article is organized as follows. In the next section, the vehicle–bridge–SD absorber coupled system is constructed, in which the axial force and an infinite series of moving loads are considered. The response curves of vehicle–bridge are derived analytically in section “Dynamic analysis of vehicle-bridge system.” SD absorber is proposed for eliminating undesired vibrations of a beam in section “Vehicle–bridge–SD absorber dynamic interactions,” and coupling effects between the vehicle–bridge system and SD absorber are analyzed. Furthermore, the efficiency of vibration reduction is detected by use of the power flow approach in section “Optimization for SD absorber.”

Vehicle–bridge–SD absorber coupled system

In a complicated real system, frequency often shifts from one mode to another, which involves a versatile dynamic absorber which can lead itself to linear dynamic dampers or nonlinear one to meet reduction demands. As is given in Tian et al., ²⁰ SD oscillator is an example of a conservative system whose natural frequency can be changed depending on the parameter $\alpha =l_0/l$ (see Figure 1). In fact, from Figure 1, we know that if the springs of SD absorber are vertical and $\alpha =0$ , SD absorber can be regarded as a linear absorber (TMD). When $\alpha \ne 0$ , it can be a nonlinear one. Moreover, its natural frequency changes relying on the value of $\alpha$ .

Figure 1.

SD absorber.

The vehicles, passing through the beam bridge successively, are simplified as a moving spring mass system with the constant speed v at time $L/v$ intervals. SD absorber is hung on the mid-span of beam, as shown in Figure 2. Supposing that the wheel M keeps in touch with the beam and considering the axial force $S(x,t)=\pm 1/2\mathrm{EA}(\partial u/\partial x{)}^2$   ²¹ acting on the beam, the motion equation of the coupled system can be written as

$$\{\begin{array}{c}\mathrm{EI}\frac{{\partial }^{4}u}{\partial x^4}+\rho A\frac{{\partial }^{2}u}{\partial t^2}+S(x,t)\frac{{\partial }^{2}u}{\partial x^2}+c_1\frac{\partial u}{\partial t}=F_1(t)-F_2(t)\delta (x-\frac{L}{2})\hfill \\ m_2\stackrel{··}{z}+k_2(z-u{|}_{x=\mathrm{vt}})+c_2(\stackrel{·}{z}-\stackrel{·}{u}{|}_{x=\mathrm{vt}})=0\hfill \\ m_3\stackrel{··}{y}-F_2(t)=0\hfill \end{array},t\ge 0$$

(1)

where

$$\begin{array}{c}\multicolumn{1}{c}{F_1(t)=\sum _{i=0}^{\infty }[M(g-\frac{{\partial }^{2}u}{\partial t^2})+m_2(g-\stackrel{··}{z})]}\\ \delta [x-(\mathrm{vt}-\mathrm{iL})]U(i,t)\hfill \end{array}$$

(2)

$$U(i,t)=H\left[(t-\frac{\mathrm{iL}}{v})(\frac{L}{v}-(t-\frac{\mathrm{iL}}{v}))\right]$$

(3)

$$H(t)=\{\begin{array}{c}0,t<0\hfill \\ \multicolumn{1}{c}{1,t\ge 0}\end{array}$$

(4)

$$\begin{array}{l}{F}_2(t)=2k_3(u{|}_{x=\frac{L}{2}}-y)\left(1-\frac{l}{\sqrt{{(u{|}_{x=\frac{L}{2}}-y)}^2+{l_0}^2}}\right)\\ +c_3\left(\frac{\partial u{|}_{x=\frac{L}{2}}}{\partial t}-\stackrel{.}{y}\right)\end{array}$$

(5)

Figure 2.

Vehicle–bridge–SD absorber model.

Moreover, EI, $u(x,t)$ , $\rho A$ , and $c_1$ are the flexural rigidity, the vertical displacement of beam, the mass of unit length, and the damping coefficient of beam, respectively. M, $m_2$ , $k_2$ , and $c_2$ are the mass of the wheel, the mass of the suspension, the spring stiffness, and the damping coefficient of the spring–mass system. $m_3$ , $k_3$ , and $c_3$ are the mass, the spring stiffness, and the damping coefficient of SD absorber. The use of $\delta$ function is to accommodate a point-wise concentrated load.

Our attention is focused on the effectiveness of SD absorber in this article. Therefore, only the first vibration mode of beam is interested. Let

$$\{\begin{array}{c}u=\overline{u}\sin \frac{\pi x}{L}\hfill \\ z=\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\overline{z}\hfill \\ y=\overline{u}-\overline{y}\hfill \end{array}$$

(6)

Furthermore, multiplying the first equation of equation (1) and integrating over $[0,L]$ , we have

$$\{\begin{array}{c}\frac{\mathrm{EI}{\pi }^4}{2L^3}\overline{u}+\frac{\rho \mathrm{AL}}{2}\stackrel{··}{\overline{u}}\pm \frac{\mathrm{EA}{\pi }^4}{16L^3}{\overline{u}}^3+\frac{c_{1}L}{2}\stackrel{·}{\overline{u}}={\overline{F}}_1-{\overline{F}}_2\hfill \\ m_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\stackrel{··}{\overline{z}}+k_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\overline{z}+c_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\stackrel{·}{\overline{z}}\hfill \\ =k_2\overline{u}\sin \frac{\pi \mathrm{vt}}{L}+c_2(\stackrel{·}{\overline{u}}\sin \frac{\pi \mathrm{vt}}{L}+\frac{\pi v}{L}\overline{u}\cos \frac{\pi \mathrm{vt}}{L})\hfill \\ m_3\stackrel{··}{\overline{y}}+{\overline{F}}_2=m_3\stackrel{··}{\overline{u}}\hfill \end{array}$$

(7)

where

$$\begin{array}{c}{\overline{F}}_1=[(M+m_2)g-m_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\stackrel{··}{\overline{z}}]\sum _{i=0}^{\infty }U(i,t)\sin \frac{\pi (\mathrm{vt}-\mathrm{iL})}{L}\hfill \\ -M\stackrel{··}{\overline{u}}\sum _{i=0}^{\infty }U(i,t){\sin }^2\frac{\pi (\mathrm{vt}-\mathrm{iL})}{L}\hfill \end{array}$$

(8)

$${\overline{F}}_2=2k_3\overline{y}(1-\frac{l}{\sqrt{{\overline{y}}^2+l_0^2}})+c_3\stackrel{·}{\overline{y}}$$

(9)

Notice

$$\sum _{i=0}^{\infty }U(i,t)\sin (\frac{\pi \mathrm{vt}}{L}-i\pi )=\left|\sin \frac{\pi \mathrm{vt}}{L}\right|$$

(10)

$$\sum _{i=0}^{\infty }U(i,t){\sin }^2(\frac{\pi \mathrm{vt}}{L}-i\pi )={\sin }^2\frac{\pi \mathrm{vt}}{L}$$

(11)

Therefore, we can obtain

$$\{\begin{array}{c}\frac{\rho \mathrm{AL}}{2}\stackrel{··}{\overline{u}}+\frac{\mathrm{EI}{\pi }^4}{2L^3}\overline{u}\mp \frac{\mathrm{EA}{\pi }^4}{16L^3}{\overline{u}}^3+\frac{c_{1}L}{2}\stackrel{·}{\overline{u}}=(\mathrm{Mg}+m_{2}g-m_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\stackrel{··}{\overline{z}})\left|\sin \frac{\pi \mathrm{vt}}{L}\right|\hfill \\ -M\stackrel{··}{\overline{u}}{\sin }^2\frac{\pi \mathrm{vt}}{L}-[2k_3\overline{y}(1-\frac{l}{\sqrt{{\overline{y}}^2+{l_0}^2}})+c_3\stackrel{·}{\overline{y}}]\hfill \\ m_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\stackrel{··}{\overline{z}}+k_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\overline{z}+c_2\sqrt{\frac{\rho \mathrm{AL}}{2m_2}}\stackrel{·}{\overline{z}}\hfill \\ =k_2\overline{u}\sin \frac{\pi \mathrm{vt}}{L}+c_2\stackrel{·}{\overline{u}}\sin \frac{\pi \mathrm{vt}}{L}+c_2\frac{\pi v}{L}\overline{u}\cos \frac{\pi \mathrm{vt}}{L}\hfill \\ m_3\stackrel{··}{\overline{y}}+2k_3\overline{y}(1-\frac{l}{\sqrt{{\overline{y}}^2+{l_0}^2}})+c_3\stackrel{·}{\overline{y}}=m_3\stackrel{··}{\overline{u}}\hfill \end{array}$$

(12)

Let

$$\begin{array}{c}q=\frac{\overline{u}}{l},p=\frac{\overline{z}}{l},R=\frac{\overline{y}}{l},T\to t,{\omega }_1^2=\frac{\mathrm{EI}{\pi }^4}{\rho A{L}^4},\hfill \\ t=\frac{T}{{\omega }_1},\omega =\frac{\pi v}{L{\omega }_1}\hfill \\ \beta =\pm \frac{E{\pi }^4{l}^2}{8\rho L^4{{\omega }_1}^2},{\gamma }_1=\frac{c_1}{\rho A{\omega }_1},{\gamma }_2=\frac{c_2}{m_2{\omega }_1},\hfill \\ {\gamma }_3=\frac{c_3}{m_3{\omega }_1},{\omega }_2^2=\frac{k_2}{m_2{{\omega }_1}^2}\hfill \\ {\omega }_3^2=\frac{2k_3}{m_3{{\omega }_1}^2},{\mu }_1=\frac{2M}{\rho \mathrm{AL}},{\mu }_2=\sqrt{\frac{2m_2}{\rho \mathrm{AL}}},{\mu }_3=\frac{2m_3}{\rho \mathrm{AL}},\hfill \\ {\phi }_1=\frac{g}{l{{\omega }_1}^2},\alpha =\frac{l_0}{l}\hfill \end{array}$$

(13)

Equation (12) can be written as

$$\{\begin{array}{c}\stackrel{··}{q}+q+\beta q^3+{\gamma }_1\stackrel{·}{q}=[{\phi }_1({\mu }_1+{\mu }_2^2)-{\mu }_2\stackrel{··}{p}]\left|\sin \omega t\right|-{\mu }_1\stackrel{··}{q}{\sin }^2\omega t\hfill \\ -{\mu }_3({\omega }_3^{2}R(1-\frac{1}{\sqrt{R^2+{\alpha }^2}})+{\gamma }_3\stackrel{·}{R})\hfill \\ \stackrel{··}{p}+{\omega }_2^{2}p+{\gamma }_2\stackrel{·}{p}={\mu }_2[({\omega }_2^{2}q+{\gamma }_2\stackrel{·}{q})\sin \omega t+\omega {\gamma }_{2}q\cos \omega t]\hfill \\ \stackrel{··}{R}+{\omega }_3^{2}R(1-\frac{1}{\sqrt{R^2+{\alpha }^2}})+{\gamma }_3\stackrel{·}{R}=\stackrel{··}{q}\hfill \end{array}$$

(14)

In the following, we will analyze the dynamical characteristics of the coupled system without SD absorber.

Dynamic analysis of vehicle–bridge system

The motion equation of the coupled system without SD absorber can be mathematically modeled by

$$\{\begin{array}{c}\stackrel{··}{q}+q+\beta q^3+{\gamma }_1\stackrel{·}{q}=[{\phi }_1({\mu }_1+{{\mu }_2}^2)-{\mu }_2\stackrel{··}{p}]\left|\sin \omega t\right|-{\mu }_1\stackrel{··}{q}{\sin }^2\omega t\hfill \\ \multicolumn{1}{c}{\stackrel{··}{p}+{\omega }_2^{2}p+{\gamma }_2\stackrel{·}{p}={\mu }_2[({\omega }_2^{2}q+{\gamma }_2\stackrel{·}{q})\sin \omega t+\omega {\gamma }_{2}q\cos \omega t]}\end{array}$$

(15)

Let

$$\{\begin{array}{c}{\stackrel{·}{q}}_1=q_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{q}}_2=\stackrel{··}{q}}\\ \multicolumn{1}{c}{{\stackrel{·}{p}}_1=p_2}\\ \multicolumn{1}{c}{{\stackrel{·}{p}}_2=\stackrel{··}{p}}\end{array}$$

(16)

we have

$$\{\begin{array}{c}{\stackrel{·}{q}}_1=q_2\hfill \\ \multicolumn{1}{c}{(1+{\mu }_1{\sin }^2\omega t){\stackrel{·}{q}}_2+q_1=-{\gamma }_1{q}_2-\beta q_1^3+[{\phi }_1({\mu }_1+{\mu }_2^2)-{\mu }_2{\stackrel{·}{p}}_2]\left|\sin \omega t\right|}\\ \multicolumn{1}{c}{{\stackrel{·}{p}}_1=p_2}\\ \multicolumn{1}{c}{{\stackrel{·}{p}}_2+{\omega }_2^2{p}_1=-{\gamma }_2{p}_2+{\mu }_2[({\omega }_2^2{q}_1+{\gamma }_2{q}_2)\sin \omega t+\omega {\gamma }_2{q}_1\cos \omega t]}\end{array}$$

(17)

In order to discuss the amplitude–frequency response function of system equation (15), we will use the average method and let

$$\{\begin{array}{c}{q}_1=a_1\cos (2\omega t+{\theta }_1)\hfill \\ \multicolumn{1}{c}{q_2=-2\omega a_1\sin (2\omega t+{\theta }_1)}\\ \multicolumn{1}{c}{p_1=a_2\cos (\omega t+{\theta }_2)}\\ \multicolumn{1}{c}{p_2=-\omega a_2\sin (\omega t+{\theta }_2)}\end{array}$$

(18)

we can obtain

$$\{\begin{array}{c}-\frac{{\gamma }_1{a}_1}{(2+{\mu }_1)}+\frac{2{\phi }_1({\mu }_1+{\mu }_2^2)}{3\pi \omega (2+{\mu }_1)}\sin {\theta }_1-\frac{{\mu }_2\omega a_2}{4(2+{\mu }_1)}\cos ({\theta }_1-{\theta }_2)=0\hfill \\ -\omega a_1+\frac{a_1}{2\omega (2+{\mu }_1)}+\frac{3\beta a_1^3}{8\omega (2+{\mu }_1)}+\frac{2{\phi }_1({\mu }_1+{\mu }_2^2)}{3\pi \omega (2+{\mu }_1)}\cos {\theta }_1+\frac{{\mu }_2\omega a_2}{4(2+{\mu }_1)}\sin ({\theta }_1-{\theta }_2)=0\hfill \\ -\frac{{\gamma }_2{a}_2}{2}+\frac{{\mu }_2{\omega }_2^2{a}_1}{4\omega }\cos ({\theta }_1-{\theta }_2)-\frac{{\mu }_2{\gamma }_2{a}_1}{2}\sin ({\theta }_1-{\theta }_2)+\frac{{\mu }_2{\gamma }_2{a}_1}{4}\sin ({\theta }_1-{\theta }_2)=0\hfill \\ -\frac{\omega a_2}{2}+\frac{{\omega }_2^2{a}_2}{2\omega }+\frac{{\mu }_2{\omega }_2^2{a}_1}{4\omega }\sin ({\theta }_1-{\theta }_2)+\frac{{\mu }_2{\gamma }_2{a}_1}{2}\cos ({\theta }_1-{\theta }_2)-\frac{{\mu }_2{\gamma }_2{a}_1}{4}\cos ({\theta }_1-{\theta }_2)=0\hfill \end{array}$$

(19)

which leads to

$$\{\begin{array}{c}{[{\gamma }_1\omega a_1+\frac{{\omega }^5{\gamma }_2{a}_2^2}{2({\omega }_2^4+{\omega }^2{{\gamma }_2}^2)a_1}]}^2+{\left\{\begin{array}{c}[{\omega }^2(2+{\mu }_1)-\frac{1}{2}]a_1\hfill \\ -\frac{3\beta a_1^3}{8}-\frac{({\omega }^2{\omega }_2^2-{\omega }_2^4-{\omega }^2{\gamma }_2^2){\omega }^2{a}_2^2}{2({\omega }_2^4+{\omega }^2{\gamma }_2^2)a_1}\hfill \end{array}\right\}}^2={\left[\frac{2{\phi }_1({\mu }_1+{\mu }_2^2)}{3\pi }\right]}^2\hfill \\ 4{({\omega }^2{\omega }_2^2-{\omega }_2^4-{\omega }^2{\gamma }_2^2)}^2{a}_2^2+4{\omega }^6{\gamma }_2^2{a}_2^2={({\omega }_2^4+{\omega }^2{\gamma }_2^2)}^2{\mu }_2^2{a}_1^2\hfill \end{array}$$

(20)

By solving equation (20), the amplitude–frequency response curves of system equation (15) shown in Figures 3–5 are obtained and the influence of speed and other various parameters is examined as follows.

1. Figure 3 shows the amplitudes of beam via the speeds v of moving spring–mass under different damping coefficients. Other parameters are chosen as $\mathrm{EI}=1\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =9000\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=60\mathrm{m}$ , $m_2=1600\mathrm{kg}$ , and $k_2=900\mathrm{N}/\mathrm{m}$ . Two peaks appear in three curves and the changes of the amplitudes with the speeds are identical. Furthermore, when the damping of SD absorber increases, the values of speed for the vehicle become larger and larger.

2. Figure 4 illustrates the amplitudes of beam with the speeds under different wheel masses. The wheel masses are $M=7000,9000,\mathrm{and}10,000\mathrm{kg}$ , respectively, and the other parameters are $\mathrm{EI}=1\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =9000\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=60\mathrm{m}$ , $m_2=1600\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , and $k_2=900\mathrm{N}/\mathrm{m}$ . As the wheel mass increases, the amplitude of beam increases and there exists a peak near $v=11.5\mathrm{m}/\mathrm{s}$ which is the resonance speed.

3. The changes in the amplitude with the speed and different mass of the suspension are shown in Figure 5. The oscillator masses are $m_2=1500,1550,\mathrm{and}1630\mathrm{kg}$ and the other parameters are $\mathrm{EI}=1\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =9000\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=60\mathrm{m}$ , $M=7000\mathrm{kg}$ , $c_2=20\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , and $k_2=900\mathrm{N}/\mathrm{m}$ . The amplitude becomes larger when the oscillator mass increases. But, when the speed is up to the specific value, three curves have the same change tendency.

Figure 3.

Influence of v and damping coefficient on the amplitude.

Figure 4.

Influence of v and wheel mass on the amplitude.

Figure 5.

Influence of v and mass of the suspension on the amplitude.

As a consequence of the above, the results indicate that the mid-span deflection of the beam subjected to an infinite sequence of regularly spaced concentrated moving loads has rich dynamic behaviors, which may lead to the potential hazard of beam. Hence, we will propose SD absorber for eliminating undesired vibrations of the beam.

Vehicle–bridge–SD absorber dynamic interactions

In order to reveal the efficiency of SD absorber, theoretical analysis and numerical simulations are carried out to investigate the maximum amplitude for the vehicle–bridge–SD absorber coupled system.

Let

$$\{\begin{array}{c}{\stackrel{·}{R}}_1=R_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{R}}_2=\stackrel{··}{R}}\end{array}$$

(21)

From equations (16) and (21), equation (14) can be rewritten as

$$\{\begin{array}{c}{\stackrel{·}{q}}_1=q_2\hfill \\ (1+{\mu }_1{\sin }^2\omega t+{\mu }_3){\stackrel{·}{q}}_2+q_1+\beta q_1^3+{\gamma }_1{q}_2=[{\phi }_1({\mu }_1+{{\mu }_2}^2)-{\mu }_2{\stackrel{·}{p}}_2]\left|\sin \omega t\right|+{\mu }_3{\stackrel{·}{R}}_2\hfill \\ {\stackrel{·}{p}}_1=p_2\hfill \\ {\stackrel{·}{p}}_2+{\omega }_2^2{p}_1+{\gamma }_2{p}_2={\mu }_2[({\omega }_2^2{q}_1+{\gamma }_2{q}_2)\sin \omega t+\omega {\gamma }_2{q}_1\cos \omega t]\hfill \\ {\stackrel{·}{R}}_1=R_2\hfill \\ {\stackrel{·}{R}}_2+{\omega }_3^2{R}_1(1-\frac{1}{\sqrt{R_1^2+{\alpha }^2}})+{\gamma }_3{R}_2={\stackrel{·}{q}}_2\hfill \end{array}$$

(22)

In order to discuss the amplitude–frequency response function of system equation (22), we will use the average method and assume

$$\begin{array}{l}{R}_1=a_3\cos (2\omega t+{\theta }_3)\\ R_2=-2\omega a_3\sin (2\omega t+{\theta }_3)\end{array}$$

(23)

Using equations (18) and (23) and the average method, we have the amplitude–frequency response function of 3-degree-of-freedom coupled system as follows

$$\{\begin{array}{c}{[{\gamma }_1\omega a_1+\frac{{\omega }^5{\gamma }_2{a}_2^2}{2({\omega }_2^4+{\omega }^2{\gamma }_2^2)a_1}+\frac{{\mu }_3\omega {\gamma }_3{a}_3^2}{a_1}]}^2+\{[{\omega }^2(2+{\mu }_1+2{\mu }_3)-\frac{1}{2}]a_1\hfill \\ -\frac{3\beta a_1^3}{8}-\frac{({\omega }^2{\omega }_2^2-{\omega }_2^4-{\omega }^2{\gamma }_2^2){\omega }^2{a}_2^2}{2({\omega }_2^4+{\omega }^2{\gamma }_2^2)a_1}+\frac{({\omega }_3^2-4{\omega }^2){\mu }_3{a}_3^2}{2a_1}+\frac{{\mu }_3{\omega }_3^2}{2\pi a_1}\hfill \\ {[\frac{{\alpha }^2}{\sqrt{a_3^2+{\alpha }^2}}F(\sqrt{\frac{a_3^2}{a_3^2+{\alpha }^2}},2\pi )-\sqrt{a_3^2+{\alpha }^2}E(\sqrt{\frac{a_3^2}{a_3^2+{\alpha }^2}},2\pi )]\}}^2={\left[\frac{2{\phi }_1({\mu }_1+{\mu }_2^2)}{3\pi }\right]}^2\hfill \\ 4{({\omega }^2{\omega }_2^2-{\omega }_2^4-{\omega }^2{\gamma }_2^2)}^2{a}_2^2+4{\omega }^6{\gamma }_2^2{a}_2^2={({\omega }_2^4+{\omega }^2{\gamma }_2^2)}^2{\mu }_2^2{a}_1^2\hfill \\ \frac{{\gamma }_3^2{a}_3^2}{4{\omega }^2{a}_1^2}+{\{\frac{({\omega }_3^2-4{\omega }^2)a_3}{4{\omega }^2{a}_1}+\frac{{\omega }_3^2}{4\pi {\omega }^2{a}_1{a}_3}[\frac{{\alpha }^2}{\sqrt{a_3^2+{\alpha }^2}}F(\sqrt{\frac{a_3^2}{a_3^2+{\alpha }^2}},2\pi )-\sqrt{a_3^2+{\alpha }^2}E(\sqrt{\frac{a_3^2}{a_3^2+{\alpha }^2}},2\pi )]\}}^2=1\hfill \end{array}$$

(24)

where $F(\sqrt{a_3^2/(a_3^2+{\alpha }^2)},2\pi )\mathrm{and}E(\sqrt{a_3^2/(a_3^2+{\alpha }^2)},2\pi )$ are the first kind of elliptic integral and the second one.

Figure 6 shows the effects of the speed on the amplitudes of beam and SD absorber, where $\mathrm{EI}=1\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =9000\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=60\mathrm{m}$ , $M=7000\mathrm{kg}$ , $m_2=1600\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_2=500\mathrm{N}/\mathrm{m}$ , $c_3=0.1\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_3=500\mathrm{N}/\mathrm{m}$ , $l=10\mathrm{m}$ , $m_3=5000\mathrm{kg}$ , and $\alpha =0.1$ . It can be seen form Figure 6 that the amplitudes of beam and SD absorber reach a maximum when $v=11\mathrm{m}/\mathrm{s}$ . Furthermore, comparing the results of Figures 3–6, we can see that the amplitudes of vehicle–bridge system with SD absorber are simplified, which indicates that the vibrations of this coupling system are suppressed by the SD absorber. In order to apply SD absorber on the vehicle–bridge system, we will detect the influence of different parameters on the efficiency of vibration reduction.

Figure 6.

Influence of v on $a_1$ and $a_3$ .

Optimization for SD absorber

In this section, the power flow approach^22–27 is used to detect the efficiency of vibration reduction of SD absorber. The portion of the input energy dissipated by the viscous damper and the kinetic energy of SD absorber is computed, which shows the effect of SD absorber.

Calculation of power flow

Let

$$\tilde{u}=\frac{u}{l}$$

(25)

From equations (6) and (13) and $x=\mathrm{vt}$ , we obtain

$$\tilde{u}=q\sin \omega t$$

(26)

Hence, the instantaneous input power flow density $p_{\mathrm{in}}$ at any point $x=\mathrm{vt}$ is defined by the dot product of the velocity and the force

$$F=[{\phi }_1({\mu }_1+{{\mu }_2}^2)-{\mu }_2\stackrel{··}{p}]\left|\sin \omega t\right|-{\mu }_1\stackrel{··}{q}{\sin }^2\omega t$$

caused by the moving spring mass and the axial force, that is

$$p_{\mathrm{in}}=F·\stackrel{·}{\stackrel{~}{u}}$$

(27)

Thus, the time-averaged input power flow $P_{\mathrm{in}}$ yields

$$\begin{array}{c}{P}_{\mathrm{in}}=\frac{1}{T}\underset{0}{\overset{T}{\int }}{p}_{\mathrm{in}}\mathrm{d}t\hfill \\ =\frac{{\phi }_1({\mu }_1+{\mu }_2^2)\omega a_1}{4}\sin {\theta }_1-\frac{2{\mu }_2{\omega }^3{a}_1{a}_2}{15\pi }\hfill \\ (7\cos {\theta }_1\cos {\theta }_2+8\sin \theta \sin {\theta }_2)-\frac{16{\mu }_1{\omega }^3{a}_1^2}{105\pi }\sin 2{\theta }_1\hfill \end{array}$$

(28)

where $T=\pi /\omega$ .

The instantaneous and time-averaged dissipated power flow absorbed by SD absorber can be written as

$$p_{\mathrm{d}}=\left[{\omega }_3^{2}R\left(1-\frac{1}{\sqrt{R^2+{\alpha }^2}}\right)+{\gamma }_3\stackrel{.}{R}\right]\stackrel{.}{R}$$

(29)

and

$$P_{\mathrm{d}}=2{\gamma }_3{\omega }^2{a}_3^2$$

(30)

Efficiency of SD absorber

We introduce

$$\eta =\frac{P_d}{P_{\mathrm{in}}}$$

(31)

as the efficiency parameter of vibration reduction. The input power is converted into dissipative energy as much as possible to minimize the vibration of the bridge when $\eta$ is close to 1.

Figure 7 shows the effect of the wheel mass M on $\eta$ under three different values of $\alpha$ which are $0,0.3,\mathrm{and}0.8$ , respectively. The other parameters are $\mathrm{EI}=3.0996\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =5500\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=40\mathrm{m}$ , $m_2=5000\mathrm{kg}$ , $m_3=3000\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $c_3=0.1\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_2=500\mathrm{N}/\mathrm{m}$ , $k_3=500\mathrm{N}/\mathrm{m}$ , $l=10\mathrm{m}$ , and $v=32.1\mathrm{m}/\mathrm{s}$ . It can be seen that the wheel mass M and the smooth parameter $v=32.1\mathrm{m}/\mathrm{s}$ have the significant effects on $\eta$ . When the wheel mass is small, red line and blue one coincide, which means the efficiency of nonlinear SD absorber and linear one has no difference. But as the wheel mass increases, the effect of vibration reduction of nonlinear SD absorber is up to a peak. Especially, $\eta$ is up to 0.9 when $\alpha =0.3$ and linear SD absorber has little effect, which shows the distinct advantage of nonlinear SD absorber.

Figure 7.

Influence of M on $\eta$ .

Figure 8 shows the effect of the ratio $\mu$ between the masses of SD absorber and beam on $\eta$ under three different values of $\alpha$ which are $0,0.7,\mathrm{and}0.9$ . The other parameters are $\mathrm{EI}=3.0996\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =5500\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=40\mathrm{m}$ , $m_2=30,000\mathrm{kg}$ , $M=30,000\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $c_3=0.0001\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_2=500\mathrm{N}/\mathrm{m}$ , $k_3=500\mathrm{N}/\mathrm{m}$ , $l=10\mathrm{m}$ , and $v=32.1\mathrm{m}/\mathrm{s}$ .

Figure 8.

Influence of the ratio $\mu$ on $\eta$ .

As shown in Figure 8, the mass ratio $\mu =m_3/\rho \mathrm{AL}$ and the smooth parameter have the remarkable effect on $\eta$ . Nonlinear SD absorbers when $\alpha =0.7\mathrm{and}0.9$ have better efficiency of vibration reduction than linear one when $\alpha =0$ . Furthermore, as the ratio $\mu$ changes, the effects of nonlinear absorber corresponding to $\alpha =0.7$ and $\alpha =0.9$ also change, respectively. When $\mu$ is small, the effect of nonlinear absorber corresponding to $\alpha =0.9$ is better than the other one corresponding to $\alpha =0.7$ . But when $\mu$ is large enough, the nonlinear absorber corresponding to $\alpha =0.7$ has better effect of vibration reduction, which shows the importance of the smooth parameter $\alpha$ in SD absorber.

Figure 9 shows the joint effects of the smooth parameter $\alpha$ and the wheel mass M on $\eta$ . The other parameters are $\mathrm{EI}=3.0996\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =5500\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=40\mathrm{m}$ , $m_2=5000\mathrm{kg}$ , $m_3=3000\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $c_3=0.1\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_2=500\mathrm{N}/\mathrm{m}$ , $k_3=500\mathrm{N}/\mathrm{m}$ , $l=10\mathrm{m}$ , and $v=32.1\mathrm{m}/\mathrm{s}$ . It can be seen that two parameters have the distinct effects on $\eta$ . When $M=15,000\mathrm{kg}$ , $\alpha =0.15$ should be chosen to obtain the best effect of vibration reduction and when $M=20,000\mathrm{kg}$ , $\alpha =0.225$ is very proper, which reveals the advantages of the frequency-variable SD absorber. Hence, the smooth parameter $\alpha$ can be adjusted to solve the problems of frequency variation of the vehicle–bridge coupled system.

Figure 9.

Influence of $\alpha$ and M on $\eta$ .

In Figure 10, the effects of the smooth parameter and the mass ratio on $\eta$ are illustrated. The parameters are $\mathrm{EI}=3.0996\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =5500\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=40\mathrm{m}$ , $m_2=30,000\mathrm{kg}$ , $M=30,000\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $c_3=0.001\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_2=500\mathrm{N}/\mathrm{m}$ , $k_3=500\mathrm{N}/\mathrm{m}$ , $l=10\mathrm{m}$ , and $v=32.1\mathrm{m}/\mathrm{s}$ . As shown in Figure 10, two parameters play an obvious role in $\eta$ . When $\mu =0.0048$ , $\alpha =0.2$ are chosen for the best effect of vibration reduction and the wider adjustable range of the smooth parameter $\alpha$ , which is convenient to adjust SD absorber.

Figure 10.

Influence of $\alpha$ and $\mu$ on $\eta$ .

The coupling effects of the smooth parameter and the speed on $\eta$ are examined in Figure 11 and other parameters are fixed as $\mathrm{EI}=3.0996\times 10^{10}\mathrm{N}{\mathrm{m}}^2$ , $\rho =5500\mathrm{kg}/{\mathrm{m}}^3$ , $A=6.8{\mathrm{m}}^2$ , $c_1=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $L=40\mathrm{m}$ , $m_2=30,000\mathrm{kg}$ , $M=30,000\mathrm{kg}$ , $m_3=5000\mathrm{kg}$ , $c_2=0.02\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $c_3=0.1\mathrm{N}/(\mathrm{m}/\mathrm{s})$ , $k_2=500\mathrm{N}/\mathrm{m}$ , $k_3=500\mathrm{N}/\mathrm{m}$ , and $l=10\mathrm{m}$ . We can know that under different speeds, $\eta$ changes with the smooth parameter $\alpha$ . The smooth parameter can be adjusted to obtain the best efficiency of vibration reduction under different speeds, which shows the flexibility of SD absorber.

Figure 11.

Influence of $\alpha$ and v on $\eta$ .

In conclusion, the smooth curves in Figures 8–11 reveal that SD absorber can be used to control the complex vibrations of the coupled vehicle–bridge system. ²⁵ The natural frequency of SD absorber can be changed depending on the values of the smooth parameter $\alpha$ . Hence, the SD absorber can be the linear absorber or the nonlinear one, which can adapt itself to meeting different vibration reduction demands.

Conclusion

The 3-degree-of-freedom vehicle–bridge–SD absorber–coupled system was constructed and the application of SD absorber in the vibration reduction of beam was discussed. The moving vehicles are simplified as an infinite series of moving spring masses. By use of a series of elliptic integrals of the first and second kind, the amplitude–frequency response functions of the coupled system were obtained. The influence of different parameters on $\eta$ was investigated, especially $\alpha$ and v. The beam subjected to an infinite sequence of moving spring masses had rich dynamic behaviors via the speed of vehicle and SD absorber was proposed for eliminating its undesired vibrations. The results show that SD absorber can adapt itself to effectively reducing the vibration energy of beam subjected to an infinite series of moving spring masses. Most important of all, depending on the value of the smoothness parameter $\alpha$ , the efficiency parameter of vibration reduction was optimized, which laid a theoretical foundation for the design of beam.