J. Pac .Sci. Tech.

Vol.19No.2（2010）

Originals〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜〜

Equivalent Drop Test Modification for Determinationof Cushioning Performance
Chen ZHONG* and Katsuhiko SAITO**
There are two main testing methods for cushioning performance of the transport packaging: One is the
dynamic compression test which uses simple shapes of material and the other is the equivalent drop test---the shock
test, which is based on the traditional equivalent drop theory. Because of limitations of the traditional theory, thetraditional equivalent drop test causes errors in many cases. For this reason, by introducing the concept of the velocity
correction coefficient that can be expressed by a parameter of the damping ratio, a new attenuation equivalent drop
theory (modified theory in this paper) based on the traditional theory is proposed in this paper. Subsequently, twodifferent correcting methods that are based on the modified theory are proposed. In order to verify the viability of the
new theory, expanded polyethylene is used as a test material, and the dynamic compression test and the shock test
based on the traditional theory and the modified theory respectively are performed. Finally, the results of two tests arecompared. Verification experiments show that the modified theory can improve the equivalent precision of the two
tests, and one of the two correcting methods is superior to the other.
Keywords：equivalent drop test, equivalent free-fall height, dynamic compression test, shock test, correction
coefficient

1. INTRODUCTION
Peak acceleration is an important indicator when evaluating the shock fragility of packaged products,
and the dynamic compression test and the shock test are often performed to evaluate cushioning
performance of transport packaging.
In general, the impact acceleration can be detected by the dynamic compression test for materials of
simple shapes; however, the dynamic compression test is not suitable for cushioning materials such as
molded pulp because of the materials' complex shape. In such cases, the shock test should be conducted.
For the drop test, there is a recommendation in JIS-Z-0240-2002[1][Structural Cushioning Materials for
Packaging---Determination of Cushioning Performance] (this standard corresponds with
ASTM-D4168-95[2] [Standard Test Methods for Transmitted Shock Characteristics of Foam-in-Place
Cushioning Materials]): When the velocity of the free-fall test is equal to the velocity change occurring on
the shock table of the shock test that uses the half-sine wave, the height of the free-fall test is called theequivalent free-fall height. This recommendation is based on the assumption that the cushioning materials
can be modeled as a one-dimensional non-attenuated spring-quality model (traditional model in thispaper). On the basis of this assumption, we can conclude that the dynamic compression test and the shock
test are equivalent. In other words, if the drop heights applied in the dynamic compression test and the* Graduate School of Kobe University, Fukaeminami, Higashinada, Kobe 658-0022, JAPAN** corresponding auther; Professor, Kobe University, Transport Packaging Laboratory,
5-1-1, Fukaeminami, Higashinada, Kobe 658-0022, JAPAN. E-mail: ksaito@maritime.kobe-u.ac.jp− 123 −

Equivalent Drop Test Modification for Determination of Cushioning Performance

shock test are equal, the peak accelerations of those two tests should yield the same theoretical result.
However, experimental data shows that for such tests with equal drop heights, the peak accelerations arecertainly not equal. The prime reason is that the ubiquitous phenomenon of friction in the realenvironment is not considered under the assumption mentioned above.

2. MATERIALS AND METHODS
2.1 Traditional equivalent drop theory

Fig. 1. Traditional Model
The structural cushioning materials are assumed to follow a linear spring-mass model under the
traditional equivalent drop theory (traditional theory in this paper) (Fig. 1). In this assumption, the peakacceleration occurring on a weight dummyof the dynamic compression test is shown as

follows:
(1)

where

is the natural angular frequency with no attenuation,is the drop height, andis the mass,is the spring constant,is the velocity of the weight dummy.

Fig. 2. Shock response spectrum (SRS)For the shock test, when an extremely short half-sine shock pulse is applied to the system, the shock(whereis theresponse spectrum (SRS) of the system is recorded, as shown in Fig. 2. Whennatural frequency with no attenuation, andis the effective impact duration) is smaller than, themodel is regarded as a "soft spring" and the SRS curve can be expressed as linear with slope. Hence,the shock transmissibility
is equal to, andcan be represented as follows [3]:- 124 -

J. Pac .Sci. Tech.

Vol.19No.2（2010）

(2)
(3)

is the peak acceleration,

whereshock table, andis the input acceleration,is the velocity change on theis the impact duration.

Comparing Eqs.(1) and (3), if we let

when we perform the dynamic compression test and theshock test, the peak accelerations of those two tests should be the same theoretically. In other words, we
can draw the conclusion that the two tests are equivalent.

2.2 Tests based on the traditional theory
First of all, the dynamic compression test and the shock test, which are based on the traditional
equivalent drop theory, are performed.

2.2.1

Decision of equivalent free-fall height

According to JIS-Z-0240-2002, equivalent free-fall height should be set at, which is adopted as

described in Section. 2.1.

2.2.2

Low-pass filtering

In order to remove noise from the target pulse and the inherent vibration of the weight dummy,
low-pass filtering must be correctly set up. According to JIS-Z-0240-2002,or

low-pass

filtering can often be employed in a test. With respect to structural cushioning materials, the acceleration
may become very complicated due to their different shape. Thus,

low-pass filtering is

recommended for the test of structural cushioning materials.

2.2.3

Test equipment and materials

The test equipment and materials used in this paper are shown as follows:

Illustration of the dynamic compression test

Illustration of the shock test

Fig. 3. Illustration of the dynamic compression test and the shock test

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Equivalent Drop Test Modification for Determination of Cushioning Performance1)

2)

Equipment (Fig. 3):

Dynamic compression tester
Shock machine"Shock manager" measuring and test instrument.
Test materials:

Weight dummy (Fig. 3)[Mass: 4kg]Expanded polyethylene from Asahi Kasei Chemicals Corporation (Fig. 4)[Expansion rate:, Size:]Drop height: 60cm
(Average of 2~5 times dropping)
T=2cm

100

T=3cm

Deceleration (×9.8 m/s2)

100

100
30

T=4cm
T=5cm
50

3.9 KPa

Unit: mm
0

2

5

10

20

40

Static Stress (KPa)

Expanded Polyethylene (EPE-25)

Static Stress-acceleration Curve of EPE-25

Fig. 4. Expanded polyethylene (EPE-25)
The weight dummy is selected using the following procedure: As shown in Fig. 4, according to thethickness of the test material and the official static stress-peak acceleration curve, when the drop height is, we draw a line from the lowest point of the curve described as
to theaxis. Then wefind the static stress acting on the material to be. It is easy to calculate that the mass of thewith the formula, whereis the static stress,is the load, andweight dummy isis the compression area.

2.2.4

Setup of the test condition

The test materials should be placed in a controlled atmospheric condition where the temperature is
23℃ and the relative humidity isforhours before the dynamic compression test and the shock
test.

Fig. 5. Parameter setting in the dynamic compression test
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J. Pac .Sci. Tech.

Vol.19No.2（2010）

Fig. 6. Parameter setting in the shock test
The mass of the weight dummies used in both tests is the same (), and an acceleration sensor used
for measuring the peak acceleration is fixed in the center above the weight dummy.
Because the equivalent free-fall height is, we can calculate that the theoretical velocity is
according to the formula. The parameter of the dynamic compression test and theshock test are set by the test control software separately, as shown in Figs. 5 and 6.
When the dynamic compression test is performed, the set-up drop height is not the same as theequivalent free-fall height (), but is equal to the corresponding drop height for the real velocity of
the weight dummy to the same as the theoretical velocity (). Therefore, the test has to be
adjusted repeatedly until the velocity of the weight dummy becomes.
On the other hand, when the shock test is performed, in order to adjust the velocity change of the shock
waveform occurring on the shock table to, an adjusting function in the controlling software(Fig. 6) needs to be used by repetition, too.
For two tests, the results are effective only when
or
.

2.2.5

Result of two tests

For both tests, the effective results are shown in Fig. 7.

×9.8 m/s2

y

y=x

x
×9.8 m/s2

Fig. 7. Result of two tests based on the traditional theory− 127 −

Equivalent Drop Test Modification for Determination of Cushioning Performance

As the expanded polyethylene's cushioning performance decreases as the test proceeds, the peakaccelerations of the two tests gradually increase and there is a linear relationship between theaccelerations observed under the two tests. the result shown in Fig. 7 is not the same as a theoreticalresult that all the equivalent points of the two tests should be exactly on the line
under thetraditional theory, but a majority of them are scattered around the line. This establishes the limitation of
the traditional equivalent drop theory.

2.3 Modified equivalent drop theory [3]Fig. 8. Attenuation ModelFirst, the structural cushioning material is assumed to obey a one-dimensional attenuation model (Fig.
8), and the peak acceleration occurring on the weight dummyof the dynamic compression test

is shown as follows:
(4)
(5)
where

is the correction coefficient of the dynamic compression test, andis the damping ratio.

For the shock test, the response acceleration of the attenuation model in which input acceleration is thehalf-sine wave can be obtained theoretically. Subsequently,
response acceleration. If we define

can be calculated using the input andas the ratio between

transmissibility based on traditional theory

and the approximation of the shock, we can obtain(6)

then the peak acceleration of the shock test

can be as follows:
(7)

where

is the correction coefficient of the shock test, andis the velocity change on the shock

table.
With respect to the preceding information, a condition---correction coefficient---must be considered if
we want the dynamic compression test and the shock test to be equivalent under the attenuation model,
and the expression of the specific condition is(8)- 128 -

J. Pac .Sci. Tech.

Vol.19No.2（2010）80

AffmaxAcceleration (×9.8 m/s2)

70
60
50
40
30
20

γ

10
0
-10

0

5

10

15

20

25

Time (ms)Fig. 9. Illustration of parameter
It must be pointed out that the attenuation model'sandmust be known in order to be able tocalculate the velocity change correction coefficient. These two parameters can be obtained from thetime-acceleration curve that contains the peak accelerationand the half-duration fromto(Fig. 9). The method mentioned above is the correct shock test procedureof the shock pulse

for deriving the velocity change correction coefficient based on the cushioning materials' dynamic
compression test.

Function Value

2.4 Correction of the velocity changeVn ff

The dynamic compression test
basing on the traditional theory

Accelaration (G)

Vc =

Tr =

Acs max
Ain

n ff
ncs

V

ncs

Time (s)The shock test basing on the
traditional theorycs

Fig. 10. Flowchart for correcting the velocity change
Under the new equivalent drop theory, a correction of the velocity change follows the flowchart in
Fig. 10.
We can get the response acceleration-time curve after the dynamic compression test is performed(Fig. 9), andcan be expressed with

and

as:
(9)

− 129 −

Equivalent Drop Test Modification for Determination of Cushioning Performancethen, we combine Eq.(9) with Eqs.(4) and (5), Eq.(10) is derived:(10)
where " " is valid for ξ <0.707, and " " for ξ >0.707.

y

y=

2 2-1
exp(
2(1 - 2 )
2 1-

2

)

Fig. 11. Calculation of the damping ratio
For the right-hand side of Eq.(10), a curve that relates tocan be plotted (Fig. 11). Furthermore, aspecific value can be calculated if the metadata of the dynamic compression test is imported into theleft-hand side of Eq.(10). Next we match these specific values with the curve graphics; the corresponding
damping ratio

is available when both of them match. At last,

can be calculated afterisimported into Eq.(5).
On the other hand, the response acceleration of the shock test can be expressed as follows:

(11)

where

is the initial impact duration. The maximum value of the response acceleration can be

determined by means of the shock response analysis (SR analysis) [4] (Fig. 12), and thencalculated using Eq.(2). The corrected velocity changeEq.(8).

- 130 -

can becan eventually be calculated according to

J. Pac .Sci. Tech.

No.2（2010）

Vol.19Fig. 12. SR analysis
Using the corrected velocity change, the shock test is re-performed, and the new test data are

obtained. The comparison of the result based on the traditional theory and the result after being correctedby the correcting method-1 (The method in which the input velocity change of the shock test isdetermined after solving the model'sandaccording to the dynamic compression test is called thecorrecting method-1 in this paper.) is shown in Fig. 13.

×9.8 m/s2

y

(

,

,

…):

The result basing on the traditional theory( , , …):
The result basing on the correcting method-1

y=xx×9.8 m/s2

Fig. 13. Comparison of results of the tests based on the traditional method
and the correcting method-1

2.5 Equivalent drop height
The equivalent drop height can also be determined according to the shock test. First, the frequency is
calculated using the response acceleration waveform of the weight dummy during the shock test. Second,
SR analysis is performed and the desired damping ratio is obtained.
calculated using the input and response acceleration. Next,

can be determined afterisis obtained using Eq.(5). Finally, theequivalent drop height corresponding with the free-fall test can be calculated according to the following
equation:

(12)
(11)

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Equivalent Drop Test Modification for Determination of Cushioning Performance

2.6 Correction of the velocity
Calculation ofn ff

V = 2 gh

SR Analysis

=

ncs
Vc
n ff

ncs
The shock test basing on
the traditional theory

Acs max

Fig. 14. Flowchart for correcting the velocity
Fig. 14 shows the flowchart for correcting the velocity. First, the shock test based on the traditional
theory is performed, and the input and response accelerations are measured. The reciprocal number of 2
times the period of a half cycle in the response acceleration is the model's natural frequency.

Subsequently, SR analysis is performed using the data of the input acceleration to compute the desired
damping ratio.

can be calculated using the input acceleration and the response acceleration, andis determined afteris imported into Eq.(6), subsequently,

Finally, the corrected velocity

is obtained when we use Eq.(5).

can be calculated according to Eq.(12).

Using the corrected velocity, the dynamic compression test is re-performed, and the new test data

are obtained. The comparison of the result based on the traditional theory and the result after being
corrected by the correcting method-2 (the method in which the drop height of the dynamic compression
test is determined after solving the model'sandaccording to the shock test is called the correcting

method-2 in this paper.) is shown in Fig. 15.

×9.8 m/s2y():
The result basing on the traditional theory( 1 2 3 ):The result basing on the correcting method-2
7
5

9

3

8

2

6

1

4

y=x

x
×9.8 m/s2

Fig. 15. Comparison of results of the tests based on the traditional method
and the correcting method-2- 132 -

J. Pac .Sci. Tech.

Vol.19No.2（2010）

3. RESULTS AND DISCUSSION
3.1 Correlation Coefficient
In statistics, correlation and dependence are two of a broad class of statistical relationships between two
or more random variables or observed data values. Correlations are useful to indicate a predictive
relationship that can be exploited in practice. There are several correlation coefficients, but the sample
correlation coefficient, denoted, which is mainly sensitive to a linear relationship between two

variables, is in common use. The sample correlation coefficientwith expected values

andand standard deviations

between two random variables
and

and

is defined as
(13)

where

is the expected value operator.

The value of the correlation coefficient must satisfy the inequality, indicating the degreeof linear dependence between the variables. The closer the coefficient is to either

or , the stronger[5]

the correlation between the variables.

3.2 Discussion of results
Table 1. Comparison of the cushioning material evaluation in the peak acceleration (G-factor)
Traditional Theory

Order

a

a

DC Test (G)Modified Theory

Shock Test (G)Shock Test (G)DC Test (G)The dynamic compression test.

The values in Table 1 are the peak accelerations (G-factor) of the weight dummy. The values from theleft to the right are: the result of the dynamic compression test when the weight dummy's velocity is equal
to the velocity in which the corresponding drop height is, the result of the shock test that uses thehalf-sine wave based on the traditional theory (the velocity change determined by the half-sine wave
equals the velocity of the weight dummy), the result of the shock test after being corrected by thecorrecting method-1, and the result of the dynamic compression test after being corrected by thecorrecting method-2.

− 133 −

Equivalent Drop Test Modification for Determination of Cushioning PerformanceFig. 16. Comparison of the correlation coefficient

Between the above data, three comparisons (Fig. 16) are performed to calculate the correlation
coefficient, and the result is shown in Fig. 16. It is easy to see that the correlation relationship between the
results becomes gradually stronger from the traditional method to the correcting method-1 to thecorrecting method-2. Therefore, the equivalent precision of the tests based on the modified theory is
increased over those based on the traditional theory, and we can see that the correcting method-2 issuperior to the correcting method-1. The reason is considered to be as follows: For the spring-mass model,
the peak acceleration of the weight dummy[4] is
(14)
(12)
where

and

denote time. For the correcting method-1, the input acceleration is assumed as thehalf-sine wave when the dynamic compression test is performed, whereas real pulse is applied as the
input acceleration when the correcting method-2 is adopted; therefore, the response acceleration based on
the actual waveform, according to Eq.(14), is more accurate.

4. CONCLUSIONS
Based on the preceding analysis, the traditional theory is limited because it does not consider an effectof attenuation phenomenon. Consequently, the result of the equivalent drop test based on the traditional
theory has errors. In this paper, the concept of velocity correction coefficient is introduced, and the new
equivalent model considering the attenuation is established. The experiment shows that the equivalence
accuracy of two tests is further improved after correction; thus, it can be seen that the velocity correctionis the key to reducing the error in the equivalent drop test.

References[1] JIS-Z-0240-2002, Structural Cushioning Materials for Packaging---Determination of Cushioning
Performance.
[2] ASTM-D4168-95, Standard Test Methods for Transmitted Shock Characteristics of Foam-in-Place
Cushioning Materials.
[3] Katsuhiko SAITO and Kazuaki KAWAGUCHI, 23rd International Association of Packaging- 134 -

J. Pac .Sci. Tech.

Vol.19No.2（2010）

Research Institutes Symposium, 3-5 September 2007.
[4] Cyril M. Harris and Allan G. Piersol, Harris' Shock and Vibration Handbook (Fifth Edition),23.1-23.26.
[5] Robert V. Hogg, Joseph W. McKean and Allen T. Craig, Introduction to Mathematical Statistics(Sixth Edition), 101-107.

（原稿受付 2010 年 1 月 26 日）（審査受理 2010 年 3 月 8 日）

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Equivalent Drop Test Modification for Determination of Cushioning Performance

緩衝性能確認等価落下試験の革新
仲 晨*、斎藤勝彦**

落下衝撃から包装内容品を保護するための緩衝材の性能は、動圧縮試験によって評価されるが、複雑な形状の構造体緩衝材では、動圧縮試験ができないため、衝撃試験装置を用いた等価落下試
験で評価される。等価落下試験では、発生させる衝撃台の正弦半波パルスの速度変化と等しい衝
突速度となる落下高さが等価自由落下高さであると規定されており、緩衝系が減衰のない線形バ
ネ-質量１自由度系とみなせる場合には実用上問題ないが、実際の緩衝系は減衰の影響が無視できない。本論文では、緩衝材性能評価としての動圧縮試験と等価落下試験の結果が必ずしも等価とみなせるほどではないことを実験的に明らかにするとともに、減衰の影響を考慮した理論に基づいた等価落下試験の方法を提案する。さらに、減衰等価落下モデルによる等価落下試験を行い、両試験の等価度を実験的に確認する。
キーワード: 等価落下試験、等価自由落下高さ、動圧縮試験、衝撃試験、修正係数- 136 -