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No.4（2022）

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

Measurement and Analysis of Vibrations on the Trailer Bed, with
Particular Attention to Velocity Kurtosis and Kurtosis Response
Spectrum
Daichi NAKAI *, ** and Katsuhiko SAITO *The packaging vibration test in the laboratory is important to confirm the safety against vibration during
distribution. Traditionally, the power spectral density is used as the test conditions. However, the realvibration during distribution is non-Gaussian, and many studies propose the vibration test methods
considering its non-Gaussian. Non-Gaussian is conventionally represented by acceleration kurtosis or the
probability density distribution of acceleration. According to current research, velocity kurtosis and kurtosis
response spectrum are also crucial elements in vibration testing. However, there has been few research on
the velocity kurtosis and there have been only a few researches on the kurtosis response spectrum of a realvibration. This study aims to measure the acceleration on the trailer bed and determine the non-Gaussian of
the genuine vibration. Seven different routes, including highways and nonhighways, were used for driving
tests. As the indicator of non-Gaussian, acceleration kurtosis, velocity kurtosis, kurtosis response spectra,
and so on were calculated. Acceleration and velocity were both leptokurtic, with velocity kurtosis near toacceleration kurtosis. The values of kurtosis response varied with frequency. The contour of the kurtosis
response spectrum became flattered as the damping factor was raised.
Keywords: Vibration Measurement, Acceleration Kurtosis, Velocity Kurtosis, Response spectrum, nonGaussian Distribution

1. Introduction

The packaging vibration test in the laboratory is important to confirm the safety against vibration on thetruck bed or the railways. It is necessary to assess the actual vibration measured on the truck bed to define
the test conditions for the vibration test. The vibration levels in shipping have been studied in the UnitedStates 1, 2), China 3), Japan 4, 5), South-East Asia 2, 6-8), India 9, 10), Europe 11-13), and Brazil 14). The feature
quantities are calculated from the acceleration measured on the truck bed and are used as the vibration test
condition. The Power Spectral Density (PSD) of acceleration is the most important vibration test condition.
A random vibration test is often performed using only a PSD (traditional method). As a result, the PSD androot mean square (RMS) of acceleration, the integral of the PSD, have been reported in the majority ofprevious investigations.
It is pointed out that there are differences of acceleration kurtosis (𝐾 ) and the probability density
distribution of acceleration between the traditional method and the vibration measured on the truck bed 15,
16). The probability density distribution of acceleration in a traditional method is Gaussian, and its
acceleration kurtosis (𝐾 ) is 3. Conversely, the probability density distribution of the real vibration has alarger skirt than the Gaussian distribution, and the real vibration is leptokurtic (𝐾 is greater than 3). The
discrepancies are thought to be caused by the unevenness of the road surface as well as the vehicle’s
acceleration and deceleration. As a result, traditional methods may result in underpackaging. Several studies* Transport Packaging Laboratory, Kobe University, 5-1-1, Fukaeminami, Higashinada, Kobe 658-0022, Japan** Sankyu Inc. 46-51 Sakinohama, Nakabaru, Tobata, Kitakyushu, 804-0002, Japan
Corresponding author: Daichi Nakai. Tel: 093-883-8561 E-mail: d.nakai@sankyu.co.jp－ 227 －

Measurement and Analysis of Vibrations on the Trailer Bed, with Particular Attention
to Velocity Kurtosis and Kurtosis Response Spectrum

have been carried out to improve the vibration test. In addition to PSD, these vibration generation studies
take acceleration kurtosis, the probability density distribution of acceleration, or the RMS distribution into
account 16-21).
Recently, the idea of the kurtosis response spectrum has been proposed 22). The response is calculatedusing input acceleration using a Single-Degree of Freedom (SDOF) model and Fig. 1 shows schematic of
the kurtosis response spectrum. Since the previous studies used the absolute acceleration response and thisstudy uses the acceleration applied inside the SDOF structure (proportional to the relative displacement
response), Fig. 1 is reconstructed from the cited paper. It is assumed that the higher the kurtosis response,
the greater the risk of product damage, even if the PSD and the acceleration kurtosis are the same 23). Thesestudies show that the kurtosis response is not constant and changes depending on natural frequency values
and damping factors 22, 23).
Kurtosis Response

Fig. 1 Schematic of kurtosis response spectrum [based on Hosoyama et al.22)].
Our previous study focused on velocity (integration of acceleration waveform), which is sometimes usedas criteria of impact strength in shock fragility test or vibration strength in earthquake engineering 24, 25).
Moreover, the study proposed a method generating the time series acceleration with arbitrary acceleration
kurtosis and velocity kurtosis (𝐾 ) 26). The closer the velocity kurtosis is to that of the real vibration, thecloser to the kurtosis response spectrum is to that of the real vibration, even if the acceleration kurtosis is
the same as that of the real vibration 27). The vibration with the same 𝐾 as that of the real vibration and
𝐾 is smaller than that of the real vibration, the kurtosis response is significantly smaller than that of thereal vibration at a low natural frequency range (20 Hz or less). However, no decrease in the kurtosis
response spectrum was detected in the low natural frequency range in the vibration with the same 𝐾 and
𝐾 as the real vibration. In addition, our earlier research suggests a strategy for regulating the kurtosis
response spectrum 27). This method is characterized by the fact that the kurtosis is changed for eachfrequency component. Using this method makes it possible to generate vibrations with the sameacceleration and velocity kurtosis but different response spectrum.

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As described above, the studies about vibration test methods for packaging considering non-Gaussian
nature have been advanced. However, there are not many studies of real transport measurements that report
non-Gaussian nature. For example, only some studies reported 𝐾 which is the most major statistic for
non-Gaussian nature 3, 13, 19). Furthermore, the previous studies about kurtosis response spectra measured
on the roads were by a small van only 22, 23, 28). Moreover, no studies that we are aware of have examinedseveral real vibrations about the 𝐾 . As a result, the non-Gaussian nature of genuine vibration remains
unknown.
The aims of this study are as follows:To clarify 𝐾 and the probability density distribution of velocity.

To clarify the kurtosis response spectrum on the trailer bed.

To report 𝐾 and the probability density distribution of acceleration on the trailerbed.

2. Experiment
2.1 Test vehicle
Fig. 2 shows the trailer (maximum load: 13.2 ton, Air
Suspension) used in this study. The surface of the trailer
bed was made of wood. Dummy weights (5.6 tonf) wereeight steel members, each with a size of 7000×300×
300 mm. The dummy weights were placed in front of
the trailer and were placed in two stacks. Fig. 3 shows
the positions of the dummy weights on the truck bed.
Fig. 2 The photo of the trailer anddummy weight.
12,900
9,600
900

300

Trailer Head

Truck Bed

2,400

1,500

1,300

7,000

Accelerometer
Dummy Weights (0.7 tonf×8 = 5.6 tonf)

Fig. 3 Positions of dummy weights and the accelerometer on the truck bed.
2.2 Accelerometer
The accelerometer was DER-1000 (SHINYEI Technology Co., LTD.). The bottom of the DER-1000
was attached to the trailer bed with double-sided tape and was fixed with adhesive tape to prevent theaccelerometer from moving. The accelerometer data logger was attached in the rear center of the trailer.
Fig. 3 shows a schematic of the position where the accelerometer was attached.

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to Velocity Kurtosis and Kurtosis Response Spectrum

The accelerometer was recorded on three axes: lateral, longitudinal, and vertical. The sampling
frequency of the accelerometer was 1,000 Hz. The greatest acceleration was 490 m/s2. There was no
impact or time trigger set, and all acceleration data were recorded throughout driving testing, including
when the car was paused due to traffic signals or the like. After the test was completed, high-frequency
components were removed from the acceleration waveform using a 100 Hz low-pass filter 27).
2.3 Test route
Table 1 shows the test routes and conditions. Experiments were conducted in Tokyo (the largest cityin Japan) and its suburbs. In this study, we ran both highway and nonhighway and analyzed themseparately. Route (a) and (b) are the outbound route, and routes (c) and (d) are the inbound route.
Therefore, routes (c) and (d) are, respectively, in the opposite lane of the route (a) and (b). Times inTable 1 show the analyzed time preprocessed by the method described in Chapter 3.2 with the stopping
section removed. Average speeds in Table 1 were calculated from the Distances and Times in Table 1. The
detailed test routes are shown in Fig. 4 and 5. The lines added on the map show the routes the trailer ran
during driving tests.
Table 1 Test routes and conditions.
Route
(a)
(b)

Departure point

Arrival point

Road

Distance(km)

Time (s)Higashi-Ogishima
Interchange
Chiba-kita
Interchange

Chiba-kita
Interchange

Dummy
Weight(tonf)

Average
Speed(km/h)

Highway

0

57

2,856

72

Yachiyo city

Nonhighway

0

17

1,129

54

Nonhighway

5.6

17

1,176

52

Highway

5.6

57

3,034

68

Highway

5.6

21

861

88

Highway

5.6

53

2,950

65

Nonhighway

5.6

101

10,04036

Rinkai-fukutoshin
Interchange
Tokyo
Interchange

Chiba-kita
Interchange
HigashiOgishima
Interchange
Tokyo
Interchange
Nakai
Parking Area

Gotenba city

Kawasaki city(c)Yachiyo city(d)Chiba-kita
Interchange

(e)
(f)
(g)

5 km

Route (b)Route (c)Route (e)Fig. 4 The test route from (a) to (e) 29).

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5 km

Fig. 5 The test route from (f) to (g) 29).

3. Calculation
3.1 Estimation of velocity
The Fourier transform of velocity 𝑉 𝑓 was calculated by equation (1):, 1
𝑉 𝑓
where, 𝐿 𝑓 , and 𝐴 𝑓 are respectively the low-cut filer, and the Fourier transform of acceleration 30).
𝐿 𝑓 is a third-order Butterworth filter, and the cutoff frequency is 0.5 Hz. The time series velocity 𝑣 𝑡
was calculated from 𝑉 𝑓 using an inverse Fourier transform.
3.2 Moving RMS and the estimation of the stopping section
In the case of nonhighway, there are stop sections such as traffic lights even when the trailer is on
the road. While the time when the trailer is halted is included, the RMS value of the entire vibration is
reduced, and the vibration during transportation may be underestimated. As a result, it is vital todistinguish between the time the trailer is running and the paused sections.
In this study, the moving RMS (vertical direction) was used to determine the stopping section. First,
we divided the vibration in the vertical axis direction into 0.5 second intervals. The RMS value wasthen determined for each 0.5 second. Segments with low RMS values in the vertical axis direction wereconsidered halted and were eliminated from the study in all directions: lateral, longitudinal, and vertical.
As a stop segment, a value of 0.08 m/s2 was set. This is due to the probability density distribution ofmoving RMS having a local minimum value of about 0.08 m/s2, as explained in Chapter 4.
3.3 Response spectrum
The same way calculated the response spectrum as in our previous study 26, 27). A SDOF system as shownin Fig. 1 is expressed by equation (2):
𝑚

𝑎 𝑡

𝑐

𝑘𝑥 𝑡

0,

2

where 𝑡, m, 𝑐, 𝑘, 𝑎 𝑡 and 𝑥 𝑡 respectively, time, the mass, viscosity coefficient, spring constant, thetime series acceleration on the truck bed, and the relative displacement between the mass and the trailer
bed. An impulse response function of the relative displacement 𝑓 𝑡 is expressed by equation (3):－ 231 －

Measurement and Analysis of Vibrations on the Trailer Bed, with Particular Attention
to Velocity Kurtosis and Kurtosis Response Spectrum

𝑓 𝑡𝑒𝑠𝑖𝑛𝜔 𝑡,3where 𝜔 , 𝜔 , and ℎ are, respectively, undamped natural angular frequency, and damped natural angularfrequency of the response and the damping factor. In this system, a natural angular frequency 𝜔 equals
to 𝑘⁄𝑚 and a natural frequency 𝑓 is 𝜔 ⁄2𝜋. The damping factor, ℎ is expressed by equation (4):ℎ.
4The damped natural angular frequency, 𝜔 is expressed by equation (5):𝜔
𝜔 1 ℎ .
5𝑥 𝑡 is expressed by the equation (6):6𝑥 𝑡𝑎 𝜏 𝑓 𝑡 𝜏 𝑑𝜏,
where 𝜏 is a parameter. Equation (6) is a convolution integral; therefore, 𝑥 𝑡 can be calculated byequation (7):𝑥 𝑡
𝐴 𝑓 𝐹 𝑓 𝑒𝑑𝑓,
7
where 𝐴 𝑓 and 𝐹 𝑓 are the Fourier transform of 𝑎 𝑡 and 𝑓 𝑡 , respectively. The acceleration
applied inside the SDOF structure 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 0.5 Hz. The RMS and kurtosis of 𝑎 𝑡 calculated for eachnatural frequency 𝑓 by fixing the damping factor with ℎ, are the RMS response spectrum 𝑅 𝑓 andthe kurtosis response spectrum 𝐾 𝑓 .

4. Results and Discussion

Fig. 6 shows the probability density distribution of moving RMS on the vertical axis. The dotted line
represents the threshold of moving RMS. Fig. 6 (a), (d), (e), and (f) were the results when the trailer ran on
the highway. There are no traffic signals because of driving on highways, and few sections with RMS values
below 0.08 m/s2 are considered stop sections. Moreover, Fig. 6 (b), (c), and (g) are the results when the
trailer ran on nonhighway. The local minimum of the probability density distribution of the moving RMS
was recorded around 0.08 m/s2 for all the nonhighway routes. Therefore, it is statistically valid to set thethreshold value at 0.08 m/s2 in this study.

Fig. 6 The probability density distribution of moving RMS (Vertical Axis).

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Table 2 displays the acceleration and velocity statistics that were preprocessed using the method outlined
in Chapter 3.2. The RMS levels of acceleration and velocity in the longitudinal direction were the lowestof all directions. The lateral direction has the highest acceleration RMS levels of all directions. The verticaldirection has the greatest velocity RMS levels of all directions. Both skewnesses of acceleration andvelocity were close to 0, and no clear asymmetry was observed. 𝐾 and 𝐾 were leptokurtic and muchlarger than 3 for all values. For example, Zhou et al. have reported the 𝐾 between 3.94-24.3 3). Therefore,
the results of the 𝐾 in this study are within the range reported in the previous studies. On the other hand,since there are few previous studies about 𝐾 , it is difficult to compare with previous studies. But it isreasonable that 𝐾 are also leptokurtic because 𝐾 are leptokurtic.
Routes (a) and (d) are outward and inward on the same route, although there is a difference in the presenceof the dummy weights or not. Since there was no clear difference between these two tests, it is assumed
that the vibration levels in this test mainly depended on the road surface condition and driving speed rather
than the presence of the dummy weights.
The difference between highway and nonhighway was not clear in terms of RMS levels, as the smallest
RMS level for vertical acceleration was Route (e) [highway], and the largest RMS level for verticalacceleration was route (a) [highway]. 𝐾 in the vertical direction for nonhighway [Routes (b), (c) and (g)]was larger than that for highway [Routes (a), (d), (e) and (f)]. Further, there was no clear difference between
𝐾 in the vertical direction of the nonhighway and that of the highway. There were different trends between
the acceleration and velocity statistics because the vibration in the low-frequency region has a larger effecton the velocity than the acceleration.
Table 2 Statistical Values of Vibrations
Stats
RMS
(m/s2)

Acceleration

Skewness

Kurtosis

RMS
(m/s)

Velocity

Skewness

Kurtosis

Direction

Route
(a)

Route
(b)

Route
(c)

Route
(d)

Route
(e)

Route
(f)

Route
(g)

Lateral

0.99

0.74

0.62

0.91

0.52

0.89

0.50

Longitudinal

0.18

0.15

0.16

0.19

0.14

0.19

0.15

Vertical

0.63

0.49

0.51

0.60

0.42

0.60

0.51

Lateral

−0.01

0.06

−0.10

−0.07

0.01

0.09

0.01

Longitudinal

0.04

−0.01

0.02

0.07

0.00

−0.03

−0.03

Vertical

−0.02

0.11

0.12

0.02

−0.01

0.03

0.05

Lateral

12.99

12.35

13.36

12.93

8.75

16.99

14.53

Longitudinal
Vertical

9.20
8.77

9.81
10.97

8.44
12.38

9.86
8.89

6.64
10.25

7.59
10.05

14.69
12.14

Lateral0.01270.01020.00830.01100.00630.01060.0065Longitudinal0.00160.00130.00130.00150.00100.00130.0011

Vertical

0.01800.01140.01110.01500.00950.01450.0109Lateral

0.08

0.03

0.16

0.03

−0.01

0.13

−0.04

Longitudinal

0.13

0.00

−0.01

0.07

−0.17

−0.03

−0.01

Vertical

0.08

0.03

0.16

0.03

−0.01

0.13

−0.04

Lateral

11.05

11.34

12.87

11.08

8.65

15.01

10.47

Longitudinal

11.04

6.25

8.30

7.69

8.47

8.73

12.83

Vertical

9.18

8.42

9.03

8.65

10.28

9.13

10.02

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to Velocity Kurtosis and Kurtosis Response Spectrum

Fig. 7 shows the PSDs of acceleration. In the lateral axis, there was a peak around 15 Hz. In thelongitudinal axis, there was almost no vibration component below 10 Hz. In the vertical axis, there werepeaks at 5 Hz and 12 Hz. 12 Hz was assumed to be the peak caused by the vibration of the tires 31). In
general, the natural frequency caused by suspension and tire vibration decreases with cargo loading.
However, no discernible difference was seen in this investigation between the routes (a) and (b) without thedummy weight and the routes (d) and (c) with the dummy weight.

Fig. 7 The power spectral densities of acceleration.
The probability density distributions of acceleration are depicted in Fig. 8. The probability density
distribution of acceleration in the lateral direction is substantially wider than that in the vertical direction inroutes (b), (d), and (f). However, in the routes (e) and (g), there was no discernible variation between theprobability density distributions in the vertical and lateral directions.

Fig. 8 The probability density distributions of acceleration.

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Fig. 9 shows the probability density distributions of velocity. The spread of the probability density
distribution was narrower for the longitudinal axis than for the other orientations. Unlike the probability
density distribution of acceleration, the vertical axis had the greatest spread in the probability density
distribution of velocity.

Fig. 9 The probability density distributions of velocity.
Fig. 10 shows the 𝑅 𝑓 of vertical direction. The local maximum of the 𝑅 𝑓 for most of the routes
was around 12 Hz. As ℎ increased, the 𝑅 𝑓 values decreased, and the effect of natural frequency on
the 𝑅 𝑓 became smaller.
Fig. 11 shows the 𝐾 𝑓 of the vertical direction. All 𝐾 𝑓 values are above 3, and the response
vibrations were leptokurtic. Previous studies which used the small van have also reported that the absolute
acceleration response is leptokurtic 22, 28). Therefore, the results of 𝐾 𝑓 in this study are also assumed
to be valid. The local maximum of 𝐾 𝑓 around 12 Hz became smaller as ℎ became larger. In the caseof the natural frequency values where 𝐾 𝑓 was smaller than the surrounding frequencies, 𝐾 𝑓
became larger as ℎ became larger. As a result, the effect of the natural frequency on 𝐾 𝑓 became
smaller as ℎ increased. Previous study which used the small van has also reported the same effect ofnatural frequency and ℎ 22).
Comparing Fig. 10 and Fig. 11, some of the natural frequencies of the peak of 𝑅 𝑓 and the peak of
the natural frequency of 𝐾 𝑓 coincided with each other. The most obvious oscillation is the tire-induced
peak around 11-13 Hz; the local maximum values around 11-13 Hz of 𝐾 𝑓 for the nonhighway [Routes(c) and (g)] are 13.0 and 12.5 Hz, respectively. In the case of highway [Routes (d), (e) and (f)], the peaks
around 11-13 Hz of 𝑅 𝑓 and 𝐾 𝑓 are 11.5, 12.5, and 13.0 Hz, respectively. There were also cases
where 𝐾 𝑓 was a local minimum at the natural frequency where 𝐾 𝑓 was a peak. For example, inroute (f), 𝐾 𝑓 reached a peak around 87.0 Hz (Fig. 10), and 𝐾 𝑓 reached a local minimum value
around 88.0 Hz (Fig. 11).
On the other hand, some of the natural frequencies of the peak of 𝑅 𝑓 and the peak of the natural
frequency of 𝐾 𝑓 did not coincided with each other. For example, in the case of routes (a) and (b), theRMS response spectra have the peaks around 12.0 Hz as shown in Fig. 10(a) and (b), but there were no
local maxima or minima around 12.0 Hz for the kurtosis response spectrum as shown in Fig. 11(a) and (b).
Although routes (a) and (b) were without dummy weights, it is not possible to determine whether dummy
weights have the effect on 𝐾 𝑓 because of the insufficient number of the experiments in this study.

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Measurement and Analysis of Vibrations on the Trailer Bed, with Particular Attention
to Velocity Kurtosis and Kurtosis Response Spectrum(A) ℎFig. 10 The RMS response spectra (vertical axis),0.04, (B) ℎ 0.08, (C) ℎ 0.12, (D) ℎ 0.16, (E) ℎ

0.20.

(A) ℎ

Fig. 11 The Kurtosis response spectra (vertical axis),0.04, (B) ℎ 0.08, (C) ℎ 0.12, (D) ℎ 0.16, (E) ℎ

0.20.

5. Conclusions

In this study, the vibrations were measured on the trailer bed and analyzed. The study concludes thefollowing:
Acceleration and velocity were both leptokurtic. The velocity kurtosis was nearly
identical to the acceleration kurtosis.
The probability density distribution of acceleration tended to be widest in the lateral
direction. On the other hand, the probability density distribution of velocity tended to be
widest in the vertical direction. This is because the vibration in the low-frequency region
has a larger effect on the velocity than the acceleration.

The values of kurtosis response were leptokurtic and varied with frequency. This
result agrees with the previous studies by the small van.

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As the damping factor increased, the shape of the kurtosis response spectrum became
flatter. This result agrees with the previous study by the small van.
Some frequencies at which the local maximum or minimum values of the kurtosis
response spectrum coincided with the frequencies at which the maximum local valuesof the RMS response spectrum were.

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Measurement and Analysis of Vibrations on the Trailer Bed, with Particular Attention
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（Received 13January 2022）（Accepted 10June 2022）－ 238 －

J. Pac .Sci. Tech.

Vol.31

No.4（2022）

速度の尖度と尖度応答スペクトルに注目した
トレーラー荷台振動の測定と分析
中 井太 地 *, **,

斎 藤

勝 彦*

包装貨物振動試験は、流通時の振動に対する安全性を確認するために実施される。一般的に試験条件としてパワースペクトル密度が使用されている。それに加えて非ガウス性を考慮した
振動試験方法が研究されおり、非ガウス性は加速度の尖度で評価されている。近年、速度の尖度と尖度応答スペクトルも振動試験の重要な要素であることが示されているが、これらの指標の実振動計測事例はほとんど存在しない。実振動の加速度を測定し、非ガウス性を評価することが本研究の目的である。輸送試験は、高速道路と一般道を含む 7 つのルートで行った。非ガ
ウス性の指標として、加速度の尖度、速度の尖度、尖度応答スペクトルなどで評価した。加速度と速度の尖度は、ガウス分布の値である 3 より大きく、両方の値は近かった。尖度応答の値
は周波数に応じて変化し、減衰係数が大きくなるに従い、尖度応答スペクトルの形状は平坦になった。
キーワード: 振動測定, 加速度の尖度, 速度の尖度, 応答スペクトル, 非ガウス分布－ 239 －

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