The Statistical Distribution of the Root Mean Square of Road Vehicle
Vibrations
Manuel-Alfredo Garcia-Romeu-Martinez1, Vincent Rouillard2, Vicente-Agustin Cloquell-Ballester3
1
ITENE, Instituto Tecnológico del Embalaje, Transporte y Logística
Polígono Industrial D’Obradors, C/Soguers 2, 46110 Godella – Valencia, Spain
[email protected]
2
Victoria University, Melbourne Australia
PO Box 14428 MCMC, Melbourne 8001, Australia
[email protected]
3
Valencia University of Technology, Valencia, Spain
Camino de Vera s/n, 46021, Valencia, Spain
[email protected]
Abstract
This paper presents the initial results of a study aimed at improving the method by which the vibrations produced by
transport vehicles are characterised and simulated. More specifically, this paper focuses on the rigid body vibrations
generated by road transport vehicles in the context of distribution of packaged goods and produce. The research uses a
variety of vibration data, collected from various vehicle types and routes in Spain and Australia with high-capacity vibration
recorders. Vehicles used range from small transport vehicles to large truck-trailers with both airbags and steel spring
suspensions while the routes travelled include suburban streets, main roads and motorways. The paper discusses the
significance and limitations of the average power spectral density (PSD) and explains why the average PSD is not always
adequate as the sole descriptor of road vehicle vibrations as the process generally tends to be non-stationary and nonGaussian. The paper adopts an alternative analysis method, based on the statistical distribution of the moving root-meansquare (RMS) vibrations, as a supplementary indicator of overall ride quality. The measured data was used to compute
the statistical distribution of each vibrations record, the shape of which was compared for the entire set of records. The
suitability of various mathematical models, based on the Weibull and Rayleigh distributions were investigated for
describing the probability distribution function (PDF) of road vehicle vibrations. The paper proposes a single mathematical
model that can accurately describe the statistical character of the random vibrations generated by road vehicles in
general. It shows that the model can also effectively describe the statistical parameters of the process namely the mean,
median, standard deviation, skewness and kurtosis.
Introduction
In order to develop optimum packaging it is important that engineers are not only aware, but have a thorough
understanding of the expected mechanical hazards to which packages are subjected during shipping and handling. This
information allows them to engineer the optimum amount of protective packaging needed to suitably protect the
consignment against the risk of damage. To assist designers in reducing cost, either by avoiding wasting packaging
materials due to over-packaging or avoiding damage due to under-packaging, distribution vibrations need to be simulated
in the laboratory in order to test and validate protective package designs. Because verification of design by trial shipments
have been shown to be both impractical and inadequate [1], performance testing of packaging systems in the laboratory
has become increasingly the more adopted tool in the optimisation of package designs. Testing of package designs under
controlled laboratory conditions usually involves the simulation of vibrations expected to be encountered during
transportation. Assumptions regarding the nature and level of vibrations are sometimes adopted and make it difficult to
optimise protective packaging without experimental verification. These unsophisticated and approximate simulation
methodologies promote the adoption of a conservative approach to packaging design which, in many cases, lead to over
packaging.
Vibrations that occur in vehicles during transportation are complex and play a significant role in the level of damage
experienced by products during shipment. Vehicle vibrations have a random nature and their character and level is
dependent on the type of vehicle, suspension type, payload, vehicle speed and road condition. Because of these
variabilities, it is not always possible to represent transport vibrations with a simple function such as the power spectral
density (PSD) function. With the advent of sophisticated vibration recorders in the past decade, packaging engineers have
been able to measure and analyze increasing volumes of vibrations that occur in commercial shipments. Recently,
numerous studies have been undertaken with the aim of measuring and evaluating the vibrations in various distribution
environments around the globe and using particular vehicle types to enable packaging engineers to develop packaging
solutions to meet world-wide distribution challenges [2][3][4][5][6]. The main purpose of these exercises is to generate
effective laboratory test schedules for evaluating the performance of package systems when subjected to vehicle
vibrations during distribution. Unfortunately, the prevailing trend is to characterise these complex vibrations with a single
function namely, the average power spectral density (PSD). The now well-established and widely adopted procedure for
the laboratory simulation of vehicle vibrations is to synthesize random vibrations from the average PSD of measured
vibration data (which is assumed to wholly describe the transport environment) using random vibration controllers. These
controllers synthesize normally-distributed vibrations by continually computing the Inverse Fourier Transform of the PSD
coupled with a uniformly distributed random phase array. Unfortunately, by virtue of the fact that these systems use solely
the average PSD to synthesize a normally-distributed random signal, the simulated vibrations turn out to be stationary
hence not capable of emulating the excursions in vibration amplitude which are found to occur in the field. One interesting
characteristic of road vehicle vibrations is that the shape of the PSD remains largely unchanged for the duration of each
transport event [6]. In effect, the non-stationarity of the process is manifested through fluctuations in RMS level [7]. Most
vibration controllers can be programmed so that the RMS level of the synthesised random signal is made to vary as a
function of time. However, there is no established technique to determine how this modulation of amplitude should be
implemented.
The main objective of this paper is to establish whether a single mathematical model can be used to describe the
statistical distribution of the moving RMS of vibrations generated by road vehicles in general, and whether the model can
be used to characterise the overall ride quality as well as be of use in determining laboratory test schedules that include
the generation of random vibrations of varying RMS levels.
Modelling the RMS distribution
Rouillard & Sek [8], studied the non-stationary behaviour of road vehicle vibrations and proposed a statistical model for
characterising what they term the “vibration intensity”. Their model is a modified version of the Rayleigh distribution that
includes an exponent parameter and a scale parameter. Their model applied to the vibration intensity which can only be
computed by an elaborated algorithm based on the Hilbert transform. Further work aimed at using the RMS distribution of
vehicle vibrations to design laboratory tests schedules was undertaken by Rouillard & Sek [7]. This work shows how more
realistic vibrations can be synthesized by recognizing that road vehicle vibrations are non-stationary and by making use of
the RMS distribution. One of the most elementary approaches to characterising non-stationarities is to compute the RMS
(or mean-square) of the vibration record over relatively short segments [9]. The length of the segments and the
incremental step for computing the moving RMS are critical to the analysis. The moving RMS of a function x(t) can be
written as:
x̂i ( t ) =
1 i+w 2
∑ x ( j)
n j =i
for i = 0, δ ,2δ ,3δ ......N δ
(1)
Where w is the segment length, δ is the incremental step and N is the total number of segments in the sample.
The effects of the window width and the incremental step (overlap) on the moving RMS of non-stationary vibration signals
have been illustrated by Rouillard [10]. It shows that care must be taken in selecting the parameters for computing the
RMS time history of non-stationary signals.
In order to validate the proposed model, a number of sample vibration records were collected from a wide range of
vehicles and routes. The vibration data were collected using self-contained data recorders (Saver® by Lansmont)
configured to record vibrations for predetermined sub-record lengths of 8 seconds at a sampling rate of 1024 Hz. The
recorders were configured to initiate recording at specific periods varying from 9 seconds to one minute. A total of thirteen
measurements were undertaken using various vehicles including small utility, vans, rigid trucks and semi-trailers with
various suspension types and payloads. Routes included poorly maintained local roads, country roads, urban roads, and
highways located in Victoria, Australia and Spain. The RMS time history of each vibration record was computed using (1)
with w = 8 seconds and no overlap (δ = w + 1/fs, where fs = sampling frequency in Hz). A typical example of a vibration
record along with the moving RMS is shown in Fig. 1. It also includes a plot the moving crest factor which indicates the
non-stationary character of the process; for a Gaussian process of 8192 samples, the likelihood that the crest factor
exceeds 3.65 is 0.012%. This is a strong indication that the data recorded are non-Gaussian and non-stationary [10].
The Probability Density Function (PDF) of the RMS time history of each of the thirteen vibration records was computed
with the aim of developing a generic mathematical model that can be used to characterise the statistical characteristics of
the process regardless of vehicle type, payload or route.
A range of statistical distributions were studied and a model given in (3) was developed, based on the three-parameter
Weibull distribution given in (2), of which the Rayleigh and Gaussian distributions are special cases.
γ  x − x0 
P ( x) = 
α  α 
γ −1
γ
 x − x0 
−

⋅e  α 
∀ x ≥ x0
(2)
6
10
4
5
2
0
0
-5
-2
-10
crest factor
2
a(m/s )
15
-4
RMS
Crest Factor
-15
-6
0
30
60
90
120
150
180
event number
210
240
270
300
330
Figure 1. Typical road vehicle vibration record along with the RMS and crest factor time histories.
The proposed modified Weibull distribution model was developed to afford additional control on various aspects of the
shape of the distribution function. It includes an exponent parameter, β, which enable the control of the slope of the righthand tail of the distribution and increases the scope of the model for characterising a wider range of distribution functions.
The model, given in (3), was found to be generic enough to be able to produce a range of well-known distributions for
which the parameters are given in Table 1.
∀x ∈ ]−∞, xi // xi ≥ x0 ]

0

β
γ −1 − x − x0 



−
x
x
β


α
0


P ( x) = 
 ⋅e
β 


 xi − x0    α 
γ
α ⋅ Γ  β , 
 
 α  


(3)
∀x ∈ [ xi // xi ≥ x0 , +∞[
Where x is the moving RMS, α, β, γ and x0 are the modified Weibull parameters and xi is the left hand domain limit.
Distribution
Rayleigh
xi
α
β
0
σ 2
2
γ
x0
Γ  γ β , xi − xo 
2
0
1
0
Γ [k 2]
k
Chi-Square
0
2
1
Exponential
0
1
1
1
0
1
Gamma
0
1
k
0
Γ[k]
Pearson Type III
α
1
p
α
Γ [ p]
Weibull two-parameters
Weibull three-parameters
0
0
γ
γ
γ
γ
0
x0
1
1
λ
θ
β
α
α
2
Table 1. Parameters values for typical distributions.
The single-parameter statistics for the model, namely, the mean, µ, the median, mdn, the standard deviation, σ, the
skewness, ν (Sk) and the kurtosis, Kt, were derived and are given as:
Mean:
Median:
Std. Dev:
µ = x0 + αΨ [1]
(4)
β
β

 mdn − x0   1  γ  xi − x0  
,
Γ γ β ,
=
⋅
Γ


β 

 
α

  2 
 α  

( )
where E ( x ) = α
σ = E x2 − µ 2
2
(5)
(6)
2
2
Ψ [ ] + 2 x0 µ − x02
Skewness:
ν = Sk =
( )
where E x
Kurtosis:
( )
3
(7)
3
2
= α Ψ [ ] + 3 x0α 2 Ψ [ ] + 3 x02 µ − 2 x03
3
( )
where E ( x ) = α
Kt =
( )
1 
E x 3 − 3µ ⋅ E x 2 − 2 µ 3 

σ3 
( )
( )
1 
E x 4 − 4 µ ⋅ E x 3 + 6 µ 2 ⋅ E x 2 − 3µ 4 
4 

σ
4
4
(8)
4
3
2
Ψ [ ] + 4 x0α 3 Ψ [ ] + 6 x02α 2 Ψ [ ] + 4 x03 µ − 3 x04
β

 x − xo  
Γ γ + j β ,  i
 
 α  

j
and Ψ [ ] = 
β

 x − xo  
Γ γ β , i
 
 α  

In the case of RMS distribution, the left hand domain limit, xi, was chosen as greater than zero since the RMS time history
is, by definition, always positive. For the purpose of this study, in which only rigid body vibrations are of interest, xi = xo.
This has the effect of discounting the sustained, residual low level vibrations that are not caused by road – pavement
interactions [10]. Therefore (3) can be written as follows, to characterise the moving RMS PDF of road vehicle vibrations:
∀x ∈ ]−∞, x0 [

0

β
γ −1 − x − x0 
P ( x) = 
 α 
β
 x − x0 




 ⋅e
γ
 α ⋅ Γ  β , 0   α 
(9)
∀x ∈ [ x0 , +∞[
The influence of each of the four parameters on the shape of the distribution function are illustrated in Fig. 2 which shows
that each parameter alters different aspects of the distribution shape.
(Influence of α parameter)
1.2
(Influence of β parameter)
1.2
1.0
0.8
0.8
0.6
P(x)
P(x)
α = 0.75
1.0
0.4
0.6
0.4
α = 3.2
0.2
0.2
0.0
0
1
1.2
1.0
2(Influence3of
4
γ parameter)
5
β = 15
0.0
7
1.2
6
x
(Influence of x o parameter)
1.0
γ = 0.75
0.8
0.8
0.6
P(x)
P(x)
β = 0.75
γ = 15
0.4
0.6
X0 = 0
X0 = 2.25
0.4
0.2
0.2
0.0
0
1
2
3
4
x
5
6
0.0
7 0
1
2
3
4
5
6
7
x
Figure 2. Influence of parameters on the proposed four-parameter modified Weibull distribution.
Further analyses, undertaken to investigate the cross-correlation between the parameters, showed that there is no
significant inter-parameter dependence.
Results
A computer program (coded in Matlab®) was developed to determine the optimum parameter values that yield the best fit
for the proposed model with respect to the PDF of measured vibration data. Results using the sum-of-squared error (least
squares) optimisation were found to produce unstable results. This was attributed to the relatively large number (four) of
independent parameters which was found to achieve least square errors for several combinations of parameter values. In
order to address this difficulty, the code was modified to include optimisation based on the mean and standard deviation of
the errors between the fitted and measured data for the five statistics parameters namely the mean, median, standard
deviation, skewness, and kurtosis.
n=0
max_iter
k, ε, error_min
S
n=n+1
Matlab function to fit an equation to a data
by least squares optimisation
Inline ( ' Eq _ Model ', ' [α , β , γ , xo ] ', '[ Data to fit ]')
Random initial conditions
α i = ε + 1⋅ rand (1)
β i = ε + 1 ⋅ rand (1)
γ i = ε + 2 ⋅ rand (1)
False
[ Data to fit ]
xoi = 0 + 1 ⋅ rand (1)
Least squares fit
[α , β , γ , xo ]
α , β , γ > 0 and x0 ≥ 0 ?
True
E
Calculate the error of the statistic parameters
Best fit
[α , β , γ , xo ]
1 5 model_spi − data_spi
mean _ error = ⋅ ∑
5 i =1
data_spi
error < error _ min ?
True
std _ error =
5
∑
( model_spi − mean _ error )
i =1
5
error = mean _ error + std _ error
n < max_iter ?
True
False
Calculate statistic parameters
for the model
model_sp1 = model _ mean
2
model_sp2 = model _ median
model_sp3 = model _ std
model_sp4 = model _ Sk
model_sp5 = model _ Kt
error _ min = k ⋅ error _ min
n=0
Figure 3. Algorithm for optimisation of the fit based on the errors of the five statistics parameters.
This curve fitting algorithm was used to subject the proposed four-parameter modified Weibull model to validation tests
using all thirteen vibration records (Table 2) and was found to offer good agreement as shown in Fig. 4 which shows six
typical examples.
Record ID
DATA A
DATA B
DATA C
DATA D
DATA E
DATA F
DATA G
DATA H
DATA J
DATA K
DATA L
DATA M
DATA N
DATA O
Vehicle type & load
Utility vehicle (1 Tonne cap.). Load: < 5% cap.
Prime mover + Semi trailer (Air ride susp.). Load: 90% cap.
Transport van (700 kg cap.). Load: 60% cap.
Transport van (700 kg cap.). Load: 60% cap.
Transport van (700 kg cap.). Load: 60% cap.
Prime mover + Semi trailer (Leaf spring susp.). Load: < 5% cap.
Tipper truck (16 Tonnes cap., Air ride susp.). Load: 25% capacity.
Small flat bet truck (1 Tonne cap., Leaf spring susp.). Load <5% cap.
Flat bed truck (5 Tonnes cap., Leaf spring susp.). Load >95% cap.
Sedan car. Load: 1 passenger
Prime mover + Semi trailer (Air ride susp.). Load: 60% cap.
Prime mover + Semi trailer (Air ride susp.). Load: 20% cap.
Prime mover + Semi trailer (Leaf spring susp.). Load: 10% cap.
Prime mover + Semi trailer (Leaf spring susp.). Load: < 1% cap.
Country
Australia
Australia
Australia
Australia
Australia
Australia
Australia
Australia
Australia
Australia
Spain
Spain
Spain
Spain
Table 2. Summary of measured vibration record parameters and conditions.
Route Type
Suburban streets
Country roads
Suburban streets
Main suburban hwy
Motorway
Country roads
Country roads
Suburban streets
Country roads
Suburban streets
Motorway
Motorway
Motorway
Motorway
Data B
(Air Ride Truck - Load - Local Roads)
0.35
µ
Mdn
σ
Sk
Kt
µ/σ
0.30
1.4
Data / Fit: 3.09 / 3.15
Data / Fit: 2.83 / 2.89
Data / Fit: 1.46 / 1.64
Data / Fit: 0.54 / 0.96
Data / Fit: 2.73 / 4.30
Data / Fit: 0.47 / 0.52
α = 1.432
β = 1.183
0.20
γ = 3.035
x o = 0.051
0.15
1.0
γ = 8.026
x o = 0.004
0.6
0.4
0.05
0.2
Data / Fit: 1.29 / 1.29
Data / Fit: 1.27 / 1.25
Data / Fit: 0.46 / 0.42
Data / Fit: 0.58 / 0.59
Data / Fit: 4.19 / 3.48
Data / Fit: 0.36 / 0.32
α = 0.262
β = 1.190
0.8
0.10
0.00
0.0
0
1
2
3
4
5
6
2
RMS (m/s )
7
8
9
10
Data F
(Leaf Spring Truck - No Load - Local Roads)
0.20
µ
Mdn
σ
Sk
Kt
µ/σ
0.18
0.16
0.14
0.12
α = 1.789
β = 1.182
0.10
γ = 6.489
x o = 0.008
0.08
0.0
0.5
1.0
1.5
2.0
2
RMS (m/s )
µ
Mdn
σ
Sk
Kt
µ/σ
1.2
1.0
0.06
2.5
3.0
3.5
Data L
(Air Ride - Load - Highway)
1.4
Data / Fit: 7.72 / 7.48
Data / Fit: 7.27 / 7.197
Data / Fit: 2.82 / 2.70
Data / Fit: 0.68 / 0.66
Data / Fit: 3.10 / 3.61
Data / Fit: 0.37 / 0.36
P(RMS)
P(RMS)
µ
Mdn
σ
Sk
Kt
µ/σ
1.2
P(RMS)
P(RMS)
0.25
Data E
(Van - Highway)
Data / Fit: 1.38 / 1.43
Data / Fit: 1.33 / 1.36
Data / Fit: 0.38 / 0.37
Data / Fit: 1.03 / 1.10
Data / Fit: 5.03 / 4.68
Data / Fit: 0.27 / 0.26
α = 0.364
β = 1.198
0.8
γ = 2.287
x o = 0.829
0.6
0.4
0.04
0.2
0.02
0.00
0.0
0
2
4
6
8
10
2
RMS (m/s )
14
16
18
Data N
(Leaf Spring - Load - Highway)
0.8
0.6
0.5
Data / Fit: 2.59 / 2.70
Data / Fit: 2.52 / 2.59
Data / Fit: 0.71 / 0.77
Data / Fit: 1.03 / 1.15
Data / Fit: 6.51 / 5.29
Data / Fit: 0.27 / 0.25
1.0
1.5
2.0
2
RMS (m/s )
µ
Mdn
σ
Sk
Kt
µ/σ
0.30
0.1
0.05
0.0
3.5
Data / Fit: 3.73 / 4.13
Data / Fit: 3.27 / 3.54
Data / Fit: 1.99 / 2.12
Data / Fit: 2.75 / 2.50
Data / Fit: 15.15 / 14.70
Data / Fit: 0.54 / 0.51
γ = 4.460
x o = 1.665
0.15
0.10
3.0
α = 0.001
β = 0.345
0.20
0.2
2.5
Data O
(Leaf Spring - No Load - Highway)
0.35
P(RMS)
γ = 11.6
x o = 1.187
0.3
0.5
0.25
α = 0.001
β = 0.452
0.4
0.0
0.40
µ
Mdn
σ
Sk
Kt
µ/σ
0.7
P(RMS)
12
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0 3.5 4.0
2
RMS (m/s )
4.5
5.0
5.5
6.0
6.5
0
2
4
P(RMS) Data N
P(RMS) Fit
Mean Data
Mean Fit
Median Data
Median Fit
6
8
10
2
RMS (m/s )
12
14
16
18
Figure 4. Validation of four-parameter modified Weibull model for six typical cases.
The goodness of fit between the distribution of the measured data and the model are best revealed graphically as shown
in Fig. 5 which shows plots of the main statistical parameters for all thirteen cases. It can be seen that very good
agreement is achieved (R2 = 0.99) for the first and second order statistics (mean, median and standard deviation) while
2
reasonably good agreement is achieved (R = 0.96) for the third and fourth order statistics, represented here by the
skewness and Kurtosis.
8
16
4.0
14
7
14
3.5
6
12
4
10
1.5
1.0
2
4
1
2
Kurtosis
0
2
4
0.5
Std. Deviation
0
0
6
8
10
12
2
Experimental Data, RMS (m/s )
14
2.5
6
4
Median
co
2.0
3
2
1:1
3.0
n
tio
8
6
Mean
1:1 c
orr ela
5
2
r
la
rr e
Skewness
0
16
Fit Data, Skewness
8
r
co
Fit Data, Kurtosis
cor
r
1 :1
n
Fit Data, σ (m/s )
tion
ela
10
tio
ela
1 :1
2
Fit Data, RMS (m/s )
12
tion
16
0.0
0
2
4
6
8
10
Experimental Data
12
14
16
Figure 5. Goodness of fit plots for the main statistical parameters.
The modelled PDFs for all thirteen vibration records, normalized in both axes, are shown in Fig. 6. It helps to illustrate the
various shapes of the RMS distribution for the vibration records. It also reveals how the model is capable of representing
RMS distributions consisting of various values of kurtosis, skewness and standard deviations. Fig. 5 confirms the finding
that twelve of the thirteen distributions are leptokurtic, Kt > 3, while all are positively skewed, Sk > 0 as seen in Fig. 4.
1.2
DATA A
DATA B
DATA C
DATA D
DATA E
DATA F
DATA G
DATA H
DATA K
DATA L
DATA M
DATA N
DATA O
1.1
1.0
P(RMS) Normalitzed
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
RMS Normalitzed
Figure 6. PDF of the thirteen vibration records, normalized in both axes.
Conclusions
This paper has presented the initial results of a study aimed at improving the method by which the rigid body vibrations
produced by road transport vehicles are characterised. Vibration data, collected from various vehicle types and routes in
Spain and Australia, was used to develop and validate a mathematical model, based on the Weibull distribution, to
describe the probability density function of the moving RMS time histories of the process. The paper has addressed the
limitations of the average power spectral density (PSD) and explains why the average PSD is not always adequate as the
sole descriptor of road vehicle vibrations as the process generally tends to be non-stationary and non-Gaussian. The
paper adopts an alternative analysis method, based on the statistical distribution of the moving root-mean-square (RMS)
vibrations, as a supplementary indicator of overall ride quality. The paper proposes a single mathematical model that can
accurately describe the statistical character of the random vibrations generated by road vehicles in general.
The proposed modified Weibull distribution model was developed to afford additional control on various aspects of the
shape of the distribution function. The model was found to be generic enough to be able to produce a range of well-known
distributions. Curve fitting results using the sum-of-squared error (least squares) optimisation were found to produce
unstable results which required inclusion of the mean, median, standard deviation, skewness, and kurtosis in the
optimisation algorithm. Validation tests using all thirteen vibration records and was found to offer good agreement in
general. The paper also shows how the model is capable of accurately describing the statistical parameters of the process
namely the mean, median, standard deviation, skewness and kurtosis.
This result is relevant not only for the characterisation of ride quality but also for the accurate synthesis of road vehicle
vibrations in the laboratory. The results can be used to assist in developing a novel method for simulating non-stationary
(modulated) vibration in the laboratory. The RMS distribution function can be used to create an RMS level schedule that
will enable the synthesis of random vibrations with varying RMS level to better represent the road transport vibration
process.
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