J. Pac .Sci. Tech.

Vol.30No.4（2021）

Original Paper～～～～～～～～～～～～～～～～～～～～～～～～～～～～A Method for Generating Random Vibration
Considering Kurtosis Response Spectrum
Daichi NAKAI*, ** and Katsuhiko SAITO*
A random vibration test is important for confirming the safety of packaging when transporting goods ona truck bed. There are differences between the traditional random vibration method and real vibration
measured on the truck bed. These differences may result in overpackaging or underpackaging. Therefore,
improving the input vibration during vibration testing is an important theme. In this study, we focused on
the kurtosis response spectrum. A previous study showed that the response kurtosis under the real vibration
varies depending on the natural frequency. Therefore, it is assumed that a random vibration with the same
kurtosis response spectrum as the real vibration is similar to the real vibration. We propose a new method
for generating random vibration considering the kurtosis response spectrum. Vibrations with the same
power spectral density, root mean square of acceleration, acceleration kurtosis, and velocity kurtosis (butdifferent kurtosis response spectra) were generated by the proposed method and the previous study method.
Comparing the kurtosis response spectra between the real and generated vibrations, the proposed method
showed a particular improvement in the input vibration, especially when the damping factor was small.
Keywords: Transportation, Vibration test, Response Spectrum, Kurtosis

1. Introduction
A random vibration test is important for confirming the safety of packaging when transporting goods ona truck bed or railway car. In the traditional vibration method, the power spectral density (PSD) of theacceleration is only considered as a test condition. However, it has been pointed out that there aredifferences between real vibration on the truck bed and random vibration generated by the traditional
method. Real vibrations on a truck bed are non-stationary because there are fluctuations in the truck driving
speed and road roughness 1). The traditional vibration method cannot reproduce this non-stationary
condition. These differences may result in overpackaging or underpackaging because of inappropriate
evaluation of antivibration requirements. Therefore, it is important to improve the random vibration during
vibration testing.
Many studies have focused on the acceleration kurtosis (𝐾 ) and the probability density of acceleration.
The 𝐾 value of a random vibration generated by the traditional method is 3, and the probability densityof acceleration is the Gaussian distribution. On the other hand, the 𝐾 values of real vibrations are usually
leptokurtic (i.e., greater than 3) 2, 3). Therefore, it is assumed that a random vibration with the same 𝐾 as
the real vibration is more similar to the real vibration than the one generated by the traditional method.
Rouillard proposed a method for generating random vibration with a non-Gaussian probability density
distribution by combining Gaussian oscillations with different intensities 4). Furthermore, Rouillard et al.
proposed a method that considered the moving RMS of acceleration 5). Winterstein proposed a method totransform a Gaussian vibration into non-Gaussian vibration using a Hermite polynomial 6). Hosoyama et al.
proposed a method for generating leptokurtic random vibration based on the phase differential 7). Our
previous study proposed a method for controlling not only 𝐾 but also the velocity kurtosis 𝐾 . It showsthe effect of 𝐾 on packaging damage using a single degree of freedom (SDOF) model 8).
* Transport Packaging Laboratory, Kobe University, 5-1-1, Fukaeminami, Higashinada, Kobe 658-0022, Japan** Sankyu Inc. 16-1 Tsukiji, Yahatanishi, Kitakyushu, 806-0001, Japan
Corresponding author: Daichi Nakai. Tel: 093-645-7262 E-mail: d.nakai@sankyu.co.jp－ 261 －A Method for Generating Random Vibration Considering Kurtosis Response Spectrum

As mentioned above, the proposed methods for improving random vibration mainly considered 𝐾 , 𝐾 ,the probability density of the acceleration, the probability density of the velocity, and the moving RMS.
Hosoyama et al. proposed the kurtosis response spectrum as a characteristic of vibration for packaging 9).
Fig. 1 shows the schematic diagram of the response spectrum. The response spectrum assumes the SDOF
model and is used to investigate the effects of vibrations at various natural frequencies. To obtain the
response spectrum, the SDOF model response and its summary statistics (e.g., maximum, standard
deviation, and kurtosis) are calculated many times with varying natural frequencies and damping factors
under the random vibration or the real vibration measured on the truck bed. It is also shown that the kurtosis
response spectrum varies with the natural frequency 𝑓 of the SDOF model when the input vibration is thereal vibration measured on the truck bed.
It is assumed that a random vibration with a kurtosis response spectrum that is the same as that of thereal vibration is similar to the real vibration. In this study, we propose a method for generating the random
vibration considering the kurtosis response spectrum. The relative displacement was used as the responseof the SDOF system because it is proportional to the strain of the product. Then, the generated vibrations
were compared with the real vibrations and the proposed method was much similar to the real vibrations
than those in the previous study method, especially in the case of a small damping factor.

Fig. 1 Schematic diagram of response spectrum.
(a)

(b)

(c)

DER-1000
Fig. 2 Photos during the truck shipping experiment, (a) the truck taken from the front,(b) truck bed, (c) accelerometer.

2. Experiment
The vertical vibration data for the truck bed were used as a target. Fig. 2(a) and (b) shows the truck used
for testing and truck bed, respectively. An accelerometer (DER-1000, Shinyei Technology Co., Ltd.) was
fixed to the truck bed (Fig. 2(c)). We ran the truck on the highway for 1,200 s (low-pass filter, 100 Hz).

3. Calculation
3. 1 Method for calculating response spectrum

The acceleration applied inside the SDOF structure 𝑎 𝑡 are calculated using the same method as ourprevious study 8). A SDOF system like that in Fig. 1 is expressed by equation (1):

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J. Pac .Sci. Tech.

𝑚

𝑑 𝑥 𝑡
𝑑𝑡

𝑎 𝑡

Vol.30No.4（2021）

𝑑𝑥 𝑡
𝑑𝑡

𝑘𝑥 𝑡

𝑐

0,

1

where 𝑡, m, 𝑐, 𝑘, and 𝑥 𝑡 are, respectively, time, the mass, viscosity coefficient, spring constant, and therelative displacement between the mass and the truck bed. An impulse response function of the relative
displacement ℎ 𝑡 is expressed by equation (2):
1
𝑒
𝜔

ℎ 𝑡

𝑠𝑖𝑛𝜔 𝑡,2where 𝜔 , 𝜔 , and 𝜁 are, respectively, undamped natural angular frequency, and damped natural angularfrequency of the response and the damping factor 10). Meanwhile, 𝜔 is expressed by equation (3):
2𝜋𝑓

𝜔

3

The damping factor, 𝜁 is expressed by equation (4):4𝜁The damped natural angular frequency, 𝜔 is expressed by equation (5):
𝜔

𝜔

1

𝜁

5

𝑥 𝑡 is expressed by the equation (6):𝑥 𝑡𝑎 𝜏 ℎ 𝑡𝜏 𝑑𝜏,6where 𝜏 is a parameter. Equation (6) is a convolution integral; therefore, 𝑥 𝑡 can be calculated byequation (7):1 ∞𝐴 𝑓 𝐻 𝑓 𝑒
𝑑𝑓,
7
𝑥 𝑡
2𝜋 ∞
where 𝐴 𝑓 and 𝐻 𝑓 are the Fourier transform of 𝑎 𝑡 and ℎ 𝑡 , respectively. 𝑎 𝑡 is proportional
to the relative displacement 𝑥 𝑡 :
𝑎 𝑡

𝑥 𝑡

4𝜋 𝑓 𝑥 𝑡

8Here, 𝑓 is from 1 to 100 Hz at intervals of 1 Hz. The RMS and kurtosis of 𝑎 𝑡 calculated for eachnatural frequency 𝑓 by fixing the damping factor with 𝜁, are the RMS response spectra 𝑅 𝑓 and the
kurtosis response spectra 𝐾 𝑓 .

3. 2 Methods for generating random vibration
3. 2. 1 Previous method
The traditional method for generating random vibration is expressed by equation (9):－ 263 －A Method for Generating Random Vibration Considering Kurtosis Response Spectrum

𝑎 𝑡∑

𝐴 𝑐𝑜𝑠 2𝜋𝛥𝑓𝑡

𝜙(9)where L, 𝐴 , 𝛥𝑓, and 𝜙 are, respectively, the number of frequency components, amplitude, frequency
resolution, and the kth phase angle. 𝐴 is expressed by equation (10):𝐴

2𝜋𝛥𝑓𝑃 𝑘𝛥𝑓 ,10

where 𝑃 𝑘𝛥𝑓 is the PSD. In traditional random vibration tests, 𝜙 denotes random numbers ranging
from 0 to 2π.
were independent
When the random vibration was generated by the traditional method, 𝜙 and 𝜙from each other. The method for generating a random vibration with arbitrary 𝐾 by controlling the phaseis proposed 7). The relationship between 𝜙 and 𝜙is expressed
differences between 𝜙 and 𝜙by equation (11):𝜙2𝜋𝛥𝑓𝑡 𝑘𝛥𝑓 ,11
𝜙where 𝑡 𝑘𝛥𝑓 is a group-delay time. 𝑡 𝑘𝛥𝑓 is taken as a random number with an average value mand a standard deviation 𝜎, and 𝜎 is related to the sharpness of the envelope curve. As 𝜎 decreases, theenvelop curve becomes sharper and 𝐾 increases. As 𝜎 increases, 𝐾 converges to 3 (Gaussian). Our
previous study showed that 𝛥𝑓 also had an effect on the envelope curve and was related to the period 𝑇
during which the maximum acceleration occurred 8). The relationship between 𝛥𝑓 and 𝑇 is expressed byequation (12):
1
𝑇
12
𝛥𝑓
When 𝑇 is small, the ratio 𝐾 ⁄𝐾 is less than 1. With increasing 𝑇 , the ratio 𝐾 ⁄𝐾 converges to one.
In this study, 𝑇 was set to 16 s in the case of the proposed method and 2 or16 s in the case of the previous
method.

3. 2. 2 Proposed method

In the previous method, the standard deviation 𝜎 was a constant value, regardless of the frequency. In
this study, the standard deviation of 𝑡 𝑘𝛥𝑓 is not a constant value 𝜎 but a variable 𝜎 𝑓 thatchanges with frequency. In order to set 𝜎 𝑓 , the kurtosis response spectrum𝐾 𝑓 was transformed to𝜎 𝑓 .

Fig. 3 Power Spectral Density
transform 𝐾 𝑓 to 𝜎 𝑓 .

toFig. 4 The relationship between 𝐾 andstandard deviation 𝜎 to transform𝐾 𝑓 to 𝜎 𝑓 .

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The relationship between 𝜎 𝑓 and 𝐾 𝑓
is needed to transform 𝐾 𝑓 to 𝜎 𝑓 . We
generated a random vibration with PSD as shownin Fig. 3 using equation (9), (10), and (11).
This PSD had energy from 4.5 to 5.5 Hz and 𝛥𝑓
1/16 Hz. The standard deviation 𝜎 was from0.05 to 5 and the acceleration kurtosis 𝐾 was
calculated for each value of 𝜎 (Fig. 4).
Then, the kurtosis response spectrum𝐾 𝑓 wascalculated by equation (1) to (8). Fig. 5 shows the𝐾 . 𝑓 as an example of 𝐾 𝑓 . This 𝐾 𝑓
was converted to 𝜎 𝑓 using Fig. 4. For
example, 𝐾 . 𝑓 57 Hz is 9.0 in Fig. 5and 𝜎 is 1.53 when the kurtosis is 9.0 in Fig. 4.
Therefore, 𝜎 . 56.5 𝑓 57.5 is set to
1.53.
Fig. 6 shows an example of the standard
deviation 𝜎 . 𝑓 . It is obvious that𝜎 . 𝑓is small at frequencies where
𝐾 . 𝑓 is relatively large.
Fig. 7 shows an example of random numbers,of which the average is 8.0 and the standard
deviation is 𝜎 . 𝑓 . 𝛥𝑓 was 1/16 Hz, andrandom numbers were generated 75 times for thesame frequency.
Finally, random vibration was generated with
the PSD of the real vibration and 𝜎 𝑓 byequation (9), (10), and (11).

Vol.30No.4（2021）

Fig. 5 Kurtosis response spectrum 𝐾.

Fig. 6 Standard deviation 𝜎

𝑓 .

.

𝑓 .

4. Results and Discussion
4. 1 Effect of damping factor on theacceleration kurtosis and velocity
kurtosisFig. 8 shows the relationship between thedamping factor ζ and the kurtosis of the input
Fig. 7 Random numbers depending on 𝜎 . 𝑓 .
acceleration 𝑎 𝑡which is generated byproposed method. Both the acceleration kurtosis
𝐾 and the velocity kurtosis 𝐾 increase with increasing damping factor ζ. When ζ is 0.02, which is theminimum value as a factor of the proposed method in this study, both 𝐾 and 𝐾 are greater than 3.
Therefore, all vibrations generated by the proposed method in this study are non-Gaussian. For the entire
range of ζ in this study, 𝐾 was slightly greater than 𝐾 .
In this study, the vibration for which 𝐾 had the same value as that of the real vibration (ζ=0.07), waschosen as a vibration for evaluation.

－ 265 －A Method for Generating Random Vibration Considering Kurtosis Response Spectrum

Fig. 8 Relationship between damping factor ζ and kurtosis, (a) acceleration kurtosis, (b) velocity
kurtosis.

4. 2 Comparison of real and generated vibrations

In this study, the vibrations generated by the proposed method (𝑇 = 16 s, ζ = 0.07) and the previous
method (𝑇 = 2 or 16 s) were compared with the real vibration.
Fig. 9 shows the time series acceleration of the real and generated vibrations. All of the vibrations were1,200 s because the length of the real vibration was 1,200 s (Fig. 9 (a)). Comparing Fig. 9 (b) and (d), there
are no significant differences between the proposed method and previous methods (𝑇 = 16 s). The envelope curveof Fig. 9 (c) is different from those of Fig. 9 (b) and (d) because of the difference in 𝑇 .
Fig. 10 shows the time series velocity of the real vibration and generated vibrations. These velocities
were integrated from the acceleration with low-cut filter (0.5 Hz) 11). Almost all of the velocity peaks of thereal vibration were less than 0.5 m/s, as shown in Fig. 10 (a). Comparing Fig. 10 (b) and (d), there are no
significant differences between the proposed and previous methods (𝑇 = 16 s). The velocity peaks of the previous
method (Td=2 s) are smaller than those of the proposed method and the previous method (Td = 16 s) as shown in
Fig. 10 (b)-(d).
Fig. 11 shows the PSD of the acceleration. There were no significant differences between the real and generated
vibrations.
Table 1 shows the statistical values of the real vibration and generated vibrations. There are no significant differencesin the acceleration RMS and velocity RMS between the real vibration and generated vibrations. Also, there
were no significant differences in 𝐾 between the real and generated vibrations because they were generated
as a target of 𝐾 . The biggest difference between the vibrations was the 𝐾 . The 𝐾 value of the realvibration was 8.59 and greater than 𝐾 . The 𝐾 of the proposed method and the previous method (Td = 16s) were also slightly smaller than 𝐾 . The 𝐾 value of the proposed method (Td = 2 s) was 3.65, which
was much smaller than 𝐾 .
Table 1 Statistical value of real vibration and generated vibrations.
Velocity
Acceleration
Velocity
Acceleration
RMS(m/s)
Kurtosis
KurtosisRMS(m/s2)(a)Real Vibration
0.796
0.0804
7.24
8.59(b)Proposed (Td =16 s)
0.792
0.08007.20
6.59(c)Previous (Td = 2 s)
0.790
0.0832
7.24
3.65(d)Previous (Td = 16 s)
0.792
0.07987.18
6.41－ 266 －

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Vol.30Fig. 9 Time series acceleration,(a) real vibration, (b) proposed method,(c) previous study method (Td = 2 s),(d) previous study method (Td = 16 s).

No.4（2021）

Fig. 10 Time series velocity,(a) real vibration, (b) proposed method,(c) previous study method (Td = 2 s),(d) previous study method (Td = 16 s).

Fig. 11 Power spectral density of acceleration, (a) real Vibration, (b) proposed method,(c) previous study method (Td = 2 s), (d) previous study method (Td = 16 s).

－ 267 －A Method for Generating Random Vibration Considering Kurtosis Response Spectrum

Fig. 12 shows the probability densities of the acceleration. The dotted lines show the Gaussian
distributions. The mean parameter is zero, and the standard deviation are the values of acceleration RMS,
as shown Table 1. All of the acceleration probability densities are non-Gaussian. In particular, theprobability densities of the proposed and previous methods (Td = 16 s) are quite similar.
Fig. 13 shows the probability density of the velocity. The dashed lines show the Gaussian distribution.
The mean parameter is zero and the standard deviations are the values of the velocity RMS, as shown Table
1. The shape of the probability density of the previous method (Td = 2 s) is quite different from theprobability densities of the others. The probability densities of the real vibration, proposed, and previous
method (Td = 16 s) had the similar shapes. In particular, the probability densities of the proposed andprevious methods (Td = 16 s) were quite similar.
In terms of statistical values and probability densities, the only clear difference between real vibrationand proposed method was 𝐾 . Furthermore, there were almost no differences between the proposed andprevious methods (Td = 16 s) in terms of statistical values and probability densities.
Fig. 14 shows the RMS response spectra 𝑅 𝑓 . Since the lines almost overlapped in Fig. 14, it isconsidered that 𝑅 𝑓 had almost the same results even if the vibration changed. When the damping factor
ζ was small, 𝑅 𝑓 changed significantly depending on the natural frequency, as shown in Fig. 14(a) and(b). As ζ increased, the difference in 𝑅 𝑓 between the natural frequencies decreased.
Fig. 15 shows the kurtosis response spectra 𝐾 𝑓 . Higher values of 𝐾 𝑓 resulted in the greater thedamage to the product, even if 𝑅 𝑓 remained the same. Therefore, the closer 𝐾 𝑓 of a generated
vibration is to that of the real vibration, the more it the vibration is assumed to be to the real vibration. The
value of 𝐾 𝑓 changed significantly depending on the natural frequency, especially in the case of a small
damping factor ζ.
As shown Fig. 15(a), 𝐾 . 𝑓 of the previous method (Td = 2 s) was smaller than that of the realvibration in the low frequency region (<14 Hz). Conversely, 𝐾 . 𝑓 of the previous method (Td = 2 s)
was greater than that of the real vibration in the high frequency region (>14 Hz). This result shows that theactual vibration could not be reproduced well just by making the value of 𝐾 the same as that of the actual
vibration.
The 𝐾 . 𝑓 value of the previous method (Td = 16 s), was close to a constant value regardless of thenatural frequency except in the low frequency region. This tendency was not the same as the real vibrationand the proposed method. The 𝐾 𝑓 shape of proposed method was most similar to that of the realvibration among three generated vibrations, when damping factor ζ was 0.04 and the natural frequency
𝑓 was around 10 Hz or above 50 Hz. For example, in the case of the real vibration, 𝐾 . 𝑓 10 𝐻𝑧
was 12.12, which was the maximum value in the range of natural frequencies from 1 to 100 Hz. In the caseof the previous method (Td = 2 or 16 s), 𝐾 . 𝑓 10 𝐻𝑧 were 5.23 and 6.84, respectively, which weremuch smaller than those of the real vibration. In the case of the proposed method, 𝐾 . 𝑓 10 𝐻𝑧
was 12.19, which was close to that of the real vibration.
When the damping constant was large (ζ = 0.12, 0.16), there was a small difference of 𝐾 𝑓 between
the real vibration and generated vibrations. Hence, the proposed method had the advantage in the case of asmall damping constant (ζ = 0.04).

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Fig. 12 Probability density of acceleration, (a) real vibration, (b) proposed method,(c) previous method (Td = 2 s), (d) previous method (Td = 16 s).

Fig. 13 Probability density of velocity, (a) real Vibration, (b) proposed method,(c) previous study method (Td = 2 s), (d) previous method (Td = 16 s).

－ 269 －A Method for Generating Random Vibration Considering Kurtosis Response Spectrum

Fig. 14 Root Mean square response spectra of 𝑎 𝑡 , (a) ζ=0.04, (b) ζ=0.08, (c) ζ=0.12, (d) ζ=0.16.

Fig. 15 Kurtosis Response spectra of 𝑎 𝑡 , (a) ζ=0.04, (b) ζ=0.08, (c) ζ=0.12, (d) ζ=0.16.

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5. Conclusions
Random vibration considering the kurtosis response spectrum was generated by the proposed methodof changing the random number of the phase difference for each frequency. There were almost no
differences between proposed and previous methods in terms of statistical values, probability densities, andRMS response spectrum. However, there were differences in the kurtosis response spectrum between the
proposed and previous methods. From the point of view of the kurtosis response spectrum, the proposed
method was an improvement compared with the existing one, especially when the damping constant ζ was
0.04. The proposed method needs to be closer to the real vibration when the damping constant is large (ζ =0.12 and 0.16).
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP18K04608.
References
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for packaging testing purposes, Packag. Technol. Sci., 28(8), p. 672 (2015).
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logistics transportation in South China, Packag. Technol. Sci., 31(10), p665 (2018).
3） P. Böröcz1 and S. P. Singh, Measurement and analysis of delivery van vibration levels tosimulate package testing for parcel delivery in Hungary, Packag. Technol. Sci., 31(5),p342 (2018).
4） V. Rouillard, On the non-Gaussian nature of random vehicle vibrations, World Congress
on Engineering, London, UK, p1219 (2007).
5） V. Rouillard and M. A. Sek, Synthesizing nonstationary, non-Gaussian random
vibrations, Packag. Technol. Sci., 23(8), p423 (2010).
6） S. R. Winterstein, Nonlinear vibration models for extremes and fatigue, Eng. Mech.,
114(10), p1772 (1988).
7） A. Hosoyama and T. Nakajima, The Method of Generating Non-gaussian Random
Vibration Using Kurtosis, Packag. Sci. Technol., Jpn, 20(1), p27 (2011).
8） D. Nakai and K. Saito, A method for generating random vibration using acceleration
kurtosis and velocity kurtosis. J. Appl. Packag. Res., 11(2), p64 (2019).
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（Received 16Mach 2021）（Accepted 8June 2021）－ 271 －A Method for Generating Random Vibration Considering Kurtosis Response Spectrum

尖度応答スペクトルを考慮した
ランダム振動の生成方法
中井 太地*, **、 斎藤

勝彦*

ランダム振動試験は、包装にとってトラック荷台での安全性を確認するため重要である。従来のランダム振動試験方法とトラック荷台の実際の振動には違いが存在する。この違いにより、過剰包装や過少包装が発生しうる。故に振動試験の入力振動を改善することは重要なテーマである。本研究では、尖度の応答スペクトルに焦点を当てた。先行研究で、実際の振動の応答尖度は
固有振動数によって異なることが示されている。故に実振動と同じ尖度応答スペクトルを持つ試験振動は、実環境により近いと想定される。本研究では、尖度応答スペクトルを考慮して振動を生成させる方法を提案する。提案方法と今までの振動生成方法によって、PSD、RMS、加速度
尖度、速度尖度は同じであるが、尖度応答スペクトルのみが異なる振動を生成した。実際の振動と生成された振動の間の尖度応答スペクトルを比較すると、提案手法は、特に減衰係数が小さい
場合に改善効果が大きかった。
キーワード: 輸送、振動試験、尖度、応答スペクトル－ 272 －