Beneficial stiffness design of a high-static-low-dynamic-stiffness vibration isolator based on static and dynamic analysis

International Journal of Mechanical Sciences 142–143 (2018) 235–244

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Beneficial stiffness design of a high-static-low-dynamic-stiffness vibration isolator based on static and dynamic analysis Xiaojie Wang, Hui Liu∗, Yinqi Chen, Pu Gao School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, PR China

a r t i c l e

i n f o

Keywords: Vibration isolation High-static-low-dynamic-stiffness Stiffness design Averaging method

a b s t r a c t A high-static-low-dynamic-stiffness (HSLDS) isolator has good application prospects in low-frequency vibration and micro-vibration isolation. The nonlinear stiffness of the vibration isolation system strongly influences its dynamic responses and vibration isolation performance. In this paper, for a HSLDS isolator model, a stiffness range parameter d and a static equilibrium position stiffness Ksep are introduced to indicate the different stiffnesses of the nonlinear isolator. The averaging method is employed to solve the nonlinear dynamic equation of the proposed system at steady state for each excitation frequency. The effects of the parameters d and Ksep on the dynamic responses and isolation effectiveness of the HSLDS isolator are clearly analysed. The results show that an increase in d and a reduction in Ksep improve the isolation performance of the system. Considering the application of the HSLDS isolator in micro-vibration environments, the beneficial stiffness design under various excitation amplitudes and damping conditions is discussed on the basis of the limited maximum response amplitude. Several useful guidelines for the stiffness design of the HSLDS isolator are developed, which help to improve the vibration isolation performance of the system. The results provide useful insight for the design, analysis and application of HSLDS vibration isolators.

1. Introduction Vibrations that occur at low frequency are challenging to isolate. Many forms of structural damage in practical engineering, such as fatigue and failure induced by vibration, often occur at a low excitation frequency [1–2]. In the ideal case of a mass m supported by a linear stiffness k on a rigid foundation, as shown in Fig. 1(a), isolation does √ not occur until the excitation frequency 𝜔 is greater than 2𝑘∕𝑚 [3]. To develop a means to isolate low frequencies, many nonlinear isolators have been studied in recent years. A typical nonlinear isolator combines positive-stiffness and negative-stiffness elements at an equilibrium position under a designed loading weight, as shown in Fig. 1(b). Regarding the nonlinear isolator in Fig. 1(b), the task of the vertical spring k0 is to support the weight of the mass m, and low dynamic stiffness at the equilibrium position can be obtained. This is the so-called high-staticlow-dynamic-stiffness (HSLDS) vibration isolator. By careful choice of system parameters, it is possible to achieve a system with zero dynamic stiffness [4]. This is the quasi-zero-stiffness (QZS) characteristic of the HSLDS isolator. Many previous studies have shown that HSLDS isolators have prospective application in low-frequency vibration and microvibration isolation [5–6]. In the last decade, the theoretical analysis, design, and experimental study of HSLDS or QZS isolators have been widely reported. Carrella

∗

Corresponding author. E-mail address: [email protected] (H. Liu).

et al. [4,7] proposed a simple model of a QZS isolator comprising two oblique springs in parallel with a vertical spring and studied the restoring force and force transmissibility characteristics of the system. Subsequently, a more in-depth dynamic analysis of this nonlinear mechanism with three springs was conducted [8–9]. Thanh et al. [10] studied the nonlinear stiffness of a vibration isolator using negative stiffness structure and obtained the HSLDS characteristics of the system, then finished the experimental investigation with various values of stiffness. Using a linear spring and a permanent magnet to design negative stiffness structures, Xu et al. [11–12] studied two different designs of QZS vibration isolators; the theoretical and experimental results showed that the designed isolators offer favourable performance in low-frequency vibration isolation. Huang et al. [13] designed a HSLDS isolator using Euler buckled beam and performed theoretical and experimental studies of the isolator. Further, the effects of the imperfections of this isolation system on the dynamic response was also studied [14]. In addition to the above single-stage isolator, two-stage and multi-degree HSLDS isolators have been studied. Zhou et al. [15] theoretically studied the force transmissibility of a two-stage HSLDS isolator with cam-roller-spring mechanisms. Lu et al. [16] experimentally studied a novel design of a two-stage nonlinear vibration isolation system, and the negative stiffness was realized via a bistable carbon fibre-metal composite plate. Furthermore, Wang et al. [17] compared the dynamic performance of one- and two-stage QZS vibration isolators; the comparison demonstrated that the two-stage vibration isolator can be tuned to achieve improved isolation performance in the higher isolation frequency region than the baseline vibra-

https://doi.org/10.1016/j.ijmecsci.2018.04.053 Received 14 November 2017; Received in revised form 20 April 2018; Accepted 29 April 2018 Available online 30 April 2018 0020-7403/© 2018 Elsevier Ltd. All rights reserved.

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International Journal of Mechanical Sciences 142–143 (2018) 235–244

Fig. 1. Schematic diagram of the isolators: (a) linear isolator and (b) nonlinear isolator. Fig. 2. Schematic of the HSLDS vibration isolator device.

tion isolators. That and another study showed that these two-stage isolators exhibit better isolation performance than the corresponding singlestage isolators [17–18]. Multi-degree-of-freedom HSLDS vibration systems have also been studied. Zhou [19–20] and Zhu [21] studied sixdegrees-of-freedom vibration isolators in which the negative stiffness values were implemented based on cam-roller-spring mechanisms and magnetic levitation. In addition to the above reports, there have been many other design and dynamic studies on HSLDS isolators [22–25]. Moreover, review articles [5–6] have also summarized novel designs of HSLDS isolators. Regarding the theoretical study of the HSLDS nonlinear isolation system, Yang et al. [26] studied the dynamic and power flow characteristics of a QZS isolator and analysed the sub-harmonic response of the system. Hao et al. [27] studied the frequency-response characteristics of an archetypal QZS dynamical model, and a novel concept of initial value control was proposed; this proposed control promises to be an effective method to address the frequency jump phenomena and multi-solution coexistence by inducing the system to perform the desired motion with a small vibration amplitude. The force and displacement transmissibility of two different HSLDS isolators have been studied, and the results showed that these two parameters are different for these nonlinear vibration isolation systems [28–29]. In addition, the dynamic characteristics of systems with imperfect design [14,30–31] and design parametric errors [32] have also been studied. The nonlinearity of the QZS isolation system affects the safety and stability of the system. Thus, some researchers have added an active control unit to the original HSLDS isolator to improve the stability and vibration isolation performance of the system. Sun et al. [1] developed a QZS isolator with a simple linear time-delayed active control strategy in which beneficial isolation performance and stability were obtained when the system was operated under the conditions of an external impact load and external harmonic load. Next, a QZS isolation system combined with chaotification was studied; excellent performance in both the vibration isolation and the suppression of vibration signal features was obtained [33]. Moreover, time-delayed feedback control has also been studied [34–35]. The resulting active-control HSLDS isolators all obtained better isolation performance than the corresponding passive nonlinear isolation systems. The above findings show that the study of HSLDS isolation systems has become increasingly detailed including static analysis, dynamic analysis and active control research. Due to the nonlinear stiffness characteristics of the HSLDS isolator, it has many differences in terms of isolation performance from a linear system. For example, the force and displacement transmissibility expressions are different in a HSLDS vibration isolation system [28–29]. In addition, the excitation amplitude has a strong influence on the transmission characteristics, different from the case of a linear isolation system [36]. Summarizing the static and dynamic characteristics of HSLDS vibration isolation systems, we conclude that the nonlinear stiffness, excitation amplitude and damping have a great and intuitive effects on the dynamic performance. Therefore, considering the nonlinearity of the stiffness of a HSLDS system, for a variety

Fig. 3. Force principle of the horizontal spring.

of vibration environments, designing a reasonable stiffness will be beneficial to improve the vibration isolation performance of the system. In study of the stiffness design and analysis, Carrella et al. optimized the stiffness of the HSLDS isolators from static analysis [4,7,9], Le et al. studied the influence of different stiffnesses on the system dynamics [10,37]. These studies and others have inspired much of our research. In this paper, the beneficial stiffness design of a HSLDS isolator based on static and dynamic analysis will be studied. Different from previous studies, we introduce the stiffness range parameter d and static equilibrium position stiffness Ksep to intuitively represent various stiffnesses of the vibration isolation system and analyse their effects on the system dynamics characteristics. The purpose of this paper is to find the optimal stiffness with good vibration isolation performance in several different stiffness curves, and provide a basis stiffness design for the HSLDS isolator operated in various vibration environments. This study provides a useful insight for the design, analysis and application of HSLDS isolators, which is also the original contribution of this paper. This paper is organized as follows. In Section 2, a simple isolator with negative stiffness structure is proposed, and the nonlinear stiffness characteristics are analysed. Then the stiffness range parameter d and static equilibrium position stiffness Ksep are introduced to indicate the different stiffnesses of the isolator. In Section 3, the nonlinear dynamic equation is analytically solved via the averaging method, and then the effects of the parameters d and Ksep on the steady state response and vibration isolation performance of the HSLDS isolator are clearly analysed. In Section 4, the beneficial stiffness design is discussed by combining static and dynamic analysis. Finally, in Section 5, some conclusions are given. 2. Isolation system and static analysis 2.1. Model description The model used in this study is based on the design concept of [1,10], as shown in Fig. 2. Fig. 3 shows the force principle of the horizontal spring. The mass M, which is the equipment to be isolated from the base, is connected to the base through two horizontal springs of linear stiffness k1 with a connecting bar of length L and a vertical spring of 236

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International Journal of Mechanical Sciences 142–143 (2018) 235–244

stiffness k0 . One end of the horizontal spring is fixed to the base, and the other end is connected to a roller, which is free to slide in the horizontal direction [1]. The roller is linked to the connecting bar, and the bar is joined to the mass at point P. Therefore, the mass can move in the vertical direction [10]. When x = 0, the connecting bar is horizontal. At this position, the horizontal spring has pre-compression h; we designed this position as the static equilibrium position, as shown in Fig. 3, state 1. The vertical spring supports the weight of mass M. When the mass M is installed on the system, it is supported effectively only by the vertical spring, and the system is at the equilibrium position. The vibration excitation fcos 𝜔t acts on mass M. When vibration occurs, mass M undergoes a vertical displacement x, as shown in Fig. 3, state 2, the horizontal springs produce a negative stiffness force in the vertical √ direction, and the deformation of the horizontal spring is Δℎ = 𝐿 −

𝐿2 − 𝑥2 .

2.2. Static analysis Based on the model of the isolator in Fig. 2 and the force principle of the horizontal spring in Fig. 3, the negative stiffness force in the vertical direction is ( ) 𝐿−ℎ 𝐹1 = 2𝑘1 𝑥 1 − √ (1) 𝐿2 − 𝑥2

Fig. 4. The influences of parameters ℎ̄ and 𝑘̄ on 𝐾̄ 𝑠𝑒𝑝 .

dimensionless stiffness are shown in Fig. 5(a). When 𝑘̄ = 0.3 and ℎ̄ = 1.2, the nonlinear stiffness curve is a convex parabola on which the stiffness achieves a maximum at the static equilibrium position, and the stiffness of the system will decrease to negative as the mass moves away from the static equilibrium position. Therefore, this case cannot be used for designing a system to improve the isolation performance. When 𝑘̄ = 0.3 and ℎ̄ = 0.5, the value of the stiffness of the system is always greater than zero, so this case meets the basic conditions of stiffness design. In particular, when ℎ̄ = 0, the stiffness of the system at the static equilibrium position is one. The stiffness of this system will always be greater than the corresponding linear system without a negative stiffness structure. In this case, the system cannot effectively isolate low-frequency vibrations [10]. In addition, when ℎ̄ = 1, the stiffness does not change with displacement, and its value is determined by the stiffness ratio 𝑘̄ . When select the parameters ℎ̄ and 𝑘̄ on the curve 𝐾̄ 𝑠𝑒𝑝 = 0, the QZS characteristic of the nonlinear isolator can be obtained. Through the above analysis, we see that the region 𝐾̄ 𝑠𝑒𝑝 ≥ 0with ℎ̄ ≤ 1 is the parameter selection region for stiffness design, and these are the basic conditions to design the stiffness of this nonlinear vibration isolator. Fig. 5(b) shows stiffness curves with ℎ̄ = 0.5 and various 𝑘̄ . In this condition, the curves are concave parabolas, and two common intersection points exist at the dimensionless stiffness 𝐾̄ = 1; this can be obtained by solving Eq. (5). To describe the static characteristics of the system more clearly and intuitively, the dimensionless stiffness range parameter 𝑑̄ is introduced, defined as the mass displacement from the static equilibrium position over which the dimensionless nonlinear stiffness 𝐾̄ of the HSLDS system is smaller than one [10]. By solving Eq. (5) for 𝐾̄ = 1, based on the values of ℎ̄ and 𝑘̄ , the parameter 𝑑̄ can be obtained √ 2∕3 𝑑̄ = 2 1 − (1 − ℎ̄ ) (7)

Therefore, the relationship between the restoring force F and the displacement x can be derived. ( ) 𝐿−ℎ 𝐹 = 𝑘0 𝑥 − 2𝑘1 𝑥 1 − √ (2) 𝐿2 − 𝑥2 We introduce the dimensionless parameters as follows 𝑘̄ =

𝑘1 ℎ 𝑥 𝐹 , ℎ̄ = , 𝑥̄ = , 𝐹̄ = 𝑘0 𝐿 𝐿 𝑘0 𝐿

(3)

where 𝑘̄ is the spring stiffness ratio, ℎ̄ is the dimensionless precompression deformation, and 𝑥̄ and 𝐹̄ are the dimensionless displacement and the dimensionless restoring force, respectively. Thus, the dimensionless restoring force can be given by ( ) 1 − ℎ̄ ̄ ̄ 𝐹 = 𝑥̄ − 2𝑘𝑥̄ 1 − √ (4) 1 − 𝑥̄ 2 By differentiating Eq. (4) with respect to the dimensionless displacement 𝑥̄ , the dimensionless nonlinear stiffness 𝐾̄ of the system is obtained as ( ) 1 − ℎ̄ 𝐾̄ = 1 − 2𝑘̄ 1 − (5) 3∕2 (1 − 𝑥̄ 2 ) According to Eq. (5), the stiffness 𝐾̄ is nonlinear about the dimensionless parameter 𝑥̄ and is influenced by dimensionless parameters 𝑘̄ and ℎ̄ . By substituting 𝑥̄ = 0 into Eq. (5), the dimensionless nonlinear stiffness at the static equilibrium position is found to be 𝐾̄ 𝑠𝑒𝑝 = 1 − 2𝑘̄ ⋅ ℎ̄

(6)

It can be seen that the value of 𝐾̄ 𝑠𝑒𝑝 is influenced by parameters 𝑘̄ and ℎ̄ , as shown in Fig. 4. Fig. 5 depicts the dimensionless nonlinear stiffness curves for various values of 𝑘̄ and ℎ̄ . In Fig. 4, two regions exist, divided by the curve corresponding to 𝐾̄ 𝑠𝑒𝑝 = 0. It is known that a system with negative stiffness cannot support the load. Thus, in order to support the load, the values of parameters ℎ̄ and 𝑘̄ must be selected suitably so that the nonlinear stiffness is always positive. Considering that the isolation system oscillates near the static equilibrium position, the stiffness at the equilibrium position cannot be negative. Thus, in the region 𝐾̄ 𝑠𝑒𝑝 < 0, the parameters ℎ̄ and 𝑘̄ cannot be selected to design the isolator. From Eq. (5) it can be obtained that the dimensionless stiffnesses of ℎ̄ > 1 and ℎ̄ < 1 are expressed as a convex function and a concave function, respectively. Select the parameters ℎ̄ and 𝑘̄ in region 𝐾̄ 𝑠𝑒𝑝 > 0, the

Eq. (7) indicates that parameter 𝑑̄ is related only to parameter ℎ̄ , which is also shown in Fig. 5(b). From Eqs. (6) and (7), for different stiffness curves of the HSLDS vibration isolation system, if parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 are fixed, then parameter ℎ̄ and 𝑘̄ can be determined; as a result, the unique stiffness curve can be obtained. Therefore, stiffness range parameter 𝑑̄ and static equilibrium position stiffness 𝐾̄ 𝑠𝑒𝑝 can be used to characterize the different stiffness curves. Fig. 6 shows the schematic description of the nonlinear stiffness represented by parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 . In the following analysis the influences of parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 on the system will be studied. Here it should be noted that the dimension parameters corresponding to dimensionless parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 are defined as d and Ksep , respectively. 237

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Fig. 5. Stiffness curves for different values of ℎ̄ and 𝑘̄ . (a) 𝑘̄ = 0.3 with different ℎ̄ , (b) ℎ̄ = 0.5 with different 𝑘̄ .

will be studied. Note that in this section, the dimensionless excitation amplitude𝑓̄ = 0.002 and the damping ratio 𝜉 = 0.04 are chosen for the following analysis. 3.1. Dynamic analysis For the nonlinear vibration isolation system shown in Fig. 2, considering the viscous linear damping of the system, the dynamic equation can be given by 𝑚𝑥̈ + 𝑐 𝑥̇ + 𝐹 = 𝑓 cos 𝜔𝑡

where m is the mass, f and 𝜔 are the amplitude and frequency of the harmonic excitation force, respectively, and F is the restoring force, as shown in Eq. (2). Introducing the following dimensionless expressions √ 𝑘0 𝑓 𝑐 𝜔 𝜔𝑛 = ,𝜉 = , 𝑓̄ = ,Ω = , 𝜏 = 𝜔𝑛 𝑡 (11) 𝑚 2𝑚𝜔𝑛 𝑘0 𝐿 𝜔𝑛

Fig. 6. Schematic description of the nonlinear stiffness with 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 .

To simplify the subsequent dynamic analysis, the relationship between the restoring force and displacement is approximated by its seventh-order Taylor expansion at the static equilibrium position 𝐹̄ = 𝛼1 𝑥̄ + 𝛼2 𝑥̄ 3 + 𝛼3 𝑥̄ 5 + 𝛼4 𝑥̄ 7

and combining Eqs. (3), Eq. (10) can be rewritten in dimensionless form 𝑥̄ ′′ + 2𝜉 𝑥̄ ′ + 𝐹̄ = 𝑓̄ cos(Ω𝜏)

(8)

where 𝛼1 = 1 − 2𝑘̄ ⋅ ℎ̄ , 𝛼2 = 𝑘̄ (1 − ℎ̄ ), 𝛼3 = 34 𝑘̄ (1 − ℎ̄ )and 𝛼4 = 58 𝑘̄ (1 − ℎ̄ ). The seventh-order Taylor expansion of the nonlinear stiffness is 𝐾̄ = 𝛼1 + 3𝛼2 𝑥̄ 2 + 5𝛼3 𝑥̄ 4 + 7𝛼4 𝑥̄ 6

(10)

(12)

where the prime (·)’ represents differentiation with respect to 𝜏, and 𝐹̄ is the dimensionless restoring force, as shown in Eq. (4). The averaging method can be used to solve Eq. (12). Consider the following expressions

(9)

A one-dimensional form of Eq. (5) and its approximation given by Eq. (9) are illustrated in Fig. 7(a). We see that the seventh order form (short dash lines) closely follows the original restoring force curve (solid lines). Fig. 7(b) shows the accuracy of the seventh-order Taylor series expansion in fitting the dimensionless stiffness. For dimensionless displacements 𝑥̄ < 0.55, the percentage error between Eqs. (5) and (9) is < 5%. Obviously, the error is extremely small in the neighbourhood of the static equilibrium position, but it would rise steeply when the displacement is far away from the static equilibrium position. In other words, this approximate but concise expression of the stiffness is effective as long as the system does not undergo large amplitude oscillations. Fig. 8 shows the errors when 𝑥̄ = 0.55 with different parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 . It can be seen that the percentage error between Eqs. (5) and

𝑥̄ = 𝑋̄ cos(Ω𝜏 + 𝜑)

(13a)

𝑥̄ ′ = −Ω𝑋̄ sin(Ω𝜏 + 𝜑)

(13b)

where 𝑋̄ is the response amplitude of the system and 𝜑 is the corresponding phase angle. According to the theory of the averaging method, 𝑋̄ and 𝜑 are functions of time 𝜏. Differentiating Eq. (13a) with respect to 𝜏, and comparing Eq. (13b), we obtain 𝑋̄ ′ cos(Ω𝜏 + 𝜑) − 𝑋̄ 𝜑′ sin(Ω𝜏 + 𝜑) = 0

(14)

Differentiating Eq. (13b) with respect to 𝜏 and comparing the result with Eq. (14) yields 𝑥̄ ′′ = −Ω𝑋̄ ′ sin(Ω𝜏 + 𝜑) − Ω2 𝑋̄ cos(Ω𝜏 + 𝜑) − Ω𝑋̄ 𝜑′ cos(Ω𝜏 + 𝜑)

(9) is always < 5%. As expected, the exact but complex expression of the stiffness can be replaced by the approximate expression in the subsequent dynamic analysis [15], as verified in Sections 3 and 4.

𝑥̄ ′

(15)

𝑥̄ ′′

Substituting the expressions of 𝑥̄ , and into Eq. (12) and comparing the result of Eq. (14), the following equation is obtained −Ω𝑋̄ ′ sin(Ω𝜏 + 𝜑) − Ω2 𝑋̄ cos(Ω𝜏 + 𝜑) − Ω𝑋̄ 𝜑′ cos(Ω𝜏 + 𝜑) −2𝜉 𝑋̄ Ω sin(Ω𝜏 + 𝜑) + 𝐹̄ = 𝑓̄ cos(Ω𝜏)

3. Dynamic analysis and vibration isolation performance

(16)

Substituting Eq. (14) into Eq. (16), the expressions of the time change rates of the response amplitudes and the phase angles are found

In this section, the mathematical model of the HSLDS vibration isolation system will be built, and its dynamic characteristics will be analysed using the averaging method. Furthermore, the influence of parameters d and Ksep on the dynamic responses and vibration isolation performance

1 𝑋̄ ′ = 𝐻 sin(Ω𝜏 + 𝜑) Ω 238

(17a)

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International Journal of Mechanical Sciences 142–143 (2018) 235–244

Fig. 7. (a) The exact expression of stiffness and approximation. Solid line exact expression, short dash line approximation with seventh-order Taylor polynomial. (b) Percentage error between the exact and seventh-order expansions.

Eq. (21) consists of a quadratic equation in Ω2 . Solving the above quadratic equation for Ω2 gives √ √ 𝑊 1 Ω1 , 2 = − 2𝜉 2 ± 4𝜉 4 𝑋̄ 2 − 4𝑊 𝜉 2 𝑋̄ + 𝑓̄2 (22) 𝑋̄ 𝑋̄ where 𝑊 = 35 𝛼 𝑋̄ 7 + 58 𝛼3 𝑋̄ 5 + 34 𝛼2 𝑋̄ 3 + 𝛼1 𝑋̄ . Ω1 and Ω2 are the res64 4 onant and non-resonant branches in the frequency response function [9]. For small values of damping ratio, the resonant and non-resonant branches intersect at the jump down frequency, which occurs approximately when 𝑋̄ is a maximum. So the maximum displacement response amplitude 𝑋̄ max can be obtained when Ω1 = Ω2 . Thus, 4𝜉 4 𝑋̄ 2 − 4𝑊 𝜉 2 𝑋̄ + 𝑓̄2 = 0

The peak frequency can be obtained when the maximum response occurs, which is approximately equal to the jump down frequency [38]. Therefore, the peak frequency Ωd can be obtained by substituting𝑋̄ max into Eq. (22). The vibration transmissibility is the basic index to evaluate the vibration isolation performance. Assuming that the response is dominated by the fundamental harmonic response, the force transmitted to the base can be given by

Fig. 8. Percentage error between the exact and approximate expressions when 𝑥̄ = 0.55 with different values of 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 .

𝑓̄𝑡 = −2𝜉Ω𝑋̄ sin Ω𝜏 + 𝛼1 𝑋̄ + 𝛼2 𝑋̄ 3 + 𝛼3 𝑋̄ 5 + 𝛼4 𝑋̄ 7 ≈ −2𝜉Ω𝑋̄ sin Ω𝜏 + 𝑊 cos Ω𝜏 √ 2 = (2𝜉Ω𝑋̄ ) + 𝑊 2 cos(Ω𝜏 + 𝜇)

1 𝜑 = (17b) 𝐻 cos(Ω𝜏 + 𝜑) 𝑋̄ Ω where 𝐻 = 𝐹̄ − 𝑓̄ cos(Ω𝜏) − 2𝜉 𝑋̄ Ω sin(Ω𝜏 + 𝜑) − Ω2 𝑋̄ cos(Ω𝜏 + 𝜑). ′

Assuming that the response amplitudes and the phase angles are slowly varying variables of time, the left-hand sides of Eqs. (17) can be approximated by their average values over an excitation cycle. 1 𝑋̄ ′ = 2𝜋 ∫0 𝜑′ =

2𝜋 Ω

1 2𝜋 𝑋̄ ∫0

𝐻 sin(Ω𝜏 + 𝜑)𝑑𝜏 2𝜋 Ω

(18b)

The maximum force transmissibility Tmax can be obtained approximately when 𝑋̄ max and Ωd occur.

Thus, using the approximate expression of the restoring force in Eq. (8), the integrations in Eqs. (18) are calculated as 𝑓̄ sin 𝜑 𝑋̄ ′ = −𝑋̄ 𝜉 − 2Ω

3.2. Responses and isolation performance

(19a)

The steady state displacement amplitude-frequency responses of the HSLDS isolator are predicted by analytical method, as shown in Fig. 9, which is compared with numerical solutions, when 𝑓̄ = 0.002, 𝜉 = 0.04, 𝑑̄ = 0.2 and 𝐾̄ 𝑠𝑒𝑝 = 0. In Fig. 9, the solid and short dash lines denote stable and unstable analytical solutions of Eq. (12), respectively, and solid dots and hollow cycles denote numerical solutions with initial values (0,0) and (0.05,0), respectively. The numerical solutions of Eq. (12) are obtained by using a Runge–Kutta method with fourth-order accuracy. It is seen that the averaging method solutions with the seventh-order approximate of the restoring force gives close approximation to the numerical solutions. In Fig. 9, in the single-valued area of the analytical solutions, different initial value condition have the same steady-state

4 6 2 𝛼 35 𝛼4 𝑋̄ 5 𝛼3 𝑋̄ 3 𝛼2 𝑋̄ Ω 𝑓̄ cos 𝜑 (19b) + + + 1 − − 128 Ω 16 Ω 8 Ω 2Ω 2 2Ω𝑋̄ Under steady state motion, the response amplitudes and phase angles remain unchanged; thus, their derivatives will vanish, so { 𝑓̄ sin 𝜑 = −2Ω𝜉 𝑋̄ (20) 𝑓̄ cos 𝜑 = 35 𝛼4 𝑋̄ 7 + 5 𝛼3 𝑋̄ 5 + 3 𝛼2 𝑋̄ 3 + 𝛼1 𝑋̄ − Ω2 𝑋̄

𝜑′ =

64

8

(24)

where tan 𝜇=2𝜉Ω𝑋̄ ∕𝑊 , the analytical force transmissibility, defined as the ratio of the amplitude of the transmitted force to that of the excitation amplitude, can be written in the form of decibels as √ ⎛ ⎞ 2 ⎜ (2𝜉Ω𝑋̄ ) + 𝑊 2 ⎟ 𝑇 = 20log10 ⎜ (25) ⎟ 𝑓̄ ⎜ ⎟ ⎝ ⎠

(18a)

𝐻 cos(Ω𝜏 + 𝜑)𝑑𝜏

(23)

4

Cancelling out the trigonometric terms with phase angles in Eq. (20), we obtain ( )2 35 ̄ 7 5 ̄ 5 3 ̄ 3 𝛼4 𝑋 + 𝛼3 𝑋 + 𝛼2 𝑋 + 𝛼1 𝑋̄ − Ω2 𝑋̄ + (2Ω𝜉 𝑋̄ )2 = 𝑓̄2 (21) 64 8 4 239

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vibration isolation system is studied in detailed in [27], so this is not shown here. By solving Eq. (23), the maximum response amplitudes with different values of 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 are obtained, as shown in Fig. 10(a). Fig. 10(a) shows that a larger 𝑑̄ and smaller 𝐾̄ 𝑠𝑒𝑝 correspond to a larger 𝑋̄ max under the same values of excitation amplitude and damping. For different stiffness parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 , a large maximum response amplitude results in a low peak frequency and a low maximum transmissibility, as shown in Fig. 10(b) and (c). Dynamic stiffness 𝐾̄ 𝑠𝑒𝑝 has a great influence on the peak frequency and maximum force transmissibility. In particular, when 𝐾̄ 𝑠𝑒𝑝 = 1, the peak frequency is equal to one because at this point, the HSLDS vibration isolation system is equivalent to a linear system with a dimensionless stiffness of one. Similarly, there is the same maximum transmissibility when 𝐾̄ 𝑠𝑒𝑝 = 1. Fig. 11 is the relationship between the peak frequency Ωd and dynamic stiffness 𝐾̄ 𝑠𝑒𝑝 under different parameters 𝑑̄. It also can be seen that the curves have the same intersection at point 𝐾̄ 𝑠𝑒𝑝 = 1. In addition, for most of the values of parameter 𝑑̄, the values of peak frequency will increase when the dynamic stiffness 𝐾̄ 𝑠𝑒𝑝 increases. But when 𝑑̄ = 0.02, the peak frequency curve will decrease when the dynamic stiffness 𝐾̄ 𝑠𝑒𝑝 increase, the sys-

Fig. 9. Steady state displacement amplitude-frequency response of the HSLDS isolator. Solid and short dash lines denote stable and unstable analytical solutions of Eq. (12), respectively, and solid dots and hollow cycles denote numerical solutions with different initial values, when 𝑓̄ = 0.002, 𝜉 = 0.04, 𝑑̄ = 0.2 and 𝐾̄ 𝑠𝑒𝑝 = 0.

tem loses its low-frequency vibration isolation capability. In Fig.11 it can be seen that when 𝑑̄ < 0.1, increasing 𝑑̄ can effectively reduce the peak frequency. But when 𝑑̄ > 0.3, for relatively large 𝐾̄ 𝑠𝑒𝑝 , the effect of the parameter 𝑑̄ upon the peak frequency is small. As shown in Fig. 11, for the same value of 𝐾̄ 𝑠𝑒𝑝 , a larger parameter 𝑑̄ corresponds to a smaller peak frequency. Furthermore, a larger 𝑑̄ yields a larger maximum response amplitude and wider isolation region

response. In the three-valued area of the analytical solutions, the short dash line is an unstable solution, and simulation calculation cannot be realized. At this point, the steady state of the HSLDS vibration isolation system is sensitive to the initial value conditions, different initial value conditions make the steady state of the vibration isolation system be located in one of the other two solutions. The other influence of the initial value conditions on dynamic characteristics of the HSLDS nonlinear

Fig. 10. The influences of parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 on the dynamic characteristics for 𝑓̄ = 0.002 and 𝜉 = 0.04. (a) maximum response amplitude (b) peak frequency (c) maximum force transmissibility. 240

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the static analysis in Section 2.2, the corresponding selectable stiffness design can be obtained, as shown in Fig. 14. The curves in Fig. 14 are critical choices that exactly meet the designed maximum response amplitude 𝑋̄ max . For example, when the designed maximum response amplitude is 0.1, the solid line𝑋̄ max = 0.1 divides the plot into two areas, namely a area and b area. The stiffness selection in the b area does not meet the design conditions because the response amplitude exceeds the designed maximum response amplitude. In the a area, such as, point E, although its response amplitude is smaller than the designed maximum response amplitude, the vibration isolation region will be smaller than the points C and D on the critical curve𝑋̄ max = 0.1, this result was already analysed in Section 3.2, as shown in Figs. 12(b) and 13(b). As also can be seen from Fig. 14, under the premise that the system response does not exceed the designed maximum response amplitude, as the designed maximum response amplitude increases, there will be a wider parameters selection area for the stiffness design of the HSLDS isolator. In Fig. 14 on the critical curve, many stiffness choices make the system response amplitude exactly equal to the designed maximum response amplitude. Here we take the maximum response amplitude of 0.1 as an example, select four points on the critical curve (in Fig. 14, the coordinates of the four points A, B, C and D are (0.39, 0), (0.46, 0.02), (0.60, 0.04) and (0.75, 0.05), respectively.), and analyse the stiffness and vibration isolation performance of these four points. The results are shown in Fig. 15. Fig. 15(a) shows the dimensionless stiffness curves of the points A, B, C and D, which have the same maximum response amplitude. There is an interesting phenomenon here. The stiffness curves have two common approximate intersections at 𝑥̄ = ±0.05. This can be proved as follows. Substituting W into Eq. (23), Eq. (23) can then be rewritten as

Fig. 11. The peak frequency Ωd -dynamic stiffness 𝐾̄ 𝑠𝑒𝑝 characteristic under different values of 𝑑̄ for 𝑓̄ = 0.002 and 𝜉 = 0.04.

of the isolation system, as shown in Fig. 12, in which the solid lines denote stable solutions and the short dashed lines unstable ones. It can be observed that the responses are sensitive to parameter 𝑑̄in the resonance frequency region, the jump unstable phenomenon disappears for comparatively large 𝑑̄. But in the effective vibration isolation region, as parameter 𝑑̄changes, there is no significant effects on the transmissibility. We can see that the isolation region, the vibration attenuation rate, and the dynamic stability of the HSLDS system are improved when the parameter 𝑑̄is increased. As shown in Fig. 11, for the same value of 𝑑̄, a larger parameter 𝐾̄ 𝑠𝑒𝑝 corresponds to a larger peak frequency. The peak frequency is sensitive to 𝐾̄ 𝑠𝑒𝑝 , especially when parameter 𝑑̄ is comparatively large. Fig. 13 show the displacement amplitude-frequency response curves and force transmissibility curves under different values of 𝐾̄ 𝑠𝑒𝑝 . It can be seen that as 𝐾̄ 𝑠𝑒𝑝 decreases, the effective vibration isolation region increases and the force transmissibility decreases both in the resonance frequency region and in the effective vibration isolation region. Meanwhile, we find that as the 𝐾̄ 𝑠𝑒𝑝 decreases, the response amplitude increase and the jump unstable phenomenon occurs as shown in Fig. 13(a). In summary, the system responses and vibration isolation performance are sensitive to the parameter 𝑑̄and 𝐾̄ 𝑠𝑒𝑝 . Increasing 𝑑̄ and reducing 𝐾̄ 𝑠𝑒𝑝 , improves the vibration isolation effectiveness of the HSLDS vibration isolation system in terms of both increasing the vibration isolation region and reducing the force transmissibility. However, this will create two problems: First, the response amplitude increases, as shown in Figs. 12(a) and 13(a), which challenges the installation and application of the isolator; Second, the jump unstable phenomenon occurs as 𝐾̄ 𝑠𝑒𝑝 decreases, as shown in Fig. 13(a) and (b).

𝑓̄2 35 ̄ 6 5 ̄ 4 3 ̄ 2 𝛼 𝑋 + 𝛼3 𝑋 + 𝛼2 𝑋 + 𝛼1 = + 𝜉2 64 4 8 4 4𝜉 2 𝑋̄ 2

(26)

From Eq. (9) and combined with Eq. (26),we can obtain 3 5 7 2 𝐾̄ (0.5𝑋̄ max ) = 𝛼1 + 𝛼2 𝑋̄ max + 𝛼 𝑋̄ 4 + 𝛼 𝑋̄ 6 4 16 3 max 64 4 max 𝑓̄2 5 7 = + 𝜉2 − 𝛼 𝑋̄ 4 − 𝛼 𝑋̄ 6 16 3 max 16 4 max 4𝜉 2 𝑋̄ 2

(27)

max

Considering the small vibration response amplitudes of the microvibration system, its higher order terms can be ignored. Therefore, Eq. (27) can be rewritten approximately as 𝐾̄ (0.5𝑋̄ max ) ≈

𝑓̄2

2 4𝜉 2 𝑋̄ max

+ 𝜉2

(28)

This approximate expression does not include the parameters 𝛼 1 , 𝛼 2 , 𝛼 3 , and 𝛼 4 , which represent different stiffnesses. Thus, for the same excitation amplitude, damping and designed maximum response amplitude, the stiffness curves can be approximately intersected at two points ̄2 𝑥̄ = ±0.5𝑋̄ max , and the value of the stiffness is 𝑓 + 𝜉2 .

4. Discussion As mentioned in Section 3, the dynamic responses, vibration isolation performance and stability are sensitive to the stiffness of the HSLDS vibration isolation system. Therefore, in the application of the HSLDS vibration isolation system, it should be studied how to design a system with beneficial stiffness according to the vibration environments. As described in [6], in micro-vibration environments, the HSLDS vibration isolation system can be very helpful for the design of novel isolators to achieve high stability and excellent isolation performance at both low and high frequencies. In this section, we mainly consider using the HSLDS isolator in micro-vibration isolation environments. In the application of the HSLDS isolator, the installation problem should be considered first under specific environments, thus, the displacement should be limited for a realistic design [17]. This means that the maximum vibration displacement responses should be limited so the isolator does not affect other devices. As shown in Fig. 10(a), when the maximum response amplitude 𝑋̄ max of the system is set to a certain value, corresponding to multiple settings of stiffness design, substituting the certain maximum response amplitude into Eq. (23) and considering

2 4𝜉 2 𝑋̄ max

Fig. 15(b) shows the force transmissibility of the four points A, B, C and D. It can be seen that the curves have the same maximum transmissibility and peak frequency. In the effective vibration isolation region, the stiffness represented by point A has less force transmissibility than the other three points. In addition, the stiffnesses represented by point A and point B make the system unstable, and the stiffnesses represented by point C and point D make the system stable. This also reflects that increase in 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 are beneficial to the stability of the HSLDS vibration isolation system as mentioned in Section 3.2. In application, the excitation amplitude and damping will have a major effects on the HSLDS vibration isolation system [36,38].Therefore, we will discuss the beneficial stiffness design of the HSLDS system under different excitation amplitudes and damping. Before designing the stiffness of the isolator, it is necessary to consider the installation environment, structural design and other factors, to limit the maximum 241

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Fig. 12. (a) Displacement amplitude-frequency response and (b) force transmissibility influenced by different values of 𝑑̄ when 𝐾̄ 𝑠𝑒𝑝 = 0 for 𝑓̄ = 0.002 and 𝜉 = 0.04.

Fig. 13. (a) Displacement amplitude-frequency response and (b) force transmissibility influenced by different values of 𝐾̄ 𝑠𝑒𝑝 when 𝑑̄ = 0.7 for 𝑓̄ = 0.002 and 𝜉 = 0.04.

analysis, we limit the maximum dimensionless response amplitude to 𝑋̄ max = 0.1, which means that the maximum vibration displacement of the system is 0.1 L, L is the length of a connecting bar as shown in Fig. 2. Based on this defined maximum response amplitude, the beneficial stiffness design of the system with different excitation amplitudes and damping are shown in Fig. 16, in which solid lines indicate that its stiffness will give a stable solution to Eq. (12), short dashed lines indicate unstable solutions. The solid dots denote the critical stiffness between stability and instability, which can be verified by Eq. (21). As can be seen in Figs. 14 and 15(b), for the stiffness curve with same maximum response amplitude, smaller values of parameter 𝐾̄ 𝑠𝑒𝑝 yield a lower vibration transmissibility in the effective isolation region. Therefore, to ensure a good vibration isolation performance and stability of the system, the stiffness near solid dots can be used as a reference for beneficial stiffness design of the system with different excitation amplitudes and damping. Fig. 16 shows that the excitation amplitude and damping have a great effects on the stiffness design of the system. As shown in Fig. 16(a), with different excitation amplitudes, the parameters 𝑑̄ and 𝐾̄ 𝑠𝑒𝑝 at the critical point are different. As the excitation amplitude increases, 𝑑̄ de-

Fig. 14. Stiffness curves corresponding to maximum response amplitude for 𝑓̄ = 0.002 and 𝜉 = 0.04.

response amplitude of the isolator, in order to avoid collisions with surrounding equipment or other negative effects. As mentioned earlier, in the micro-vibration environment, only the case of small oscillation amplitude is considered here. Without loss of the generality, in the later

Fig. 15. (a) The dimensionless stiffness of four points and (b) the corresponding force transmissibility for 𝑓̄ = 0.002 and 𝜉 = 0.04. 242

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Fig. 16. Stiffness curves based on 𝑋̄ max = 0.1 and (a) various excitation amplitudes with 𝜉 = 0.04 and (b) various damping values with 𝑓̄ = 0.002. Solid lines and short dashed lines denote stable solutions and unstable solutions, respectively. The solid dots donate the critical stiffness between stability and instability.

creases and 𝐾̄ 𝑠𝑒𝑝 increases. Fig. 16(b) shows that the values of 𝑑̄ at the critical point are not much different under various damping conditions, but the values of 𝐾̄ 𝑠𝑒𝑝 are quite different. Thus, in the design of stiffness, the parameter 𝐾̄ 𝑠𝑒𝑝 should be seriously considered. In Fig. 16(b), it can be seen that, when 𝜉 = 0.06, there is no unstable region in the stiffness design curve, so in this case the QZS characteristics can be selected to design the HSLDS isolator with parameter 𝐾̄ 𝑠𝑒𝑝 = 0. It should be noted that, the above stiffness design analysis is based on the response amplitude of the system being exactly equal to the designed maximum response amplitude. If the system stiffness is designed on the premise that the vibration amplitude does not exceed the designed maximum displacement response amplitude or the maximum displacement response amplitude is designed for larger values, there will be more stiffness design choices for the HSLDS isolator for smaller excitation amplitudes and larger damping.

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5. Conclusions In this paper, a HSLDS vibration isolator is studied from static and dynamic analysis perspectives. In the static analysis, based on the isolator model, the restoring force equation is established and its nonlinear stiffness characteristics are analysed. Furthermore, the stiffness range parameter d and static equilibrium position stiffness Ksep are introduced to represent various stiffnesses of the vibration isolation system. In dynamic analysis, the dynamic equation of the HSLDS vibration isolation system is established and solved by the averaging method, which have been evidenced in numerical simulations that it was valid to evaluate the performance of vibration isolation. The influences of parameters d and Ksep on the dynamic response and vibration isolation performance of the system are clearly analysed. The results show that an increase in d and a reduction in the value of Ksep , the isolation performance of the system are improved, including increasing the vibration isolation region and reducing the force transmissibility. Considering the application of the HSLDS isolator in micro-vibration environments, the maximum response amplitude of the isolator is limited, and the beneficial stiffness design of the isolator under different excitation amplitudes and damping is discussed. Based on good vibration isolation performance and system stability, the beneficial stiffness design for the HSLDS vibration system is given. The study results provide a useful insight for the design, analysis and application of HSLDS vibration isolators. Acknowledgements The authors would like to gratefully acknowledge the support from the National Natural Science Foundation of China under Grant No.51775040 and 51375047. References [1] Sun XT, Xu J, Jing XJ, Cheng L. Beneficial performance of a quasi-zero-stiffness vibration isolator with time-delayed active control. Int J Mech Sci 2014;82:32–40. 243

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