Design and analysis of a vibration isolation system with cam–roller–spring

Abstract

The vibration isolation system using a pair of oblique springs or a spring-rod mechanism as a negative stiffness mechanism exhibits a high-static low-dynamic stiffness characteristic and a nonlinear jump phenomenon when the system damping is light and the excitation amplitude is large. It is possible to remove the jump via adjusting the end trajectories of the above springs or rods. To realize this idea, the article presents a vibration isolation system with a cam–roller–spring–rod mechanism and gives the detailed numerical and experimental studies on the effects of the above mechanism on the vibration isolation performance. The comparative studies demonstrate that the vibration isolation system proposed works well and outperforms some other vibration isolation systems.

Keywords

Vibration isolation system negative stiffness mechanism cam–roller mechanism high-static low-dynamic stiffness characteristic jump phenomenon

1. Introduction

Mechanical vibrations often deteriorate the performance and durability of equipment in engineering. The techniques of vibration control, thus, have received great attention over the past decades. Compared with active and semi-active techniques of vibration control (Hu, 1996; Li et al., 2019), the passive technique of vibration isolation has simpler structures and higher reliability and is the most prevalent choice in vibration control (Franchek et al., 1996). The simplest vibration isolation system has the model of a linear oscillator involving a mass $m$ and a spring of stiffness coefficient $k$ and works for the vibration isolation when the excitation frequency is higher than $\sqrt{2 k / m}$ . To expand the lower bound of the excitation frequency, the stiffness coefficient $k$ of the spring needs to be a small value and then reduces the static load capacity. A solution to this problem is the so-called the high-static low-dynamic stiffness (HSLDS) vibration isolation system equipped with a negative stiffness mechanism (NSM) in the above linear oscillator. The HSLDS vibration isolation system is a nonlinear vibration system, which can achieve a low-dynamic stiffness level or even a quasi-zero-stiffness (QZS) level near the static equilibrium of the system but keeps the same static stiffness as the linear oscillator. Finding an efficient and appropriate NSM, hence, is a key issue to the design of HSLDS vibration isolation systems.

The most common NSM consists of two springs inclined to the major spring supporting the isolated object so as to provide an NSM. Carrella et al. (2007, 2009, 2012) and Kovacic et al. (2008) made comprehensive studies on the mechanism, analyzing the dynamic characteristics and optimizing the vibration isolation system. Hu et al. (2018) designed an adjustable NSM using springs and rods to act as an NSM and studied the impact of pre-compression of a horizontal spring on the vibration isolation performance. Liu and Yu (2018) connected a small-mass auxiliary system to the HSLDS vibration isolation system to eliminate the jump phenomenon because of the stiffness nonlinearity. Huang et al. (2014) replaced the inclined springs with Euler buckling beams to provide an NSM.

Recent years have also seen many attempts to replace traditional springs and rod mechanisms in the design of an NSM. For example, magnetic springs have served as an efficient NSM because of their advantages such as no contact and quick response (Li et al., 2013; Xu et al., 2013; Zheng et al., 2018). Besides, cam–roller mechanisms have also been a choice of an NSM. Liu et al. (2019) proposed an NSM of a cam–roller–spring and analyzed the influence of the radius ratio of a cam and roller on vibration isolation performance. Han et al. (2018) designed a base spring–roller vibration isolation system with curved edges as a negative stiffness element. Sun et al. (2019) investigated the effectiveness of the parabolic cam–roller mechanism under an asymmetric spring support structure.

It is noted that in the abovementioned vibration isolation systems using oblique springs or a spring–rod mechanism, the dynamic stiffness level can only be adjusted by changing the stiffness or pre-compression of the spring. Moreover, most HSLDS vibration isolation systems exhibit nonlinear jump phenomena when the system damping is light and the excitation amplitude is large, where the transmissibility changes suddenly for a small change in the frequency of excitation (Ho et al., 2012). The isolation effect vanishes at frequencies between the two jump frequencies, which causes the isolation performance to deteriorate. The current work is motivated by the fact that the surface of the cam in the cam–roller mechanism can be designed following special curves, which can be used for varying the end trajectories of the springs or rods. It will provide different restoring forces and stiffness (Liu et al., 2014) and make it possible to remove the jump. The cam is usually designed as a circular arc and approximated in a quadratic polynomial form for simplifying the calculation (Zhou et al., 2015). A parabolic cam is adopted in this work, which can provide the stiffness in an exact quadratic polynomial form without error in calculation (Sun et al., 2019).

This article presents the design and analysis of a vibration isolation system with a cam–roller–spring–rod (CRSR) mechanism shown in Figure 1, where four vertical springs with total stiffness coefficient $k_{1}$ support the isolated object and the NSM includes a pair of cam–roller mechanisms and two spring–rod mechanisms. Such a pair has two asymmetric cams of parabolic shapes and two rollers connected by a horizontal spring of stiffness coefficient $k_{2}$ . In each spring–rod mechanism, one end of the horizontal spring of stiffness coefficient $k_{3}$ is fixed to the base, and the other end of the spring is connected to the roller through a rod of length $L$ . The two ends of the rod are hinged to the spring and the roller, respectively. The pre-compressions of the springs of stiffness coefficients $k_{1}$ , $k_{2}$ , and $k_{3}$ are $δ_{1}$ , $δ_{2}$ , and $δ_{3}$ , respectively. Figure 1(b) shows the static equilibrium position (solid line) and the dynamic position (dashed line) of the vibration isolation system.

Figure 1.

Prototype model and its schematic diagram of the cam–roller–spring–rod isolator: (a) prototype model and (b) schematic diagram.

2. Static analysis

2.1. Analysis of NSM

In view of the symmetry of the system in Figure 1, only the vertical motion of the isolated object should be taken into account such that a half of an NSM in Figure 2 is analyzed. Now, the cam surface yields $δ_{x} = a Y^{2}$ , where $a$ is the parabolic coefficient, $Y$ is the vertical displacement of the roller center, and $δ_{x}$ is the horizontal displacement of the roller center.

Figure 2.

Force analysis of the negative stiffness mechanism.

When the roller moves along the cam surface from the equilibrium position to the position in Figure 2, the compression of the horizontal spring is given by

δ = a Y^{2} + \sqrt{L^{2} - Y^{2}} - L + δ_{3}

(1)

where

δ_{3}

is the pre-compression of the spring. Balancing the vertical forces and horizontal forces, respectively, yields

{\begin{matrix} F_{3} + F_{N} \cos φ = k_{2} (δ_{2} - 2 a Y^{2}) \\ F_{nsm} = - (F_{N} \sin φ + F_{3} \tan θ) \end{matrix}

(2)

where

F_{3}

is the horizontal force exerted by the lateral spring,

F_{N}

is the reaction force provided by the cam,

φ

is the angle between

F_{N}

and the horizontal line,

F_{nsm}

is the restoring force of the unilateral NSM, and

θ

is the angle between the axis of the rod and the horizontal line.

It is easy to derive the following relations

{\begin{matrix} F_{3} = k_{3} δ \\ θ = \arcsin (Y / L) \\ φ = \arctan (2 a Y) \end{matrix}

(3)

Substituting equation (3) into (2) and eliminating $F_{N}$ give the restoring force

F_{nsm} (Y) = - Y [2 k_{2} a (δ_{2} - 2 a Y^{2}) + k_{3} δ (\frac{1}{\sqrt{L^{2} - Y^{2}}} - 2 a)]

(4)

By introducing the dimensionless parameters, $y = Y / L$ , $\hat{a} = a L$ , $λ_{2} = k_{2} / k_{1}$ , $λ_{3} = k_{3} / k_{1}$ , ${\hat{δ}}_{2} = δ_{2} / L$ , ${\hat{δ}}_{3} = δ_{3} / L$ , and $\hat{δ} = δ / L$ , the dimensionless restoring force of the whole NSM is $f_{nsm} = 2 F_{nsm} / (k_{1} L)$ and can be written as

f_{nsm} (y) = - 2 y [2 λ_{2} \hat{a} ({\hat{δ}}_{2} - 2 \hat{a} y^{2}) + λ_{3} \hat{δ} (\frac{1}{\sqrt{1^{2} - y^{2}}} - 2 \hat{a})]

(5)

where

y

is the dimensionless vertical displacement,

\hat{a}

is the dimensionless parabolic coefficient,

λ_{2}

λ_{3}

{\hat{δ}}_{2}

, and

{\hat{δ}}_{3}

are the spring stiffness ratios and dimensionless pre-compression ratios of the two types of horizontal springs, respectively.

\hat{δ} = \hat{a} y^{2} + \sqrt{1 - y^{2}} - 1 + {\hat{δ}}_{3}

denotes the dimensionless compression of the lateral spring.

The dimensionless stiffness function can be obtained by differentiating equation (5) with respect to the dimensionless displacement $y$ as follows

k_{nsm} (y) = - 2 [2 λ_{2} \hat{a} ({\hat{δ}}_{2} - 6 \hat{a} y^{2}) + λ_{3} \hat{δ} (\frac{1}{{(1 - y^{2})}^{3 / 2}} - 2 \hat{a}) - λ_{3} y^{2} {(\frac{1}{\sqrt{1 - y^{2}}} - 2 \hat{a})}^{2}]

(6)

2.2. Conditions for cam–roller contact and compressed springs

Equation (2) holds true when the roller always keeps in contact with the cam surface as $F_{N} \geq 0$ , which is also equivalent to

k_{3} (a Y^{2} + \sqrt{L^{2} - Y^{2}} - L + δ_{3}) \leq k_{2} (δ_{2} - 2 a Y^{2})

(7)

The dimensionless form of equation (7) reads

λ_{3} (\hat{a} y^{2} + \sqrt{1 - y^{2}} - 1 + {\hat{δ}}_{3}) \leq λ_{2} ({\hat{δ}}_{2} - 2 \hat{a} y^{2})

(8)

For a small range of motion, both $\hat{a}$ and $y$ are less than one such that the higher-order terms can be dropped off, and equation (8) can be simplified as

λ_{3} {\hat{δ}}_{3} \leq λ_{2} {\hat{δ}}_{2}

(9)

Besides, all horizontal springs are designed in compression during motions. If $Y_{0}$ is taken as the upper limit of the absolute value of $Y$ , then the conditions for pre-compressions can be written as

{\hat{δ}}_{3} > 1 - \sqrt{1 - y_{0}^{2}} - \hat{a} y_{0}^{2}, {\hat{δ}}_{2} > 2 \hat{a} y_{0}^{2}

(10)

where

y_{0}

is the dimensionless form of

Y_{0}

2.3. Analysis of the CRSR isolator

As shown in Figure 1(b), the vertical spring provides the positive stiffness and supports the isolated object. If the mass of the isolated object is $m$ , then the gravitational effect will be balanced by the pre-compression of the vertical spring at the equilibrium position, namely

k_{1} δ_{1} = m g

(11)

Therefore, the restoring force of the vibration isolator can be expressed as

F = k_{1} Y - 2 Y [2 k_{2} a (δ_{2} - 2 a Y^{2}) + k_{3} δ (\frac{1}{\sqrt{L^{2} - Y^{2}}} - 2 a)]

(12)

Using the aforementioned dimensionless parameters, the dimensionless restoring force $f$ and stiffness function $k$ become

f (y) = y {1 - 2 [2 λ_{2} \hat{a} ({\hat{δ}}_{2} - 2 \hat{a} y^{2}) + λ_{3} \hat{δ} (\frac{1}{\sqrt{1 - y^{2}}} - 2 \hat{a})]}

(13)

k (y) = 1 - 2 [2 λ_{2} \hat{a} ({\hat{δ}}_{2} - 6 \hat{a} y^{2}) + λ_{3} \hat{δ} (\frac{1}{{(1 - y^{2})}^{3 / 2}} - 2 \hat{a}) - λ_{3} y^{2} {(\frac{1}{\sqrt{1 - y^{2}}} - 2 \hat{a})}^{2}]

(14)

It is noted that the stiffness function reaches its minimum at the equilibrium position. The vibration isolation system in Figure 1 exhibits a low-dynamic stiffness property when $k (0) < 1$ holds and achieves the QZS property when $k (0) = 0$ holds. It is easy to determine the following relationship between the parameters of cam–roller and springs for achieving QZS from equation (14) by setting $y = 0$

λ_{3} {\hat{δ}}_{3} + 2 \hat{a} (λ_{2} {\hat{δ}}_{2} - λ_{3} {\hat{δ}}_{3}) = \frac{1}{2}

(15)

which is the so-called zero-stiffness condition.

However, it is impossible to realize the above condition because of manufacturing errors (Liu and Yu, 2018), even though the zero stiffness is an ideal design of vibration isolation. Besides, the system with zero stiffness will be unstable if the vertical stiffness becomes negative. For this reason, the condition in equation (15) is relaxed as

λ_{3} {\hat{δ}}_{3} + 2 \hat{a} (λ_{2} {\hat{δ}}_{2} - λ_{3} {\hat{δ}}_{3}) \leq \frac{1}{2}

(16)

3. Dynamic analysis

3.1. Dynamic modeling of the CRSR isolator

Figure 3 presents the schematic diagram of the CRSR vibration isolation system under a harmonic base excitation

X = X_{0} \cos Ω t

(17)

where

X_{0}

is the excitation amplitude and

Ω

is the excitation frequency.

Figure 3.

Cam–roller–spring–rod vibration isolation system under a harmonic base excitation.

From the static equilibrium condition of equation (11), the dynamic equation of the above system in Figure 3 is given as

m \ddot{Y} + c \dot{Y} + F = X_{0} Ω^{2} \cos Ω t

(18)

In the dynamic analysis as usual, the following dimensionless parameters are introduced, that is

x_{0} = X_{0} / L, ω_{n} = \sqrt{k_{1} / m}, ζ = c / (2 m ω_{n}), ω = Ω / ω_{n}, τ = ω_{n} t

(19)

Accordingly, equation (18) can be recast as

y^{″} + 2 ζ y^{'} + f = x_{0} ω^{2} \cos ω τ

(20)

where the prime represents the differentiation with respect to

τ

To simplify the dynamic analysis, the restoring force $f$ is approximated by using the truncated Taylor series of the first three orders at the static equilibrium position

\bar{f} (y) = α_{1} y + α_{2} y^{3}

(21)

where

\begin{array}{l} α_{1} = 1 - 2 [λ_{3} δ_{3} + 2 \hat{a} (λ_{2} {\hat{δ}}_{2} - λ_{3} δ_{3})] \\ α_{2} = 8 λ_{2} {\hat{a}}^{2} - λ_{3} {\hat{δ}}_{3} + λ_{3} {(1 - 2 \hat{a})}^{2} \end{array}

(22)

Figure 4(a) presents both exact expression and approximate expression for the restoring force, and Figure 4(b) shows that the difference between the two expressions is less than 7% for $| y | \leq 0.5$ . This fact indicates the high accuracy of the approximation within a small range of relative displacement.

Figure 4.

Comparison and difference between the exact and approximate restoring forces ( $a = 0.1$ , $λ_{2} = 1$ , $λ_{3} = 1$ , ${\hat{δ}}_{2} = 0.4$ , and ${\hat{δ}}_{3} = 0.3$ ): (a) exact and approximate restoring forces and (b) difference between the exact and approximate restoring forces.

Thus, according to equations (21) and (22), (20) can be approximately written as

y^{″} + 2 ζ y^{'} + α_{1} y + α_{2} y^{3} = x_{0} ω^{2} \cos ω τ

(23)

3.2. Analytical solution of primary resonance

According to the averaging method, the primary resonance of equation (23) can be assumed as

y = \bar{Y} \cos (ω τ + ϕ)

(24)

y^{'} = - \bar{Y} ω \sin (ω τ + ϕ)

(25)

where

\bar{Y}

and

ϕ

are the amplitude and relative phase angle varying with

τ

. By taking the derivative of equation (24) with respect to

τ

and comparing it with equation (25), one obtains the following equation

{\bar{Y}}^{'} \cos (ω τ + ϕ) - \bar{Y} ϕ^{'} \sin (ω τ + ϕ) = 0

(26)

Then, by taking the derivative of equation (25), one obtains

y^{″} = - {\bar{Y}}^{'} ω \sin (ω τ + ϕ) - \bar{Y} ω^{2} \cos (ω τ + ϕ) - \bar{Y} ω ϕ^{'} \cos (ω τ + ϕ)

(27)

Substituting equations (24), (25), and (27) into (23) and then comparing the result with equation (26), one derives the expressions of the amplitude and phase angle as follows

{\begin{matrix} {\bar{Y}}^{'} = \frac{U}{ω} \sin (ω τ + ϕ) \\ ϕ^{'} = \frac{U}{\bar{Y} ω} \cos (ω τ + ϕ) \end{matrix}

(28)

where

U = \bar{f} (y) - x_{0} ω^{2} \cos ω τ - \bar{Y} ω^{2} \cos (ω τ + ϕ) - 2 ζ \bar{Y} ω \sin (ω τ + ϕ)

Under the slow-varying assumption, the right-hand side of equation (28) can be replaced by an average over a period of excitation as follows

{\begin{matrix} {\bar{Y}}^{'} = \frac{1}{2 π} \int_{0}^{2 π / ω} U \sin (ω τ + ϕ) d τ \\ ϕ^{'} = \frac{1}{2 π} \int_{0}^{2 π / ω} \frac{U}{\bar{Y}} \cos (ω τ + ϕ) d τ \end{matrix}

(29)

After the integration, equation (29) can be expressed as

{\begin{array}{l} {\bar{Y}}^{'} = - \frac{1}{2 ω} (x_{0} ω^{2} \sin ϕ + 2 ω ζ \bar{Y}) \\ ϕ^{'} = \frac{1}{2 ω \bar{Y}} (\frac{3 α_{2} {\bar{Y}}^{3}}{4} + α_{1} \bar{Y} - ω^{2} \bar{Y} - x_{0} ω^{2} \cos ϕ) \end{array}

(30)

To derive the steady-state response, the left-hand side of equation (30) is set to be zero. Eliminating the phase angle $ϕ$ yields the so-called amplitude–frequency response equation expressed as follows

{(\frac{3 α_{2} {\bar{Y}}^{3}}{4} + α_{1} \bar{Y} - ω^{2} \bar{Y})}^{2} + {(2 ω ζ \bar{Y})}^{2} = x_{0}^{2} ω^{4}

(31)

Detailed stability analysis for the steady-state response is available in Hao and Cao (2015) and Yang and Cao (2018).

Figure 5 shows the amplitude–frequency relation of the primary resonance obtained by using the average method compared with the numerical solution by using the fourth-order Runge–Kutta algorithm for the following system parameters $\hat{a} = 0.1$ , $λ_{2} = 1$ , $λ_{3} = 1$ , ${\hat{δ}}_{2} = 0.4$ , ${\hat{δ}}_{3} = 0.3$ , $x_{0} = 0.12$ , and $ζ = 0.05$ . The figure verifies the effectiveness of the average method.

Figure 5.

Analytical and numerical solutions of frequency responses.

3.3. Jump phenomenon

The amplitude–frequency relation of primary resonance of a hardening Duffing oscillator often exhibits a jump phenomenon induced by the saddle-node bifurcation. Such a jump will deteriorate the dynamic performance of a hardening Duffing-type system of vibration isolation and has stimulated many studies on the jump conditions and jump avoidance (Jazar et al., 2006; Sun et al., 2018). For example, the jump-up and jump-down frequencies of a hardening Duffing-type system of vibration isolation under a harmonic force excitation were determined (Brennan et al., 2008; Kevorkian and Cole, 1981).

This subsection focuses on the jump frequencies and the corresponding response amplitudes of a hardening Duffing-type system under a harmonic base excitation described by equation (23). Following the study in Subsection 3.2, the amplitude of primary resonance in equation (31) can be reformulated as a polynomial in $ω$

({\bar{Y}}^{2} - x_{0}^{2}) ω^{4} + (4 ζ^{2} - 2 α_{1} - \frac{3 α_{2} {\bar{Y}}^{2}}{2}) {\bar{Y}}^{2} ω^{2} + {(α_{1} \bar{Y} + \frac{3 α_{2} {\bar{Y}}^{3}}{4})}^{2} = 0

(32)

Solving the above equation gives

ω_{1,2} = \frac{1}{2} \sqrt{\frac{(3 α_{2} {\bar{Y}}^{2} - 8 ζ^{2} + 4 α_{1}) {\bar{Y}}^{2} \pm \bar{Y} \sqrt{A {\bar{Y}}^{4} + B {\bar{Y}}^{2} + C}}{{\bar{Y}}^{2} - x_{0}^{2}}}

(33)

where

A = 9 x_{0}^{2} α_{2}^{2} - 48 ζ^{2} α_{2}

B = 64 ζ^{4} - 64 ζ^{2} α_{1} + 24 x_{0}^{2} α_{1} α_{2}

, and

C = 16 x_{0}^{2} α_{1}^{2}

. The two solutions of equation (33) are resonant and nonresonant branches and will intersect to each other if

A {\bar{Y}}^{4} + B {\bar{Y}}^{2} + C = 0

holds. The resonant frequency and the corresponding maximal response are

ω_{p} = \frac{1}{2} {\bar{Y}}_{\max} \sqrt{\frac{3 α_{2} {\bar{Y}}_{\max}^{2} - 8 ζ^{2} + 4 α_{1}}{{\bar{Y}}_{\max}^{2} - x_{0}^{2}}}

(34)

{\bar{Y}}_{\max} = \sqrt{\frac{32 ζ^{4} - 32 ζ^{2} α_{1} + 12 x_{0}^{2} α_{1} α_{2} + 16 \sqrt{{(2 ζ^{4} - 2 ζ^{2} α_{1})}^{2} + 3 x_{0}^{2} ζ^{4} α_{1} α_{2}}}{48 ζ^{2} α_{2} - 9 x_{0}^{2} α_{2}^{2}}}

(35)

where

ω_{p}

is the resonant frequency and

{\bar{Y}}_{\max}

is the maximal amplitude. If the jump phenomenon exists, then the jump-down frequency

ω_{jd}

is equal to

ω_{p}

. To avoid unbounded response, the following inequality is required

x_{0} < 4 ζ \sqrt{\frac{1}{3 α_{2}}}

(36)

The jump-up frequency weakly depends on the damping coefficient (Brennan et al., 2008). Thus, by setting $ζ$ to be zero for $d ω_{1} / d \bar{Y} = 0$ , the approximate jump-up amplitude can be obtained. The jump-up amplitude ${\bar{Y}}_{up}$ is one of the solutions of the following equation

P_{5} {\bar{Y}}^{5} + P_{4} {\bar{Y}}^{4} + P_{3} {\bar{Y}}^{3} + P_{2} {\bar{Y}}^{2} + P_{1} \bar{Y} + P_{0} = 0

(37)

where

\begin{array}{l} P_{5} = 6 α_{2}, P_{4} = 3 x_{0} α_{2}, P_{3} = - 12 x_{0}^{2} α_{2} \\ P_{2} = - 9 x_{0}^{3} α_{2} - 4 x_{0} α_{1}, P_{1} = - 8 x_{0}^{2} α_{1}, P_{0} = - 4 x_{0}^{3} α_{1} \end{array}

(38)

Remove impossible solutions of equation (37) (two of which are equal to $- x_{0}$ and the other two are complex numbers) such that the solution of practical significance can be derived as follows

{\bar{Y}}_{up}^{*} = \frac{1}{6} (\frac{9^{1 / 3} H}{α_{2}} + \frac{9^{2 / 3} x_{0}^{2} α_{2}}{H}) + \frac{x_{0}}{2}

(39)

where

H = {[x_{0} α_{2}^{2} (3 x_{0}^{2} α_{2} + 8 α_{1} + 4 \sqrt{4 α_{1}^{2} + 3 x_{0}^{2} α_{1} α_{2}})]}^{1 / 3}

In fact, only when ${\bar{Y}}_{up}^{*} < {\bar{Y}}_{\max}$ holds, the jump phenomenon occurs and ${\bar{Y}}_{up} = {\bar{Y}}_{up}^{*}$ makes sense. Therefore, the condition for jump avoidance is ${\bar{Y}}_{up}^{*} \geq {\bar{Y}}_{\max}$ as shown in Figure 6, where the upper bound of $x_{0}$ is 0.055 for system parameters $\hat{a} = 0.1$ , $λ_{2} = 1$ , $λ_{3} = 1$ , ${\hat{δ}}_{2} = 0.4$ , ${\hat{δ}}_{3} = 0.3$ , and $ζ = 0.05$ .

Figure 6.

Jump phenomenon: (a) jump phenomenon with $x_{0} = 0.14$ and (b) jump phenomenon with the increase of $x_{0}$ .

4. Comparative analysis of vibration isolation systems

The proposed CRSR vibration isolation system is compared in this section with other two vibration isolation systems shown in Figure 7. Model 1 is a common HSLDS isolation system with two spring–rod mechanisms. Model 2 uses a similar NSM with the isolator in Sun et al. (2019) that a horizontal spring is combined with a pair of cam–roller mechanisms. The effectiveness of the proposed CRSR system is verified by comparing the negative stiffness and absolute displacement transmissibility with these systems.

Figure 7.

Simplified models of two isolators: (a) Model 1: a spring–rod isolator and (b) Model 2: a cam–roller–spring isolator.

4.1. Negative stiffness

The dimensionless negative stiffness function of the CRSR system is given in equation (6), and the negative stiffness function of Model 1 can be obtained by setting $\hat{a}$ , $λ_{2}$ , and ${\hat{δ}}_{2}$ to be zero in equation (6)

k_{M 1} (y) = 2 λ_{3} [\frac{1 - {\hat{δ}}_{3}}{{(1 - y^{2})}^{3 / 2}} - 1]

(40)

which is the same as the expression in Liu and Yu (2018). The negative stiffness function of Model 2 is calculated by

k_{M 2} (y) = 2 λ_{3} [\frac{1 - {\hat{δ}}_{3} - \hat{a} y^{2} (5 - 4 y^{2})}{{(1 - y^{2})}^{3 / 2}} - 1 + 2 \hat{a} (3 \hat{a} y^{2} + \sqrt{1 - y^{2}} + {\hat{δ}}_{3} - 1)]

(41)

Assume that the two kinds of horizontal springs have the same stiffness, and the negative stiffness of the three systems is compared as shown in Figure 8. A wider range of relative displacement for achieving negative stiffness enables the vibration isolation system to withstand a stronger excitation. And the static carrying capacity will be expanded by combing a stiffer vertical spring if the NSM can provide a lower peak value of negative stiffness. Figure 8 shows that the CRSR vibration isolation system has the two advantages simultaneously.

Figure 8.

Dimensionless negative stiffness provided by three isolators ( $\hat{a} = 0.1$ , $λ_{2} = 1$ , $λ_{3} = 1$ , ${\hat{δ}}_{2} = 0.4$ , and ${\hat{δ}}_{3} = 0.3$ ).

4.2. Vibration isolation performance

The vibration isolation performance can be evaluated by comparing the absolute displacement transmissibility, which is defined as the ratio of the amplitude of absolute displacement of the isolated object to the excitation amplitude of the base

T = \frac{| x_{0} + \bar{Y} |}{x_{0}} = \frac{\sqrt{x_{0}^{2} + {\bar{Y}}^{2} + 2 x_{0} \bar{Y} \cos ϕ}}{x_{0}}

(42)

Under small excitation amplitude and high damping, the vibration isolation performance is quite different from that under large excitation amplitude and light damping. Therefore, the vibration isolation performance of the three systems is analyzed in the two cases mentioned above. Besides, a linear isolation system is also included for comparison in the following.

When the damping is high enough and the base excitation amplitude is small enough, corresponding to $ζ = 0.1$ and $x_{0} = 0.05$ , the jump will not occur. In the other case, $ζ$ is set to be 0.05, which is relatively small, and $x_{0}$ is set to be 0.12, which is relatively large. The absolute displacement transmissibility curves of four systems in the two cases are presented in Figure 9.

Figure 9.

Absolute displacement transmissibility of four vibration isolation systems ( $\hat{a} = 0.1$ , $λ_{2} = 1$ , $λ_{3} = 1$ , ${\hat{δ}}_{2} = 0.4$ , and ${\hat{δ}}_{3} = 0.3$ ): (a) $ζ = 0.1$ and $x_{0} = 0.05$ and (b) $ζ = 0.05$ and $x_{0} = 0.12$ .

It can be seen from Figure 9 that in the first case, Model 1 has a lower peak frequency and a lower peak transmissibility than Model 2. However, under large excitation amplitude and light damping, the transmissibility curve of Model 1 will bend to the right, which will make the isolation effect even worse than the linear model. The CRSR vibration isolation system can effectively eliminate the jump phenomenon in Model 1 without affecting the vibration isolation performance in the high frequency band. The hardening nonlinearity of the isolation system is weakened by the introduction of the cam–roller mechanism. The cubic coefficient $α_{2}$ of the restoring force in equation (22) has a quadratic form with respect to the parabolic coefficient of the cam, and it will decrease as the parabolic coefficient increase within a certain range. Therefore, the spring–rod mechanism in the CRSR isolation system can move in a nonlinear geometric trajectory by designing the parabolic coefficient, thereby weakening the nonlinear effects to avoid the jump phenomenon.

5. Prototype and experiments

5.1. Prototype and experimental setup

A prototype of the proposed CRSR system is developed using the parameters in Table 1. The parameters of springs are chosen considering the size of springs and the isolation performance practically. The supporting stiffness is provided by four springs each with a stiffness coefficient of 0.38686 N/mm. The mass of the isolated object is the sum of the upper platform and the additional weight for attaining the static equilibrium position. By replacing the cams with ones without curvature, the prototype of Model 1 in Figure 7 can be achieved, in which the cam–roller mechanism does not work as an NSM. Furthermore, removing the spring–rod mechanisms of the CRSR isolator, one obtains Model 2. The linear model without any NSM can also be realized.

Table 1.

Parameters of the prototype of the cam–roller–spring–rod isolator.

Parameter	Value	Parameter	Value
$m$	5.34 kg	$L$	50 mm
$\hat{a}$	0.15	$k_{1}$	1.5474 N/mm
$k_{2}$	1.4446 N/mm	$k_{3}$	2.3269 N/mm
$δ_{2}$	20 mm	$Y_{0}$	25 mm

Figure 10 shows the prototype of the proposed isolator and the experimental setup. The isolator is suspended using four rubber strings and subjected to a sinusoidal displacement excitation on its base. Two acceleration sensors (A and B) are installed on the base and the upper platform, respectively. Because the absolute acceleration transmissibility and displacement transmissibility have the same expression (Han et al., 2018), the transmissibility under consideration is defined as the ratio of root-mean-square values of the accelerations measured at A and B.

Figure 10.

Prototype of the cam–roller–spring–rod isolator and experiment setup: (a) prototype and (b) experiment setup: 1—the vibration isolation system, 2—vibration exciter (S 51075-M TIRA GmbH), 3—vibration controller (Spider-81 Crystal Instruments), 4—power amplifier (B&K2718 Brüel& Kjær), 5—acceleration sensor A, and 6—acceleration sensor B.

5.2. Experimental results

The linear model and Model 2 have high values of peak transmissibility. Considering the upper limit of relative displacement amplitude $Y_{0}$ , the amplitude of the excitation should be relatively small. With this consideration, the peak amplitude of the acceleration excitation in the two experiments is set to be $0.03 m / s^{2}$ , and the experimental results are shown in Figure 11. To match the experimental transmissibility curves, the values of $ζ$ are set to be 0.0308 and 0.074, respectively. Different values of $x_{0}$ at different frequencies in simulation are calculated as

x_{0} = \frac{| \ddot{X} |}{(L Ω^{2})}

(43)

where

| \ddot{X} |

is the peak amplitude of the acceleration excitation set in the vibration controller.

Figure 11.

Experimental and simulation results of the models: (a) linear model and Model 2 and (b) cam–roller–spring–rod isolator and Model 1

As seen in Figure 11(a), in the high frequency band, the experimental results are slightly higher than the simulation results. This can be explained by equation (43) that as the frequency increases, the displacement amplitude of the excitation will become smaller, which will result in a decreasing relative displacement between the base and the upper platform. Consequently, the negative stiffness effect will not be obvious, and the motion may even be affected by friction forces.

To clarify the effect of negative stiffness, the excitation amplitude should be sufficiently large (Zhou et al., 2015). In this regard, the peak amplitude of the excitation is set to increase from 0.04 to 0.8

{m/s}^{2}

in the frequency range of 1–10 Hz, which can be seen in Table 2. The excitation amplitude in each frequency band increases in the form of a fixed logarithmic slope. The values of

x_{0}

vary with the change of frequency and can be calculated by equation (43). Figure 11(b) presents the isolation performance of the CRSR isolator and Model 1. In Figure 12, the transmissibility curves of the CRSR isolator with different

δ_{3}

are depicted.

ζ

is set to be the same value as that of Model 2 for similar contact conditions.

Table 2.

Acceleration amplitude of excitation.

$f /Hz$	$\| \ddot{X} \| / m \cdot s^{- 2}$
1–2.5	0.04–0.06
2.5–5	0.06–0.4
5–10	0.4–0.8

Figure 12.

Experimental and simulation results of the cam–roller–spring–rod isolator with different $δ_{3}$ .

Vibration isolation refers to the frequency region with the transmissibility less than 1. Table 3 lists the critical frequency at which the transmissibility of the isolator reduces to 1. The results indicate that the proposed CRSR system is superior to the other vibration isolation systems under comparison. It is worth noting that the performance of the CRSR isolator can be further improved by increasing the compression of the lateral spring, just as analyzed in the previous sections.

Table 3.

Comparison of isolation frequencies.

Isolation system	Frequency/Hz
Linear	4.415
Model 1 $δ_{3} = 8.7 mm$	2.935
Model 2	4.021
CRSR $δ_{3} = 3.7 mm$	3.291
CRSR $δ_{3} = 8.7 mm$	2.416
CRSR $δ_{3} = 10.7 mm$	2.165

Note: CRSR: cam–roller–spring–rod.

6. Conclusions

A vibration isolation system with an HSLDS characteristic is proposed by combining a pair of cam–roller mechanisms and two spring–rod mechanisms (called CRSR for short) as an NSM. The static analysis is made to give the analytic expressions of restoring force and negative stiffness of the system, along with the contact and compression conditions for cam–rollers and springs. Then, the primary resonance of the system is solved by using the averaging method, and the jump conditions of the amplitude–frequency relation are derived. The proposed system is compared with some other vibration isolation systems from the aspects of the negative stiffness and absolute displacement transmissibility. The numerical and experimental studies indicate that the proposed system is superior to some other vibration isolation systems studied before.

The above studies show that the NSM can not only reduce the resonant frequency and resonant transmissibility of the cam–spring isolation system under high damping and small excitation amplitude but also alleviate the transmissibility jump of the slightly damped cam–spring isolation system under strong base excitation. Furthermore, adjusting the pre-compression of the spring can improve the vibration isolation performance. However, the selection of system parameters is also subject to multiple constraints, such as contact constraints and the upper bound of relative displacement. The optimization of the system parameters is the topic of our future work.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 11732006) and the Equipment Pre-research Foundation (grant number 6140210010202).

ORCID iDs

Yonglei Zhang

Guo Wei

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Design and analysis of a vibration isolation system with cam–roller–spring–rod mechanism

Abstract

Keywords

1. Introduction

2. Static analysis

2.1. Analysis of NSM

2.2. Conditions for cam–roller contact and compressed springs

2.3. Analysis of the CRSR isolator

3. Dynamic analysis

3.1. Dynamic modeling of the CRSR isolator

3.2. Analytical solution of primary resonance

3.3. Jump phenomenon

4. Comparative analysis of vibration isolation systems

4.1. Negative stiffness

4.2. Vibration isolation performance

5. Prototype and experiments

5.1. Prototype and experimental setup

5.2. Experimental results

6. Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iDs

References