Damped response analysis of nonlinear cushion systems by a linearization method

Computers and Structures 83 (2005) 1584–1594 www.elsevier.com/locate/compstruc

Damped response analysis of nonlinear cushion systems by a linearization method Yuqi Wang, K.H. Low

*

School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 30 June 2004; accepted 16 February 2005 Available online 18 April 2005

Abstract Dynamic loading often plays a critical role in the functional performance and mechanical reliability of packaged components and devices. The prediction of the drop-impact response of the structural element of electronic components becomes one of the important concerns in product design. The interaction response between the cushion system and its packed products is also focused. In the analysis of these dynamic responses, the cushion system is often simplified as a linear and un-damped system. However, in reality, the stiffness of the cushion system is nonlinear and the damping effect cannot be ignored in the drop-impact process. Therefore, an algorithm for the nonlinear response analysis of viscous damping models is presented in this study. The whole cushion buffer is first represented by a multi-degree-of-freedom system in this model. Nonlinear characteristics and viscous damping are included in the system. The developed mathematical model provides a practical and reliable method to predict the dynamic behaviors of nonlinear systems with viscous damping. The effects of nonlinear characteristics and viscous damping in the protection of packaged products are both discussed in this paper. The discussion provides a useful insight that both nonlinear characteristics and viscous damping can reduce ‘‘rigid impact’’ to certain level.
2005 Elsevier Ltd. All rights reserved. Keywords: Dynamic loading; Nonlinear analysis; Viscous damping model; Packaged products; Local linearization method; Cushion buffer

1. Introduction Dynamic loading often plays a critical role in the function performance and mechanical reliability of electronic devices. Such a loading can be caused by accidental mishandling or misuse of the equipment in service, or

*

Corresponding author. Tel./fax: +65 7910200. E-mail addresses: [email protected] (Y. Wang), [email protected] (K.H. Low).

can occur during its manufacturing, testing, or shipment (transportation). Therefore one of the most important functions of a buffer is to protect an article against damage when it is dropped or subjected to impact [1,2]. This is achieved by the deformation of the cushioning element, which absorbs and reduces the force acting on the unit. The prediction of the dynamic response of packaged electronic devices subjected to impulsive loads can be used to provide a design recommendation under various influencing factors [3–5]. Based on this concern, a

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2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.02.004

Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

mathematical model focused on the drop-impact is provided by James et al. [6]. In their model, the dropped object was idealized into an ideal single-degree-of-freedom (SDOF) system. The basic equation based on this model was described as M€x þ Kx ¼ Mg

ð1Þ

where M is the mass of the object, while K is the stiffness coefficient of the cushioning material, which is assumed constant. In this ideal model, the damping coefficient and the nonlinear characteristic of cushion materials were both ignored in the analysis. In reality, damping always exists in the cushion material and the stiffness coefficient of buffer shows obvious nonlinear characteristics corresponded to the high nonlinearity in the stress–strain curve [7]. In order to predict the dynamic response of packaged goods in the dropimpact process, Suhir [8,9] described two characteristics in his mathematical models: a viscous damping model with linear stiffness characteristic [8] and an un-damped model with nonlinear stiffness characteristic [9]. We next discuss briefly the two models. 1. Viscous damping characteristic.

model

with

linear

stiffness

For this model, Suhir presented the following equation for a SDOF system [8], M€x þ C x_ þ Kx ¼ 0

ð2Þ

where M is the mass of the system, C is the damping coefficient, and K is the spring stiffness coefficient, which is assumed constant. Therefore, the model is only applicable to the linear system. 2. Un-damped model with nonlinear stiffness characteristic.

vibration system is provided [12–14]. The equation of this type is written as: €x þ 2fxn x_ þ x2n xð1 þ ax2 Þ ¼ F ðtÞ €x þ 2fxn x_ þ x2n x þ ax3 ¼ F ðtÞ

ð5Þ

where F(t) is the applied excitation, xn is the un-damped natural frequency, and f is the damping ratio. Because xn is a constant, the damping term 2fxn x_ is only linearly proportional to the velocity. The objective of the present study is to predict the maximum acceleration and related dynamic behaviors of the packaged components with a cushion buffer in the drop-impact case. The cushion buffer will be modeled as a non-linear stiffness system with nonlinear damping. All the input coefficients to the mathematical model are set to obtain easily in reality, while the output results should predict the dynamic behavior quantitatively with reasonable accuracy. Through this study, the authors propose to find an effective method to solve some practical problems addressed in Ref. [15].

2. Mathematical model A mathematical model of packaged electronic device is depicted in Fig. 1, in which the buffer is idealized as a vibration system assembled with multi-unit nonlinear springs with unit stiffness coefficient kc and damping ratio f. This system with multi-components provides sufficient flexibility to describe the arbitrary shapes of cushion buffer. The electronic device carried by the buffer system is modeled as a rigid block. Note that X is the drop direction in the drop-impact model, and the

ð3Þ

As mentioned, the damping is ignored in this model. Furthermore, K remains a constant in Kx, only the parameter a is applied to reflect the nonlinear behavior in ax3. This approach introduces a nonlinear feature with respect to displacement, and stiffness coefficient as a constant in its linear part [10,11]. It is generally insufficient to detect the dynamic behavior quantitatively by virtue of this nonlinear model practically, because the parameter of nonlinearity a cannot be obtained directly from the nonlinear stress–strain curve of cushion buffer. With further consideration of damping elements and nonlinear stiffness, the linear damping in the nonlinear

ð4Þ

or

In order to describe the nonlinear characteristics of cushion buffer, Suhir [9] suggested another form of equation for the same vibration model: €x þ Kx þ ax3 ¼ 0

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Fig. 1. Mathematical model of packaged product.

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deformation along the YZ plane is ignored in this study. In the following analysis, we evaluate the effects of the cushion material on the maximum displacement and the maximum acceleration of the packaged components subjected to shock loads during drop tests [16]. The cushion effect of the nonlinear damping system will also be discussed. At first, by virtue of a single unit spring, its motion can be described as mc €xc þ cc x_ c þ k c xc ¼ 0

ð6Þ

or €xc þ 2fxc x_ c þ x2c xc ¼ 0

ð7Þ

in which the following notations are used: sffiffiffiffiffiffi kc cc ; f ¼ pffiffiffiffiffiffiffiffiffiffi xc ¼ mc 2 k c mc

1. Combination of unit springs along the drop direction, X, in series [17]: X0

dX X 1 1 ¼ K X ðY ; ZÞ k ðX ; Y ; ZÞ ci i¼1

ð9Þ

where KX(Y, Z) is the combination of unit stiffness coefficient kci along X-direction. Note that X0 = X0(Y, Z) is the vertical dimension of cushion X0 buffer along X-direction, while dY describes the number of unit springs along X-direction. Stiffness coefficient of the unit spring kci(X, Y, Z) is distributed over the arbitrary XYZ positions of buffer. 2. Combination of integrated springs KX(Y, Z) along Ydirection in parallel [17]: Y0

ð8Þ

where xc is the displacement of the unit spring after touching the floor; mc is the mass of each identical unit spring; xc is the un-damped frequency of the unit spring; cc is the damping coefficient of the unit spring. Next, the multi-degree-of-freedom system needs to be integrated into one degree of freedom system. As shown in Fig. 1, the buffer is taken as an assembly of unit nonlinear springs with stiffness coefficient kc and damping ratio f; dX, dY and dZ are the dimensions of the unit spring. The resultant or equivalent stiffness coefficient of the cushion is set as K and equivalent damping coefficient of the whole system is set as C. At first, the equivalent stiffness coefficient K is obtained through the combination of this multi-spring system. The integration process via three steps is shown in Fig. 2.

K X Y ðZÞ ¼

dY X

K Xj ðY ; ZÞ

ð10Þ

j¼1

where KX_Y(Z) is the combination of the stiffness coefficient KX along Y-direction. Note that Y0(Z) is the dimension of cushion buffer along Y-direction, Y0 while dY describes the number of integration springs KX(Y, Z) along Y-direction. Stiffness coefficient KXj(Y, Z) is distributed over arbitrary positions of YZ plane. 3. Combination of integrated springs KX_Y(Z) along Zdirection in parallel [17]: Z0

K ¼ KX

Y Z

¼

dZ X

KX

Ym ðZÞ

ð11Þ

m¼1

where KX_Y_Z is the resultant or equivalent stiffness coefficient K. Note that Z0 is the dimension of cushZ0 ion buffer along Z-direction, while dZ is the number

Fig. 2. The integration of unit springs.

Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

1587

of the integration springs KX_Y(Z) along Z-direction. Stiffness coefficient KX_Ym(Z) is distributed over arbitrary positions of Z-axis. Eqs. (9)–(11) are the generic equations to arbitrary shape buffer and material properties. For simplicity, we restrict the analysis to the case of standard cubic cushion model and the isotropic cushion material property. Since the material property of the buffer is taken as isotropic case, r and e are the same in each element in the buffer model. Moreover, their derivative kci(X, Y, Z) is a constant for every unit spring, k ci ðX ; Y ; ZÞ
k c ¼

r dZ dY e dX

ð12Þ

On the other hand, we have the deformation of cushion buffer along X-direction, Z X0 Z X0 x ¼ DX 0 ¼ DdX ¼ e dX ¼ eX 0 ð13Þ 0

0

and the total force applied to the cushion buffer, Z Z Y 0 Z Z0 r dY dZ F ¼ df ¼ 0

ð14Þ

0

Note that the unit force df is applied to every single unit spring, which can be obtained from stress in r dY dZ; DdX is the deformation of the unit spring, which equates to e dX. Since the cushion buffer is set as a standard cubic structure, the dimensions of cushion buffer X0, Y0 and Z0 in Fig. 1 are all taken as constants, X 0
d;

Y 0
w;

Z0
l

ð15Þ

By virtue of the above derivations, we have 1 d k c dX ) KX ¼ ¼ K X k c dX d KX

Y

¼

K ¼ KX

w KX dY Y Z

¼

Z Y 0Z 0

ð16Þ ð17Þ

l KX dZ

Y

x ¼ eX 0 ¼ ed F ¼

ð19Þ

Z0

r dY dZ ¼ rY 0 Z 0 ¼ rwl ¼ rA

ð20Þ

0

¼

Y Z

¼

rwl rA ¼ ed ed

cause different stiffness coefficient, K [15]. The equivalent stiffness coefficient K can be set as K = SrA/ed, where S is structural coefficient accounted for different rib configuration. As the cushion buffer is a standard cubic in the present case, S = 1. By combining Eqs. (8) and (21), we can obtain the equivalent damping coefficient, rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi rAM C ¼ 2f KM ¼ 2f ð22Þ ed where M is the total mass of packaged block attached to the nonlinear cushion buffer. Through the developed model, the original multi-degree-of-system has been combined into one-degree-offreedom system with stiffness coefficient K and damping coefficient C, as shown in Fig. 3.

ð18Þ

By combining Eqs. 12 and (16)–(18), we obtain K ¼ KX

Fig. 3. Single-degree-of-freedom drop-impact system.

k c wl dX rdZ dY wl dX ¼  d  dZ dY e dX d  dZ dY ð21Þ

where A is the contact area as shown in Fig. 1. Note that for the cushion buffers with the same contact area A and cushion thickness d, different rib configurations might

3. Analytical analysis In this work, the cushion package system is simplified as a nonlinear SDOF system. It can be found from Eq. (21) that the nonlinearity is performed in the nonlinear stress–strain curve of cushion buffer [7], since the ratio between stress and strain is directly proportional to the stiffness K in the SDOF system in this equation. So far, the procedure available for an exact determination of nonlinear system is not easy to obtain. An exact solution exists, however, for linear system [18]. Therefore, in order to analyze this nonlinear system quantitatively, the global nonlinear system is linearized in the local working point. In this approach, the global nonlinear

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where f / ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi 1f

ð29Þ

Note that, in the local linear working point, the time coefficient t in Eqs. (27) and (28) do not cover the whole nonlinear system, it can only be used to describe the phase state of the local oscillation. In order to avoid confusion in the following analysis, the local time coefficient is replaced as s, tlocal ¼ s Fig. 4. Nonlinear stress–strain curve.

system is divided into a series of linear stages, then each stage is treated as a locally linearized working point. As shown in Fig. 4, at each working point on the nonlinear stress–strain curve, the linearized stiffness coefficient Ki is applied. Note that Ki is a variable corresponding to different deformation xi. As the objective of this work is to predict the maximum acceleration and related dynamic behaviors, the point of maximum acceleration will be traced in the global nonlinear system, and the local linearized analysis will be performed at this working point. The initial displacement and velocity conditions of the one-degree-of-freedom system in Fig. 3 are given by pffiffiffiffiffiffiffiffi xð0Þ ¼ 0; x_ ð0Þ ¼ vð0Þ ¼ 2gh ð23Þ where x is the deformation of the buffer when the packaged model free-drops to the ground and h is the drop height. By neglecting the gravity in the drop-impact force, the differential equation of motion of the mass M is M€x þ CðxÞ_x þ KðxÞx ¼ 0

ð24Þ

ð30Þ

where s is set as the phase state of local working point in the whole nonlinear system. Therefore, Eqs. (27) and (28) are rewritten as pffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi  2gh x_ ðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi efxs cos x 1  f2 s þ /1 ð31Þ 1f pffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi  2gh €xðsÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffi2ffi xefxs sin x 1  f2 s þ 2/1 1f

ð32Þ

In the following work, the point of maximum acceleration will be investigated. At this point, the velocity of packaged achieve zero in the free-drop vibration ð_xmax acc ¼ 0Þ. At the same time, all the system energy is absorbed and dissipated by the cushion buffer, and the energy performance can be calculated in the global nonlinear system. Therefore, two conditions at the point of maximum acceleration will be considered in the analysis: (1) dynamic behavior in the local linear working point; (2) energy reservation in the global nonlinear system. Damping effect will also be considered in the analysis. 3.1. Dynamic behavior in the local linear working point

or €x þ 2fxðxÞ_x þ x2 ðxÞx ¼ 0 where the following notations are used rffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi KðxÞ xðxÞ ¼ and CðxÞ ¼ 2f KðxÞM M

ð25Þ

ð26Þ

in which x is the frequency of the local linear oscillation. Note that the stiffness coefficient K and the related damping C are variables in the global nonlinear system as shown in Fig. 4. The solution of Eqs. (23)–(26) yields the velocity x_ ðtÞ and acceleration €xðtÞ of the SDOF system as given by pffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi  2gh fxt 2 x_ ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi e cos x 1  f t þ /1 ð27Þ 1f pffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi  2gh €xðtÞ ¼  pffiffiffiffiffiffiffiffiffiffiffiffi2ffi xefxt sin x 1  f2 t þ 2/1 1f

ð28Þ

When the velocity of the packaged system becomes zero, the drop-impact process achieves its rebound instant, and acceleration achieves its maximum point ð€xmax acc Þ. Substituting x_ ¼ 0 into Eq. (31), we obtain the phase state,smax_acc, at the point of rebound, pffiffiffiffiffiffiffiffi 2gh pffiffiffiffiffiffiffiffiffiffiffiffiffi efxmax acc smax acc _xðsmax acc Þ ¼ xmax acc 1  f2   qffiffiffiffiffiffiffiffiffiffiffiffiffi  cos xmax acc 1  f2 smax acc þ /1 ¼0 ð33Þ where smax_acc is the phase state as the acceleration achieves its maximum value, and xmax_acc is the local linear frequency at the phase state smax_acc. Through the solution of Eq. (33), we can find the phase state at where the maximum acceleration takes place

Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

xmax

acc ðnÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1  f2 smax

acc ðnÞ

þ /1 ¼

p þ ðn  1Þp; 2

€xmax

n ¼ 1; 2; . . . ; N or p þ ðn  1Þp  /1 pffiffiffiffiffiffiffiffiffiffiffiffiffi smax acc ðnÞ ¼ 2 xmax acc ðnÞ 1  f2 where n is the number of cycles for the oscillation. It was found from Refs. [2,15] that the maximum acceleration occurred in the first cycle in a free-drop process under the protection of cushion buffer. Therefore, in our study, we need only to consider the first cycle (n = 1). Hence, we can find ¼

acc

p 2

xmax

 /1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  f2 acc

ð35Þ

Till this step, the phase state at the rebound point is obtained, which is the key to derive the formulation of maximum acceleration. Substituting Eqs. (29) and (35) into Eq. (32), the maximum acceleration €xmax is expressed as €xmax ¼ j€xðsmax accÞ j pffiffiffiffiffiffiffiffi 2gh ¼ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi xmax acc efxmax acc smax acc 1f   qffiffiffiffiffiffiffiffiffiffiffiffiffi  sin xmax acc 1  f2 smax acc þ 2/1 pffiffiffiffiffiffiffiffi 2gh ¼ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi xmax 1f ¼

pffiffiffiffiffiffiffiffi 2ghxmax

fðp/1 Þ

2 ffiffiffiffiffiffi p 1f2

acc e

sin

p 2

þ /1

pffiffiffiffiffiffi 1f2

where W is the weight of package components, which is equal to Mg. Eq. (40) shows a formulation to achieve maximum acceleration by tracing the change of stress and strain in cushion buffer. Here, the ratios h/d and W/A will both affect the strain at the instant of maximum acceleration. However, there are still two unknown coefficients in Eq. (40), the stress rmax_acc and the strain emax_acc at the instant of maximum acceleration. As a conclusion of this section, through the analysis of the phase state at the instant of rebound instant smax_acc, the local linear formula of maximum acceleration is obtained for the whole nonlinear system. However, it is found from Eq. (40) that in order to obtain the maximum acceleration, it is necessary to trace the stress and strain at the time of rebound, which will be covered in the energy analysis. 3.2. Energy reservation in the global nonlinear system



fðp/1 Þ  2

acc e

ð40Þ

A

ð34Þ

smax

rffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðp2/1 Þ 2gh A rmax acc  pffiffiffiffiffiffi 2 ¼ e 1f M d emax acc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   fðp2/1 Þ max acc h  pffiffiffiffiffiffi 2 remax 2 acc d W  e 1f ¼g

1589

ð36Þ

By virtue of Eq. (26), the local linear frequency xmax_acc can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K max acc ð37Þ xmax acc ¼ xðsmax acc Þ ¼ M

At the instant of maximum deflection and acceleration, there is an energy balance at a moment when the kinetic energy at the initial condition is completely transferred to the deformation energy and the energy dissipated by viscous damping. It simply follows the principle of energy balance, Z xmax acc 1 M x_ 2 ð0Þ ¼ KðxÞx dx þ Ed ð41Þ 2 0 where Ed is the energy dissipated by damping, substituting Eqs. (23) into (41), we obtain Z xmax acc Mgh ¼ KðxÞx dx þ Ed ð42Þ 0

where Kmax_acc is the local linear stiffness coefficient when the maximum acceleration takes place. Substituting Eqs. (37) into (36), the solution of maximum acceleration can also be expressed as follows:

€xmax

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðp2/1 Þ 2ghK max acc  pffiffiffiffiffiffi 2 e 1f ¼ M

ð38Þ

At the instant of maximum acceleration, the stiffness coefficient Kmax_acc can be expressed through Eq. (21) as K max

acc

¼

A rmax  d emax

acc

ð39Þ

acc

where rmax_acc and emax_acc are the stress and strain of buffer element at the instant of maximum acceleration. Substituting Eqs. (39) into (38), the final expression of maximum acceleration is

By virtue of Eqs. (19), (21) and (42), we find that Z emax acc 1 Ed þ rAd  de ¼ M x_ 2max ¼ Mgh ¼ Wh ð43Þ 2 0 or Z emax 0

acc

r  de ¼

Wh Ed  Ad Ad

ð44Þ

For convenience, the term r · de = Et is defined as unit energy of the buffer element as shown in Fig. 5, and the curve in Fig. 5 is obtained from Fig. 6. In Fig. 6, the area below the stress–strain curve can be expressed by the unit energy r · de. As the stress is a variable with respect to strain, the relation between unit energy and strain can be expressed R e by Et  e curve as shown in Fig. 5. The definition 0 max acc r  de ¼ Et max is the unit

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Fig. 5. Unit energy of a nonlinear damping system at the instant of maximum acceleration and deflection of EPS 20 buffers.

where C is the damping coefficient that can be obtained from Eq. (22). By substituting Eqs. (22), (30) and (31) into (46), the energy dissipation Ed can be expressed as follows:  qffiffiffiffiffiffiffiffiffiffiffi  Z smax acc pffiffiffiffiffiffiffiffi 2gh 2fxs 2 2 sþ/ dt 2f MK e cos x 1f Ed ¼ 1 1f2 0 Z smax acc pffiffiffiffiffiffiffiffi 2gh ¼ f MK e2fxs 2 1f 0   qffiffiffiffiffiffiffiffiffiffiffi   1þcos 2x 1f2 sþ2/1 dt " # " # fðp2/1 Þ f  pffiffiffiffiffiffi pffiffiffiffiffiffi ðp2/1 Þ 2 2 ð47Þ ¼ Mgh 1e 1f ¼ Wh 1e 1f Next, the substitution of Eqs. (47) into (46) yields the unit energy at the instant of rebound, ðp2/1 Þ Wh Ed Wh pffiffiffiffiffiffi 2  e 1f ¼ Ad Ad Ad Z emax acc ¼ r  de f

Et

max

¼

ð48Þ

0

Fig. 6. Elasto-plastic strain–stress curve of the EPS buffers.

energy at the instant of rebound. Therefore, Eq. (44) can be rewritten as Et

max

¼

Wh Ed  Ad Ad

The obtained unit energy from Eq. (48) will be used to find the corresponding strain emax_acc through Fig. 5. The corresponding stress can then be found through stress–strain curve in Fig. 6. Finally by substituting the strain and stress obtained into Eq. (40), the maximum acceleration under various conditions can be achieved. Note that the maximum acceleration depends on the bufferÕs material property (damping ratio f and stress–strain curve) and the initial conditions of h/d and W/A. Fig. 7 illustrates the solution scheme to obtain the maximum acceleration in a nonlinear damping system.

ð45Þ

in which Ed is still an unknown coefficient that can be evaluated by Z tmax acc Z Z tmax acc   dx dt ¼ C x_ C x_ 2 dt Ed ¼ C x_ dx ¼ dt 0 0 ð46Þ

4. Mathematical example An example is now presented to illustrate the practical application of the nonlinear damping mathematical model. As shown in Fig. 8, a packaged component with the protection of cushion undergoes the free drop. The

Fig. 7. Solution algorithm of the nonlinear mathematical model.

Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

1591



Gnumerical  Gexpermental

errorð%Þ ¼



Gexperimental



26.16 g  27 g

 100% ¼ 3.1% ¼



27 g

ð53Þ

which is reasonable in the mathematical analysis.

5. Comparison with experimental results

Fig. 8. Drop-impact model.

following characteristics are assumed in the example: weight of packaged material W of 400 N, cushion area A of 400 cm2, cushion thickness d of 10 cm, drop height h of 1 m, density of 20 kg/m3 and damping ratio of 0.1 for EPS 20 buffer. Note that the strains of the buffer are all less than 72% as shown in Fig 6. In this example, the static surface load W/A, the ratio between drop height and cushion thickness h/d, and damping ratio f are assumed W =A ¼ 1 N=cm2 ;

h=d ¼ 10;

and

f ¼ 0.1

ð49Þ

The angle factor /1 is obtained through Eq. (29), f 0.1 /1 ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0.1 1f 1  0.12

ð50Þ

Substituting the values of W/A, h/d, and /1 into Eq. (48), we obtain the maximum unit energy Et_max Z emax acc Et max ¼ r  de 0 ðp2/1 Þ Wh pffiffiffiffiffiffi 2  74.41 kPa e 1f Ad

In order to verify the validity of the proposed model, a series of experimental results from BASF [19] are to compare the mathematical results of the mathematical model. For comparison, the value of h/d is fixed as 10 as shown in Fig. 9, and the damping ratio is fixed as 0.1. The maximum acceleration is plotted with respect to the value of W/A. We found from this comparison that the mathematical and experimental results matched each other quite well. It is proven that the mathematical model is indeed successful.

6. Discussion 6.1. Discussion of nonlinear characteristics The concept of ‘‘soft spring’’ and ‘‘hard spring’’ was discussed in SuhirÕs work [9]. The effect of these two nonlinear systems was compared with that of linear spring in his study. (1) Application of a nonlinear spring with a hard characteristic. It results in an appreciable decrease in the maximum displacement and a substantial increase in the maximum acceleration, in comparison with the linear case. The accumulative effect of a hard spring is such that the product of the maximum displacement and the maximum acceleration increases with an increase in the degree

f

ð51Þ

We can find from Fig. 5 that the strain emax_acc according to the unit energy of 74.41 kPa is 46%. From the stress–strain curve in Fig. 6, the corresponding maximum stress rmax is 212 kPa. Substituting the value of emax and rmax into Eq. (39), the maximum acceleration is found to be rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rmax acc h sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 emax   fðp2/1 Þ acc ðd Þ rmax acc h  p ffiffiffiffiffiffi ¼0 . 8626 g 2 emax acc d W 2 ð 1f AÞ   €xmax ¼ g e W A

¼ 26.16 g ¼ Gnumerical

ð52Þ

180

maximum G-force, G

¼

160

mathematical result

140

experimental result

120 100 80 60 40 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4 2

The corresponding experimental result of maximum G-force (acceleration/g) provided by BASF is 27 g (Gexperimental) [19]. The error is thus

static surface load W/A, N/cm

Fig. 9. Comparison of mathematical results and experimental results with h/d = 10.

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Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

of nonlinearity. Therefore, employment of such a spring can become feasible for elements that are able to withstand high accelerations, while the maximum deformation has to be made small by any means. (2) Application of a nonlinear spring with a soft characteristic. It results in an appreciable increase in the maximum displacement and a substantial decrease in the maximum acceleration, in comparison with the linear case. The accumulative effect of a soft spring is such that the product of the maximum displacement and the maximum acceleration decreases with an increase in the degree of nonlinearity. Therefore, employment of such a spring can become feasible for elements that cannot withstand high accelerations, while the maximum deformation allowed being larger than that of the linear case. Next, we would like to investigate the effect of nonlinear model while compared to the linear system. In the study, a linear model is developed based on the YoungÕs modulus (E = 4.6 MPa) of the cushion buffer as shown in Fig. 6, thus Eq. (21) can be rewritten as K linear ¼

Ar A ¼ E de d

ð54Þ

in which the stiffness coefficient Klinear is a constant value. Substituting Klinear of Eq. (54) into Eqs. (36) and (37), we obtain the maximum acceleration under different conditions of h/d and W/A, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðp2/1 Þ fðp/1 Þ 2 ffiffiffiffiffiffi p pffiffiffiffiffiffiffiffi 2ghK linear pffiffiffiffiffiffi 2 2 €xmax ¼ 2ghxlinear e 1f ¼ e 1f M sffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðp2/1 Þ   fðp2/1 Þ 2E dh pffiffiffiffiffiffi 2ghAE pffiffiffiffiffiffi 1f2   e 1f2 ¼ e ¼g ð55Þ W Md A The maximum acceleration of the linear system is obtained as shown in Fig. 10. In order to compare the effect of nonlinear characters with that of linear system,

the maximum accelerations for the nonlinear system are also shown in the same figure. In comparison, the value of h/d is fixed as 10. Note that the damping ratio f is 0.1 and the maximum acceleration becomes a variable with respect to the value of W/A. We found from Fig. 10 that the maximum acceleration of the linear system is obviously higher than that of the nonlinear model. Next, the maximum strain is used to verify the soft spring characteristics of nonlinear cushion system. The energy balance Eq. (42) can be applied to the linear system as, Z xmax acc 1 Mgh ¼ KðxÞx dx þ Ed ¼ K linear x2max þ Ed ð56Þ 2 0 where K is a constant. Substituting Eqs. (47) and (54) into Eq. (56), it is found that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðp2/1 Þ 2Wh pffiffiffiffiffiffi 2 e 1f xmax ¼ K linear sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðp2/1 Þ 2Whd pffiffiffiffiffiffi 2 e 1f ¼ ð57Þ AE In the linear system, the maximum strain also occurs simultaneously with the maximum acceleration and can be evaluated by substituting Eqs. (19) into (57) as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðp2/1 Þ xmax 2Wh pffiffiffiffiffiffi 2 ¼ e 1f emax ¼ d AEd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    pffiffiffiffiffiffi f ðp2/1 Þ 2 W h 2 e 1f ¼ ð58Þ E A d As a similar comparison of maximum acceleration under linear system and nonlinear system with damping ratio f of 0.1, the maximum strain in the two models is compared as shown in Fig. 11. The value of h/d is set as 10, again. Note that, before the yield point 2% as indicated in Fig. 6, the stress–stress curve is in the linear stage and has the same material property to the linear system.

70

300

nonlinear system

250

linear system

200 150 100 50

nonlinear system

60 maximum strain, %

maximum G-force, G

350

linear system

50 40 30

yield strain 2% 20 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4

static surface load W/A, N/cm2

Fig. 10. Comparison of maximum acceleration under linear system and nonlinear system with h/d = 10 and f = 0.1.

0 0.005

0.18

0.38 0.58 0.78 0.98 static surface load W/A, N/cm2

1.18

1.38

Fig. 11. Comparison of maximum strain under linear system and nonlinear system with h/d = 10.

Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

6.2. Discussion on damping ratio In the previous section we discussed the nonlinear characteristics in the cushion system. Let us now assess whether viscous damping can have an effect on the dynamic response. The maximum acceleration and maximum strain are still of concerns. In the next comparison study, the values of damping ratio are taken as 0, 0.05, 0.10 and 0.15 as shown in Fig. 12. In order to illustrate these trends clearly, the range of static surface load W/A is only varied from 0.6 to 0.7. It is found from Fig. 12 that the maximum acceleration in the nonlinear cushion system decreases along with the increase in damping ratio under the same conditions, h/d and W/A. Therefore in the case of EPS 20 cushion system, the increasing damping ratio can decrease the possibility of undesired ‘‘rigid impact’’. Furthermore, through Eq. (54), the error between mathematical results under different damping ratios (0, 0.05, 0.10 and 0.15) and experimental results is obtained and shown in Fig. 13. The figure illustrates that the error of the un-damped model is always bigger than that of the damping model. This implies that the damping into

mathematical result without damping mathematical result with damping ratio 0.05 mathematical result with damping ratio 0.1 mathematical result with damping ratio 0.15 experimental result

maximum G-force, G

44 42 40 38 36 34

error without damping error with damping ratio 0.05 error with damping ratio 0.10 error with damping ratio 0.15

18 16 14 Error(%)

12 10 8 6 4 2 0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 0.8 0.9 static surface load W/A , N/cm2

1

1.1

Fig. 13. Acceleration error of mathematical models under various damping ratios with h/d = 10.

the mathematical model will affect the accuracy of the mathematical model. Fig. 13 also shows that the error curves of the damping models are constantly ‘‘crisscrossing’’ each other, which means that the damping ratio is not a constant value under different conditions. In the low static surface load stage, the error of high damping models is always less than that of the low damping models; while in the high static load stage, the results of low damping model is closer to the test results than that of high damping models. As a conclusion, the damping ratio decreases with the increase of static load, deformation and impact time; while it decreases with the decrease of maximum acceleration. Next, the effect of viscous damping on the maximum strain is evaluated as shown in Fig. 14. The damping ratio is still set for four values: 0, 0.05, 0.10 and 0.15, respectively. As evident from the illustrated curves, the damping can substantially reduce the maximum strain. Through the above discussions of nonlinear effect and damping effect, the following observations can be concluded: (1) Application of cushion with soft nonlinear characteristics results in an increase in the maximum 80 70 maximum strain, %

Therefore as shown in Fig. 11, the strain for both the linear and the nonlinear systems have the same value when the strain is less than 2%. We found from the figure that the maximum strain beyond the yield point in the nonlinear mathematical model is obviously higher than that of the linear system. In view of these two comparisons, it is seen that the nonlinear mathematical model in the study retains all the characteristics of soft spring: (1) lower maximum acceleration and (2) larger maximum strain, if compared to the linear model. With the decrease of maximum acceleration under the protection of soft spring, a cushion buffer provides the effect to the mitigation of ‘‘rigid impact’’.

1593

60

nonlinear models without damping nonlinear models with damping ratio 0.05 nonlinear models with damping ratio 0.10 nonlinear models with damping ratio 0.15

50 40 30 20 10

32 0.6

0.62

0.64

0.66

static surface load W/A , N/cm

0.68

0.7

2

Fig. 12. Comparison of maximum acceleration under various damping ratios with h/d = 10.

0 0.005

0.18

0.38

0.58

0.78

0.98

1.18

1.38

static surface load W /A , N/cm2

Fig. 14. Comparison of maximum train under various damping ratios with h/d = 10.

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Y. Wang, K.H. Low / Computers and Structures 83 (2005) 1584–1594

deformation and a decrease in the maximum acceleration (rigid impact) in comparison with that of the linear model. (2) Application of cushion with damping results in a decrease in the maximum deformation and maximum acceleration (rigid impact) in comparison with that of the un-damped model. (3) The damping ratio depicts nonlinear characteristics in the drop-impact process.

7. Concluding remarks The objective of this paper is to develop a relatively reliable and practical method to predict the response of nonlinear system under impact. In addition, the coefficients that influence these behaviors are taken into account in the analysis, such as damping ratio in the cushion system. Through a serial of comparisons with experimental results and various mathematical models, the effects of nonlinear characteristics and viscous damping have been investigated. The application of nonlinear characteristic and viscous damping can both decrease the ‘‘rigid impact’’ on certain level, while damping can decrease the maximum deformation at the same time. Through a successful analysis of the nonlinear damping system, the designers can better predict the crashworthiness of the packaged goods quantitatively, not limited in the qualitative analysis. It is believed that the multi-spring system is possible to provide a broad analysis of buffers with arbitrary shapes, not necessarily with buffers with cubic shape. In the present work, the authors only consider the dynamic behavior at the instant of rebound. The time coefficient, t or s, in the local linearized working point can only be used as phase state, whereas it cannot be used to evaluate the time response in the whole nonlinear system. Therefore, the transient response is not covered in this method. However, the maximum acceleration and related coefficients is important from a technical point of view. And these values can provide reliable boundary information in the future analysis of transient response for the nonlinear systems.

Acknowledgments The authors are grateful to Ms. Agnes Tan in the Robotics Research Centre of the Nanyang Technological University, for her kind assistance with the use of equipment and facilities. Thanks are also due to the reviewers for their constructive comments and suggestions, which have improved the quality of the manuscript.

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