Dropping shock response of corrugated paperboard cushioning packaging system

Abstract

In this paper, a dynamic model was proposed for a corrugated paperboard cushioning packaging system. The variational iteration method was applied and the multiplier was identified. The approximate solution for the nonlinear equations was obtained and compared with the numerical simulation results with ordinary differential equation solver in MATLAB, showing good agreement. Then resonance conditions were obtained, which should be avoided in the cushioning packaging design.

Keywords

Corrugated paperboard cushion nonlinear oscillator resonance

1. Introduction

Newton’s damage boundary concept (Newton, 1968) and succeeding modified damage evaluation approaches, such as fatigue damage boundary concept (Burgess, 1988), displacement damage boundary concept (Wang et al., 1998) and damage boundary surface concept (Wang and Wang, 2008) were widely applied in cushioning packaging design. These previous works discussed the products fragility evaluation under the action of base excitation. However, packaged products can be potentially dropped in the distribution circle, during which the packaged products may be seriously damaged by the inner-resonance, especially for those large and fragile products (Wang et al., 2010). Therefore, it is essential to obtain the inner-resonance conditions for dropped packaging systems.

However, the oscillations in most systems are of inherent nonlinearity, including the packaging system (Maleki and Ahmadi 2010; White et al., 2010; Wang, et al., 2011a,b; Zhang and Dupuis, 2011), and it remains a problem to obtain the resonance condition for nonlinear packaging systems, especially for multi-freedom degree nonlinear cushioning packaging systems. Most recently, various analytical approaches for solving nonlinear differential equations were widely applied in the analysis of engineering practical problems, such as homotropy pertubation method (HPM) (Ganji et al., 2010), energy balance method (EBM) (Mehdipouret al., 2010), parameter-expanding method (PEM) (Kimiaeifar et al., 2010) and variational iteration method (VIM) (He, 1999). The VIM does not need the assumption of linearization or weak nonlinearity, and depends totally on the Langrange multiplier theory, which means that this method will only fail if the Langrange multiplier for any partial differential equation/ordinary differential equation (PDE/ODE) or coupled PDE/ODE does not exist. Fortunately, the Langrange multiplier for nonlinear equations can be simply obtained in most cases (Wazwaz, 2009; Faraz et al., 2010; He et al., 2010). In this paper, the VIM was suggested to obtain the inner-resonance condition for a strong nonlinear packaging system.

2. Modeling and equations

The dynamic model for a dropped cushioning packaging is depicted in Figure 1. Here the coefficients m denotes the mass of the product, while $f (x)$ denotes the restoring force of cushioning pad. And h is the dropping height.

Corrugated paperboard is widely applied in packaging design because of its excellent protective performance and energy absorbing capacity (Ge, 2009). The compression stress-strain curve for corrugated paperboard is illustrated in Figure 2, from which we can obtain some general laws as below:

$σ (ɛ)$ is continuous on the whole interval;

The number of the points where stress-strain function derivative equals to 0 depends on the layer of the corrugated paperboard, to be specific, the n-layer corrugated paperboard has 2n points with stress-strain function derivative equaling to 0;

$σ (ɛ) > 0$ , when $0 < ɛ < ɛ_{b}$ , while $σ (0) = 0$ ;

$σ (ɛ) = E ɛ$ , when $0 < ɛ < ɛ_{L}$ , where E is the initial elasticity modulus of corrugated paperboard.

To reflect these features of corrugated paperboard, a nonlinear model was suggested as (Gao, 1992; Gao and Xi, 1995; Gao et al., 1995)

σ (ɛ) = a_{1} ɛ + a_{2} \sin (a_{3} ɛ) + a_{4} \tan (a_{5} ɛ) .

(1)

Here, a_i denote respectively the characteristic constants of corrugated paperboard which could be obtained by compression test.

A typical compression stress-strain curve for U-shaped B flute corrugated paperboard with a thickness of 3.2 mm was compared with the modeling results as shown in Figure 3. The compression experiments were conducted under the environmental conditions of 23°C and 50% humidity. The results show good agreement, implying that the proposed model is suitable for corrugated paperboards. It should be noted that this proposed model is not reliant on the flute type or the core layer and is suitable for nearly all kinds of corrugated paperboards. The main difference for different kinds of corrugated paperboard is the deviation of the model parameters (Gao, 1992).

By assuming that there is no restriction to the deformation of the cushioning pad, the governing equations of the corrugated paperboard packaging system can be expressed as (Gao, 1992; Wang and Wang, 2011)

{\begin{matrix} m \overset{··}{x} + β_{1} x + β_{2} \sin (β_{3} x) + β_{4} \tan (β_{5} x) + c \overset{\cdot}{x} = 0 \\ x (0) = 0 \\ \overset{\cdot}{x} (0) = \sqrt{2 gh} \end{matrix}

(2)

Here β _i can be respectively obtained by the material characteristic parameters a_i together with the dimension of the pad, and c is the damping ratio of the corrugated paperboard.

By introducing these parameters: $T_{0} = m / β_{1}$ , $L = 1 / β_{3}$ and let $X = \frac{x}{L}$ , $τ = \frac{t}{T_{0}}$ , $λ_{1} = \frac{β_{2} β_{3}}{β_{1}}$ , $λ_{2} = \frac{β_{3} β_{4}}{β_{1}}$ , $λ_{3} = \frac{β_{5}}{β_{3}}$ , $ζ = \frac{c}{2 \sqrt{m β_{1}}}$ , equation (2) can be written in the following forms

{\begin{matrix} \overset{··}{X} + X + λ_{1} \sin X + λ_{2} \tan (λ_{3} X) + 2 ζ \overset{\cdot}{X} = 0 \\ X (0) = 0 \\ \overset{\cdot}{X} (0) = V = \frac{T_{0}}{L} \sqrt{2 gh} = \sqrt{\frac{2 β_{3}^{2} mgh}{β_{1}}} \end{matrix}

(3)

By using the 5-th order Taylor series for sin X and tan X, equation (3) can be equivalently written as

{\overset{··}{X} + ω_{01}^{2} X + (\frac{1}{3} λ_{2} λ_{3} 3 - \frac{1}{6} λ_{1}) X^{3} + (\frac{1}{120} λ_{1} + \frac{2}{15} λ_{2} λ_{3} 5) X 5 + 2 ζ \overset{\cdot}{X} = 0 X (0) = 0 \overset{\cdot}{X} (0) = V = \frac{T_{0}}{L} \sqrt{2 gh} = \sqrt{\frac{2 β_{3}^{2} mgh}{β_{1}}}

(4)

where

ω_{01} = \sqrt{1 + λ_{1} + λ_{2} λ_{3}}

(5)

3. Variational iteration method

The VIM, first proposed by He (1999), has been widely applied in solving many different kinds of nonlinear equations, and is especially effective in solving nonlinear vibration problems with approximation. Applying the VIM, the following iteration formulae can be constructed

Y_{n + 1} (t) = Y_{n} (t) + \int_{0}^{t} λ ({LY}_{n} (s) + N {\binom{}{Y}}_{n} (s) - g (s)) d s

(6)

Here the subscript n denotes the nth-order approximation and λ is called a general Lagrange multiplier, which can be identified optimally via the variational theory. ${\binom{}{Y}}_{n} (s)$ is considered as a restricted variation, i.e., $δ {\binom{}{Y}}_{n} (s) = 0$ . Recalling equation (3), there are the following stationary conditions:

\begin{matrix} \overset{··}{λ} (s) + ω_{01}^{2} λ (s) = 0, \\ λ (s) |_{s = τ} = 0, \\ 1 - \overset{\cdot}{λ} (s) |_{s = τ} = 0 . \end{matrix}

(7)

Therefore, the multiplier, can be identified as

λ = \frac{1}{ω_{01}} \sin ω_{01} (s - τ)

(8)

And then the following iteration formula can be constructed for equation (3):

X_{n + 1} = X_{n} + \frac{1}{ω_{01}} \int_{0}^{τ} \sin ω_{01} (s - τ) {{\overset{··}{X}}_{n} + ω_{01}^{2} X_{n} + (\frac{1}{3} λ_{2} λ_{3} 3 - \frac{1}{6} λ_{1}) X_{n}^{3} + (\frac{1}{120} λ_{1} + \frac{2}{15} λ_{2} λ_{3} 5) X_{n}^{5} + 2 ζ {\overset{\cdot}{X}}_{n}} d s

(9)

4. Numerical results

Equation (9) provides us an iteration formula for equation (3), by taking the initial solutions below

X_{0} = A \sin (Ω τ)

(10)

we obtain the approximate solution of equation (3) as

\begin{matrix} X_{1} = A \sin (Ω τ) - \frac{4 {aA}^{3} + 5 {bA}^{5}}{ω_{01} (9 Ω^{2} - ω_{01}^{2})} (3 Ω \sin (ω_{01} τ) \\ + ω_{01} \sin (3 Ω τ)) + \frac{{bA}^{5}}{16 ω_{01} (25 Ω^{2} - ω_{01}^{2})} (5 Ω \sin (ω_{01} τ) \\ + ω_{01} \sin (5 Ω τ)) + \frac{A ζ Ω}{Ω^{2} - ω_{01}^{2}} (\cos (Ω τ) - \cos (ω_{01} τ)) \end{matrix}

(11)

where

a = - \frac{1}{6} λ_{1} + \frac{1}{3} λ_{2} λ_{3} 2

(12)

b = \frac{1}{120} λ_{1} + \frac{2}{15} λ_{2} λ_{3} 5

(13)

Similarly, the second-order, third-order and higher-order approximate solutions can be obtained. For most cases, the first-order or second-order is enough. Computation illustrates that for the corrugated paperboard packaging system, the first-order is enough. As shown in Figure 4, the non-dimensional dropping shock response displacement (X) of a typical cushioning packaging system is calculated and compared with the numerical integration solutions using a built-in ODE-solver in MATLAB, showing good agreement. The results are obtained for the following amounts: λ₁ = 0.1, λ₂ = 0.3, λ₃ = 0.5, $ζ = 0.2$ , and the initial dropping velocity input is $\overset{\cdot}{X} |_{τ = 0} = 3$ .

The resonance can be expected when one of the following conditions meets:

Ω = \sqrt{ω_{01}^{2} + \frac{3}{4} {aA}^{2} + \frac{5}{8} {bA}^{4}}

(14)

Ω = ω_{01}

(15)

Ω = \frac{1}{3} ω_{01}

(16)

Ω = \frac{1}{5} ω_{01}

(17)

These conditions should be avoided during the cushioning packaging design procedure.

5. Conclusion

Many commonly used packaging materials have nonlinear characteristics, and it is essential to analyse the nonlinear dynamic behavior of the packaging system. However, the solution to strong nonlinear equations is very difficult. This paper suggests the VIM to a nonlinear equation of corrugated paperboard packaging system, which depends totally on the Langrange multiplier theory. The multiplier was identified, and the approximate solution was obtained and compared with the numerical simulation results with ODE-solver in MATLAB, showing good agreement. Then the conditions for resonance, which should be avoided in the cushioning packaging design procedure, were obtained using the suggested method.

Footnotes

Funding

This work was supported by the Fundamental Research Funds for the Central Universities JUSRP111A07, and the Open Project Program of Key Laboratory of Eco-Textiles, Ministry of Education, Jiangnan University KLET1011, and supported by Opening Fund of Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, Jinan University KLPPL1102.

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