Dynamic Behavior and Isolation Performance of a Constant-Force Vibration Isolation System

Abstract

This paper will present a constant-force vibration isolator (CFVI), in which the isolated load is supported by two pulley-roller mechanisms, while the dynamic stiffness is modified by a cam mechanism with the piecewise profile redefined by the user. As a result, this model can generate the constant force-displacement response within the working region, thereby obtaining quasi-zero stiffness in this range. Because of the piecewise configuration of the cam, the system motion governed by the piecewise dynamic equation under base motion excitation will be analyzed and established. The approximate solution of the piecewise dynamic equation is derived by using the average method, from which the relative amplitude–frequency relation and the absolute amplitude transmissibility of the CFVI will be obtained. The effects of the key working parameters involving the damping coefficient, critical position, and excited amplitude on the dynamic behavior and isolation effectiveness of the CFVI are considered through numerical simulations. The simulation result reveals that the dynamic response of the CFVI offers two branches: resonance and isolation. The former is significantly affected by the working parameters, whereas the latter is weakly influenced. Furthermore, the isolation effectiveness of the CFVI will be compared with that of its linear counterpart and the quasi-zero stiffness vibration isolation model using a semicircle cam (QZSI). The results demonstrate that the CFVI outperforms the other models for base motion excitations.

1. Introduction

As is well known, the conventional vibration isolation model, including a linear spring in parallel with a damper, is one of the common methods for preventing the vibration transmission from source to isolated object, showing that the starting frequency for suppressing vibration is higher than the $\sqrt{2}{\omega }_n$ natural frequency (${\omega }_n$ was denoted as the natural frequency) [1]. Hence, there exists the dichotomy between load-bearing capacity and low stiffness, indicating limited isolation performance [2]. Recently, the growing demand for vibration isolation under diverse loading conditions and complex excitation environments has posed significant challenges for the scientific and engineering community to protect machinery, equipment, and even human health.

The quasi-zero stiffness (QZS), also referred to as high-static–low-dynamic stiffness (HSLDS) vibration isolation, is widely regarded as a promising approach to overcome the limitations of the conventional isolator, as it can extend the isolation region and enhance the capacity for vibration isolation attenuation [3]. The design principle is based on the appropriate arrangement of linear elastic elements to obtain the isolation model with QZS characteristics. For example, a vibration isolation model including three coil springs was proposed, in which one vertical spring supports the load, and a pair of oblique springs is used to modify the total stiffness. This configuration yields a symmetric stiffness curve and obtains quasi-zero stiffness at the equilibrium position as reported by Carrella et al. [4]. The dynamic analysis of this model was conducted in consideration of the effects of the configurative parameters on the system behavior, revealing the complex dynamical phenomena such as periodic, chaotic motion in [5,6]. Based on a scissor structure, an n-layer vibration isolator was proposed and experimentally evaluated by Sun et al. [7], who concluded that this mode can achieve a tunable isolation response without a resonance phenomenon. A one-DOF mechanical oscillator with quasi-zero stiffness, in which the isolated load is supported by four oblique springs arranged in a symmetric configuration, was developed and analyzed by Gatti et al. [8], showing that the system offers a softening behavior in the region of interest. Shi et al. [9] proposed and analyzed an HSLDS vibration isolation model. The system can offer low dynamic stiffness at large deflections, which remains effective for vibration isolation under complex loading conditions and large excitation amplitudes. The dynamic performance of a vibration isolation model studied by Nguebem et al. [10], which comprises a quasi-zero stiffness mechanism, a damper, and an inerter, was evaluated to mitigate vibration of a multi-span continuous beam bridge under moving load. The results showed that the resonance peak shift toward a higher frequency can be remarkably decreased by adding inertance.

Additionally, the combination of the Euler buckling beam and air spring in the QZS vibration isolation model has been considered. For example, Oyelade et al. [11] developed the negative-stiffness structure based on two Euler beams serving as variable stiffness springs. A parameter selection methodology for vehicle seat design was suggested for aiming to enhance driver comfort in the low-frequency range. A variable-stiffness vibration isolator exhibiting positive–negative stiffness characteristics was proposed by Chen et al. [12]. The system is configured with four Euler beams connected by rods through universal joints, maintaining the isolation performance by adjusting the spring stiffness according to the changing isolated mass. A quasi-zero stiffness device, which is configured by four horizontal piezoelectric buckled beams in parallel with a vertical coil spring, was proposed and successfully experimented with by Liu et al. [13]. This model can convert the mechanical energy into electrical energy and hence reduce the vibration transmission from the source to the receiver. Additionally, a vibration isolator, which is composed of four arc-shaped flexible beams, was designed and analyzed to achieve quasi-zero stiffness as reported by Zhou et al. [14]. The results indicated that the starting frequency for effective isolation of this model is higher than that of its linear counterpart. Alternatively, a quasi-zero stiffness vibration isolation system was proposed by Jiang et al. [15], which consists of an air spring in the vertical direction arranged in parallel with electromagnetic springs in the horizontal direction. This magnetic-air hybrid QZS system achieves both effective vibration isolation and adaptability to variable support loads. Palomares et al. [16] developed a vibration isolation seat for ground vehicles, in which two pneumatic cylinders providing negative stiffness are arranged in parallel with a positive-stiffness rubber air spring. A vibration isolation model for a vehicle seat using a rubber air spring was studied by Nguyen et al. [17], who showed that the system can achieve high static and low dynamic stiffness under specific operating conditions.

An alternative approach to achieving quasi-zero stiffness employs cam–roller–spring mechanisms, such as a vibration isolation system including one vertical spring with positive stiffness connected in parallel with a semicircular cam-roller mechanism generating negative stiffness, which was proposed and experimentally evaluated by Zhoiu et al. [18]. The findings showed that the initial frequency of the isolator never exceeds that of the corresponding linear model, irrespective of excitation amplitude. Sun et al. [19] proposed a high-static–low-dynamic stiffness (HSLDS) vibration isolator featuring an asymmetric load-support configuration, where negative stiffness was introduced through a cam mechanism with a parabolic profile. The effects of the negative stiffness mechanism on the system behavior were examined. The results revealed that the isolation effectiveness remained irrespective of structural asymmetry. A QZS vibration isolation system combining a rubber air spring and a semicircular cam mechanism was theoretically analyzed and experimentally validated by Vo et al. [20,21]. Although the results demonstrated superior performance compared with the linear counterpart, bifurcation phenomena were observed in the isolation region when the isolated mass departed from the optimal loading condition. As is well known, the aforementioned QZS vibration isolation models provide better isolation performance than conventional linear systems, such as extending the isolation region and enhancing vibration attenuation while maintaining load-bearing capacity. However, the transmissibility curve tends to bend toward the left. This behavior is attributed to the nonlinear dynamic stiffness characteristic. As a result, the onset frequency of isolation may increase and even exceed that of the corresponding linear counterpart, thereby reducing the overall isolation performance. To mitigate the adverse effects induced by nonlinear dynamic stiffness, a constant-force vibration isolation isolator based on a cam mechanism was proposed and experimented with by Trinh et al. [22], in which the cam profile was designed according to user requirements. The finding showed that the model can achieve a constant force-displacement relationship within the expected working region, thereby remaining quasi-zero stiffness in this range. However, the existing study mainly focused on the analysis and design of the cam profile, and no prior work has explored the system behavior as well as vibration isolation performance under base excitation. Accordingly, this paper discovers the dynamic response and isolation effectiveness of the CFVI through numerical simulation. The remainder of the paper is organized as follows. First, the vibration isolation model is presented in Section 2. Then, the governing dynamic equations are analyzed and established in Section 3. Based on this analysis, the amplitude-frequency relation is derived; subsequently, the system behavior and isolation performance are investigated through numerical simulation in Section 4. Finally, some main conclusions are drawn in Section 5.

2. Description of CFVI

Consider a vibration isolation model with a constant-force characteristic as shown in Figure 1a [22]. Herein, unlike traditional isolators, which use elastic elements such as coil mechanical springs, rubber air springs, or magnetic springs, and so on, this isolation platform utilizes three pneumatic artificial muscles (PAM) as elastic elements. Hence, the key merit of the platform is that it is easy to convert from a passive into an active working state without the addition of an actuator. Moreover, a cam-roller mechanism with the profile of cam (4) is especially designed for the purpose of obtaining the constant-force characteristic in the working region. The load plate (9) is vertically supported by two pulley-disk mechanisms, each of which includes a PAM (1) connected to a pulley (3) through the cable, a crank (7), and a disk (8). Meanwhile, the cam (4), roller (5), and PAM (2) are called the cam-roller mechanism, forming a negative stiffness structure in the vertical direction for the purpose of correcting the stiffness of the platform. The linear bearing (6) is used to guide the load plate, moving only vertically without friction.

Owing to the isolation platform having a symmetric structure, it can be simplified as a plane mechanism, as shown in Figure 1b, which expresses the front view of the platform. The mass of the load plate and isolated object is lumped into M. PAM has a complex dynamic behavior, which includes the restoring response generated by compressed air and the viscoelastic response of rubber material. Hence, each APM is denoted by a coil spring having stiffness K_pam and structure damping C_s; herein, the subscripts “1” and “2” are representative of PAM 1 and PAM 2, respectively. The four disks have the same working radius, denoted by L₁, which is hung at four corners of the load plate by the rotational joints. This disk is connected to the transmission pulley (3) through the crank (7) with the length L₂ and corresponding rotational joints. It is noted that the pulley (3) and crank (7), which are mounted on the base (10) by the connecting rod, have the same rotational speed.

As shown in Figure 1b, the static force exerted on the flat plate can be contributed by two components: one is the force due to the pulley-disk mechanism, denoted by F_pv, and the other is the force of the cam-roller mechanism, denoted by F_cv. Considering the vibration of the load plate around the equilibrium position (as shown in Figure 2a) with a small amplitude, it means that the angle a is within −10° ÷ 10°.

The restoring force F_pv is determined as follows:

$$F_{pv}={\displaystyle \frac{R}{L_2}}{F}_{pam1}$$

(1)

where R is the radius of the pulley, L₂ is the length of the crank, $F_{pam1}=K_{pam1}\left(l_1-l_{W1}\right)+F_{w1}$ with $l_1$ is the shrinkable length of the PAM1, and $l_{W1}$ is the working length of the PAM1 defined as the length at which the stiffness of the PAM1 ($K_{pam1}$) is linearized. $F_{w1}$ is the shrinkable force of the PAM1 at the working length.

According to geometric relations, we have: $y=L_1+L_2\alpha$. Herein, α is the angle of the crank with respect to the horizontal direction, and accordingly, the length of $l_1$ is expressed by:

$$l_1=l_w-{\displaystyle \frac{R}{L_2}}\left(y-L_1\right)$$

(2)

Accordingly, Equation (1) can be rewritten according to the y coordinate:

$$F_{pv}=-{\displaystyle \frac{K_{pam1}{R}^2}{L_2^2}}y-{\displaystyle \frac{K_{pam1}{R}^2}{L_2^2}}{y}_c$$

(3)

The structure of the cam profile is divided into two regions, including the effective region and the non-effective region, as shown in Figure 2b. If the roller contacts the cam surface in a non-effective region, the force acting on the load plate is only contributed by the F_pv. However, the load plate is supported by the forces F_pv and F_cv when the contact between the roller and the cam surface occurs in the effective region.

In the effective region, the reaction between the cam surface and roller is generated due to the pulling force (F_pam2) of PAM2 being F_c. This projection of this force in the vertical and horizontal directions includes two components, which are F_cv and F_ch, respectively, as shown in Figure 2b. The relation of F_cv and F_ch is expressed as follows:

$$F_{cv}=F_{cv}\tan \theta$$

(4)

where θ is the angle of the tangent line of the cam profile with respect to the horizontal direction. The component F_ch is equal to the value of F_pam2. In order to obtain a constant force-displacement characteristic during the expected working region, the force F_cv is determined as follows:

$$F_{cv}={\displaystyle \frac{K_{pam1}{R}^2}{L_2^2}}y+b$$

(5)

With b as a constant, it is determined through the expected working.

Based on this ideal, the design principle of the cam profile was presented clearly in [23]. The equation of the cam profile can be expressed approximately as follows:

$$x=b_1+b_{2}y+b_3{y}^2+b_4{y}^3+b_5{y}^4$$

(6)

Herein, b_i (i = 1 ÷ 4) are coefficients of the polynomial.

Therefore, the force acting on the load plate is:

$$F=\left\{\begin{array}{cc}2\left(-{\displaystyle \frac{K_{pam1}{R}^2}{L_2^2}}y+{\displaystyle \frac{K_{pam1}{R}^2{L}_1}{L_2^2}}+{\displaystyle \frac{R}{L_2}}{F}_{w1}\right)& \left|y\right|>y_d\\ 2\left({\displaystyle \frac{K_{pam1}{R}^2{L}_1}{L_2^2}}+{\displaystyle \frac{R}{L_2}}{F}_{w1}+b\right)& \left|y\right|\le y_d\end{array}\right.$$

(7)

where y_d denotes the critical position; when the vertical displacement of the roller center exceeds this value, the contact state of the roller transitions from the effective region into the non-effective region of the cam profile.

3. Dynamic Modeling of Platform

In the case in which the base is excited vertically by a signal of z_e, the absolute motion of the load plate is z (as denoted in Figure 1c), and the relative motion between the load plate and the base is y = z − z_e. The kinetic energy T of the system includes the translational energy of the load plate and the rotational energy of pulleys, cranks, and disks, which can be written as follows:

$$T={\displaystyle \frac{1}{2}}\left(M+4M_d\right)v^2+{\displaystyle \frac{1}{2}}2J_{o3}{\omega }_3^2+{\displaystyle \frac{1}{2}}4J_{o2}{\omega }_2^2+{\displaystyle \frac{1}{2}}4J_{o1}{\omega }_1^2$$

(8)

where M is the mass of the load plate and the isolated object (called isolated mass), M_d is the mass of the disk, J_oi (i = 1, 2, 3) is the moment of inertia of the disk, crank, and pulley, respectively, ω_i is the corresponding angular velocities, and $v=\dot{z}$ is the translational velocity of the load plate.

Considering a small displacement vibration around the equilibrium position and based on geometrical and kinematic relations, we have:

$$\begin{array}{l}{\omega }_1={\displaystyle \frac{\left(y-L_1\right)}{L_1{L}_2}}\dot{y}\\ {\omega }_2={\omega }_3={\displaystyle \frac{1}{L_2}}\dot{y}\end{array}$$

(9)

with $J_{o1}={\displaystyle \frac{1}{2}}{M}_d{R}_d^2; J_{o2}={\displaystyle \frac{1}{3}}{M}_c{L}_2^2; J_{o3}={\displaystyle \frac{1}{2}}{M}_p{R}_p^2,$ M_c being the mass of the crank. Equation (8) is rewritten as follows:

$$T={\displaystyle \frac{1}{2}}\left(M+4M_d\right){\dot{z}}^2+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{M_p{R}_p^2}{L_2^2}}+{\displaystyle \frac{4}{3}}{M}_c+2M_d{R}_d^2{\left({\displaystyle \frac{y-L_1}{L_1{L}_2}}\right)}^2\right){\dot{y}}^2$$

(10)

The potential energy V of the system is the elastic energy, which can be expressed as follows:

$$V=\left\{\begin{array}{cc}\left(M+4M_d\right)gy+2M_{c}g(y-L_1)& \left|y\right|\le y_d\\ \begin{array}{l}2\left({\displaystyle \frac{K_{pam}^P{R}^2}{2L_2^2}}{y}^2-{\displaystyle \frac{K_{pam}^P{R}^2{L}_1}{L_2^2}}y-{\displaystyle \frac{K_{pam}^P{R}^2{L}_1}{L_2^2}}y\right)\\ +\left(M+4M_d\right)gy+2M_{c}g(y-L_1)\end{array}& \left|y\right|>y_d\end{array}\right.$$

(11)

where g is the gravitational acceleration.

The damping forces generated by viscous damper (C) and structure damper (C_s)1 and 2 are defined as F_d, F_sd1, and F_sd2 as follows:

$$\begin{array}{l}{F}_d=C\dot{y}\\ F_{sd1}={\displaystyle \frac{C_{s1}{R}_p}{L_2}}\dot{y}\\ F_{sd2}=2C_{s2}\left|b_2+2b_{3}y+3b_4{y}^2+4b_5{y}^3\right|\dot{y}\end{array}$$

(12)

The generalized force Q is defined as follows:

$$Q={\displaystyle \sum \overrightarrow{F}\frac{\partial {\overrightarrow{r}}_i}{\partial y}}=\left\{\begin{array}{cc}-C\dot{y}-{\displaystyle \frac{2C_{s1}{R}_p^2}{L_2^2}}\dot{y}-4C_{s2}{\left(b_2+2b_{3}y+3b_4{y}^2+4b_5{y}^3\right)}^2\dot{y}& \left|y\right|\le y_d\\ -C\dot{y}-{\displaystyle \frac{2C_{s1}{R}_p^2}{L_2^2}}\dot{y}& \left|y\right|>y_d\end{array}\right.$$

(13)

Therefore, the Lagrange equation is written as follows:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{y}}}\right)-\left({\displaystyle \frac{\partial L}{\partial y}}\right)=\left\{\begin{array}{cc}-C\dot{y}-{\displaystyle \frac{2C_{s1}{R}_p^2}{L_2^2}}\dot{y}-4C_{s2}{\left(b_2+2b_{3}y+3b_4{y}^2+4b_5{y}^3\right)}^2\dot{y}& \left|y\right|\le y_d\\ -C\dot{y}-{\displaystyle \frac{2C_{s1}{R}_p^2}{L_2^2}}\dot{y}& \left|y\right|>y_d\end{array}\right.$$

(14)

At the static equilibrium position, the isolated mass is determined as follows:

$$M=2{\displaystyle \frac{1}{g}}\left({\displaystyle \frac{K_{pam1}{R}^2{L}_1}{L_2^2}}+{\displaystyle \frac{R}{L_2}}{F}_{w1}^P+b\right)-2M_c-4M_d$$

Substituting kinetic energy (10) and potential energy (11), the piecewise dynamic equation of the vibration platform can be attained as follows:

$$\begin{array}{l}{\mathcal{M}}_{eq}\ddot{y}+\left(-{\displaystyle \frac{4M_d{R}_d^2}{L_1{L}_2^2}}+{\displaystyle \frac{4M_d{R}_d^2}{L_1^2{L}_2^2}}y\right){\dot{y}}^2+C\dot{y}\\ +{\displaystyle \frac{2C_{s1}{R}_p^2}{L_2^2}}\dot{y}+{\mathcal{F}}_d(y,\dot{y})+{\mathcal{F}}_r(y)=-\left(M+4M_d\right){\ddot{z}}_e\end{array}$$

(15)

where ${\mathcal{F}}_d(y,\dot{y})$ is the equivalent damping force, and ${\mathcal{F}}_r(y)$ is the equivalent restoring force.

$$\begin{array}{l}{\mathcal{F}}_d(y,\dot{y})=\left\{\begin{array}{cc}4C_{s2}{\left(b_2+2b_{3}y+3b_4{y}^2+4b_5{y}^3\right)}^2\dot{y}& \left|y\right|\le y_d\\ 0& \left|y\right|>y_d\end{array}\right.\\ {\mathcal{F}}_r(y)=\left\{\begin{array}{cc}0& \left|y\right|\le y_d\\ K_{eq}y& \left|y\right|>y_d\end{array}\right. \mathrm{with} K_{eq}=2{\displaystyle \frac{K_{pam1}{R}^2}{L_2^2}}: \mathrm{the} \mathrm{equivalent} \mathrm{stiffness}\end{array}$$

Meanwhile ${\mathcal{M}}_{eq}$ is the equivalent mass, defined by

$$\begin{array}{l}{\mathcal{M}}_{eq}=M_1+M_2 \mathrm{in} \mathrm{which}\\ M_1=M+4M_d+{\displaystyle \frac{M_p{R}_p^2}{L_2^2}}+{\displaystyle \frac{4}{3}}{M}_c; M_2=2M_d{R}_d^2{\left({\displaystyle \frac{y-L_1}{L_1{L}_2}}\right)}^2\end{array}$$

(16)

Considering the dimensionless form of the equivalent mass, defined as follows: ${\widehat{\mathcal{M}}}_{eq}={\displaystyle \frac{{\mathcal{M}}_{eq}}{M_1}}=1+{\displaystyle \frac{M_2}{M_1}}$, we have:

$${\displaystyle \frac{M_2}{M_1}}={\displaystyle \frac{2M_d{R}_d^2}{M_1{L}_2^2}}{\left(1-{\displaystyle \frac{y}{L_1}}\right)}^2\le {\displaystyle \frac{2M_d{R}_d^2}{M_1{L}_2^2}}$$

(17)

The condition for which the nonlinear term M₂ can be ignored can be obtained as follows:

$${\displaystyle \frac{2M_d{R}_d^2}{M_1{L}_2^2}}\ll 1 \mathrm{or} M_d\ll {\displaystyle \frac{M_1{L}_2^2}{2R_d^2}}$$

Therefore, if the mass (M_d) of the disk is designed to be much smaller than the parameter $M_1{L}_2^2/2R_d^2$, the equivalent mass can be approximately considered as a constant value of M₁, while the nonlinear component M₂ can be ignored. For instance, if the value of M = 19 kg, M_d = 0.1 kg, M_p = 0.4 kg, and M_c = 0.2 kg, as shown in Figure 3, the component M₂ is too small compared with the component M₁.

4. Dynamic Analysis and Numerical Simulation

4.1. Displacement Amplitude-Frequency Relation

Due to the piecewise dynamic model of the proposed system, the Average method (AM) [23,24] is a good selection to attain the relation between displacement amplitude and the fundamental frequency.

Supposing that the base is excited by a harmonic vibrating displacement $z_e=Z_e\cos \omega t,$ simultaneously, the dimensionless transform is given by:

$$\begin{array}{l}\tau ={\omega }_{n}t; \hat{\mathrm{y}}= {\displaystyle \frac{y}{Z_E}}; {\omega }_n=\sqrt{{\displaystyle \frac{K_{eq}}{{\mathcal{M}}_{eq}}}}; \xi ={\displaystyle \frac{C_1}{{\mathcal{M}}_{eq}{L}_2^2{\omega }_n}}; \\ {\xi }_1={\displaystyle \frac{2M_d{L}_1{Z}_E}{{\mathcal{M}}_{eq}{L}_2^2}}; {\xi }_2={\displaystyle \frac{2M_d{Z}_e^2}{{\mathcal{M}}_{eq}{L}_2^2}}; {\xi }_3={\displaystyle \frac{2C_1{Z}_e^2{R}^2}{{\mathcal{M}}_{eq}{L}_2^2{\omega }_n}}\\ {\eta }_1={\displaystyle \frac{4C_2{b}_2^2}{{\mathcal{M}}_{eq}{\omega }_n}}; {\eta }_2={\displaystyle \frac{16C_2{b}_2{b}_3{Z}_e}{{\mathcal{M}}_{eq}{\omega }_n}}; {\eta }_3={\displaystyle \frac{4C_2\left(4b_3^2+6b_2{b}_4\right)Z_e^2}{{\mathcal{M}}_{eq}{\omega }_n}}; \\ {\eta }_4={\displaystyle \frac{4C_2\left(8b_2{b}_5+12b_3{b}_4\right)Z_e^3}{{\mathcal{M}}_{eq}{\omega }_n}}; {\eta }_5={\displaystyle \frac{4C_2\left(16b_3{b}_5+9b_4\right)Z_e^4}{{\mathcal{M}}_{eq}{\omega }_n}};\\ {\eta }_6={\displaystyle \frac{96C_2{b}_4{b}_5{Z}_e^5}{{\mathcal{M}}_{eq}{\omega }_n}}; {\eta }_7={\displaystyle \frac{64C_2{b}_5^2{Z}_e^6}{{\mathcal{M}}_{eq}{\omega }_n}}; {\widehat{Z}}_e={\displaystyle \frac{\left(M+4M_d\right)}{{\mathcal{M}}_{eq}}}; \mathsf{\Omega }={\displaystyle \frac{\omega }{{\omega }_n}}\end{array}$$

The dynamic equation can be rewritten in terms of dimensionless form as follows:

$${\widehat{y}}^{″}={\widehat{Z}}_e\mathsf{\Omega }\cos \omega t-\left(\left({\delta }_1+{\delta }_2\widehat{y}\right){\widehat{y}}^{\prime 2}+2\xi y^{\prime }+{\widehat{\mathcal{F}}}_d(\widehat{y},{\widehat{y}}^{\prime })+{\widehat{\mathcal{F}}}_r(\widehat{y})\right)$$

(18)

where ${(\cdot )}^{\prime }$ denotes the derivation versus dimensionless time (τ)

• and $\begin{array}{l}{\widehat{\mathcal{F}}}_d(\widehat{y},{\widehat{y}}^{\prime })=\left\{\begin{array}{cc}\left({\eta }_1+{\eta }_2\widehat{y}+{\eta }_3{\widehat{y}}^2+{\eta }_4{\widehat{y}}^3+{\eta }_5{\widehat{y}}^4+{\eta }_6{\widehat{y}}^5+{\eta }_7{\widehat{y}}^6\right){\widehat{y}}^{\prime }& \left|\widehat{y}\right|\le 1\\ 0& \left|\widehat{y}\right|>1\end{array}\right.\\ {\widehat{\mathcal{F}}}_r(\widehat{y})=\left\{\begin{array}{cc}0& \left|\widehat{y}\right|\le 1\\ {\omega }_n^2\widehat{y}& \left|\widehat{y}\right|>1\end{array}\right. \end{array}$

Adding $\widehat{y}$ to two sides of Equation (18) and introducing a dummy perturbation parameter ε, we have:

$${\widehat{y}}^{″}+\widehat{y}=\epsilon \left({\widehat{Z}}_e\mathsf{\Omega }\cos \omega t+\widehat{y}-\left(\left({\delta }_1+{\delta }_2\widehat{y}\right){\widehat{y}}^{\prime 2}+2\xi y^{\prime }+{\widehat{\mathcal{F}}}_d(\widehat{y},{\widehat{y}}^{\prime })+{\widehat{\mathcal{F}}}_r(\widehat{y})\right)\right)$$

(19)

And then, to reflect the difference between the natural frequency and excitation, the tuning parameter σ is introduced through ${\mathsf{\Omega }}^2=1+\epsilon \sigma ,$ Equation (18) can be rewritten as follows:

$$\begin{array}{l}{\widehat{y}}^{″}+{\mathsf{\Omega }}^2\widehat{y}=\epsilon \mathcal{P}(\widehat{y},{\widehat{y}}^{\prime })\\ \mathcal{P}(\widehat{y},{\widehat{y}}^{\prime })={\widehat{Z}}_e{\mathsf{\Omega }}^2\cos \mathsf{\Omega }t+\left(1+\sigma \right)\widehat{y}-\left(\begin{array}{l}\left({\delta }_1+{\delta }_2\widehat{y}\right){\widehat{y}}^{\prime 2}+2\xi y^{\prime }\\ +{\widehat{\mathcal{F}}}_d(\widehat{y},{\widehat{y}}^{\prime })+{\widehat{\mathcal{F}}}_r(\widehat{y})\end{array}\right)\end{array}$$

(20)

The harmonic solution of the relative motion is assumed as follows:

$$\widehat{y}(\tau )=\widehat{A}(\tau )\cos \left(\mathsf{\Omega }\tau +\phi (\tau )\right)$$

(21)

where $\widehat{A}(\tau )$ and $\phi (\tau )$ are the amplitude and phase difference, which are functions of time τ.

Utilizing the AM, the autonomous differential equation of the amplitude and phase versus time is as follows:

$$\begin{array}{l}{\overline{A}}^{\prime }=-{\displaystyle \frac{\epsilon }{2\pi \mathsf{\Omega }}}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\mathcal{P}(\overline{y},{\overline{y}}^{\prime })\sin \psi }d\psi \\ {\phi }^{\prime }=-{\displaystyle \frac{\epsilon }{2\pi \overline{A}\mathsf{\Omega }}}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\mathcal{P}(\overline{y},{\overline{y}}^{\prime })\cos \psi d\psi }\end{array}$$

(22)

In the case of $\overline{A}\le 1$, substituting the second Equation of (20) into Equation (22), then taking integration in the region [0, 2π], obtaining the following:

$$\begin{array}{l}{\widehat{A}}^{\prime }=-{\displaystyle \frac{\epsilon }{2\mathsf{\Omega }}}\left({\widehat{Z}}_e{\mathsf{\Omega }}^2\sin \phi +{\mathcal{R}}_1(\widehat{A})\mathsf{\Omega }\right)\\ {\phi }^{\prime }=-{\displaystyle \frac{\epsilon }{2\widehat{A}\mathsf{\Omega }}}\left({\widehat{Z}}_e{\mathsf{\Omega }}^2\cos \phi -{\displaystyle \frac{1}{4}}{\delta }_2{\widehat{A}}^3{\mathsf{\Omega }}^2+\widehat{A}{\mathsf{\Omega }}^2\right)\end{array}$$

(23)

with ${\mathcal{R}}_1(\widehat{A})=\left(2\xi \widehat{A}+{\eta }_1\widehat{A}+{\displaystyle \frac{1}{4}}{\eta }_3{\widehat{A}}^3+{\displaystyle \frac{1}{8}}{\eta }_5{\widehat{A}}^5+{\displaystyle \frac{5}{64}}{\eta }_7{\widehat{A}}^7\right)\mathsf{\Omega }$

In the case of $\overline{A}>1$, the second equation of (20), which is a piecewise function, the segment point is ${\widehat{y}}_d=\widehat{A}\cos {\psi }_a$, integrating Equation (22) in the region [0, 2π] is divided into four segments, including

$\left[-{\psi }_a,{\psi }_a\right]; \left[{\psi }_a,\pi -{\psi }_a\right]; \left[\pi -{\psi }_a,\pi +{\psi }_a\right]; \left[\pi +{\psi }_a,2\pi -{\psi }_a\right].$ We have:

$$\begin{array}{l}{\overline{A}}^{\prime }=-{\displaystyle \frac{\epsilon }{2\pi \mathsf{\Omega }}}\left({\widehat{Z}}_e{\mathsf{\Omega }}^2\pi \sin \phi +{\mathcal{R}}_2(\widehat{A})\right)\\ {\phi }^{\prime }=-{\displaystyle \frac{\epsilon }{2\pi \widehat{A}\mathsf{\Omega }}}\left({\widehat{Z}}_e{\mathsf{\Omega }}^2\pi \cos \phi +{\mathcal{R}}_3(\widehat{A}){\mathsf{\Omega }}^2+{\mathcal{R}}_4(\widehat{A})\right)\end{array}$$

(24)

in which:

$$\begin{array}{l}{\mathcal{R}}_2(\widehat{A})=2\left(\xi +{\xi }_o\right)\widehat{A}\pi +{\eta }_1\widehat{A}\mathsf{\Omega }\left(\pi -2{\psi }_a+\sin 2{\psi }_a\right)+{\displaystyle \frac{1}{4}}{\eta }_3{\widehat{A}}^3\left(\pi -2{\psi }_a+{\displaystyle \frac{1}{2}}\sin 4{\psi }_a\right)\\ +2{\eta }_5{\widehat{A}}^5\left({\displaystyle \frac{\pi -2{\psi }_a}{16}}-{\displaystyle \frac{\sin 2{\psi }_a}{32}}+{\displaystyle \frac{\sin 4{\psi }_a}{32}}+{\displaystyle \frac{\sin 6{\psi }_a}{96}}\right)\\ +2{\eta }_7{\widehat{A}}^7\left({\displaystyle \frac{5\left(\pi -2{\psi }_a\right)}{128}}-{\displaystyle \frac{\sin 2{\psi }_a}{32}}+{\displaystyle \frac{\sin 4{\psi }_a}{64}}+{\displaystyle \frac{\sin 6{\psi }_a}{96}}+{\displaystyle \frac{\sin 8{\psi }_a}{512}}\right)\\ {\mathcal{R}}_3(\widehat{A})=\widehat{A}\pi -{\displaystyle \frac{1}{4}}\pi {\delta }_2{\widehat{A}}^3\\ {\mathcal{R}}_4(\widehat{A})=-\widehat{A}\left(2{\psi }_a+\sin 2{\psi }_a\right)\end{array}$$

By setting ${\overline{A}}^{\prime }={\phi }^{\prime }=0$ and the condition ${\sin }^2\phi +{\cos }^2\phi =1,$ the amplitude-frequency relation is as follows:

$$\left\{\begin{array}{cc}\left({\widehat{Z}}_e^2-{\left({\displaystyle \frac{1}{4}}{\delta }_2{\widehat{A}}^3-\widehat{A}\right)}^2\right){\mathsf{\Omega }}^2={\mathcal{R}}_1^2(\widehat{A})& \widehat{A}\le 1\\ \left({\mathcal{R}}_3^2(\widehat{A})-{\pi }^2{Z}_e^2\right){\mathsf{\Omega }}^4+\left(2{\mathcal{R}}_3(\widehat{A}){\mathcal{R}}_4(\widehat{A})+{\mathcal{R}}_2^2(\widehat{A})\right){\mathsf{\Omega }}^2+{\mathcal{R}}_4^2(\widehat{A})=0& \widehat{A}>1\end{array}\right.$$

(25)

The absolute displacement of the load plate is obtained as follows:

$$\widehat{z}=\widehat{y}+{\widehat{z}}_e=\widehat{A}\cos (\mathsf{\Omega }\tau +\phi )+{\widehat{Z}}_e\cos (\mathsf{\Omega }\tau )$$

(26)

The amplitude $\widehat{Z}$ of $\widehat{z}$ is determined as follows:

$$\widehat{Z}=\sqrt{{\widehat{A}}^2+{\widehat{Z}}_e^2+2\widehat{A}{\widehat{Z}}_e\cos (\phi )}$$

(27)

with:

$$\left\{\begin{array}{cc}\cos \phi ={\displaystyle \frac{1}{4{\widehat{Z}}_e{\mathsf{\Omega }}^2}}{\delta }_2{\widehat{A}}^3{\mathsf{\Omega }}^2-\widehat{A}{\mathsf{\Omega }}^2& \widehat{A}\le 1\\ \cos \phi =-{\displaystyle \frac{1}{{\widehat{Z}}_e{\mathsf{\Omega }}^2\pi }}\left({\mathcal{R}}_3(\widehat{A}){\mathsf{\Omega }}^2+{\mathcal{R}}_4(\widehat{A})\right)& \widehat{A}>1\end{array}\right.$$

The absolute displacement transmissibility T_d is defined as follows:

$$T_d={\displaystyle \frac{\widehat{Z}}{{\widehat{Z}}_e}}=\sqrt{1+{\displaystyle \frac{{\widehat{A}}^2}{{\widehat{Z}}_e^2}}+2{\displaystyle \frac{\widehat{A}}{{\widehat{Z}}_e}}\cos (\phi )}$$

(28)

Solving the quadratic equation of (18), the frequency ratio determined via the excited force is expressed as below:

$$\left\{\begin{array}{cc}{\mathsf{\Omega }}_1={\displaystyle \frac{{\mathcal{R}}_1(\widehat{A})}{\sqrt{{\widehat{Z}}_e^2-{\left(\frac{1}{4}{\delta }_2{\widehat{A}}^3-\widehat{A}\right)}^2}}}& \widehat{A}\le 1\\ {\mathsf{\Omega }}_{12}=\sqrt{{\displaystyle \frac{-\left(2{\mathcal{R}}_3(\widehat{A}){\mathcal{R}}_4(\widehat{A})+{\mathcal{R}}_2^2(\widehat{A})\right)\pm \sqrt{\begin{array}{l}{\left(2{\mathcal{R}}_3(\widehat{A}){\mathcal{R}}_4(\widehat{A})+{\mathcal{R}}_2^2(\widehat{A})\right)}^2\\ -4\left({\mathcal{R}}_3^2(\widehat{A})-{\pi }^2{Z}_e^2\right){\mathcal{R}}_4^2(\widehat{A})\end{array}}}{2\left({\mathcal{R}}_3^2(\widehat{A})-{\pi }^2{Z}_e^2\right)}}}& \widehat{A}>1\end{array}\right.$$

(29)

4.2. Parameter Effects on the Displacement-Frequency Response

This subsection will consider the effects of the parameters on the dynamic response as well as the displacement isolation effectiveness of the CFVI. To maintain the profile of the cam surface that provides the constant-force characteristic of the CFVI, the configuration parameters for simulation, including L₁, L₂, R, and working pressure P_W, are adopted from [22] as listed in

Table 1. Parameter used in simulation.

Configurative Parameters	Original Value
The working radius of the disk denoted (L1)	24.5 mm
The length of the crank (L2)	140 mm
The radius of the pulley (R)	38 mm
The working pressure of the PAM1 (Pw1)	3.1 bar
The working pressure of the PAM2 (Pw2)	2.5 bar
Viscous damping coefficient (C)	43 Ns/m
Structure damping coefficient (C1)	20 Ns/m

. Meanwhile, the mass of the model, including isolated mass (M1), disk, crank, and pulley, is the same as in Figure 3.

As a result, when the framework is excited by a harmonic signal with the displacement amplitude of 12 mm while the frequency ratio varies from 0 to 5, and simultaneously the critical position of the cam profile and the damping ratio are set to y_d = 15 mm and ξ = 0.16, the piecewise dynamic response of the model is divided into three branches as shown in Figure 4. The upper solid-line branch is representative of the resonance response, while the lower dashed-line branch corresponds to the vibration-transmission mitigation from the framework to the load plate. This indicates that the upper branch exhibits the reciprocal movement between the non-effective and effective regions of the cam, whereas the lower branch exhibits oscillation only within the effective region. The remaining curve, marked by the dotted line, is an unstable branch. This indicates that Equation (19) will have three solutions, including two stable solutions and one unstable solution. Indeed, when the frequency ratio lies within the region of the unstable branch, the dynamic response of the CFVI may evolve on either the upper or the lower branch, and even a jump phenomenon between the upper and the lower branch may occur, both depending on the initial condition involving the initial dimensionless position (${\widehat{y}}_o$) and velocity (${\widehat{v}}_o$). Outside this region, the system response exists only on the isolation branch. It is noted that the isolation branch asymptotically trends toward unity as the frequency ratio is developed.

In order to clarify the effects of the initial condition on the dynamic response of the isolator, the original differential Equation (18) was integrated directly by using the Matlab R2024 ODE solver ODE45 with a time step of 0.01 s and default tolerance settings. As observed in Figure 4, the dynamic response can exist in the upper or lower branch as the frequency ratio occurs within 0.4–1.4. Specifically, when the frequency ratio is 0.8 and 1.2, the displacement response of the load plate with respect to time is shown in Figure 5 and Figure 6, respectively. It can be seen that, when the initial dimensionless position and velocity are set to 0 and −0.2, respectively, the dynamic response is on the upper branch, as shown in Figure 5a. In contrast, the initial dimensionless position is set to 0.3, the initial dimensionless velocity is the same as in Figure 5a, and the displacement response evolves on the lower branch as shown in Figure 5b. Similarly, for the case with the frequency ratio of 1.2, the displacement response evolves on the upper branch when both the initial conditions are set to zero, as shown in Figure 6a. However, the response that appears on the lower branch appears when the initial position and velocity are set to 0 and −0.2 m/s, respectively.

In addition, Figure 4, Figure 5 and Figure 6 reveal that the steady-state response of the isolator at a frequency ratio of 0.8 and 1.2 obtained from the direct numerical integration matches well with those determined by the AM as marked by the purple dots in Fig. 4. Specifically, when the frequency ratio is 0.8, the dimensionless amplitude on the lower and upper branches is approximately 0.95 and 2.23, respectively, by ODE45, whereas by using the AM, the corresponding amplitude response is obtained as 0.96 and 2.2.

For the case with a frequency ratio of 1.2, the dimensionless amplitudes obtained using ODE45 are 0.96 and 2.69 for the lower and upper branches, respectively. The amplitudes predicted by the AM at this frequency ratio are 0.97 and 2.67. These results confirm the reliability of the analytical solution obtained from the AM.

To provide a further insight into the influence of the effective-region size of the cam profile, the dimensionless displacement-frequency response of the CFVI will be simulated for three critical positions involving y_d = 12 mm, 15 mm, and 18 mm. As shown in Figure 7, the kinds of lines are annotated in the right-upper panel. The numerical simulation results reveal that the isolation branch is insensitive to variation in the critical position. On the other hand, the resonance branch as well as the unstable branch are influenced by the effective region size. Indeed, increasing the value of y_d leads to a contraction of the resonance branch, and simultaneously the unstable branch shifts upward, meaning that the frequency region in which a reciprocal jump can occur between the resonance and isolation branches is narrowed. However, in this case, the peak response, including peak frequency and amplitude, has almost remained unchanged regardless of the contraction or extension of the resonance branch.

Considering the effects of the damping coefficient on amplitude-frequency behavior shown in Figure 8 (the types of the lines are indicated in the upper-right corner of the figure), it can be seen clearly that on the resonance branch, the peak amplitude is reduced according to the increase in the damping ratio, meanwhile the peak frequency and the unstable branch remain almost unchanged in spite of variation in the damping ratio. Although there is a leftward shift in the isolation branch when the damping ratio is lessened, the tendency of the isolation branch to asymptotically approach a value of one as the frequency ratio increases remains unchanged.

Finally, the variation tendency of the amplitude-frequency curve as excitation displacement amplitude is increased from 9 mm to 15 mm will be taken into account, while the other parameters remain the same as in Figure 4. The numerical simulation result is shown in Figure 9. It is noteworthy that as the excitation amplitude is reduced, the amplitude-frequency behavior of the CFVI tends to be similar to that investigated with increasing critical position. Specifically, the unstable branch will be shrunk as the excitation amplitude is reduced, meaning that the frequency region displaying the reciprocal jump phenomenon between the resonance and isolation branches is reduced. However, both peak amplitude and frequency, as well as the isolation branch, are not sensitive to the changes in the excitation amplitude.

4.3. Isolation Effectiveness

The isolation effectiveness of the CFVI is compared with that of its linear counterpart. In this case, the critical position is set at 20 mm, while other parameters are the same as in Figure 4. The excitation amplitude is 10 mm. It is noted that the linear counterpart retains the pulley-disk mechanism similar to that of the CFVI, but without the cam mechanism, meaning that this linear model has the same isolated mass as the CFVI. As seen in Figure 10 (detailed annotation is provided in the upper-right corner of the figure), the CFVI offers better isolation effectiveness than that of the linear counterpart. Indeed, the proposed isolation model has the isolation frequency region (≥15 rad/s), in which the transmissibility characteristic (T) is smaller than unity, wider than that of the linear counterpart (≥19 rad/s) simultaneously, and the vibration attenuation ratio of the CFVI is significantly higher. Additionally, it is noteworthy that the bending of the peak response observed in the CFVI is absent and similar to that of the linear counterpart. As indicated by the dashed line, which presents the resonance response of the CFVI, reciprocal movement between the effective and non-effective regions of the cam surface may appear within this frequency area. When the roller is outside the effective region, both the CFVI and linear counterpart are only governed through the pulley-disk mechanism.

For further evaluation, the CFVI will be compared with a quasi-zero stiffness isolation model (QZSI) studied by the authors [21] in which the semicircle cam mechanism is utilized to correct the dynamic stiffness of the QZSI. The dynamic characteristic of this isolator is that the quasi-zero stiffness is only attained at the equilibrium position; as the isolated object moves away from this position, the dynamic stiffness increases.

To ensure a fair comparison, the isolated load and the viscous damping coefficient of the QZSI are kept the same as those employed in the CFVI. Additionally, the external excitation conditions are identical for both models. As a result, the transmissibility curve of the QZSI trends toward leftward bending, and the peak frequency correspondingly shifted to approximately 19 rad/s, as shown in Figure 10. Although the initial frequency at which T < 1 is 13.8 rad/s, reciprocal jump phenomena between the resonance and isolation behavior may appear within 13–19 rad/s. Accordingly, the effective isolation frequency of the QZSI is recommended as being higher than the peak frequency. In this case, the practical starting frequency for the effective isolation is bigger than 19 rad/s. Compared with the CFVI, it can be seen that the CFVI can provide a wider isolation region and faster vibration attenuation capacity than the QZSI.

Furthermore, when the critical position is increased while the other parameters remain similar to those in Figure 4, the isolation region will be broadened to low frequency, as shown in Figure 11, and the peak resonance will be almost unchanged. This is attributed to the extension of the cam’s effective working region, within which the CFVI maintains quasi-zero dynamic stiffness. In the same manner of variation in the critical position, reducing the excitation amplitude enlarges the effective isolation region, as shown in Figure 12. When the excited amplitude is decreased from 15 mm to 9 mm (the remaining parameters are the same as in Figure 4), the isolation region becomes wider. Notably, when the excited amplitude is further lessened to a reasonable value, for example, Z_e = 5 mm, the transmissibility curve of the CFVI only exists in the isolation branch, as shown in Figure 13. For this case, the CFVI can effectively isolate over the entire frequency range. In contrast, the QZSI can only prevent vibration having an excited frequency exceeding approximately 12 rad/s.

5. Conclusions

This work introduced an innovative vibration isolation model using a piecewise cam mechanism in which the cam’s working surface is separated into two distinct regions involving effective and non-effective regions. The cam profile within the effective region has been predefined by the user. During the vertical movement of the cam, the rollers always remain in contact with the cam surface during the vertical movement of the cam via pneumatic artificial muscles, thereby generating constant force behavior within the working range. Then, the piecewise dynamic model of the CFVI was established, and the relative displacement-frequency relation and the amplitude transmissibility were analyzed and derived by using the average method.

Subsequently, numerical simulation was conducted to investigate the tendency of the piecewise dynamic behavior as well as the absolute displacement transmissibility of the CFVI when the working parameters of the model are varied. The results confirmed that the dynamic response of the CFVI includes the resonance and isolation branches. The dynamic response of the resonance branch is remarkably affected by the damping coefficient, critical position, and excited amplitude, whereas the isolation branch is only weakly influenced by these parameters. Specifically, the effective isolation region will be increased according to the development of the critical position, while the resonance peak is almost unaffected regardless of the variation in the critical position. This effect is due to the extension of the cam’s effective surface corresponding to the increase in the critical position. In contrast, the isolation region is also enlarged as there is a reduction in the excited amplitude. Especially, this model can isolate over the entire frequency range as the excitation amplitude is reduced to a moderate value.

A comparison of the isolation effectiveness among the CFVI, its linear counterpart, and the QZSI under base motion excitation was performed by numerical simulation. As a result, the CFVI can provide the best isolation effectiveness. Specifically, the effective isolation region of the CFVI is larger than that of the other two models. Simultaneously, the vibration attenuation capability of the CFVI is significantly higher. In future work, a prototype of this model and the corresponding experimental apparatus will be fabricated and established to further assess the validity of the theoretical model.