On the shock performance of a nonlinear vibration isolator with high-static-low-dynamic-stiffness

International Journal of Mechanical Sciences 81 (2014) 207–214

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

On the shock performance of a nonlinear vibration isolator with high-static-low-dynamic-stiffness Bin Tang a,n, M.J. Brennan b a b

Institute of Internal Combustion Engine, Dalian University of Technology, Dalian 116023, China Departamento de Engenharia Mecânica, UNESP Ilha Solteira, SP 15385-000, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 6 May 2013 Received in revised form 9 February 2014 Accepted 17 February 2014 Available online 22 February 2014

The vibration isolation characteristics of a high-static-low-dynamic-stiffness (HSLDS) isolator, which has geometrically nonlinear stiffness, have been well established both theoretically and experimentally in the recent literature. However, the shock isolation characteristics of such an isolator subject to base excitation are not currently known. In this paper, these characteristics are determined for two illustrative inputs, which are a rounded step and a versed sine displacement, using a simple model of the isolator comprising a vertical spring coupled to two horizontal springs. The isolator is configured to reduce the dynamic stiffness of the isolator and hence increase the frequency range of isolation. The shock responses of the isolator are determined analytically for low levels of excitation, and numerically for high levels of excitation. It is found that when the shock amplitude is small, the nonlinearity is beneficial, and that the quasi-zero stiffness isolator has the best shock performance in terms of the smallest displacement and acceleration of the suspended mass. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Shock isolation Geometric nonlinear stiffness Rounded step displacement Versed sine displacement

1. Introduction A nonlinear isolator with high-static-low-dynamic-stiffness (HSLDS) can have an arbitrarily small dynamic stiffness. It can thus achieve vibration isolation at very low frequencies, but without the excessive static displacement that is unavoidable with a linear isolator tuned to the same isolation frequency, supporting the same load [1–5]. The application of such a nonlinear isolator is ubiquitous in several fields of engineering, many of which have been described by Ibrahim [6] and Rivin [7]. For relatively small levels of vibration, the dynamic characteristics of the HSLDS discussed in [2–5] can be captured by the Duffing equation allowing physical insight into the effects of the nonlinearity [8]. Following the theoretical formulation for the HSLDS isolator, several researchers have realised practical HSLDS vibration isolators; for example Carrella et al. [9] combined a mechanical spring in parallel with a magnetic spring, Zhou and Liu [10] implemented a semi-active isolation system with HSLDS, Le and Ahn [11,12] proposed a vehicle seat with two symmetric negative stiffness structures for improving the low frequency vibration isolation performance, Liu et al. [13] used Euler buckled beams

n

Corresponding author. Tel.: þ 86 411 84708246x8302. E-mail addresses: [email protected] (B. Tang), [email protected] (M.J. Brennan). http://dx.doi.org/10.1016/j.ijmecsci.2014.02.019 0020-7403 & 2014 Elsevier Ltd. All rights reserved.

to realise a nonlinear isolator with a negative stiffness corrector, and Xu et al. [14] developed a passive HSLDS nonlinear isolator using permanent magnet springs. In all the work cited above, the focus has been on the vibration isolation performance of a nonlinear HSLDS system under harmonic excitation. However, in many cases, such as isolation in road vehicles and aircraft, the shock performance of an isolation system is equally important [15–17]. Accordingly, in this paper the shock performance of an HSLDS isolator is studied. Two illustrative shock inputs applied to the base of the system are considered: a rounded-step displacement [15] and a versed-sine displacement [16,17]. These are described in terms of a severity parameter [15–17]. When this parameter is very large, the inputs are asymptotic towards the Heaviside function and a pure impulse respectively. The basic concepts of linear shock isolation using such inputs have been given in several books, for example [15–18]. Various approximate methods have been used for transient analysis of the nonlinear systems excited by a force step function applied to the mass, such as Lighthill's extension of Poincaré's perturbation method [19], the ultraspherical polynomial approximation method [20,21], and the linearization method [22]. Chandra Shekhar et al. [23,24] have considered the performance of several kinds of nonlinear isolators under transient base shock excitation. They combined the perturbation method and the Laplace transform to determine the transient response of the nonlinear isolation system; they also provided a review of the literature on analytical

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B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

x

cv

O

kh kv

y

l Fig. 1. Model of the base-excited high-static-low-dynamic-stiffness nonlinear isolator in the static equilibrium position.

1.2 1.0

VI V

0.8

2. General model

IV III

0.6

II

I

0.4

where z ¼ x  y is the relative displacement. Introducing the nondimensional parameters kv c ; ω ¼ ; τ ¼ ωn t; ζ ¼ pffiffiffiffiffiffiffiffiffi; u¼ ymax m 2 kv m qffiffiffiffiffiffiffiffiffiffiffiffi ymax ^l ¼ l ; k^ ¼ kh ; xs ¼ l2  l2 ; y^ max ¼ 0 l0 kv xs 2 n

0.0

ð2Þ

where the primes denote derivatives with respect to nondimensional time τ, and gðτÞ ¼  y^ ″, in which y^ ¼ y=ymax . The two shock inputs applied to the base are a rounded step displacement [15] and a versed sine impulse [16,17], which are respectively given by y^ R ¼ 1 ð1 þ β R τÞe  βR τ ;

0rτ

ð3aÞ

1 y^ V ¼ ð1  cos ð2βV τÞÞ; 2

0 r τ r τp

ð3bÞ

where the subscripts R and V denote the rounded step displacement and the versed sine impulse respectively, β R ¼ T=2t pR is the

0

2

4

6

τ

8

10

12

4

5

6

1.2

VI V IV III 1.0

II

I

0.8 0.6 0.4 0.2 0.0

in which xs is the static deflection of the isolator when the mass is placed onto it, and ymax is the maximum amplitude of the base motion, Eq. (1) can be written as 0 1 1 B C u″ þ 2ζ u0 þ u þ 2k^ @1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAu ¼ gðτÞ ^l2 þ u2 y^ 2 ð1  ^l2 Þ max

0.2

yˆV

The nonlinear isolation system of interest in this paper is shown in Fig. 1. It consists of a rigid mass m suspended on a vertical spring of stiffness kv in parallel with a viscous damper c. Two springs of stiffness kh (horizontal springs) with original length l0, and compressed length l, orientated at 901 to the vertical spring are also connected between the mass and the base. All the elements in the system are linear. However, the isolation system is nonlinear because of the horizontal springs, which act as a negative stiffness for small perturbations from the equilibrium position [2]. The vibration isolation characteristics of this system have been studied extensively [2–5], using the equation of motion given by ! l0 mz€ þcz_ þ kv z þ 2kh 1  pffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼  my€ ð1Þ 2 l þ z2

z

m

kh

yˆR

methods for nonlinear shock isolation problems. Recently, a passive shock isolator with nonlinear damping or stiffness [25– 28] and several active or semi-active control techniques [29–33] have been used in shock isolation in an attempt to obtain a better performance than that of a linear isolator. Balandin et al. [34] have provided an excellent review on the theory of optimal shock and impact isolation. The aim of this paper is to determine whether the HSLDS system described in [2–5] and validated in [9–14], which can outperform a linear isolator in terms of vibration isolation, is better at isolating shock than a linear isolator. To achieve this aim, a simple three spring model of the isolator [2] is used together with linear viscous damping. As mentioned above, the isolator is subject to base excitation, and it supports a lumped mass. The effects of increasing nonlinearity and input excitation are considered. Three evaluation parameters are used to judge the performance of the nonlinear isolation system: the Shock Acceleration Ratio (SAR), the Shock Displacement Ratio (SDR) and the Relative Displacement Ratio (RDR). Numerical results are presented, as are some approximate closed-form solutions for the shock performance when the input excitation is small.

0

1

2

3

τ

Fig. 2. Types of shock input applied to the base of the system in Fig. 1, (a) rounded step displacement, (b) versed sine displacement; βR (or βV) ¼ 0.5 (I), 1.0 (II), 2.0 (III), 5.0 (IV), 10.0 (V), and 50.0 (VI).

ratio of the period of natural vibration of the linear isolation system T to twice the rise time t pR , which is the time required for the displacement to reach 82% of its final value [15], τp ¼ ωn t pV , in which t pV is the period of the versed sine pulse, and βV ¼ T=2t pV . Eqs. (3a and 3b) are plotted in Fig. 2 for six values of βR (or βV), ranging from 0.5 to 50. When βR is small the rise time is large and the rounded step input is less abrupt, and when βR is large the input tends towards the Heaviside step-function. When βV is small the versed sine shock input is less severe, and when it is large the shock input tends to an impulse as can be seen in Fig. 2. Using Eqs. (3a and 3b) the shock inputs to the system for a rounded step displacement and a versed sine impulse can be written as g R ðτÞ ¼  β R ð1  β R τÞe  βR τ ; 2

0rτ

ð4aÞ

B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

g V ðτÞ ¼  2β V cos ð2βV τÞ; 2

0 r τ r τp

209

ð4bÞ

Three performance indices, the SAR, the SDR and the RDR, which are described in Appendix A, are used to compare the shock performance of the HSLDS isolation system in Fig. 1 for three different values of ^l. Simulations and analysis of the results are presented in Section 4.

3. Response of the system to low amplitude input When the relative displacement z is less than about 40% of the length of the horizontal spring l [3], Eq. (1) can be approximated by mz€ þcz_ þ k1 z þ k3 z3 ¼ my€

ð5Þ 3

where k1 ¼ kv  2ðl0 =l  1Þkh and k3 ¼ kh l0 =l . This enables some analysis to be undertaken which facilitates physical insight into the shock isolation behaviour of the HSLDS isolator. Eq. (5) describes a base-excited Duffing oscillator [8]. Note that the effect of the horizontal springs is to reduce the linear stiffness of the isolator k1 and to add an additional hardening cubic nonlinear stiffness k3 . The effects of these terms on the performance of a vibration isolation system, which have been discussed comprehensively in Ref. [3], are investigated here for shock isolation. Using the non-dimensional parameters given in Eq. (2) and 0 1 ! ^2 1  ^l ^ 2 @1  l A ^ k α ¼ 1  2 ^ k; γ ¼ y^ max ^l3 l Eq. (5) can be written as u″ þ 2ζ u0 þ αu þ γ u3 ¼ g R ðτÞ ðor g V ðτÞÞ

ð6Þ

Note that ^l is the parameter that controls the linear stiffness and the cubic nonlinearity. Three values of this are given in Table 1 together with the corresponding linear ðαÞ and cubic ðγ Þ stiffness parameters for small displacements when k^ ¼ 1 and y^ max ¼ 1. When ^l ¼ 2=3, the linear stiffness term in Eq. (6) α ¼ 0, and the system becomes a quasi-zero stiffness (QZS) system. For a small amplitude (jy^ max j r 0:1), Eq. (6) gives an adequate description of the system dynamics, but for larger amplitudes, it is necessary to use Eq. (2). To understand the behaviour of the nonlinear shock isolator it is helpful to examine Eq. (6) in more detail. When jy^ max j 5 1, such that jγ u3 j 5 jαuj, the nonlinear effect is negligible and the dominant effect of the horizontal springs is to reduce the linear stiffness and hence the natural frequency. Eq. (6) can then be approximated as u″ þ 2ζ u þ αu ¼ g R ðτÞ ðor g V ðτÞÞ 0

ð7Þ

The magnitude of α compared to ζ is crucial for the type of 2 behaviour exhibited by the isolator. In particular, when α 4 ζ the 2 system behaves as an under-damped system, and when α o ζ the system is over-damped. Moreover, as mentioned above, α and ^l are ^ The way in which α, ζ and ^l affect related by α ¼ 1  2ðð1  ^lÞ=^lÞk. the behaviour of the isolator is illustrated in Fig. 3. Three areas are 2 depicted: A corresponds to the under-damped case when α 4 ζ , B 2 corresponds to the over-damped case when α o ζ and C corre2 sponds to the more heavily over-damped case when α 5 ζ . Also

Fig. 3. Different damped cases for the system in Fig. 1. Solid line, α ¼ ζ 2 ; dashdotted line, α ¼ 0:1ζ 2 . A or circle , under-damped case, α 4 ζ 2 ; B or multiplication , over-damped case 1, α o ζ 2 ; and C or plus , over-damped case 2, α 5ζ 2 .

plotted in Fig. 3 are the specific values of ^l given in Table 1 together with ^l ¼ 1, which corresponds to α ¼ 1 when the horizontal springs do not act a negative stiffness. From Fig. 3 it can be seen that when ^l decreases from 1 to 0.7, the system will be underdamped for the majority of damping values. However, when ^l decreases from 0.7 to 0.667 the stiffness changes dramatically and, depending on the damping in the system, it can behave as an under-damped or an over-damped system. When the system tends towards a QZS isolator ð^l-2=3Þ, the system becomes over-damped for the majority of values of damping depicted in the figure. 3.1. Base displacement is a Heaviside function (βR b1) In the limit when βR b 1, the shock input to the system y^ R ðτÞ tends to a Heaviside function and g R ðτÞ tends to the second derivative of this function. The response of a system described by Eq. (7) to a base excitation of the form of a Heaviside function is not currently available in the literature according to the knowledge of the authors. It is thus derived in this section. The equation of motion for a force-excited, single-degree-freedom, mass-springdamper system supported on a rigid base should be considered first. It is given by w″ þ 2ζ w0 þ αw ¼  f ðτÞ

where f ðτÞ is the Heaviside step function and w is the nondimensional displacement of the mass. The solutions to Eq. (8) when α ¼ 1 for both the under-damped and over-damped cases are given in Refs. [16,17]. Following the procedure given in these references, the solutions for the case when α a 1 can be determined. They are given by " !# pffiffiffiffi 1 α  ζτ w¼ 1þ e cos Ωd τ  ϕ1 α 4 ζ2 ð9aÞ

α

Parameters

α and γ. α

^l ¼ 0.8 ^l ¼ 0.7

^l ¼ 0.667

γ

0.5

0.703125

0.14286

1.48688

0.001499

1.87069

Ωd

"

pffiffiffiffi

α  ζτ 1þ e sinh iΩd τ þ ϕ2 w¼ iΩd α 1

Table 1

ð8Þ

!#

α o ζ2

ð9bÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffi α  ζ 2 and ϕ1 ¼ arctan ðζ =Ωd Þ. Now, because pffiffiffiffi pffiffiffiffi sinh ϕ2 ¼ iΩd = α and cosh ϕ2 ¼ ζ = α, ϕ2 ¼ arctanh ðiΩd =ζ Þ, pffiffiffiffiffiffiffiffi where i ¼  1. Differentiating Eq. (8) twice with respect to non-dimensional time τ results in an equation which has the same form as Eq. (7). Setting w″ ¼ u and f ″ðτÞ ¼  g R ðτÞ, the where Ωd ¼

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B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

Table 2 Relative displacement for step and impulsive shock excitation.

ζ2 o α

Step

Impulse

pffiffiffi  αe  ζτ

pffiffiffi  ζτ  π 2αβe sin Ωd τ V  pffiffiffi π α  4β 1  iΩζ e  ððζ  iΩd Þτ þ ϕ2 Þ

cos ðΩd τ þ ϕ1 Þ Ωd pffiffiffi α  ððζ  iΩd Þτ þ ϕ2 Þ  i2Ω e d  e  2ζτ

ζ2 4 α ζ2 b α

d

V

 2ζτ

 πζ eβ

3.2. Base displacement is an impulse (β V b 1)

V

qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2 where Ωd ¼ α  ζ , ϕ1 ¼ arctan ðζ =Ωd Þ, ϕ2 ¼ arctanh ðiΩd =ζ Þ, i ¼  1.

Table 3 Maximum shock response parameters when

βR (or βV)-1.

Step

Impulse

SAR SDR

0

pffiffiffi 1 þ e  ðζπ = αÞ

0 0

RDR

1

1

response of Eq. (7) is found to be pffiffiffiffi  ζτ αe 2 cos ðΩd τ þ ϕ1 Þ α 4 ζ u¼ 

Ωd

u¼ 

When βV b 1, the shock input to the system y^ V ðτÞ tends to a pure impulse and g V ðτÞ tends to the second derivative of this. Using two Heaviside functions, with the second function being a delayed and inverted version of the first, approximate solutions for the case when the input is an impulse can be obtained using a similar method as that discussed in the previous section. They are given by pffiffiffiffi  ζτ (    ) ζπ αe π 2β V u¼  þ ϕ1 cos ðΩd τ þ ϕ1 Þ  e cos Ωd τ  Ωd 2β V

α 4 ζ2 ð10aÞ

α o ζ2

ð10bÞ

Eqs. (10a and 10b) can be used to determine the shock performance indices given in Appendix A. For a displacement input described by a Heaviside function, the acceleration of the base is infinite. However, because the absolute acceleration of the mass is finite, the SAR is zero. To determine the SDR, it is noted that the non-dimensional absolute displacement x=ymax ¼ 1 þ u. It is possible to determine the RDR directly by calculating jujmax . The results are given in the left-hand column of Tables 2 and 3 for convenience, and can be compared with the numerical results in Section 4. 3.1.1. Under-damped case (α 4 ζ ) 2 When the isolation system is under-damped, such that α 4 ζ , pffiffiffiffi then Ωd  α, and the response given by Eq. (10a) can be approximated by 2

ð11Þ

The maximum value of the non-dimensional absolute displacement occurs approximately when cos ðÞ ¼  1, which is when pffiffiffiffi τ  π = α as ϕ1 5 1. Thus, using x=ymax ¼ 1 þ u, the shock displacement ratio is found to be pffiffiffi ð12Þ SDR  1 þ e  ðζπ = αÞ The maximum value of relative displacement umax   1 occurs at the beginning of the movement, when the mass is almost static and the relative displacement is roughly equal to the displacement of the base. When the mass begins to move, the oscillation will decay away governed by the exponential term in Eq. (11). Thus, the relative displacement ratio RDR  1. 3.1.2. Over-damped case (α o ζ ) In the over-damped case, Eq. (10b) can be approximated by pffiffiffiffi α  ððζ  iΩd Þτ þ ϕ2 Þ u e ð13Þ i2Ωd 2

because the term e  ððζ þ iΩd Þτ  ϕ2 Þ decays rapidly with time. As Eq. (13) is only approximate, there is a small overshoot, which 2 will be seen in Section 4. When α 5 ζ ,  iΩd  ζ and e  ϕ2

ð14aÞ

pffiffiffiffi u¼ 

pffiffiffiffi  ζτ αe sinh ð  iΩd τ þ ϕ2 Þ iΩd

u   e  ζτ cos ðΩd τ þ ϕ1 Þ

pffiffiffiffi ¼ cosh ϕ2  sinh ϕ2  2ζ = α, so Eq. (13) can be approximated by u   e  2ζτ . In the over-damped case, the maximum value of the relative displacement also occurs at the beginning of the movement and the displacement of the mass then decays exponentially and does not oscillate, which means that the SDR  1 and the RDR  1.

(







ζπ αe  ζτ π þ ϕ2 sinh ð  iΩd τ þ ϕ2 Þ  e2βV sinh  iΩd τ  iΩd 2β V

α o ζ2

)

ð14bÞ

When β V c 1 the input tends to a delta function, and in this case the following approximations can be made   ζπ ζπ πΩd  1; ; cos e2βV  1 þ 2β V 2β V       πΩd πΩd iπΩd iπΩd iπΩd ; cosh   1; sinh  sin 2β V 2β V 2β V 2β V 2β V Substituting these  into Eqs.  (14a and 14b), and omitting the 2 higher order terms of 1=βV , Eqs. (14a and 14b) become pffiffiffiffi π αe  ζτ 2 u sin Ωd τ α 4 ζ ð15aÞ 2β V pffiffiffiffi

u



π αe  ζτ ζ sinh ð  iΩd τ þ ϕ2 Þ þ cosh ð  iΩd τ þ ϕ2 Þ iΩ d 2βV



α o ζ2 ð15bÞ

It can be seen that the larger the severity parameter βV, the smaller the relative amplitude. Expanding the hyperbolic functions into exponential functions in Eq. (15b) and omitting the rapidly decaying term e  ððζ þ iΩd Þτ  ϕ2 Þ , Eq. (15b) can be approximated by pffiffiffiffi  π α ζ 2 u 1 ð16Þ e  ððζ  iΩd Þτ þ ϕ2 Þ α o ζ iΩd 4β V pffiffiffiffi 2 Further, if α 5 ζ , e  ϕ2  2ζ = α and  iΩd  ζ , then the response given by Eq. (16) can be simplified to u  πζ e  2ζτ =β V . These results are given in the right-hand column of Table 2 for convenience. The three shock isolation performance indices can be determined easily and are listed in Table 3.

4. Simulations and discussion The three performance indices SAR, SDR, RDR are evaluated for the system in Fig. 1 for the values of βR (or βV) given in Fig. 2, and for three different values of ^l given in Table 1. The results are shown in Figs. 4–6 and are compared with those of the linear system, in which there are no horizontal springs. Although the responses of the linear system are determined analytically as in

101

101

100

100 SAR

SAR

B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

10-1

10-1 10-2

10-2 10-3 0.1

1

10

10-3 0.1

100

1

101

100

100

SAR

SAR

101

10-1 10-2

10

100

10

100

10-2

1

10

10-3 0.1

100

1

βV

101

101

100

100 SAR

SAR

100

10-1

βR

10-1

10-1 10-2

10-2 10-3 0.1

10 βV

βR

10-3 0.1

211

1

10

100

βR

10-3 0.1

1

βV

Fig. 4. SAR curves for the linear and nonlinear systems with (a) and (d) ^l ¼ 0:8, (b) and (e) ^l ¼ 0:7, (c) and (f) ^l ¼ 0:667, and ζ¼0.1, under (a–c) rounded step displacement, (d–f) versed sine-shape displacement. Solid line , linear system (numerical and closed-form solutions); dashed line , nonlinear system with y^ max ¼ 0:1; dotted line , nonlinear system with y^ max ¼ 1:0; and dash-dotted line , nonlinear system with y^ max ¼ 5:0.

Refs. [15,32], the responses of the nonlinear system are obtained numerically. The fourth-order Runge–Kutta method with a step size control algorithm is used to calculate the responses of the nonlinear isolation systems. To illustrate the behaviour, the damping ratio was set to ζ ¼ 0.1 and the non-dimensional stiffness of the system k^ ¼ 1. Results are shown for three different amplitudes of excitation y^ max ¼ 0:1, 1.0, and 5.0. If y^ max r 0:1, Eq. (6) can be used to obtain the shock response of the nonlinear isolators and if y^ max 4 0:1 then Eq. (2) needs to be used to give an accurate response to the shock excitation. 4.1. Rounded step displacement Fig. 4(a–c) gives the response of the system when subject to a rounded step displacement, for increasing nonlinearity (^l ¼ 0:8, 0:7 and 0:667 respectively). Each graph contains four plots, one for the linear system and three for the nonlinear system with y^ max ¼ 0:1, 1:0; and 5.0. The corresponding plots for the SDR and RDR are in

Figs. 5(a–c) and 6(a–c) respectively. From Fig. 4(a–c) it can be seen that, in general, with the nonlinear system for a large range of βR , as the amplitude of the shock input increases the SAR also increases. In all cases, when the input tends to a step function βR b 1, which has been discussed in Section 3.1, the response of the nonlinear system (for any degree of nonlinearity) tends towards that of the linear system which will tend to zero when βR b 1. For the case when the nonlinearity is relatively weak as in Fig. 4(a), this is also true for the quasi-static case when β R 5 1. Apart from these extremes, when the shock amplitude is small ðy^ max o 1Þ the shock isolation performance of the nonlinear isolation system is marginally better than that of the linear system. The nonlinearity, however, has a detrimental effect compared to the linear system when the shock amplitude is large ðy^ max 41Þ. When jy^ j o 1 the force transmitted through the vertical and horizontal springs to the mass is less than that for the vertical linear spring alone, when jy^ j 41 the transmitted force through the vertical and horizontal springs is greater than that through the vertical spring alone.

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B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

2.0

1.6 1.4

1.8

1.2 SDR

SDR

1.6 1.4

0.8 0.6

1.2

0.4 0.2

1.0 0.1

1.0

1

10

0.0 0.1

100

1

βR 2.0

SDR

SDR

100

10

100

1.2

1.4

1.0 0.8 0.6

1.2

0.4 0.2

1.0 1

10

0.0 0.1

100

1

βV

βR 2.0

1.6 1.4

1.8

1.2 SDR

1.6 SDR

10

1.4

1.6

1.4

1.0 0.8 0.6

1.2

0.4 0.2

1.0 0.1

100

1.6

1.8

0.1

10

βV

1

10

100

βR

0.0 0.1

1

βV

Fig. 5. SDR curves for the linear and nonlinear systems with (a) and (d) ^l ¼ 0:8, (b) and (e) ^l ¼ 0:7, (c) and (f) ^l ¼ 0:667, and ζ¼ 0.1, under (a–c) rounded step displacement, (d–f) versed sine-shape displacement. Solid line , linear system (numerical and closed-form solutions); dashed line , nonlinear system with y^ max ¼ 0:1; dotted line , nonlinear system with y^ max ¼ 1:0; and dash-dotted line , nonlinear system with y^ max ¼ 5:0.

Examining Fig. 5(a–c), it can be seen that as the nonlinearity increases, the response of the system decreases dramatically for low amplitude excitation ðy^ max ¼ 0:1Þ when βR 41. In particular, when the nonlinearity is such that the natural frequency of the system is almost zero, as in Fig. 5(c), the SDR is nearly constant, and less than that of the linear system when the severity parameter βR is larger than about 0.6. It can also be seen in Fig. 5(a–c) that the system is under-damped except when ^l ¼ 0:667 and is excited by low amplitude excitation (y^ max ¼ 0:1). From Eq. (12), the asymptotic value for the SDR for under-damped cases can be easily obtained, while the over-damped case is approximated to unity with a small overshoot as discussed in Section 3.1.2. Fig. 6(a–c) gives the RDR curves of the linear and nonlinear isolation systems. It can be seen that, in general, the RDR of the nonlinear systems is a little higher than that of the linear system. The RDR of the nonlinear QZS system is the highest within these three nonlinear isolators, which is the penalty for having the best SDR performance. The larger the shock excitation amplitude y^ max

and nonlinearity ^l, the closer the values of the RDRs of the nonlinear isolator to that of the linear isolator. When the severity parameter βR is very large, the relative displacement ratio is unity as discussed in Section 3.1. From all the response curves in Figs. 4(a–c), 5(a–c) and 6(a–c), it can be seen that if the SAR is used as the design criterion, the nonlinear shock isolation system is only marginally better than the linear system when y^ max ¼ 0.1 or 1.0. If the SDR is used as the design criterion, the QZS isolation system fares much better than the linear system when the severity parameter βR is larger than 0.6 and y^ max ¼0.1. If the RDR is the design criterion for the isolator, all the nonlinear isolation systems are a little worse than the linear system. 4.2. Versed sine displacement The SAR, SDR, and RDR curves of the linear and nonlinear systems for versed sine displacement excitation are shown in

B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

1.4

1.0

1.2 1.0

0.6

RDR

RDR

0.8

0.4

0.8 0.6 0.4

0.2 0.0 0.1

0.2 1

10

0.0 0.1

100

1

βR

100

10

100

1.0

0.6

RDR

RDR

10

1.2

0.4

0.8 0.6 0.4

0.2

0.2 1

10

0.0 0.1

100

1

βR

βV 1.4

1.0

1.2

0.8

1.0

0.6

RDR

RDR

100

1.4

0.8

0.4

0.8 0.6 0.4

0.2 0.0 0.1

10

βV

1.0

0.0 0.1

213

0.2 1

10

100

0.0 0.1

βR

1

βV

Fig. 6. RDR curves for the linear and nonlinear systems with (a) and (d) ^l ¼ 0:8, (b) and (e) ^l ¼ 0:7, (c) and (f) ^l ¼ 0:667, and ζ¼0.1, under (a–c) rounded step displacement, (d–f) versed sine-shape displacement. Solid line , linear system (numerical and closed-form solutions); dashed line , nonlinear system with y^ max ¼ 0:1; dotted line , nonlinear system with y^ max ¼ 1:0; and dash-dotted line , nonlinear system with y^ max ¼ 5:0.

Figs. 4(d–f), 5(d–f) and 6(d–f), respectively. The case when βV b 1, which is when the versed sine displacement tends to an impulse input, can be easily seen in these figures. From the family of SAR curves, it can be seen that they are similar to those when the system is excited by a rounded step displacement, except around when βV  0:5, when the values are a little higher. This is because when the duration of the versed impulse matches the undamped natural period of the linear system, the response is larger. Again, it can be seen that the nonlinear systems outperform the linear system provided the excitation level is small. Moreover a greater nonlinearity gives a better performance, except when βV is either very large or very small. Considering the SDR, in Fig. 5(d–f), it can be seen that, in general, increasing the nonlinearity has a beneficial effect provided the shock input is small as seen previously. It is also evident that in this case the peak in the response changes as the input level and the nonlinearity change. As the nonlinearity increases the natural period of the system increases so the peak in the

response occurs for larger values of βV . For a given configuration, the effect of increasing the amplitude of the excitation is to increase the value of βV at which the peak occurs. It can also be seen that the nonlinear systems with a high level of excitation perform less well than the linear system, and the reason for this is the same as discussed in the previous section. It can be seen that the performance of the QZS system is the best if the SAR or SDR is the design criterion as with the case for the rounded step displacement discussed in Section 4.1.

5. Conclusions The shock performance of an HSLDS nonlinear vibration isolation system, which includes a QZS isolator, has been studied for two base excitation inputs: a rounded step displacement and a versed-sine displacement. The isolation system consists of a vertical spring providing a positive stiffness and two auxiliary

B. Tang, M.J. Brennan / International Journal of Mechanical Sciences 81 (2014) 207–214

214

springs providing a negative stiffness. The nonlinearity is due to the geometrical configuration of the linear springs and has been shown to be beneficial from the point of view of vibration isolation. To study the effects of the nonlinearity on shock isolation the lengths of the auxiliary springs were changed. The effects of different amplitudes of excitation were also studied. It was found that, in general, for both types of excitation the effects of increasing the nonlinearity were beneficial from the point of view of shock isolation, provided the amplitude of shock input was relatively small. The reason for this is that the auxiliary springs have two effects. The first is that they reduce the natural frequency and the second is that the stiffness of the system is increased for large relative displacements across the isolator. The first of these is beneficial and is the reason why the nonlinear isolator outperforms the linear isolator. The second of these is not beneficial and has a detrimental effect when the amplitude of the shock input is large. Acknowledgements The first author wishes to acknowledge financial support from the China Scholarship Council (Grant 2009821053), which enabled him to complete parts of the paper at the Institute of Sound and Vibration Research, University of Southampton, UK; the National Natural Science Foundation of China (Grant no. 11202048); and CNPq (Grant no. 374706/2012-3), which enabled him to finish this paper in UNESP, Brazil.

Appendix A. shock isolation performance indices Three quantities are generally of interest in the case of shock isolation [15–17]. I. Shock acceleration ratio (SAR) SAR ¼

jx€ jmax ju″  gðτÞjmax ¼ jy€ jmax jgðτÞjmax

ðA1Þ

This is the maximum shock acceleration transmissibility. When the shock excitation is defined by an absolute acceleration of the base or by a force applied to the mass, the response of the system can be characterised by the absolute acceleration of the mass. When the acceleration of the suspended mass is large, the elastic and damping forces within the isolator and the force transmitted to the mass are large. II. Shock displacement ratio (SDR) SDR ¼

jxjmax  ¼ u þ y^ jmax ymax

ðA2Þ

This is the maximum shock displacement transmissibility. When the SDR is large, the shock performance of the isolator is poor. III. Relative displacement ratio (RDR) RDR ¼

jx  yjmax  ¼ ujmax ymax

ðA3Þ

This represents the deformation of the isolator. A large RDR means that there is a large deformation of the elastic element of the isolator, with correspondingly large stresses.

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