Dynamic performance of an inerter vibration isolator under base excitation considering friction

Abstract

The inerter is a vibration control element related to the acceleration between its two ends, which can increase inertia through a speed-increasing mechanism. Applying the inerter to a vibration isolator can enhance its low-frequency vibration isolation performance. Frictional force always exists in the high-speed mechanism of the inerter. Even if it is not large, its influence on the dynamic performance of the low-speed end cannot be ignored. This study analyzes the force of a ball-screw inerter using the kinetic energy theorem. The inerter force is then expanded into the first-order Taylor approximation, from which the inerter coefficient and apparent friction coefficient are determined. Based on the approximate formula of the inerter force, the nonlinear dynamic model and its corresponding dynamic equation for the inerter vibration isolator under base excitation are established. The nonlinear dynamic equation is then solved by averaging method, and the approximate analytical solutions for the relative displacement, absolute displacement, and displacement transmissibility are obtained. The analysis results show that a larger apparent friction coefficient reduces the resonance peaks of the displacement transmissibility and relative displacement amplitude, but it slightly increases the initial vibration isolation frequency, as well as the resonance and valley frequencies of the displacement transmissibility and relative displacement. Increasing the inerter-mass ratio can reduce the resonance frequency and initial vibration isolation frequency, thereby broadening the vibration isolation frequency domain. When friction is considered, increasing the inerter-mass ratio can suppress the resonance peaks of the displacement transmissibility and relative displacement.

Introduction

Inerter is a kind of vibration control element related to the acceleration of its two ends, which can achieve larger inertia with smaller physical mass.^1,2 In 2002, Smith ³ first proposed the concept of “inerter” by analogy between the mechanical model and circuit model, and subsequently developed its physical implementation. In the early 2000s, the inerter was successfully applied to the suspension system of an F1 racing car, which significantly improved its operation performance. ⁴

Passive methods of reducing low-frequency vibrations are used to reduce the transmission of vibrations from the source to the component. Stosiak M et al. ⁵ indicate the sources of mechanical vibrations in machines and identify their frequencies. They also analysis the possibility of using passive vibration isolation with linear and non-linear characteristics. In general, the inerter exhibits inertia-increasing which can significantly enhance vibration isolation performance in the low-frequency range. Soong et al. ⁶ established a quarter-car model, in which the inerter was paired in parallel with passive, semi-active, and active vehicle suspension systems. They proved that the parallel inerter could improve the vibration isolation performance. Wang et al. ⁷ investigated the dynamic characteristics of the quasi-zero-stiffness inerter isolator, demonstrating that it exhibited smaller absolute displacement and acceleration transmissibility in the high-frequency band. Tang et al.^8–10 compared the vibration isolation performance of traditional vibration isolators and inerter-based isolators and proved that increasing the inerter coefficient of the inerter-based isolator would reduce both resonance and anti-resonance frequencies, thereby broadening the effective vibration isolation frequency band. Kang et al. ¹¹ investigated an integrated energy harvesting and vibration isolation system with an inertial amplification mechanism. They indicated that this mechanism could increase the system’s dynamic effective mass, and enhance its low-frequency vibration isolation performance. Dai et al. ¹² investigated a piecewise vibration isolation system incorporating an inerter. This system exhibited a smaller dynamic response peak, a smaller absolute displacement transmissibility peak, and a wider vibration isolation frequency band. Other researchers^13,14 introduced an inerter absorber into the vibration isolation system which reduced the displacement response and force transmissibility. Lou et al. ¹⁵ incorporated the inerter into the local resonance unit, further forming an acoustic metamaterial that enabled low-frequency vibration bandgap formation and blocked elastic wave propagation without increasing the system mass.

There are various nonlinear factors in practical mechanical systems that may cause nonlinear dynamic behaviors such as bifurcation and chaos. Nonlinear factors also affect the motion stability of the system. Amer T et al.^16,17 investigated the nonlinear stability analyses of a two-degrees-of-freedom dynamical systems comprising a nonlinear damping and illustrated the impact of varying parameters on the dynamic behavior of the system. They also discussed the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. Tarek S et al.^18–20 investigated and derived the nonlinear dynamic equations of spring pendulum systems. They employed the multiple scales technique to solve the dynamic equations, acquired the conditions of solvability for steady-state solutions, and outlined the stability regions for various parameters of the system. Sometimes, nonlinear devices are artificially designed to improve the performance of certain systems. For instance, the quasi-zero stiffness vibration isolators can maintain the stability of the system while achieving efficient vibration attenuation. ²¹ The obliquely symmetric arrangement of linear inerters or their combination with a linkage mechanism could achieve nonlinear inerter characteristics. Feng et al. ²² proposed a passive vibration isolator with nonlinear inertia based on an X-shaped mechanism and a rotating mass unit. They found that the introduction of nonlinear inertia could significantly enhance vibration isolation performance in the low-frequency range. Yang et al. ²³ applied the oblique inerter to the suspension damping system and proved that the integration of the oblique inerter enhanced damping performance under impact and random excitation conditions. Song et al. ²⁴ proposed a two-degree-of-freedom vibration isolation structure integrated with geometrically nonlinear inerters. This structure exhibited significantly vibration suppression performance. Tai et al. ²⁵ investigated the nonlinear inertia characteristics of the crank-connecting rod mechanism inerter and derived a simplified expression for its inertia force. The study demonstrated that the crank-connecting rod mechanism inerter exhibited superior vibration isolation control performance compared to linear inerter devices.

Some scholars had studied the influence of friction on the dynamic performance of the inerter. Brzeski et al. ²⁶ investigated the dynamics of tuned mass dampers with inerters and analyzed the influence of inerter gear friction on the dynamic characteristics of the tuned mass damper. Shen et al. ²⁷ proposed an optimized nonlinear model for fluid inerters, and analyzed the effect of inherent friction on the dynamic performance of fluid inerters under low-frequency excitation which confirmed by the experimental results. Chao et al. ²⁸ employed power flow analysis to quantitatively investigate internal vibration transmission and energy dissipation in inerter isolators with nonlinear friction. Wen et al. ²⁹ derived the nonlinear inerter coefficient considering friction, and further demonstrated the consistency between the theoretical and experimental results. Considering frictional effects, Wu et al. ³⁰ proposed a hybrid Type II inerter-based nonlinear energy sink and analyzed the friction’s impact on vibration reduction performance.

Friction is inevitable in practical mechanical systems. As a speed-increasing mechanism, the inerter can generate considerable energy dissipation even with a very small friction coefficient at its high-speed end, resulting in significant damping characteristics in the vibration system (at the low-speed end). For friction analysis, Wu et al. ³⁰ simplified the friction force to a constant value, Brzeski et al. ²⁶ simplified the friction force to a velocity-dependent force, and Wen et al. ²⁹ considered the relationship between friction force and acceleration, further deriving a nonlinear inerter coefficient that accounts for friction factors. Accurately analyzing the relationship between friction force and motion parameters, and determining the impact of inerter friction on vibration isolation systems, can offer substantial guiding value for the design and parameter selection of inerter-based vibration isolators. In this study, the force of the ball-screw inerter considering friction, is analyzed using the kinetic energy theorem. The inerter force is simplified through first-order Taylor approximation, deriving both the inerter coefficient and the apparent friction coefficient at the low-speed end. Furthermore, the nonlinear dynamic model and dynamic equation of the inerter isolator under base excitation are established. The equation is solved using the averaging method, yielding approximate analytical solutions for relative displacement, absolute displacement, displacement transmissibility and base force. The effects of the apparent friction coefficient and inerter-mass ratio on the dynamic performance of the vibration isolator are systematically analyzed.

Inerter force analysis considering friction

This section introduces the mechanical model of the inerter vibration isolator considering friction under the influence of basic excitation. Based on Newton’s laws and the principle of conservation of kinetic energy, the forces borne by the inerter when considering friction are derived, and the inerter coefficient and apparent friction coefficient are obtained.

The dynamic model of an inerter vibration isolator under base excitation considering friction is shown in Figure 1. In this model, $m$ , $c$ and $k$ are the mass, damping coefficient and stiffness of the vibration isolator. The system is excited by base excitation $x_{\mathrm{s}}=x_0\hspace{0.17em}\cos \left(\omega t\right)$ , where $x_0$ and $\omega$ represents the excitation amplitude and excitation frequency, respectively. $b$ represents the inerter coefficient, and $f$ represents the apparent friction coefficient. $b$ and $f$ will be detailed hereinafter. As shown in Figure 2, the ball-screw inerter structure is composed of an upper ear plate, a retainer ring, a key, an upper cover, a ball screw, a flywheel, a shell, two bearings, and a lower ear plate. The ball screw and the flywheel form a spiral mechanism. The flywheel has large inertia. The reciprocating motion of the ball screw drives the flywheel to rotate. Bearings are installed on the upper cover and the shell to bear the axial force transmitted by the flywheel. The upper ear plate and the lower ear plate serve as connection ends.

Figure 1.

Model of an inerter vibration isolator under base excitation.

Figure 2.

Ball-screw inerter structure. 1-upper ear plate 2-retainer ring 3-key 4-upper cover 5-ball screw 6-flywheel 7-shell 8-bearing 9-lower ear plate.

Letting the force acting on the inerter be $F$ , the force exerted by the inerter on $m$ is $F^{\prime }$ . By performing a force analysis on $m$ , it can be derived

$F=-F^{\prime }=-m\ddot{x}-k\left(x-x_{\mathrm{s}}\right)-c\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)$

(1)

Supposing the lead of the ball screw be $p$ , the moment of inertia of the flywheel be $I$ , the kinetic energy of the flywheel be $W$ , the rotation angle of the flywheel be $\phi$ , the angular velocity be $\dot{\phi }$ , the angular acceleration be $\ddot{\phi }$ , the mean radius of the ball screw be $d_1$ , the mean radius of the circle where the centers of the rolling elements of the bearing are located be $d_2$ , the friction factor of the ball screw be $f_1$ , and the friction factor of the rolling elements of the bearing be $f_2$ . Let the sign of $\phi$ be the same as that of $x-x_{\mathrm{s}}$ . Considering that the friction torque always does negative work and neglecting the axial translational kinetic energy of the inerter, by applying the kinetic energy theorem to the inerter, it can be derived as

$F\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)-F\dot{\phi }\frac{f_1{d}_1+f_2{d}_2}{2}\text{sgn}\left(F\dot{\phi }\right)=\frac{dW}{dt}=\frac{d}{dt}\left(\frac{1}{2}I{\dot{\phi }}^2\right)=I\dot{\phi }\ddot{\phi }$

(2)

Taking the first and second derivatives of the geometric relationship $\frac{d\phi }{2\pi }=\frac{d\left(x-x_{\mathrm{s}}\right)}{p}$ of the ball-screw mechanism with respect to time, we obtain $\dot{\phi }=\frac{2\pi }{p}\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)$ and $\ddot{\phi }=\frac{2\pi }{p}\left(\ddot{x}-{\ddot{x}}_{\mathrm{s}}\right)$ . Substituting them into equation (2) and rearranging it yields

$F=\frac{I{\left(\frac{2\pi }{p}\right)}^2\left(\ddot{x}-{\ddot{x}}_{\mathrm{s}}\right)}{1-\frac{\pi \left(f_1{d}_1+f_2{d}_2\right)}{p}\text{sgn}\left[F\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)\right]}=\frac{b\left(\ddot{x}-{\ddot{x}}_{\mathrm{s}}\right)}{1-f\text{sgn}\left[\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)\left(\ddot{x}-{\ddot{x}}_{\mathrm{s}}\right)\right]}$

(3)

where $b=I{\left(\frac{2\pi }{p}\right)}^2$ , which has the same dimension as mass and is called an inerter coefficient. $f=\frac{\pi \left(f_1{d}_1+f_2{d}_2\right)}{p}$ is a linear combination of the friction coefficients $f_1$ and $f_2$ , and is referred to as the apparent friction coefficient. In general, p is relatively small, while d₁ and d₂ are relatively large. Therefore, the apparent friction coefficient f cannot be neglected, even if f₁ and f₂ are very small. The inerter is in operation (i.e., the reciprocating motion of the ball screw can actuate the flywheel rotation) meaning that $f<1$ . Otherwise, self-locking will occur, that is, the ball screw will not drive the flywheel to rotate. Thus, in equation (3), $F$ and $\ddot{x}-{\ddot{x}}_{\mathrm{s}}$ have the same sign.

Dynamic equation of the inerter vibration isolator under base excitation

In this section, the dynamic equation of the inerter vibration isolator under the base excitation is established. The inerter force is simplified to a first-order approximate formula. The dynamic equation is solved by numerical methods. Subsequently, the dynamic responses of the first-order approximate formula and the exact formula are compared to determine the parameter range of the apparent friction coefficient for which the first-order approximate formula is applicable.

Let the relative displacement be $y=x-x_{\mathrm{s}}$ , and the inerter-mass ratio be $b^{\prime }=\frac{b}{m}$ , and after substituting equation (3) into equation (1) one can get

$m\ddot{y}+F+c\dot{y}+ky=m\ddot{y}+\frac{b^{\prime }m\ddot{y}}{1-f\text{sgn}\left(\dot{y}\ddot{y}\right)}+c\dot{y}+ky=m{\omega }^2{x}_0\hspace{0.17em}\cos \left(\omega t\right)$

(4)

Expanding equation (3) into a Taylor polynomial at $f=0$ yields

$F=b\ddot{y}+fb\text{sgn}\left(\dot{y}\ddot{y}\right)\ddot{y}+f^{2}b\ddot{y}+f^{3}b\text{sgn}\left(\dot{y}\ddot{y}\right)\ddot{y}+\cdots$

(5)

When $f$ is small, the second order term and the higher-order terms above the second in formula (5) are relatively smaller than the zero-order and first-order terms and can be omitted (Figure 4 in the following part proves that when $f\le 0.5$ , omitting the second-order and higher-order terms has little effect on the dynamic analysis results). Neglecting the second-order and higher-order terms in equation (5), substituting into equation (4), and rearranging yields

$\ddot{y}+\frac{c\dot{y}}{m\left[1+b^{\prime }+b^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)\right]}+\frac{ky}{m\left[1+b^{\prime }+b^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)\right]}=\frac{{\omega }^2{x}_0\hspace{0.17em}\cos \left(\omega t\right)}{1+b^{\prime }+b^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}$

(6)

Expanding $\frac{1}{1+b^{\prime }+f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)]}$ in equation (3) into the first-order Taylor approximation at $f=0$ yields

$\frac{1}{1+b^{\prime }+f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}=\frac{1}{1+b^{\prime }}-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{{\left(1+b^{\prime }\right)}^2}=\frac{1}{1+b^{\prime }}\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]$

(7)

Substituting equation (7) into equation (6) and rearranging one can obtain

$\ddot{y}+\frac{k}{m\left(1+b^{\prime }\right)}y=\frac{1}{1+b^{\prime }}\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]{\omega }^2{x}_0\hspace{0.17em}\cos \left(\omega t\right)-\frac{c}{m\left(1+b^{\prime }\right)}\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]\dot{y}+\frac{k}{m}\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{{\left(1+b^{\prime }\right)}^2}y$

(8)

Compared with equation (4), the force related to the inerter in equation (8) has been simplified twice through the first-order Taylor expansion. To determine the solution deviation between equations (4) and (8), and the applicable range of the parameter $f$ , the Runge–Kutta method is used to numerically solve equations (4) and (8) for comparison. The parameters are selected as $m=100$ kg, $k=4\times {10}^4\mathrm{N}/\mathrm{m}$ , $c=100\mathrm{N}·\mathrm{s}/\mathrm{m}$ , $b^{\prime }=2$ , and the excitation frequency $\omega =11\text{rad}/\mathrm{s}$ (close to the natural frequency ${\omega }_0$ which is obtained as ${\omega }_0=\sqrt{\frac{k}{m\left(1+b^{\prime }\right)}}=\sqrt{\frac{40000}{100\times \left(1+2\right)}}=11.5\text{rad}/s$ from equation (8)). Under the conditions that the excitation amplitude is $x_0=10\text{mm}$ , the initial conditions are $y_{00}=0$ , ${\dot{y}}_{00}=0$ , and $f$ is take as 0.1, 0.25, 0.5, 0.75 and 0.95 respectively, the comparisons of the time-domain responses of equations (4) and (8) are shown in Figure 3. Let the steady-state amplitude of the relative displacement $y$ be ${y_0}^{\prime }$ , amplitude ratio of the relative displacement $\beta =\frac{{y_0}^{\prime }}{x_0}$ , and the frequency ratio $\lambda =\frac{\omega }{{\omega }_0}$ . The comparison of variations of the numerical solutions $\beta$ of equations (4) and (8) with frequency ratio $\lambda$ are shown in Figure 4.

(1) Figure 3 shows that as $f$ increases, the time from the transient state to the steady state decreases. This is a normal phenomenon caused by the increase in friction damping force due to the increase of $f$ .

(2) Figure 3(a) and Figure 3(b) indicate that when $f$ takes a small value, the numerical solutions for equations (8) and (4) are relatively close. However, the numerical solution of equation (8) is significantly larger than that of equation (4), when $f$ takes a large value, which can be shown in Figure 3(d). The reason for this phenomenon can be explained herein. In equation (4), the inerter force is not simplified, which is equivalent to the infinite-order Taylor polynomial of $F$ in equation (5). According to equation (5), it is found that the even-order terms of $F$ play a role in increasing inertia, while the odd-order terms play a role in increasing damping. However, the higher-order terms (beyond the second order) in equation (5) are omitted in equation (8), which implies a decrease in system damping and results in a larger amplitude of vibration in equation (8). This difference becomes more obvious as the $f$ increases, which is illustrated by Figure 3(c) and Figure 3(d). It is also the reason why the numerical solution of equation (8) is significantly greater than that of equation (4) near the resonance frequency in Figure 4.

(3) Figure 4 shows that the resonance frequency of the numerical solution of equation (8) is higher than that of equation (4), and the difference increases with the growth of $f$ . This is because equation (8) neglects the higher-order terms beyond the second order in equation (5), which means that the inertia is reduced, thereby increasing the natural frequency.

(4) Considering Figures 3 and 4, the dynamic characteristics reflected by equations (8) and (4) are not significantly different in terms of both time-domain response and amplitude-frequency characteristics when $f$ is less than 0.5. The subsequent analysis investigates the dynamic characteristics of the inerter vibration isolator under base excitation when $f$ is less than 0.5 by solving equation (8).

(5) From Figure 4(e), it can be found that the relative displacement is very small when the apparent friction coefficient is large. In this case, the isolated object and the foundation are approximately connected as a rigid structure through the inerter, so the isolator can hardly have any vibration isolation effect.

Figure 3.

Comparisons of the responses between numerical solutions of equations (8) and (4).

Figure 4.

Comparisons of amplitude ratio of the relative displacement $\beta$ with the frequency ratio $\lambda$ between the numerical solutions of equations (8) and (4).

Analytical solution

In this section, the dynamic equation of the inerter vibration isolator is simplified to a dimensionless form. The equation is solved using the averaging method, and the approximate analytical solution for the relative displacement response of the isolator under harmonic excitation is obtained.

Introducing the dimensionless parameters ${\zeta }_0=\frac{c}{2\sqrt{km\left(1+b^{\prime }\right)}}$ , $\omega { }_0^2=\frac{k}{m\left(1+b^{\prime }\right)}$ , $y_0=\frac{x_0}{1+b^{\prime }}$ ,and substituting them into equation (8) and rearranging it yields

$\ddot{y}+{\omega }_0^{2}y=y_0\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]{\omega }^2\hspace{0.17em}\cos \left(\omega t\right)-2{\zeta }_0{\omega }_0\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]\dot{y}+{\omega }_0^2\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}y$

(9)

To study the nonlinear vibration near the main resonance frequency, a small parameter $\epsilon$ is introduced. Here the amplitude of the base excitation is assumed with the same order as the small parameter $\epsilon$ . Meanwhile, let ${\zeta }_0=\epsilon \zeta$ and ${\omega }^2={\omega }_0^2\left(1+\epsilon \sigma \right)$ , where σ represents the detuning coefficient. Then, equation (9) becomes

$\ddot{y}+{\omega }^{2}y=\epsilon \left[y_0\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]{\omega }^2\hspace{0.17em}\cos \left(\omega t\right)-2\zeta {\omega }_0\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]\dot{y}+{\omega }_0^2\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}y+\sigma {\omega }_0^{2}y\right]$

(10)

The averaging method is used to solve equation (10). Assuming that the form of the solution to equation (10) and its derivative are as

$y=a\hspace{0.17em}\cos \left(\omega t-\theta \right)$

(11)

$\dot{y}=-\omega a\hspace{0.17em}\sin \left(\omega t-\theta \right)$

(12)

where θ is the phase difference between the excitation and the response. Differentiating equation (11) with respect to time and substituting it into equation (12) yields

$\dot{a}\cos \hspace{0.17em}\psi +a\dot{\theta }\sin \hspace{0.17em}\psi =0$

(13)

where $\psi =\omega t-\theta$ . Differentiating equation (12) with respect to time, and substituting it into equation (11) yields

$-\dot{a}\sin \hspace{0.17em}\psi +a\dot{\theta }\cos \hspace{0.17em}\psi =\frac{\epsilon }{\omega }{f}_1\left(y,\dot{y},\omega t\right)$

(14)

where

$f_1\left(y,\dot{y},\omega t\right)=f_1\left(a\hspace{0.17em}\cos \hspace{0.17em}\psi ,-\omega a\hspace{0.17em}\sin \hspace{0.17em}\psi ,\psi +\theta \right)$

$=\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]\cos \left(\psi +\theta \right){\omega }^2{y}_0+2\zeta {\omega }_0\omega a\left[1-\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\right]\sin \hspace{0.17em}\psi +{\omega }_0^{2}a\frac{f{b}^{\prime }\text{sgn}\left(\dot{y}\ddot{y}\right)}{1+b^{\prime }}\cos \hspace{0.17em}\psi +\sigma {\omega }_0^{2}a\hspace{0.17em}\cos \hspace{0.17em}\psi$

(15)

Solving equations (13) and (14) simultaneously yields

$\dot{a}=-\frac{\epsilon }{\omega }{f}_1\left(y,\dot{y},\omega t\right)\sin \hspace{0.17em}\psi =-\frac{\epsilon }{\omega }{f}_1\left(a\hspace{0.17em}\cos \hspace{0.17em}\psi ,-a\omega \hspace{0.17em}\sin \hspace{0.17em}\psi ,\psi +\theta \right)\sin \hspace{0.17em}\psi$

(16)

$\dot{\theta }=\frac{\epsilon }{\omega a}{f}_1\left(y,\dot{y},\omega t\right)\cos \hspace{0.17em}\psi =\frac{\epsilon }{\omega a}{f}_1\left(a\hspace{0.17em}\cos \hspace{0.17em}\psi ,-a\omega \hspace{0.17em}\sin \hspace{0.17em}\psi ,\psi +\theta \right)\cos \hspace{0.17em}\psi$

(17)

The solutions of equations (16) and (17) can yield the amplitudes and phases of the relative displacement response. The solution process is presented in Appendix A

$a=\epsilon {\lambda }^2{y}_0\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}$

(18)

$\sin \hspace{0.17em}\theta =\frac{AA}{\sqrt{A{A}^2+B{B}^2}}$

(19)

$\cos \hspace{0.17em}\theta =\frac{BB}{\sqrt{A{A}^2+B{B}^2}}$

(20)

where $AA=2{\pi }^2\epsilon \zeta \lambda {\left(1+b^{\prime }\right)}^2+2f{b}^{\prime }\pi \left({\lambda }^2-1+\epsilon \right)\left(b^{\prime }+1\right)-8\zeta \epsilon f^{2}b{\prime }^2\lambda$ , and $BB=\left(1-{\lambda }^2\right){\left(1+b^{\prime }\right)}^2{\pi }^2+4\epsilon f^{2}b{\prime }^2$ .

Displacement transmissibility

Based on the relative displacement solution, this section solves the absolute displacement and further derives the numerical solution for the displacement transmissibility. Then the accuracy of the approximate solution is verified by a numerical method.

Using $y=x-x_{\mathrm{s}}$ and $y_0=\frac{x_0}{1+b^{\prime }}$ , substituting equation (18) into equation (11) and rearranging it yields

$\begin{array}{l}x=y+x_s=\epsilon {\lambda }^2\frac{x_0}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\cos \left(\omega t-\theta \right)+\epsilon x_0\hspace{0.17em}\cos \left(\omega t\right)\\ =\epsilon {\lambda }^2\frac{x_0}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\cos \left(\omega t\right)\cos \hspace{0.17em}\theta +\epsilon {\lambda }^2\frac{x_0}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\sin \left(\omega t\right)\sin \hspace{0.17em}\theta +\epsilon x_0\hspace{0.17em}\cos \left(\omega t\right)\\ =\left[\epsilon {\lambda }^2\frac{x_0}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\cos \hspace{0.17em}\theta +\epsilon x_0\right]\cos \left(\omega t\right)+\left[\epsilon {\lambda }^2\frac{x_0}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\sin \hspace{0.17em}\theta \right]\sin \left(\omega t\right)\\ =\epsilon x_0\sqrt{{\left[{\lambda }^2\frac{1}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\right]}^2+2{\lambda }^2\frac{1}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\cos \hspace{0.17em}\theta +1}\cos \left(\omega t-\phi \right)\\ =\epsilon x_0\sqrt{{\left[{\lambda }^2\frac{1}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\right]}^2+2{\lambda }^2\frac{1}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}\frac{BB}{\sqrt{A{A}^2+B{B}^2}}+1}\cos \left(\omega t-\phi \right)\end{array}$

$=\epsilon x_0\sqrt{\frac{{\lambda }^4{\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]}^2+2{\lambda }^2\left(1+b^{\prime }\right)\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]BB}{{\left(1+b^{\prime }\right)}^2\left(A{A}^2+B{B}^2\right)}+1}\cos \left(\omega t-\phi \right)$

(21)

where

$\tan \hspace{0.17em}\phi =\frac{{\lambda }^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]\sin \hspace{0.17em}\theta }{{\lambda }^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]\cos \hspace{0.17em}\theta +\left(1+b^{\prime }\right)\sqrt{A{A}^2+B{B}^2}}$

$=\frac{{\lambda }^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]}{{\lambda }^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]\frac{BB}{\sqrt{A{A}^2+B{B}^2}}+\left(1+b^{\prime }\right)\sqrt{A{A}^2+B{B}^2}}\frac{AA}{\sqrt{A{A}^2+B{B}^2}}=\frac{{\lambda }^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]AA}{{\lambda }^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]BB+\left(1+b^{\prime }\right)\left(A{A}^2+B{B}^2\right)}$

(22)

For a vibration isolator under base excitation, its vibration isolation effect is determined by the displacement transmissibility η, which is the ratio of the vibration amplitude of $m$ to the amplitude of the base excitation. It can be directly obtained from equation (21) as

$\eta =\sqrt{\frac{{\lambda }^4{\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]}^2+2{\lambda }^2\left(1+b^{\prime }\right)\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]BB}{{\left(1+b^{\prime }\right)}^2\left(A{A}^2+B{B}^2\right)}+1}$

(23)

Substituting equation (3) into equation (1), and rearranging it yields equation (24) where $x_{\mathrm{s}}=x_0\hspace{0.17em}\cos \left(\omega t\right)$ , ${\dot{x}}_{\mathrm{s}}=-\omega x_0\hspace{0.17em}\sin \left(\omega t\right)$ , and ${\ddot{x}}_{\mathrm{s}}=-{\omega }^2{x}_0\hspace{0.17em}\cos \left(\omega t\right)$ . The steady-state amplitude of equation (24) can be solved using the Runge–Kutta method, and then dividing it by the amplitude of the base excitation $x_0$ will give the value η

$m\ddot{x}+\frac{b^{\prime }m\ddot{x}}{1-f\text{sgn}\left[\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)\left(\ddot{x}-{\ddot{x}}_{\mathrm{s}}\right)\right]}+c\dot{x}+kx=\frac{b^{\prime }m{\ddot{x}}_{\mathrm{s}}}{1-f\text{sgn}\left[\left(\dot{x}-{\dot{x}}_{\mathrm{s}}\right)\left(\ddot{x}-{\ddot{x}}_{\mathrm{s}}\right)\right]}+c{\dot{x}}_{\mathrm{s}}+k{x}_{\mathrm{s}}$

(24)

Let $\epsilon =1$ , $\zeta =0.1$ , $b^{\prime }=2$ and $f$ takes values of 0.1, 0.25, 0.5 and 0.75 respectively. The comparisons of the variations of the amplitude ratio $\eta$ with the frequency ratio $\lambda$ are shown in Figure 5 where the curves with the legend “NS” (Numerical Solutions) are the numerical solutions of equation (24), and the curves with the legend “AAS” (Approximate Analytical Solutions) are the approximate solutions of equation (23). It can be seen that when the friction coefficient $f\le 0.5$ , the two curves agree very well, which proves that the applicable parameter range of the approximate equation (23) is $f\le 0.5$ .

Figure 5.

Comparisons between numerical solutions and approximate solutions for displacement transmissibility $\eta$ .

Analysis of vibration isolation characteristics

In this section, the effects of the apparent friction coefficient, the mass ratio and the damping ratio on the displacement transmissibility and the relative displacement are investigated in detail.

Let $\zeta =0$ , $b^{\prime }=2$ , $\epsilon =1$ and $f$ takes the values of 0, 0.1, 0.25, and 0.5 respectively. The variations of the displacement transmissibility $\eta$ with the frequency ratio $\lambda$ are shown in Figure 6.

Figure 6.

Variation of displacement transmissibility $\eta$ with frequency ratio $\lambda$ under different apparent friction coefficients $f$ .

The following characteristics of $\eta$ can be noted from Figure 6.

(1) As $\lambda$ increases, $\eta$ shows the characteristics of first increasing, then decreasing, and then increasing again.

(2) In case $f=0$ , $\eta$ will approach to infinity as $\lambda \to 1$ . When $\lambda >1$ , the minimum value of $\eta$ is equal to 0.

(3) In case $f\ne 0$ , $\eta$ has a maximum value (resonance peak value) and a minimum value (valley value). As $f$ increases, the resonance peak value decreases, the valley value increases, and the $\lambda$ at the positions of the resonance peak value and the valley value increases slightly.

(4) The initial vibration isolation frequency ratio of the system is smaller than $\sqrt{2}$ , that is, the initial vibration isolation frequency ratio of the traditional linear vibration isolator. And the initial vibration isolation frequency ratio increases as $f$ increases.

(5) $\eta$ tends to a stable value as $\lambda$ increases at high frequencies.

In fact, equation (23) can be rearranged when $\lambda \to \infty$

$\eta =\frac{b^{\prime }}{\pi }\sqrt{\frac{{\left(1+b^{\prime }\right)}^4{\pi }^4+4f^2\left(b{\prime }^2+4b^{\prime }+1\right){\left(1+b^{\prime }\right)}^2{\pi }^2+16b{\prime }^2{f}^4}{{\left(1+b^{\prime }\right)}^4\left({\pi }^{2}b{\prime }^2+4f^{2}b{\prime }^2+2{\pi }^2{b}^{\prime }+{\pi }^2\right)}}$

(25)

Equation (25) indicates that when $\lambda \to \infty$ , $\eta$ is independent of $\lambda$ and $\zeta$ , and is solely determined by $f$ and $b^{\prime }$ . The variation of $\eta$ with $\lambda$ and $b^{\prime }$ is shown in Figure 7. It can be seen that $\eta$ increases as $b^{\prime }$ increases. When $b^{\prime }\ne 0$ , $\eta$ increases slightly as $f$ increases, but its influence is not obvious.

Figure 7.

Variation of displacement transmissibility $\eta$ with apparent friction coefficient $f$ and inerter-mass ratio $b^{\prime }$ as frequency ratio $\lambda \to \infty$ .

Let $b^{\prime }=2$ , $\epsilon =1$ . The displacement transmissibility under action of inerter friction force and linear damping force are shown in Figure 8(a). It indicates that both inerter friction force and linear damping force can effectively suppress the peak value of displacement transmissibility. Increasing the inerter friction damping or linear damping, will cause the valley value of the displacement transmissibility curve to increase or even disappear completely. From Figure 8(b), it can be seen that a larger apparent friction coefficient can slightly increase the initial vibration isolation frequency of the system and for the cases where the inerter friction damping is the same but the linear damping is different, the displacement transmissibility curves always intersect at the position $\eta =1$ .

Figure 8.

Effect of apparent friction coefficient $f$ and damping ratio $\zeta$ on displacement transmissibility $\eta$ .

The abscissa coordinate frequency ratio $\lambda$ in Figure 4 to Figure 6 represents the ratio of the excitation frequency $\omega$ to the natural frequency ${\omega }_0=\sqrt{\frac{k}{m\left(1+b^{\prime }\right)}}$ . This approach is usually applicable when $b^{\prime }$ remains unchanged. When analyzing the influence of $b^{\prime }$ on $\eta$ , since the change of $b^{\prime }$ will affect ${\omega }_0$ , ${\omega }_0$ should not be used as a reference. Therefore, a fixed and unchanging frequency should be selected as a reference. For this purpose, we can take ${\omega }_0^{\prime }=\sqrt{\frac{k}{m}}$ as a reference, define a new frequency ratio ${\lambda }^{\prime }=\frac{\omega }{{{\omega }_0}^{\prime }}$ , then we have $\lambda =\sqrt{1+b^{\prime }}{\lambda }^{\prime }$ . Substituting it into equation (23) yields

$\eta =\sqrt{\frac{{\left(1+b^{\prime }\right)}^2\lambda {\prime }^4{\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]}^2+2\lambda {\prime }^2{\left(1+b^{\prime }\right)}^2\left[{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2\right]B{B}^{\prime }}{{\left(1+b^{\prime }\right)}^2\left(AA{\prime }^2+BB{\prime }^2\right)}+1}$

(26)

where

$A{A}^{\prime }=\sqrt{1+b^{\prime }}{\lambda }^{\prime }\epsilon \zeta \left[2{\pi }^2{\left(1+b^{\prime }\right)}^2-8f^{2}b{\prime }^2\right]+2f{b}^{\prime }\pi \left[\left(1+b^{\prime }\right)\lambda {\prime }^2-1+\epsilon \right]\left(b^{\prime }+1\right),B{B}^{\prime }=\left[1-\left(1+b^{\prime }\right)\lambda {\prime }^2\right]{\left(1+b^{\prime }\right)}^2{\pi }^2+4\epsilon f^{2}b{\prime }^2$

Let $\zeta =0$ , $f=0.5$ , $\epsilon =1$ and $b^{\prime }$ takes the values of 0, 0.5, 1, and 2.0, respectively. The variation of the displacement transmissibility $\eta$ with the frequency ratio ${\lambda }^{\prime }$ is shown in Figure 9. The following characteristics of displacement transmissibility $\eta$ can be noted from Figure 9.

(1) In case $b^{\prime }=0$ , the vibration isolator is equivalent to a linear vibration isolator, and the initial vibration isolation frequency is $\sqrt{2}$ times the natural frequency. As $b^{\prime }$ increases, the initial vibration isolation frequency decreases, and the vibration isolation frequency domain is broadened.

(2) In case $b^{\prime }\ne 0$ , as ${\lambda }^{\prime }$ gradually increases, $\eta$ first increases, then decreases and then increases. $\eta$ has a maximum value (resonance peak value) and a minimum value (valley value).

(3) As $b^{\prime }$ increases, the resonance frequency of the system decreases. This is because the inertia of the system increases, resulting in a decrease in the natural frequency. At the same time, the valley frequency also decreases.

(4) The resonance peak value decreases significantly as $b^{\prime }$ increases. This is because the frictional damping force of the inerter is described as $fb\text{sgn}\left(\dot{y}\ddot{y}\right)\ddot{y}$ . The larger $b^{\prime }$ is, the larger the frictional damping force is, and the more effective it is in suppressing the resonance peak value. It can also be found that $b^{\prime }$ has little influence on the valley value.

(5) $\eta$ tends to a stable value as ${\lambda }^{\prime }$ increases, and this stable value increases as the $b^{\prime }$ increases. It is consistent with the analysis conclusion in Figure 7.

(6) When $f=0$ and ${\lambda }^{\prime }\to \infty$ , Eq.(23) is reduced to $\eta =\frac{1}{\frac{1}{b^{\prime }}+1}$ . It is indicated that when the isolator operates under high-frequency excitation, $\eta$ increases as $b^{\prime }$ increases

Figure 9.

Variation of displacement transmissibility $\eta$ with frequency ratio ${\lambda }^{\prime }$ under different inerter-mass ratio $b^{\prime }$ .

Equation (18) represents the amplitude of the relative displacement between the isolated object $m$ and the base. By using $y_0=\frac{x_0}{1+b^{\prime }}$ and dividing equation (18) by the amplitude $\epsilon x_0$ of the base excitation, the amplitude ratio of relative displacement ${\eta }_{\mathrm{R}}$ of the relative displacement is obtained as

${\eta }_{\mathrm{R}}=\frac{{\lambda }^2}{1+b^{\prime }}\frac{{\left(1+b^{\prime }\right)}^2{\pi }^2-4f^{2}b{\prime }^2}{\sqrt{A{A}^2+B{B}^2}}$

(27)

Let $\zeta =0$ , $b^{\prime }=1$ , $\epsilon =1$ , and $f$ takes the values of 0, 0.1, 0.25, and 0.5, respectively. The variation of ${\eta }_{\mathrm{R}}$ with the inerter-mass ratio $\lambda$ is shown in Figure 10. It is indicated that an increase in $f$ significantly suppresses the resonance peak value of ${\eta }_{\mathrm{R}}$ . When $f$ increases, the resonance frequency slightly increases, and ${\eta }_{\mathrm{R}}$ tends to a stable value at high frequencies.

Figure 10.

Variation of the amplitude ratio of relative displacement ${\eta }_{\mathrm{R}}$ with frequency ratio $\lambda$ under different apparent friction coefficients $f$ .

Similar to Figure 9, the abscissa axis coordinate is taken as the frequency ratio ${\lambda }^{\prime }$ . Figure 11 shows the variation of ${\eta }_{\mathrm{R}}$ with ${\lambda }^{\prime }$ when $\zeta =0$ , $f=0.5$ , $\epsilon =1$ , and $b^{\prime }$ takes the values of 0, 0.5, 1, and 2, respectively. The figure indicates that an increase in $b^{\prime }$ reduces the resonance frequency of the relative displacement, and the increase in $b^{\prime }$ significantly suppresses the resonance peak value of the relative displacement. For a determined $b^{\prime }$ , ${\eta }_{\mathrm{R}}$ tends to a stable value as ${\lambda }^{\prime }$ increases and the value decreases as $b^{\prime }$ increases.

Figure 11.

Variation of the amplitude ratio of relative displacement ${\eta }_{\mathrm{R}}$ with frequency ratio $\lambda$ under different inerter-mass ratios $b^{\prime }$ .

Substituting $\lambda \left({\lambda }^{\prime }\right)\to \infty$ into equation (27) and rearranging it yields

${\eta }_{\mathrm{R}}=\frac{{\pi }^2{\left(1+b^{\prime }\right)}^2-4f^{2}b{\prime }^2}{\pi {\left(1+b^{\prime }\right)}^2\sqrt{{\pi }^2{\left(1+b^{\prime }\right)}^2+4f^{2}b{\prime }^2}}$

(28)

Figure 12 shows the variation of ${\eta }_{\mathrm{R}}$ with $f$ and $b^{\prime }$ as ${\lambda }^{\prime }\to \infty$ . It indicates that $b^{\prime }$ has a significant influence on ${\eta }_{\mathrm{R}}$ while $f$ has little influence on ${\eta }_{\mathrm{R}}$ , which verifies the conclusions of Figures 10 and 11.

Figure 12.

Variation of the amplitude ratio of relative displacement ${\eta }_{\mathrm{R}}$ with apparent friction coefficient $f$ and inerter-mass ratios $b^{\prime }$ as ${\lambda }^{\prime }\to \infty$ 7 .

Conclusions

The force on the inerter considering internal friction is analyzed. A nonlinear dynamic model and dynamic equation of the inerter vibration isolator under base excitation are established which is solved using the averaging method. The main conclusions are as follows.

(1) For the ball-screw inerter, even if the friction coefficient of the moving parts of the inerter is very small, its apparent friction coefficient cannot be ignored.

(2) The displacement transmissibility of the inerter vibration isolator exhibits a characteristic of first increasing, then decreasing, and then increasing again as the excitation frequency increases. When the apparent friction coefficient is considered, the displacement transmissibility has a resonance peak value (maximum value) and a valley value (minimum value). The apparent friction coefficient can significantly reduce the resonance peak value, but has little influence on the valley value. Both the resonance frequency and the valley frequency increase slightly with the increase of the apparent friction coefficient. The initial vibration isolation frequency of the inerter vibration isolator increases as the apparent friction coefficient increases. Under high-frequency excitation, the displacement transmissibility tends to a stable value which increases slightly as the friction coefficient increases. To sum up, the friction of the inerter vibration isolator is conducive to improving the comprehensive performance of the isolator, and a certain friction should be ensured when designing the inerter vibration isolator.

(3) Increasing the inerter-mass ratio can reduce the initial vibration isolation frequency and broaden the vibration isolation frequency range. As the inerter-mass ratio increases, both the resonance frequency and the valley frequency of the displacement transmissibility decrease. A larger inerter-mass ratio leads to a greater friction force, resulting in more significant suppression of the resonance peak, while the inerter-mass ratio has little effect on the valley value. Under high-frequency excitation, the displacement transmissibility increases with the increase of the inerter-mass ratio. To summarize, increasing the inerter-mass ratio is beneficial for low-frequency vibration isolation. Furthermore, as the inerter-mass ratio increases, the frictional force also increases, resulting in a better effect of suppressing the resonance peak. Therefore, for low-frequency vibration isolation systems, inerter with a large inerter-mass ratio should be adopted. On the other hand, increasing the inerter-mass ratio is detrimental to high-frequency vibration isolation. Thus, for high-frequency vibration isolation systems, the inerter-mass ratio should be reduced or inerters should be avoided.

(4) The apparent friction coefficient significantly suppresses the resonance peak of the relative displacement. Increasing the inerter-mass ratio reduces the resonance frequency of the relative displacement and significantly suppresses its resonance peak. The amplitude of the relative displacement tends to a stable value as the frequency ratio increases, and this value decreases with the increase of the inerter-mass ratio.

Although the internal friction exists in inerter, there are few studies on vibration isolation considering friction. There are various nonlinear factors in practical mechanical systems that may cause nonlinear dynamic behaviors such as bifurcation and chaos. Nonlinear factors also affect the motion stability of the system. The impact of friction on the nonlinear behavior of the inerter vibration isolator will be further considered in the future.