J. Pac .Sci. Tech.

Vol.30No.4（2021）

Original Paper～～～～～～～～～～～～～～～～～～～～～～～～～～～～

Non-Gaussian Nature of the SDOF Response toGaussian Vehicle Vibrations
Akira HOSOYAMA*, Kazuki TSUDA*, and Shogo HORIGUCHI*
This study aims to verify the equivalence between a Gaussian random vibration extracted from a vehicle
vibration and that generated by a current vibration controller. The equivalence was evaluated by calculating
the kurtosis from a single degree of freedom (SDOF) response, assuming that an SDOF system serves as
the packaged product. Depending on the natural frequency of the packaged product, the SDOF response to
the Gaussian random vibration generated by the current vibration controller was found to be a Gaussian
random vibration, while the SDOF response to the Gaussian random vibration extracted from the vehicle
vibration, a non-Gaussian random vibration. The results demonstrate that both the Gaussian random
vibrations may not be equivalent. Further, solely the power spectrum density (PSD) and probability density
function (PDF) are not sufficient to completely understand the nature of the vibrations; the PDFs of the
SDOF responses also need to be considered.
Keywords: random vibration; Gaussian distribution; SDOF; vibration testing; kurtosis response spectrum

1 Introduction
In recent years, the global e-commerce market has expanded and the number of delivery parcels has
increased. As a result, the number of transportation accidents that are attributed to vibrations has increased.
In addition, the importance of vibration testing, which is performed to confirm the safety of the packaged
products against vibrations, is also increasing. Current vibration test standards1)2)3) recommend using a
vertical Gaussian random vibration test. However, it has been demonstrated that during transportation, nonGaussian random vibrations often occur. Thus, a gap arises between the vibration environment that is
reproduced by the vibration controller and the actual vibration environment.4)5) This gap can lead totransportation accidents, even if the packaged products meet the vibration test requirements. Therefore, itis necessary to improve the accuracy of vibration testing.
Several studies have been performed on non-Gaussian random vibration generation methods6) to reproduce
the non-Gaussian nature of the actual vibration environment. In this regard, the polynomial transformation,phase control, and random Gaussian sequence decomposition methods have been proposed as
representative non-Gaussian random vibration generation methods.
The polynomial transformation method is used to convert the Gaussian random vibrations into stationary
non-Gaussian random vibrations by using a polynomial function. 7)8) This method is easy to use forsimulating the non-Gaussian random vibrations; however, it has been demonstrated that the power spectral
density (PSD) is distorted during the conversion. 9)
In contrast, the phase control method is used to generate stationary non-Gaussian random vibrations bycontrolling the Fourier phase. Steinwolf10) proposed a method to control the Fourier phase by using amathematical formula that relates the Fourier phase to kurtosis. The authors11) proposed a method ofgenerating non-Gaussian random vibrations by incorporating the concept of a seismic wave simulation, in
Osaka Research Institute of Industrial Science and Technology, 2-7-1, Ayumino, Izumi-city, Osaka 594-1157, Japan
Corresponding author: Akira Hosoyama. Tel:0725-51-2703, Fax:0725-51-2639, E-mail: hosoyama@tri-osaka.jp*－ 243 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

which the shape of a seismic wave is controlled by the Fourier phase. The phase control method does notcause any distortion in the PSD in principle because the kurtosis is controlled by the Fourier phase.
Finally, the random Gaussian sequence decomposition method is used to generate non-stationary nonGaussian random vibrations. This is achieved by decomposing the non-Gaussian random vibrations into
Gaussian segments with different amplitudes. The random Gaussian sequence decomposition method isbased on the hypothesis that the peakedness of the vehicle vibration is due to the non-stationarity of thevehicle vibration caused by the changes in the vehicle speed and road surface roughness. An advantage ofthis method is that it can reproduce the non-stationarity vibration.
Many studies have focused on the random Gaussian sequence decomposition in the domain of packaging.
Rouillard et al.12) proposed a method to decompose the non-Gaussian random vibrations into Gaussian
segments with different amplitudes and generate non-Gaussian random vibrations as the sum of these
Gaussian segments. Griffiths et al. 13) proposed a method to decompose the non-Gaussian random vibrations
into Gaussian segments through an iterative process that uses the discrete wavelet transformation. Zhou etal. 14) proposed another method for a similar decomposition by detecting shocks that use a moving crest
factor, while Bonnin et al. 15) suggested an iterative process that uses the sum of the weighted Gaussians.
Thus, several decomposition methods for the non-Gaussian random vibrations have been proposed, which
contributes to improving the accuracy of vibration testing and they enable us to reproduce a vibration
environment that is close to the actual transportation.
On the other hand, the random Gaussian sequence decomposition method assumes that the Gaussian
random vibrations that are derived using an actual vehicle and those that are derived using a vibration
controller are equivalent. However, the equivalence between both these vibrations has not been sufficiently
verified thus far. Since the equivalence between these vibrations is the basis of the random Gaussiansequence decomposition method, the verification of the equivalence is crucial.
In this study, the vehicle vibration is measured using a small van and is decomposed into several Gaussian
random vibrations. In addition, the Gaussian random vibration is generated based on the PSD obtained fromthe decomposed data. Furthermore, the equivalence between the two types of Gaussian random vibrations
was evaluated by calculating the kurtosis from a single degree of freedom (SDOF) response, assuming thatan SDOF system serves as the packaged product.

2 Gaussian vehicle vibration and Gaussian shaker simulation
2.1 Gaussian vehicle vibration
The vertical acceleration that is exhibited by the vehicle bed of a small van (Fig. 1) traveling on a local
road in Japan was measured with an acceleration sensor (type 4326A, Brüel & Kjær, Denmark) that was
installed on the vehicle bed, as illustrated in Fig. 2. The details of the vehicle specifications and themeasurement conditions are presented in Tables 1 and 2, respectively. Fig. 3 depicts the acceleration datathat was measured on the vehicle bed, which also includes the data that was measured at sections where thevehicle stopped at a traffic signal. To extract the running sections, the sections where the acceleration root
mean square (RMS) was less than 0.4 m/s2 for 1 s (1280 points) were removed, and the remaining sections
were connected to remove the stopping sections. Figs. 4 and 5 depict the zero-set data when the sections
had an acceleration RMS of less than 0.4 m/s2, along with the data for these removed sections, respectively.
Table 3 presents the acceleration RMS and the kurtosis values of the acceleration data that is depicted inFig. 5. Here, a kurtosis is a measure that represents the characteristics of a non-Gaussian distribution.
Kurtosis K can be expressed as follows:
𝟏 𝑵∑𝒊 𝟏 𝒙𝒊 𝒙𝒎 𝟒
𝑵
𝑲
(1)
𝟐
𝟏 𝑵
∑𝒊 𝟏 𝒙𝒊 𝒙𝒎 𝟐
𝑵
where N is the number of data points, 𝑥 is an acceleration value, and 𝑥 is the average of acceleration
values. A Gaussian distribution has a kurtosis of 3. A kurtosis greater than 3 indicates that the distribution
peaks more and has heavier tails than a Gaussian distribution.

－ 244 －

J. Pac .Sci. Tech.

Fig. 1 Test vehicle used in this study

No.4（2021）

Vol.30Fig. 2 Acceleration sensor installedon the vehicle bed

Table 1 Vehicle specifications
Small van (Daihatsu HIJET)
Vehicle type
Total vehicle weight1380 kg
Tread (front)
1305 mm
Tread (rear)
1300 mm
Wheelbase2450 mmTable 2 Measurement conditions
Suburban road
Road type
Travel distance
10 km
0–60 km/h
Speed1280 Hz
Sampling rate
1024
Sampling size
VerticalMeasuring direction
Rear end of vehicle bed (Fig. 2)
Acceleration sensor position

Fig. 3 Acceleration data measured on
the vehicle bed

Fig. 4 Zero-set data for the sections where theacceleration RMS is less than 0.4 m/s2

Fig. 5 Acceleration data after removing the sections that have an acceleration RMS
that is less than 0.4 m/s2

－ 245 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

Table 3 Acceleration RMS and the kurtosis of the acceleration data that is shown in Fig. 5
Kurtosis
Acceleration RMS (m/s2)
1.3
8.2
The acceleration data in Fig. 5 are classified by the conditions of the acceleration RMS that are listed inTable 4 and it is decomposed into seven types of vibrations. Fig. 6 depicts the vibrations that are extracted
by the conditions of the acceleration RMS in Table 4 and the connected vibrations. Table 5 presents theacceleration RMS and kurtosis for the seven types of decomposed vibrations. In Table 5, the kurtosis of"Vibration 3" is 3.1, which is the closest value to three among the seven types of vibrations. Therefore, inthis study, "Vibration 3" is used as the Gaussian random vibration obtained from an actual vehicle.
Table 4 Decomposed conditions of the acceleration data that is shown in Fig. 5
One-second RMS, a (m/s2)
Vibration 1
0.4 ≦ a < 0.6
Vibration 2
0.6 ≦ a < 0.8
Vibration 3
0.8 ≦ a < 1.0
Vibration 4
1.0 ≦ a < 1.5
Vibration 5
1.5 ≦ a < 2.0
Vibration 6
2.0 ≦ a < 3.0
Vibration 7
3.0 ≦ a－ 246 －

J. Pac .Sci. Tech.

Vol.30(a) Extracted data

No.4（2021）(b) Connected data

Fig. 6 Gaussian decomposition of the acceleration data that is shown in Fig. 5

Table 5 Acceleration RMS and the kurtosis of the decomposed vibrations
that are shown in Fig. 6
Kurtosis
Acceleration RMS (m/s2)
Vibration 1
0.5
3.2
Vibration 2
0.7
3.2
Vibration 3
0.9
3.1
Vibration 4
1.2
3.5
Vibration 5
1.7
3.6
Vibration 6
2.3
3.6
Vibration 7
3.7
3.5－ 247 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

2.2 Gaussian shaker simulation
The Gaussian random vibration that was derived using a vibration controller was generated by using thefollowing equations10) so that they have the same PSD as the Gaussian random vibration that is derived
using an actual vehicle.
𝑎 𝑡

𝐴 𝑘 𝑐𝑜𝑠 2𝜋𝑘∆𝑓𝑡
𝐴 𝑘

𝜑 𝑘

2∆𝑓𝑆 𝑘∆𝑓(2)(3)

Acceleration (m/s2)

Fig. 7 Gaussian random vibration that isderived using an actual vehicle

Fig. 8 Gaussian random vibration that isderived using a vibration controller

PSD ((m/s2)2/Hz))

Probalility density function

Acceleration (m/s2)

Here 𝑡 is the time, 𝑛 is the number of data items, ∆𝑓 is the frequency resolution, 𝜑 𝑘 ( 𝑘1, 2, . . . , 𝑛) is the phase, and 𝑆 𝑘∆𝑓 (𝑘1, 2, . . . , 𝑛) is the PSD. The parameters that need to be definedin Equations (2) and (3) are 𝜑 𝑘 and 𝑆 𝑘∆𝑓 . These were respectively set as uniform random numbersthat range from 0 to 2π and the same PSD as "Vibration 3", generating the Gaussian random vibrations. The
generated vibration in this way is used as the Gaussian random vibration that was derived using the vibration
controller.
Figs. 7 and 8 depict the Gaussian random vibrations that are derived using an actual vehicle (“Vibration
3” is shown in Fig. 6) and a vibration controller, respectively. Meanwhile, Figs. 9 and 10 depict the PSDs
and PDFs of the Gaussian random vibrations that are derived using an actual vehicle (Fig. 7) and a vibration
controller (Fig. 8), respectively. It can be observed that both types of vibrations have almost the same PSD
and PDF.

Fig. 9 PSDs of the Gaussian random
vibrations that are derived using an actual
vehicle and a vibration controller

Fig. 10 PDFs of the Gaussian random
vibrations that are derived using an actual
vehicle and a vibration controller－ 248 －

J. Pac .Sci. Tech.

Vol.30No.4（2021）

3 Computational verification for equivalence of two types of Gaussian random
vibration
3.1 Kurtosis response spectrum analysis16)
To verify the equivalence between the Gaussian random vibrations that are derived using an actual vehicle
and a vibration controller, a kurtosis response spectrum analysis was conducted for both types of vibrations.
Here, a kurtosis response spectrum is a plot of the kurtosis of the acceleration response to the baseacceleration input for a series of SDOF systems. This spectrum is a similar concept to a shock response
spectrum17) and an RMS response spectrum18) representing the maximum response and the RMS responseof an SDOF system subjected to a certain waveform, respectively.
Fig. 11 illustrates a conceptual diagram of the kurtosis response spectrum. The horizontal axis represents
the natural frequency of the SDOF system, whereas the vertical axis represents the response kurtosis thatis obtained for each natural frequency. The procedure for obtaining a kurtosis response spectrum subjected
to a certain acceleration waveform is described below.
1. Define the damping ratio ζ or the Q value (Q ≒ 1⁄2 𝜁) of the SDOF system.
2. Define the natural frequency f of the SDOF system.
3. Calculate the response kurtosis K from the response acceleration x of the SDOF system subjected
to a certain acceleration waveform z.
4. Plot the defined natural frequency and the calculated response kurtosis on a graph.
5. Repeat the above procedure for other natural frequencies and connect the plotted points.
In kurtosis response spectrum analysis, the response kurtosis of a series of SDOF systems subjected to acertain acceleration waveform is obtained to determine whether the response kurtosis is amplified in acertain natural frequency bandwidth or not.
Kurtosis
response

Kurtosis2

Kurtosis1

Kurtosisn
f2

f1

𝑧

Kurtosisi

Acceleration response
𝑥1

𝑥2

f1, ζ

fi, ζ

fi

fn

f

𝑥i
𝑥n

f2, ζ

fn, ζ

Spring-mass-damper systems
Acceleration input

Fig. 11 Kurtosis response spectrum concept

3.2 Detailed procedure for calculation of kurtosis response spectrum
As depicted in Fig. 12, the cushioning material is assumed to be placed at the four corners of the packaged
product. The product and the cushioning material are modeled as a rigid body and a spring-damper element,
respectively. In this SDOF system, the motion of the product can be expressed as
𝑚𝑥

𝑐𝑢

𝑘𝑢

0

where 𝑥 is the absolute displacement of the product, 𝑢－ 249 －

(4)𝑥

𝑧 is the relative displacement, 𝑧 is theNon-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

absolute displacement of the vibration table, 𝑚 is the rigid mass, 𝑘 is the spring constant, and 𝑐 is thedamping coefficient. Substituting 𝑥 𝑢 𝑧 into Equation (4) yields
𝑚𝑢

𝑐𝑢

ku

m𝑧

(5)

The spring constant 𝑘 and the damping coefficient 𝑐 can be expressed as
𝑘

𝑚𝜔

(6)

𝑐

2𝑚ζ𝜔

(7)

2𝜋𝑓 is the natural angular frequency,
where 𝑓 is the natural frequency of the packaged product, ωand 𝜁 is the damping ratio. Substituting Equations (6) and (7) into Equation (5) gives𝑢2ζ 2𝜋𝑓 𝑢

2𝜋𝑓

𝑢

𝑧

(8)

The parameters that need to be defined in Equation (8) are the natural frequency 𝑓 , the damping ratio 𝜁,
and the input acceleration 𝑧.
Since this study aims to obtain the kurtosis response spectrum of the Gaussian random vibrations that arederived using an actual vehicle and a vibration controller, the input acceleration 𝒛 was set using the“Vibration 3” (Fig. 7) and the Gaussian random vibrations that are derived using a vibration controller (Fig.
8). The natural frequency was assumed to be within the range of 5–100 Hz and it was set to 5, 6 ..., and 100
Hz with intervals of 1 Hz. A plastic foam cushioning material was assumed to be used as the cushioning
material, and the damping ratio 𝜁 of the packaged product was set to 0.15 based on the damping ratios(0.06–0.29) that were investigated in a previous study. 19)
The natural frequency 𝑓 , the damping ratio 𝜁, and the input acceleration 𝑧 were set in this way, and the
response acceleration 𝑢 and its kurtosis were calculated using the Newmark 𝛽 method, which is anumerical integration method to solve differential equations. In the Newmark 𝛽 method, the value of 𝛽was set as 1∕4, which is an unconditionally stable value regardless of the time interval of the numericalintegration. In addition, the time interval Δ𝑡 was set as 1/20 ms, which is sufficiently small with respect tothe measurement sampling frequency of 1280 Hz since Δ𝑡 in the numerical integration is accompanied bya phase delay when the value is large.
Cushioning material
Response: x

Responsem

Product
Input

k

c

Input: z

Fig. 12 Analytical model of a packaged product3.3 Results
Fig. 13 compares the kurtosis response spectrum of the Gaussian random vibrations that are derived using
an actual vehicle and a vibration controller. It can be observed that the response kurtosis of the Gaussian
random vibrations that are derived using an actual vehicle varies with the natural frequency. For example,
the maximum and minimum response kurtosis of the Gaussian random vibrations are close to four at afrequency of approximately 14 Hz and three at a frequency of approximately 70 Hz, respectively. In contrast,
the response kurtosis of the Gaussian random vibrations that are derived using a vibration controller isalways constant at approximately three regardless of the frequency. Even though the Gaussian random
vibrations that are derived using an actual vehicle and a vibration controller almost have the same PSDs
and PDFs, the kurtosis of the SDOF responses may vary with the natural frequencies of the packaged
product.

－ 250 －

No.4（2021）

Vol.30

Response kurtosis

J. Pac .Sci. Tech.

Fig. 13 Kurtosis response spectrum of the Gaussian random vibrations that are derived
using an actual vehicle and a vibration controller

3.4 Discussion

Acceleration (m/s2)

Acceleration (m/s2)

We further compared the SDOF response to two types of vibrations with natural frequencies of 14 Hz and
70 Hz.
Figs. 14 and 15 depict the SDOF responses (natural frequency = 14 Hz and damping ratio = 0.15) to two
types of Gaussian random vibrations that are derived using an actual vehicle and a vibration controller,
respectively. Figs. 16 and 17 depict the corresponding PSDs and PDFs, respectively. It can be noted that
the SDOF responses to the two types of vibrations exhibit the same PSDs but they have different PDFs.
The PDF of the SDOF response to the Gaussian random vibrations that are derived using a vibration
controller remains Gaussian, whereas the SDOF response to the Gaussian random vibrations that arederived using an actual vehicle varies from Gaussian to non-Gaussian. Figs. 18 and 19 depict the SDOF
responses (natural frequency = 70 Hz and damping ratio = 0.15) to the two types of vibrations, respectively.
Figs. 20 and 21 depict the corresponding PSDs and PDFs, respectively. It can be further noted that the
SDOF responses for the two types of vibrations exhibit the same PSDs and PDFs. In general, as depicted
in Fig. 21, when the vibrations that are derived using an actual vehicle follows the Gaussian distribution,
the SDOF response also has a Gaussian distribution. However, as depicted in Fig. 17, even if the vibrations
that are derived using an actual vehicle follows the Gaussian distribution, the SDOF response does notnecessarily follow the Gaussian distribution. In other words, the PDF of the SDOF response may bedifferent, even for vibrations that have the same PSD and PDF. Therefore, to accurately replicate thevibration environment, it is necessary to consider the PSD and PDF, as well as the PDF of the SDOF
response.

Fig. 14 SDOF response to the Gaussian
random vibrations that are derived using
an actual vehicle; natural frequency = 14
Hz, damping ratio = 0.15

Fig. 15 SDOF response to the Gaussian
random vibrations that are derived using
a vibration controller; natural frequency =14 Hz, damping ratio = 0.15

－ 251 －

Probalility density function

Fig. 17 PDFs of the SDOF response to theGaussian random vibrations that arederived using an actual vehicle and avibration controller; natural frequency = 14
Hz, damping ratio = 0.15
Acceleration (m/s2)

Fig. 16 PSDs of the SDOF response to theGaussian random vibrations that are derived
using an actual vehicle and a vibration
controller; natural frequency = 14 Hz,
damping ratio = 0.15
Acceleration (m/s2)

PSD ((m/s2)2/Hz))

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

Fig. 19 SDOF response to the Gaussian
random vibrations that are derived using
a vibration controller; natural frequency =70 Hz, damping ratio = 0.15

PSD ((m/s2)2/Hz))

Probalility density function

Fig. 18 SDOF response to the Gaussian
random vibrations that are derived using
an actual vehicle; natural frequency = 70
Hz, damping ratio = 0.15

Fig. 20 PSDs of the SDOF response to
the Gaussian random vibrations that arederived using an actual vehicle and avibration controller; natural frequency = 70
Hz, damping ratio = 0.15

Fig. 21 PDFs of the SDOF response to
the Gaussian random vibrations that arederived using an actual vehicle and avibration controller; natural frequency =70 Hz, damping ratio = 0.15

4 Vibration experiments
4.1 Method
In Chapter 3, the difference between two types of Gaussian random vibrations that are derived using an
actual vehicle and a vibration controller was clarified by numerical analysis. It was indicated that the
response kurtosis of the Gaussian random vibrations that are derived using a vibration controller is always
approximately three regardless of the natural frequency, while the response kurtosis of the Gaussian random
vibrations that are derived using an actual vehicle exceeds three depending on the natural frequency. It is
not common that the response kurtosis of the Gaussian random vibrations exceeds three depending on the－ 252 －

J. Pac .Sci. Tech.

Vol.30No.4（2021）

natural frequency. In this chapter, to experimentally verify that the SDOF response of Vibration 3, which is
the Gaussian random vibrations, may not be Gaussian depending on the natural frequency, vibration
experiments are conducted. The experimental setup and the procedure are described below.
Polyethylene foam corner pads (SUNTEC FOAM™, expansion ratio: 45 times, Asahi Kasei Chemicals
Corporation, Japan) were placed at the corners of an acrylic box (11.4 kg), as shown in Figs. 22 and 23.
The acrylic box with the foam corner pads was placed in a corrugated fiberboard box to complete thedummy packaged product. Six types of corner pads with different bearing areas (Fig. 24) and two types ofweights were prepared.
The vibration experiments were conducted under the conditions listed in Table 6. The dummy packaged
product was fixed on top of a vibration table (Fig. 25), and the vibration table was vibrated vertically using
the Gaussian vehicle vibrations shown in Fig. 7. An acceleration sensor was installed at the center of theacrylic box and the vibration table (Figs. 26 and 27), and the vertical acceleration of the acrylic box andvibration table was measured using a transport environment recorder tough logger (TR-1000, IMV
Corporation, Japan), as shown in Fig. 28. The PSD was calculated from the measured acceleration, and thevibration transmissibility of the dummy packaged product was determined using the following equation:
𝑓

𝐻

(9)

where 𝐻𝑓 is the vibration transmissibility obtained in the experiment, and 𝑃
𝑓 and𝑃
𝑓 are the PSDs of the acrylic box and the vibration table, respectively.
The vibration transmissibility of the modeled packaged product (Fig. 12) can be expressed as equation (10)and is defined as a function of natural frequency 𝑓 and damping ratio 𝜁.
𝐻

𝑓

(10)

Here, 𝑓 and 𝜁 were determined as the natural frequency and damping ratio, respectively, that minimize
𝑓 and 𝐻𝑓 via the non-linear optimization
the sum of squared errors between 𝐻
method using the GRG solver built in Microsoft Excel 2016.

Fig. 22 Acrylic box

Fig. 23 Layout of the cushion corner pads

Fig. 24 Dimensions of the cushion corner pads

－ 253 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

StatusCondition 1
Condition 2
Condition 3
Condition 4
Condition 5
Condition 6
Condition 7
Condition 8

Table 6 Experimental conditions
Cushion bearing area (mm) Product weight (kg)
20 × 20
22.9
20 × 2017.1
20 × 20
11.4
30 × 30
11.4
40 × 40
11.4
50 × 50
11.4
60 × 60
11.4
80 × 80
11.4

Static load (MPa)
0.14
0.10
0.070
0.031
0.017
0.011
0.00780.0044

Fig. 25 Dummy packaged productfixed on top of the vibration table

Fig. 26 Acceleration sensor installed on theacrylic box

Fig. 27 Acceleration sensor installedon the vibration table

Fig. 28 Transport environment recorder
tough logger (TR-1000, IMV

Corporation, Japan)

4.2 Results
Fig. 29 shows the acceleration data measured on the vibration table and acrylic box. Fig. 30 shows thevibration transmissibility obtained from the vibration experiments and the vibration transmissibility
determined by the least squares approximation. Table 7 summarizes the natural frequency and damping
ratio determined by the least squares approximation, and the kurtosis and RMS of the acceleration datameasured on the vibration table and acrylic box. Fig. 31 shows the plot for the kurtosis values measured
on the acrylic box, using data summarized in Table 7 and shown in Fig. 13.
As shown in Table 7 and Fig. 31, at the natural frequencies of 15.2 Hz or 19.1 Hz, the response kurtosis
is close to 4, which is much higher than 3. On the other hand, at the natural frequencies of 37.9 Hz or 43.6
Hz, the response kurtosis is close to 3. Thus, the experimental data are in good agreement with the resultsof the kurtosis response spectrum analysis; further, the SDOF response of the Gaussian random vibration
may become a non-Gaussian random vibration, depending on the natural frequency of the packaged product.

－ 254 －

J. Pac .Sci. Tech.

Vol.30No.4（2021）

Condition 1

Condition 2

Condition 3

Condition 4

Condition 5

Condition 6

Condition 7

Condition 8(a) Vibration table(b) Acrylic box

Fig. 29 Acceleration data measured on the vibration table and on the acrylic box
for each experimental condition－ 255 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

Condition 1

Condition 2

Condition 3

Condition 4

Condition 5

Condition 6

Condition 7

Condition 8

Fig. 30 Vibration transmissibility curves for each experimental condition

Table 7 Natural frequency and damping ratio of the packaged product, and the kurtosis
and RMS of the acceleration data measured on the vibration table and acrylic box
Natural
RMS (m/s2)
Kurtosis
Kurtosis
RMS (m/s2)
Damping(Vibration(Acrylicfrequency(Acrylic(Vibration
ratio
box)
box)
table)(Hz)
table)
Condition 1
13.2
0.15
3.13
0.87
3.61
1.03
Condition 2
15.2
0.11
3.13
0.88
3.94
1.07
Condition 3
19.1
0.11
3.12
0.89
3.87
1.02
Condition 4
26.2
0.10
3.11
0.89
3.57
0.88
Condition 5
29.8
0.11
3.15
0.89
3.48
0.86
Condition 6
33.8
0.14
3.12
0.89
3.40
0.87
Condition 7
37.9
0.12
3.13
0.89
3.34
0.90
Condition 8
43.6
0.11
3.14
0.89
3.30
0.96

Fig. 31 Kurtosis response spectrum for different natural frequencies with the experimental data

－ 256 －

J. Pac .Sci. Tech.

Vol.30No.4（2021）

4.3 Discussion
Furthermore, we zoomed into the section where the difference in the SDOF response was remarkable
under each condition, particularly the section from 10 s to 12 s, and we examined the factors that caused
the SDOF response of Gaussian random vibration to be non-Gaussian. Fig. 32 shows the zoomed-in figureof the section from 10 s to 12 s in Fig. 29. The acceleration data measured on the vibration table were
almost the same shape in each experiment, and the input acceleration was the same. In contrast, the response
acceleration on the acrylic box indicates that the vibration surrounded by the dotted line considerably differs
depending on the experimental conditions. In particular, for conditions 1-3, the vibration surrounded by thedotted line is greatly amplified, and the response vibration becomes large. On the contrary, for conditions
4-8, no large amplification is observed. Thus, “Vibration 3” includes the vibration components that trigger
amplification, which causes the SDOF response of the Gaussian random vibration to be non-Gaussian.
In this experiment, the vibration table was vibrated vertically and the vertical acceleration of the acrylic
box was measured. In reality, the lateral acceleration is also considered to occur, but in this experiment,
polyethylene foam corner pads were placed at the corners of an acrylic box, so significant vibration was
not observed in the horizontal direction. Therefore, it is considered that the vertical vibration was dominant,and the dispersion of energy to other than the vertical direction was small.

Condition 1

Condition 2

Condition 3

Condition 4

Condition 5－ 257 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

Condition 6

Condition 7

Condition 8(a) Vibration table(b) Acrylic box

Fig. 32 Zoomed-in figure of the section from 10 s to 12 s in Fig. 29

5 Conclusion
In this study, a kurtosis response spectrum analysis was conducted for the Gaussian random vibrations
that are derived using an actual vehicle and a vibration controller, and the equivalence between these two
types of vibrations was verified. The findings of this study are summarized as follows:
1. The response kurtosis of the Gaussian random vibrations that are derived using a vibration controller
is close to three regardless of the natural frequency. Meanwhile, the Gaussian random vibrations thatare derived using an actual vehicle may be greater than three depending on the natural frequency.
2. Solely the PSD and PDF are not sufficient to completely understand the nature of the vibration, and
the PDFs of the SDOF responses need to be considered.
3. The kurtosis response spectrum is effective as an index to visualize the difference between theGaussian random vibrations that are derived using an actual vehicle and a vibration controller.

References
1)
2)
3)
4)
5)
6)
7)

JIS Z 0200:2020, Packaging - Complete, filled transport packages - General rules for thecompilation of performance test schedules.
ISO 4180:2019, Packaging - Complete, filled transport packages - General rules for the compilationof performance test schedules.
ASTM D4169-16, Standard Practice for Performance Testing of Shipping Containers and Systems.
Rouillard, V. and Sek, M.A. Monitoring and simulating non-stationary vibrations for package
optimization. Packag Technol Sci., 13, 149-156 (2000)
Alexander Steinwolf, W.H. Connon III. Limitations of the Fourier transform for describing test
course profiles. Sound and vibration, February, 12-17 (2005)
Lepine, J., Rouillard, V., and Sek, M. Review Paper on Road Vehicle Vibration Simulation for
Packaging Testing Purposes. Packag Technol Sci., 28, 672- 682 (2015)
Steven R. Winterstein. Nonlinear Vibration Models for Extremes and Fatigue. Journal ofEngineering Mechanics, ASCE, 114, 1772-1790 (1988)－ 258 －

J. Pac .Sci. Tech.

Vol.30No.4（2021）

8)David O. Smallwood. Generating Non-Gaussian Vibration for Testing Purposes. Sound andvibration, October, 18-24 (2005)9) Alexander Steinwolf. Closed-loop shaker simulation of non-Gaussian random vibrations; Part 2
Numerical and experimental results. Test Engineering and Management, 68, 14-19 (2006)
10) Steinwolf A, Ibrahim RA. Numerical and experimental studies of linear systems subjected to nonGaussian random excitation. Probabilist Eng Mech., 14, 289-299 (1999)11) Hosoyama A, Saito K, Nakajima T. Non-Gaussian random vibrations using kurtosis. In Eighteenth
IAPRI World Packaging Conference, DEStech Publications, Inc.: San Louis Obispo (CA), USA,2012.
12) Rouillard V, Sek MA. Synthesizing nonstationary, non‐ Gaussian random vibrations. Packag
Technol Sci., 23, 423-439 (2010)13) Griffiths KR, Hicks BJ, Keogh PS, Shires D. Wavelet analysis to decompose a vibration simulation
signal to improve pre-distribution testing of packaging. Mechanical Systems and Signal Processing,
76-77, 780-795 (2016)
14) Zhou, H., and Wang, Z. W. A New Approach for Road‐Vehicle Vibration Simulation. Packag
Technol Sci., 31, 246-260 (2018)15) Bonnin AS, Nolot JB, Huart V, Pellot J, Krajka N, Odof S, Erre D. Decomposition of the accelerationlevels distribution of a road transport into a sum of weighted Gaussians: Application to vibration
tests. Packag Technol Sci., 31, 511-522 (2018)16) Hosoyama, A, Tsuda, K, Horiguchi, S. Development and validation of kurtosis response spectrum
analysis for antivibration packaging design taking into consideration kurtosis. Packag Technol Sci.,
33, 51– 64 (2020)17) Irvine T. An introduction to the shock response spectrum. Rev P Vibrationdata, 1-73 (2012)18) Irvine T. An Introduction to the Vibration Response Spectrum. Rev D Vibrationdata, 1-20 (2009)19) Kazuaki KAWAGUCHI, Study on equivalence of simulated drop test for packaged freight, PhD
thesis, Kobe University

Acknowledgements
This work was partially supported by JSPS KAKENHI grant number JP18K13964.

（Received 1March 2021）（Accepted 8June 2021）－ 259 －

Non-Gaussian Nature of the SDOF Response to Gaussian Vehicle Vibrations

車両振動から得られるガウス型ランダム振動の一自由度系応答の非ガウス性
細山亮、津田 和城、堀口翔伍

本研究の目的は、実輸送から抽出されたガウス型ランダム振動と現状の振動制御器で再現した
ガウス型ランダム振動の等価性を検証することである。本研究では、包装品を一自由度系システ
ムと想定し、これら 2 種類のガウス型ランダム振動に対する一自由度系応答から尖度を求めることで、両者の等価性について評価を行った。その結果、現状の振動制御器で再現したガウス型
ランダム振動ではその一自由度系応答もガウス型ランダム振動となるのに対し、実輸送から得られたガウス型ランダム振動ではその一自由度系応答は、包装品の固有振動数によって、非ガウ
ス型ランダム振動となることが明らかとなった。本研究結果は、振動台でのパワースペクトル密度と確率密度分布だけでは、完全にその振動の性質を把握することができず、その一自由度系応
答の確率密度分布についても考慮する必要があることを示唆している。
キーワード: ランダム振動、ガウス分布、一自由度系、振動試験、尖度応答スペクトル－ 260 －