Vibration isolation using a bar and an Euler beam as negative stiffness for vehicle seat comfort

Abstract

In this article, the behavior and the efficiency of a nonlinear passive vibration isolator based on a bar and an Euler beam are investigated analytically. The proposed isolator is specifically for isolating vehicle seat from unwanted disturbances in the low frequency range, thereby making the seat comfortable for the driver. The static characteristics of the negative stiffness as well as the nonlinear mathematical model are presented, and its dynamic characteristics are investigated using the harmonic balance method. In addition, the effect of the initial geometric imperfection of the Euler beam is included in the study. The study shows that the performance of the negative isolator has a large isolation frequency range than that of the linear isolator. The configuration parameter choice for the seat base will be vital for the suitable application of this isolator.

Introduction

In various engineering systems, undesirable vibrations that frequently hamper the full performance of the machines are regularly encountered. ¹ Passive isolators persist to be the most commonly adopted form of isolator due to their simplicity, stability, and low cost. Their effectiveness is reliant on being able to achieve a low natural frequency when supporting an isolated object. ² From the basic theory of linear isolation, vibration attenuation occurs above $\sqrt{2}$ times the resonant frequency of the system. Therefore, the isolation frequency range can be improved by reducing the resonant frequency. This can be achieved by reducing the stiffness or increasing the mass of the system. It is a known fact that reducing the stiffness will result in a larger static deformation of the mass and degrade the stability of the isolator.^3,4 Thus, the means to achieve efficient isolation in low or ultra-low frequency band is a task in the subject of linear vibration isolation. Owing to the serious ranges of disturbances we have in real engineering application such as earthquakes and shocks in which the components of the disturbances contain low frequency, the condition for vibration isolation has become increasingly rigorous. ⁵ In recent years, nonlinear isolators have drawn increasing attention since they possess some advantages in some respects compared with linear ones. A comprehensive review of many different kinds of nonlinear isolators in recent years has been summarized by Ibrahim, ⁶ in which the resonance frequency may be shifted to the left or right of the linear resonance frequency on the transmissibility plot. Unlike the response of linear systems, the amplitude response of a nonlinear case can be multivalued. The softening system tends to shift the jump down to the left, while a hardening nonlinearity jumps down to the right.

It has been established by Liu et al. ⁷ that connecting a negative stiffness system in parallel with a positive stiffness spring is a common way to obtain a very low dynamic stiffness or even zero stiffness. Carrella et al. ⁸ demonstrated the use of two inclined springs as negative stiffness system together with a vertical spring to design a quasi-zero-stiffness (QZS) isolator. It was shown that the force–displacement characteristics of the system can be approximated by a cubic equation. Similarly, Hao and Cao ⁹ further used double inclined springs and vertical spring as Carrella et al., ⁸ but the equation of motion of the system was formulated with an originally irrational nonlinearity based upon SD oscillator in place of the usually approximate Duffing system of polynomial type. The results produced a high precision, especially for the prediction of a large displacement behavior.

By using inclined bars and springs instead of inclined springs, Le and Ahn^10,11 developed a new model of inclined rods in series with horizontal spring to attenuate the vibration of vehicle seat. Transmissibility of the proposed system for each of the excitation frequency was investigated. Also, the time responses to the sinusoidal and random excitations were also investigated by simulation and experiment. Another combination of positive and negative stiffness was investigated by Meng et al. ¹² The negative stiffness was achieved by the use of disk spring. The challenge with the variable thickness of the disk was solved by the use of the law of variable thickness. The primary resonance response of the resulting equation was solved by the harmonic balance method (HBM). Recently, Meng et al. ¹² used the curved-mount-spring-roller mechanism as a negative stiffness element. ¹³ Arrangements of the configurative parameters were optimized for a wide displacement range around the static equilibrium position (SEP) with a low dynamic stiffness and with the stiffness changing slightly.

Yang et al. ¹⁴ considered the use of two rigid bars subjected to compressive forces to elicit negative stiffness in power flow behavior. Furthermore, the corresponding power flow characteristics of the nonlinear system were investigated. Liu et al. ⁷ introduced a negative stiffness corrector through the use of Euler buckled beams. The design of the corrector followed that of inclined spring methods such that the corrector can provide negative stiffness to the isolator at the equilibrium position in order to lower the dynamic stiffness of the isolator and without sacrificing the support capacity. The results showed that the proposed QZS isolator can outperform the equivalent linear one for certain frequencies. However, due to twin challenges in the use of Euler beam: difficulty to realize the hinged conditions experimentally and unsymmetric deformation about their symmetric plane during the process of movement, further analytical and experimental studies were carried out on the Euler beam negative stiffness corrector by Huang et al.,^15,16 in which the problem of boundary condition and symmetry of the Euler beam were addressed. However, some limitations may arise in the use of the Euler beam as a negative connection in a vehicle seat. For example, connecting the Euler beam in an inclined manner may introduce an inclined force instead of purely compressional force to the beam. Therefore, an introduction of a bar as done by Le and Ahn ¹¹ is used to circumvent this problem. In addition, a buckled beam presents asymmetrical negative stiffness about the operating point, especially in an inclined position. ¹⁵

In this study, we propose an alternative means of vibration isolation with the combination of a bar and an Euler beam. The proposed car seat isolator is a system in which negative stiffness corrector is provided by a bar and an Euler beam and positive springs are connected in parallel. The primary aim of the positive stiffness springs is to provide supporting capacity for the weight of the driver, and the vehicle isolation system can widen the range of operation bands by the introduction of the negative stiffness. The analytical study involves the static and dynamic features of the vehicle seat isolator with the proposed negative stiffness. This article is organized as follows: in section “Static characteristics of the combined Euler beam and bar,” the static characteristics of Euler beam and rod in which the force–displacement and design procedure for the isolators are described; in section “Dynamic analysis,” dynamic analysis of the system is theoretically studied for different parameters of the nonlinear isolator; section “Numerical simulation” deals with numerical simulation of the proposed isolator; and concluding remarks are given in section “Conclusion.” There is also an Appendix 1 which gives the constants of the Taylor series expansion for the third and fifth order.

Static characteristics of the combined Euler beam and bar

Force–displacement characteristics of the Euler beam

Consider classical Euler beam investigated by Liu et al. ⁷ and Huang et al. ¹⁵ (Figure 1). The approximate solution for small lateral deflection (i.e. $w<0.2L$ ) for this classic Euler beam problem can be found in the study by Virgin and Davis. ¹⁷ The relation between axial load and end shortening for moderately large lateral deflections for the Euler beam is then given by^7,16,17

$$\begin{array}{cc}\multicolumn{1}{c}{P=P_e}& [1-\pi \left(\frac{q_0}{L}\right){({\left(\frac{\pi q_0}{L}\right)}^2+4\left(\frac{y}{L}\right))}^{-\frac{1}{2}}]\hfill \\ \multicolumn{1}{c}{}& [1+\frac{{\pi }^2}{8}{\left(\frac{q_0}{L}\right)}^2+\frac{1}{2}\left(\frac{y}{L}\right)]\hfill \end{array}$$ (1)

where ${\mathit{P}}_e=\mathit{EI}(1\pi /\mathit{L}{)}^2$ is the classical Euler critical load, $\mathit{P}$ is the applied load, ${\mathit{q}}_0$ is the initial imperfection at the center of the beam, $\mathit{EI}$ is the flexural rigidity of the beam, and $\mathit{L}$ is the initial length of the beam.

Figure 1.

The compressed Euler beam with hinged–hinged boundary connected with a bar. ⁷

The new model of the seat isolation system is shown in Figure 2, which consists of a vertical spring of stiffness $\mathit{k}$ , a dash pot $\mathbf{\eta }$ , a bar with length a , and Euler beam with initial length L . One end of the bar is connected to the mass and the other end to the hinge joint. Rotation is allowed at the end of the bar with the joint, and the bar can only push the Euler beam horizontally during operation. At the static equilibrium, the gravity force is opposite to the compressive force provided by the positive spring $\mathit{k}$ . When the mass is at the static state, the weight of the driver is sustained only by the vertical spring.

Figure 2.

(a) Proposed isolation system using Euler buckled beams with bar connected to the seat and (b) detailed part of the seat.

When the mass moves a distance $\mathit{x}$ from the initial position by the force F exerted by the driver, the relationship between the applied force and the displacement can be given by^11,18

$$F\partial x=2P\tan \left(\theta \right)\partial x$$ (2)

$$\tan \left(\theta \right)=\frac{h-x}{b-L_1}$$ (3)

$$L_1=b-\sqrt{a^2-{(h-x)}^2}$$ (4)

$$h=\sqrt{a^2-{(b-L)}^2}$$ (5)

The displacement of the Euler beam is related to the original length as

$$y=L-L_1$$ (6)

Therefore, the force–displacement relationship becomes

$$\begin{array}{cc}\multicolumn{1}{c}{F=2P_e}& [1-\pi \left(\frac{q_0}{L}\right){({\left(\frac{\pi q_0}{L}\right)}^2+4\left(\frac{L-L_1}{L}\right))}^{-\frac{1}{2}}]\hfill \\ \multicolumn{1}{c}{}& [1+\frac{{\pi }^2}{8}{\left(\frac{q_0}{L}\right)}^2+\frac{1}{2}\left(\frac{L-L_1}{L}\right)]\left(\frac{h-x}{b-L_1}\right)\hfill \end{array}$$ (7)

Dimensional force–displacement relationship about x gives

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{F}=}& 2[1-\pi \overline{q}{({\pi }^2{\overline{q}}^2+4\{1-{\gamma }_2+\mathrm{\Psi }\})}^{-\frac{1}{2}}]\hfill \\ \multicolumn{1}{c}{}& [1+\frac{{\pi }^2}{8}{\overline{q}}^2+\frac{1}{2}\{1-{\gamma }_2+\mathrm{\Psi }\}]\left(\frac{\overline{h}-\overline{x}}{\mathrm{\Psi }}\right)\hfill \end{array}$$ (8)

The following quantities are used to transform the equation to dimensionless parameters form as

$$\begin{array}{c}\overline{F}=\frac{F}{P_e},\overline{x}=\frac{x}{L},\overline{q}=\frac{q}{L},{\gamma }_1=\frac{a}{L},{\gamma }_2=\frac{b}{L},\hfill \\ \overline{h}=\sqrt{{\gamma }_1^2-{({\gamma }_2-1)}^2},\mathrm{\Psi }=\sqrt{{({\gamma }_2-1)}^2+2\overline{x}\overline{h}-{\overline{x}}^2}\hfill \end{array}$$

The dimensionless force–displacement characteristics for various initial imperfections $\overline{q}$ , ${\gamma }_2$ , and ${\gamma }_1$ are found by using equation (8) and plotted in Figure 3. From Figure 3, there is a non-monotone dependence of the force versus displacement (negative stiffness) relation. The turning points of the curves represent zero stiffness but are unstable at this position. This reveals that this system can elicit negative stiffness at these turning points. In Figure 3(b), the dependence of ${\gamma }_2$ on the force–displacement is shown for a constant initial imperfection of the Euler beam and ${\gamma }_1.$ Increase in ${\gamma }_2$ above one tends to reduce the maximum value of dimensional force before the instability region of negative stiffness is produced. In addition, in Figure 3(a), it can also be observed that for small initial imperfection value $\overline{q}$ , the negative region is large.

Figure 3.

Dimensionless force–deflection characteristic: (a) for ${\gamma }_1=0.9$ and ${\gamma }_2=1.4$ , (b) for initial imperfection $\overline{q}=0.05$ and ${\gamma }_1=0.8$ , and (c) for initial imperfection $\overline{q}=0.02$ and ${\gamma }_2=1.2$ .

Design method for the vehicle seat system

The schematic of the vehicle isolator is shown in Figure 2. As soon as the weight of the mass moves downward $\mathit{x}$ amounts from the initial position, the restoring force is generated by the negative Euler beam and the linear vertical spring. Therefore, displacement of the isolator is given as

$$F_s=\mathrm{kx}+F$$ (9)

where $F_s$ is the restoring force of the car seat system. Non-dimensionalizing equation (9) by dividing both sides by $\mathrm{kL}$ gives

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{F}}_s=}& \overline{x}+2\alpha [1-\pi \overline{q}{({\pi }^2{\overline{q}}^2+4(1-{\gamma }_2+\mathrm{\Psi }))}^{-\frac{1}{2}}]\hfill \\ \multicolumn{1}{c}{}& [1+\frac{{\pi }^2}{8}{\overline{q}}^2+\frac{1}{2}(1-{\gamma }_2+\mathrm{\Psi })]\left(\frac{\overline{h}-\overline{x}}{\mathrm{\Psi }}\right)\hfill \end{array}$$ (10)

where

$${\overline{F}}_s=\frac{F_s}{\mathrm{kL}},\alpha =\frac{P}{\mathrm{kL}},\overline{x}=\frac{x}{L}$$

It is worthy of note that when the mass moves down there are three restoring forces at play here: the vertical springs and the two Euler beams. When the bar is at horizontal line, then this relationship holds for dimensionless displacement $\overline{u}=\overline{h}-\overline{x}$ . From which

$$\overline{x}=\overline{h}-\overline{u}$$ (11)

The dimensionless relative displacement of the isolator in terms of $\overline{u}$ is obtained as follows

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{F}}_s=\overline{h}-\overline{u}+2\alpha }& [1-\pi \overline{q}{({\pi }^2{\overline{q}}^2+4(1-{\gamma }_2+\overline{\mathrm{\Psi }}))}^{-\frac{1}{2}}]\hfill \\ \multicolumn{1}{c}{}& [1+\frac{{\pi }^2}{8}{\overline{q}}^2+\frac{1}{2}(1-{\gamma }_2+\overline{\mathrm{\Psi }})]\left(\frac{\overline{u}}{\overline{\mathrm{\Psi }}}\right)\hfill \end{array}$$ (12)

where

$$\overline{\mathrm{\Psi }}=\sqrt{{({\gamma }_2-1)}^2+2\overline{h}(\overline{h}-\overline{u})-{(\overline{h}-\overline{u})}^2}$$

Differentiating equation (10) with respect to dimensionless displacement $\overline{\mathit{x}}$ , then substituting equation (11) into the resulting equation gives the non-dimensional stiffness for the isolator as

$${\overline{K}}_s=\frac{1}{4}[4-\frac{u^2\alpha \mathrm{\Phi }\left(\frac{1-\pi \overline{q}}{\mathrm{\Gamma }}\right)}{{\mathrm{\Delta }}^3}+\frac{4u^2\alpha \left(\frac{1-\pi \overline{q}}{\mathrm{\Gamma }}\right)}{{\mathrm{\Delta }}^2}+\frac{2\pi \overline{q}{u}^2\alpha \mathrm{\Phi }}{{\mathrm{\Gamma }}^3{\mathrm{\Delta }}^3}-\frac{\alpha \mathrm{\Phi }\left(\frac{1-\pi \overline{q}}{\mathrm{\Gamma }}\right)}{\mathrm{\Delta }}]$$ (13)

where

$$\begin{array}{c}\mathrm{\Delta }=\sqrt{{\overline{h}}^2-{\overline{u}}^2+{({\gamma }_2-1)}^2},\hfill \\ \mathrm{\Gamma }=\sqrt{4+{\pi }^2{\overline{q}}^2+4\mathrm{\Delta }-4{\gamma }_2},\mathrm{and}\hfill \\ \mathrm{\Phi }=12+{\pi }^2{\overline{q}}^2+4\mathrm{\Delta }-4{\gamma }_2\hfill \end{array}$$

From equation (12), it can be seen that the dimensionless parameters $\overline{q},\alpha ,{\gamma }_1$ , and ${\gamma }_2$ can affect the nonlinear stiffness of the isolator. Figure 4 illustrates the dimensionless nonlinear stiffness curves for initial imperfection $\overline{q}=0.01,\alpha =0.3$ and $\overline{q}=0.03,\alpha =0.1,$ various values ${\gamma }_1$ and ${\gamma }_2$ . For the two figures, stiffness achieves a maximum value at the SEP. For example, in Figure 4(a) the maximum dimensionless stiffness is −0.025 and 0.28 when ${\gamma }_2$ is 1.1 and 1.5, respectively. Therefore, when ${\gamma }_2=1.1$ , the isolator is incapable of sustaining the imposed weight of the driver.

Figure 4.

Dimensionless stiffness–deflection characteristic: (a) for initial imperfection $\overline{q}=0.01,\alpha =0.3$ , and ${\gamma }_1=0.75$ and (b) for initial imperfection $\overline{q}=0.03,\alpha =0.1,$ and ${\gamma }_2=1.9$ .

By substituting $\overline{u}=0$ in equation (12), the dimensionless nonlinear stiffness at the SEP is obtained as follows

$$\begin{array}{c}{K}_{\mathrm{sep}}=\frac{1}{4}\hfill \\ \left[\begin{array}{c}4-\frac{\alpha }{\sqrt{{\overline{h}}^2+{({\gamma }_2-1)}^2}}(1-\frac{\pi \overline{q}}{\sqrt{4+{\pi }^2{\overline{q}}^2+4\sqrt{{\overline{h}}^2+{({\gamma }_2-1)}^2}-4{\gamma }_2}})\hfill \\ \multicolumn{1}{c}{(12+{\pi }^2{\overline{q}}^2+4\sqrt{{\overline{h}}^2+{({\gamma }_2-1)}^2}-4{\gamma }_2)}\end{array}\right]\hfill \end{array}$$ (14)

The surface plot of $K_{\mathrm{sep}}$ as a function of configurative parameters ${\gamma }_1$ and ${\gamma }_2$ is plotted in Figure 5 for $\overline{q}=0.03$ and $\alpha =0.5$ , and $\overline{q}=0.03$ and $\alpha =0.2$ . For a relatively large value of $\alpha$ , most configurations will give negative stiffness values, which implies that the stiffness cannot support the mass. Therefore, a proper design is expected at this stage to choose a configuration that gives QZS. For small $\alpha$ values, for example, in Figure 5(b) when $\alpha =0.2$ all configuration parameters give positive values for $K_{\mathrm{sep}}$ .

Figure 5.

Variations in the nonlinear function of $K_{\mathrm{sep}}$ with ${\gamma }_1$ and ${\gamma }_2$ : (a) for $\overline{q}=0.03$ and $\alpha =0.5$ , and (b) for $\overline{q}=0.03$ $\mathrm{and}\alpha =0.2$ .

From equation (13), if the physical parameters of the system are chosen cautiously, the dynamic stiffness at the equilibrium position can be set to zero. This can be obtained by enforcing $K_{\mathrm{sep}}=0$ as follows

$$\begin{array}{c}{\gamma }_2=\frac{1}{12{\alpha }^2}\hfill \\ \left[\begin{array}{c}{\mathrm{\Psi }}^{1/3}+2\alpha (-4{\gamma }_1+\alpha (14+{\pi }^2{\overline{q}}^2+6{\gamma }_1))\hfill \\ \multicolumn{1}{c}{+\frac{1}{{\mathrm{\Psi }}^{1/3}}\{{\alpha }^2{(8+{\pi }^2{\overline{q}}^2)}^2+16{\gamma }_1^2+16\alpha {\gamma }_1(-4+{\pi }^2{\overline{q}}^2)\}}\end{array}\right]\hfill \end{array}$$ (15)

where

$$\mathrm{\Psi }={\left[\begin{array}{c}-{\alpha }^6{(8+{\pi }^2{\overline{q}}^2)}^3-24{\alpha }^4{\gamma }_1^2(16+5{\pi }^2{\overline{q}}^2)-24{\alpha }^5{\gamma }_1(-32+4{\pi }^2{\overline{q}}^2+{\pi }^4{\overline{q}}^4)\hfill \\ \multicolumn{1}{c}{+64{\alpha }^3{\gamma }_1^3+24\sqrt{3}\pi \sqrt{{\alpha }^7{\overline{q}}^2{\gamma }_1^2(2{\alpha }^3{(8+{\pi }^2{\overline{q}}^2)}^2+\alpha {\gamma }_1^2(96-{\pi }^2{\overline{q}}^2)+8{\alpha }^2{\gamma }_1(-24+5{\pi }^2{\overline{q}}^2)-16{\gamma }_1^3)}}\end{array}\right]}^{\frac{1}{3}}$$

Therefore, linking equations (12) and (14), the zero nonlinear stiffness features at the SEP are plotted in Figure 6 for various values of ${\gamma }_1$ . When the dimensionless ${\gamma }_1$ increases, it tends to increase the dimensionless displacement range $\overline{u}$ . For illustration, when ${\gamma }_1$ is 1.2, the displacement range is −1.0 to 1.0, while when ${\gamma }_1$ is 0.85, the displacement range is −0.40 to 0.40. Figures 7 and 8 present the dimensionless nonlinear stiffness of the system for various values of $\overline{q}$ and various values of $\alpha$ , respectively. When $\alpha =1.0$ in Figure 8, the nonlinear stiffness is always zero at any location of the mass as shown by the black solid line. Also, when the imperfection of the Euler beam is not zero the minimum stiffness will move from zero to positive value. This is illustrated in Figure 9, whereby the force is not equal to zero when $\overline{u}=0$ for other imperfection beam values except zero.

Figure 6.

Dynamic stiffness versus displacement with the zero condition at the equilibrium position with $\overline{q}=0.01\mathrm{and}\alpha =1.2$ .

Figure 7.

Dynamic stiffness versus displacement with the zero condition at the equilibrium position for various $\overline{q}$ with ${\gamma }_1=1.2\mathrm{and}\alpha =1.5$ .

Figure 8.

Dynamic stiffness versus displacement with the zero condition at the equilibrium position for various values of $\alpha$ with $\overline{q}=0.01\mathrm{and}{\gamma }_1=1.3$ .

Figure 9.

Dimensionless force–displacement of the zero stiffness system for $\overline{q}=0,\alpha =1.2,\mathrm{and}{\gamma }_1=0.8$ .

Dynamic analysis

Approximation of the restoring force

Before we proceed to develop the dynamic analysis for the vehicle seat system, we shall explain the major principle of the use of the negative stiffness mechanism in vibration isolation. When the mass is at rest, the weight is normally supported by the primary positive spring. At the state when the load is applied, the effect of the negative stiffness is induced, and the dynamic excitation also begins simultaneously in the system. The negative stiffness in this article is provided by the movement of the bar with the compression of the Euler beam. The movement of the Euler beam reduces the positive spring stiffness to zero at the equilibrium point. However, when the Euler beam is under no compression the vertical spring supports the imposed load. The exact restoring force which is given in equation (11) can be approximated under the steady-state vibration around the equilibrium position by expanding power series around the static up to the third and fifth order as given by

$${\overline{F}}_{\mathrm{app}}=\overline{h}+{\kappa }_1\overline{u}+{\kappa }_2{\overline{u}}^3$$ (16)

$${\overline{F}}_{\mathrm{app}}=\overline{h}+{\kappa }_1\overline{u}+{\kappa }_2{\overline{u}}^3+{\kappa }_3{\overline{u}}^5$$ (17)

where ${\overline{F}}_{\mathrm{app}}$ is the nonlinear restoring force acting on the mass. The terms ${\kappa }_1$ , ${\kappa }_2$ , and ${\kappa }_3$ represent the linear, cubic, and the fifth term, respectively, of the Taylor approximation of the restoring force. The expression for ${\kappa }_1,{\kappa }_2$ , and ${\kappa }_3$ terms are stated in Appendix 1. Equations (15) and (16) from the Taylor expansion of the restoring force in equation (11) are plotted in the same figure for comparison as shown in Figure 7. The approximation for the first order and third order represents the exact function for up to 80% and 65% of the plot, respectively. For this work, third-order function will be used in expressing the dynamic equation.

It should be mentioned that when the imperfection of the Euler beam is not zero, the dimensional force–displacement curve origin is not zero. As the imperfection value increases, the force–displacement shifts upward. Equation (15) can be expressed as follows

$$F_{\mathrm{app}}=\overline{h}k+k{\kappa }_{1}u+k{\kappa }_2\frac{u^3}{L^2}$$ (18)

Using D’Alembert’s principle, the dynamic equations of the system under displacement excitation $z_0\cos \omega t$ at the equilibrium position can be expressed as

$$m\frac{{\partial }^{2}u}{\partial t^2}+\eta \frac{\partial y}{\partial t}+\mathrm{ky}+k{\kappa }_{1}y+\frac{k{\kappa }_2}{L^2}{y}^3-\mathrm{mg}=0$$ (19)

The displacement of the mass relative to the base is $y$ . The relationship between $u$ and $y$ is $u=z+y$ .

When the system is subjected to displacement excitation $z=z_0\cos \omega t$ at the base, equation (19), the approximate dynamic equation of the mass at the steady state is derived as follows

$$m\stackrel{··}{y}+\eta y+k(1+{\kappa }_1)y+\frac{k{\kappa }_2}{L^2}{y}^3=-m\stackrel{··}{z}$$ (20)

It is convenient to write equation (20) in non-dimensional form as

$$\stackrel{··}{Y}+2\xi \stackrel{·}{Y}+\mu Y+\lambda Y^3={\mathrm{\Omega }}^2\cos \mathrm{\Omega }\tau$$ (21)

where

$$\begin{array}{c}Y=\frac{y}{z_0},\mu =(1+{\kappa }_1),{\omega }_n^2=\frac{k}{m},\xi =\frac{\eta }{2m{\omega }_n},\hfill \\ \tau ={\omega }_{n}t,\mathrm{\Omega }=\frac{\omega }{{\omega }_n},\lambda =\frac{{\kappa }_2{z}_0^2}{L^2}\hfill \end{array}$$

The transformed equation (20) can be solved approximately by different methods.^10,14,19 We preferred to use HBM for the steady-state approximation based on assumption that the response takes this harmonic form

$$Y=\tilde{Y}\cos \mathrm{\Omega }\tau$$ (22)

Substituting equation (21) in equation (20) gives

$$\{\begin{array}{c}-{\mathrm{\Omega }}^2\tilde{Y}+\mu \tilde{Y}+\frac{3}{4}\lambda {\tilde{Y}}^3={\mathrm{\Omega }}^2\cos \mathrm{\Omega }\tau \hfill \\ \multicolumn{1}{c}{-2\xi \mathrm{\Omega }\tilde{Y}={\mathrm{\Omega }}^2\sin \mathrm{\Omega }\tau }\end{array}$$ (23)

Squaring and adding equation (22) yield frequency–amplitude relationship as follows

$${(-{\mathrm{\Omega }}^2\tilde{Y}+\mu \tilde{Y}+\frac{3}{4}\lambda {\tilde{Y}}^3)}^2+4{\xi }^2{\mathrm{\Omega }}^2{\tilde{Y}}^2={\mathrm{\Omega }}^4$$ (24)

The frequency response curves (FRCs) for the displacement excitation systems can be obtained by solving equation (23)

$${\mathrm{\Omega }}^2=\frac{3\lambda {\tilde{Y}}^4+4{\tilde{Y}}^2(\mu -2{\xi }^2)-\sqrt{{(3\lambda {\tilde{Y}}^3+4\mu \tilde{Y})}^2-16{\tilde{Y}}^4(3\lambda {\tilde{Y}}^2+4\mu ){\xi }^2+64{\tilde{Y}}^4{\xi }^4}}{4({\tilde{Y}}^2-1)}$$ (25)

The resonant and non-resonant branches of the absolute displacement transmissibility curve are given by

$$T=|\frac{u}{z}|=\sqrt{1+2\tilde{Y}\cos \phi +{\tilde{Y}}^2}$$ (26)

where $\cos \phi$ is determined from equation (22), which gives

$$\cos \varphi =\frac{\tilde{Y}}{{\mathrm{\Omega }}^2}(-{\mathrm{\Omega }}^2+\mu +\frac{3}{4}\lambda {\tilde{Y}}^2)$$ (27)

Therefore

$$\left|T\right|=\sqrt{1+{\tilde{Y}}^2+\frac{2{\tilde{Y}}^2}{{\mathrm{\Omega }}^2}(-{\mathrm{\Omega }}^2+\mu +\frac{3}{4}\lambda {\tilde{Y}}^2)}$$ (28)

Numerical simulation

In our numerical examples, we investigate the workability and effectiveness of using Euler beam in combination with a bar for vehicle vibration isolation. The vehicle seats are modeled with the following properties: $\mathrm{mass}=2.14\mathrm{kg},{\omega }_n^2=2.162,\xi =0.02,z_0=10\mathrm{mm}$ . The transmissibility is a measure of the reduction in the vibrations transmitted from a source to a receiver, that is, it is an index of the performance of an isolator. All the transmissibility results are plotted in dB, that is, as $20\mathrm{lo}{\mathrm{g}}_{10}(T)$ . Figure 10 shows the transmissibility curve for various values of $\alpha$ , which is a function of ${\kappa }_1$ with the numerical results, showing excellent agreement. As the values of $\alpha$ reduce, it tends to lower the natural frequency of the system. For illustration, when $\alpha =0.21$ the peak transmissibility is 25 dB, whereas for $\alpha =0.5$ , the peak transmissibility is 28 dB. In addition, the frequency isolation range of system with $\alpha =0.21$ is wider than that of $\alpha =0.5$ , as can be seen in Figure 10. When $\alpha >0.5$ , then ${\kappa }_1\to 0$ , which makes the contribution of the negative stiffness to be zero. Hence, $\alpha =0.5$ in Figure 10 represents the proposed system without negative stiffness.

Figure 10.

Transmissibility frequency response graph of the zero stiffness system for $\overline{q}=0,{\gamma }_1=0.75$ .

The effect of ${\gamma }_1$ value with constant $\overline{q}$ and $\alpha$ is shown in Figure 11. The nonlinearity of the system increases as the value of ${\gamma }_1$ reduces. This can be observed from the nonlinear part in equation (20). The $\lambda$ part is a function of ${\kappa }_2$ , which largely determines the response of the curve. For the negative values of $\lambda$ as seen, the response curves lean toward the lower frequencies, resulting in a softening response. Accordingly, choosing suitable parameters for the vehicle seat, the driver can experience comfortable ride by the greater frequency range isolation. Figure 12 shows the transmissivity curve for various damping at zero stiffness system. It is found that transmissivity of the vehicle seat is sensitive to the damping, although its effect is only observed around the resonance frequency. ²⁰

Figure 11.

Transmissibility frequency response graph of the zero stiffness system for $\overline{q}=0\mathrm{and}\alpha =1.2$

Figure 12.

Transmissibility frequency response graph of the zero stiffness system for different damping.

Conclusion

An alternative method to obtain negative stiffness for single degree of freedom nonlinear oscillators has been proposed in this study. The proposed characteristics of the nonlinear isolator consist of three springs; the mechanical spring provides a positive stiffness and the two secondary springs (a bar and Euler beam) act as a negative stiffness. The principles and fundamental characteristics of the combination of the mechanical springs and the nonlinear stiffness of the Euler beam at static and dynamics states were studied. This study essentially focused on the mathematical ways to obtain low dynamic stiffness of the vehicle seat in order to improve the comfort of the driver in low frequency. The proper selection of the configuration parameters for design purpose of the vehicle seat needs to be further addressed.

The concept presented may as well be applied to the design of other advanced vibration absorption components in engineering. Hence, the work serves as a potential for various applications in mechanical engineering.