Random data are any type of data occurring especially in vehicle tyre–road induced noise and vibration that do not have an explicit mathematical formula to describe their properties. It is impossible to predict the precise level of the disturbance at any given time and hence it is impossible to express such disturbances as continuous functions in the time domain – only statistical representations are possible. Any time-history record represents only one record out of a collection of different time-history records that might have occurred. From the vibration point of view, the frequency content of a random signal is very important. For example, the frequency spectrum of a road input to a vehicle is a function of the spatial random
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profi le of the road surface and the speed of the vehicle. For a given set of conditions, it results in a large number of frequency components distributed over a wide band of frequencies.
5.1.1 Defi nition of terms
As shown in Fig. 5.1, an ensemble is defi ned as a collection of records.
A random process is defi ned as a process which is represented by the ensemble and defi ned by analysing various statistical properties over the ensemble.
Stationary random data is defi ned as data whose ensemble-averaged statistical properties are invariant with time. For such data, ensemble- averaged mean values are the same at every time. For vehicle applications, stationary processes include idle conditions, constant-speed driving or cruise control driving conditions. Ergodic data is stationary random data where one long-duration average on any arbitrary time-history record gives results that are statistically equivalent to associated ensemble averages over a large collection of records. In practice, stationary random data will auto- matically be ergodic if there are no sine waves or other deterministic phe- nomena in the records. Classifi cation of random systems is shown in Fig.
5.2. For example, a white noise, wideband random signal is stationary and ergodic. For such data, one long-duration experiment is suffi cient to obtain useful information, as a stationary random record should never have a beginning or an end.
xN(t)
x3(t)
x2(t)
x1(t)
t1 + t t1
t t
t t
5.1 Ensemble of time-history records defi ning a random process.
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Non-stationary random data is defi ned as data whose ensemble-averaged statistical properties change with time. Transient random data is a special class of non-stationary random data with a clearly defi ned beginning and end to the data, for example vehicle second/third gear slow acceleration, fi rst gear wide open throttle (WOT) acceleration, overrun/coastdown deceleration processes, braking, cornering, etc.
When stationary random data pass through constant-parameter linear systems, the output data will also be stationary. When transient random data pass through constant-parameter linear systems, the output will be transient random data. However, when stationary or transient random data pass through time-varying linear systems, the output will be non-stationary.
In general, techniques for analysing stationary random data are not appro- priate for analysing non-stationary random data.
5.1.2 Time-averaging and expected value
In random data, we often encounter the concept of time-averaging over a long period of time:
x t x t
T x t t
T T
( ) =< ( ) >= ( )
→∞
∫
lim 1
0
d (5.1)
or the expected value of x(t), which is:
E x t
T x t t
T T
[ ( )] = ( )
→∞
∫
lim 1
0
d (5.2)
In the case of discrete variables xi, the expected value is given by:
E x n x
n i
i
[ ] = →∞ n
∑
=lim1
1
(5.3)
Random
Stationary Non-stationary
Non-ergodic Special types of non-stationary Ergodic data
5.2 Classifi cation of random systems.
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5.1.3 Mean square value
The mean square value, designated by
( )
x2 or E[x2(t)], is found by integrat- ing x2(t) over time interval T and taking its average value according to:E x t x
T x t t
T T
2 2 2
0
( ) 1
[ ]
=( )
= ( )→∞
∫
lim d (5.4)
5.1.4 Variance and standard deviation
An important property describing the fl uctuation in ensemble is the vari- ance σ2, which is the mean square value about the mean, given by:
σ2 2
0
= 1 ( − )
→∞
∫
lim
T T
T x x dt (5.5)
By expanding, you can determine that:
V2=(x2)− ( )x 2 (5.6)
so that the variance is equal to the mean square value minus the square of the mean. The positive square root of the variance is the standard devia- tion σ.
5.1.5 Probability distribution
Refer to the random data in Fig. 5.3. A horizontal line at the specifi ed value x1 is drawn and the time interval Δti (i = 1, 2, 3 . . .) during which x(t) is less than x1 is summed and divided by the total time, which represents the frac- tion of the total time that x(t) < x1. The probability density that x(t) will be found less than x1 is:
p x P x x P x x
P x x ( ) = x ( + ) − ( ) = ( )
lim→ Δ
Δ Δ
0
d
d (5.7)
1.0
0
0
x x1
Δt1 Δt2 Δt3
t P(x)
x(t)
5.3 Calculation of cumulative probability.
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From Fig. 5.4 you can see that p(x) is the slope of the cumulative probability distribution P(x):
P x p x x
x 1
0 1
( ) =
∫
( )d (5.8)The area under the probability density curve of Fig. 5.4(b) between two values of x represents the probability of the variable being in this interval.
Because the probability of x(t) being between x =±∞ is certain:
P(+∞ =) p x( ) x=
−∞
∞
∫
d 1 (5.9)and the total area under p(x) must be unity.
The mean and mean square value defi ned in terms of the time average are related to this probability density function in the following manner. The mean value x coincides with the centroid of the area under the probability density curve p(x), as shown in Fig. 5.4(b), and can be determined by the fi rst moment:
x= xp x( ) x
−∞
∞
∫
d (5.10)The mean square value is determined from the second moment:
x2 x p x2 x
( )
= ( )−∞
∞
∫
d (5.11)Δx ΔP
p(x) (a)
(b)
0 0 1.0 P(x)
x x
x
5.4 (a) Cumulative probability; (b) probability density.
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which is analogous to the moment of inertia of the area under the probabil- ity curve about x = 0.
The variance σ2 is defi ned as the mean square value about the mean:
σ2 2 2
2 2
= ( − ) ( ) = ( )
− ( ) + ( ) ( )
−∞
∞
−∞
∞
−∞
∞
−∞
∞
∫ ∫
∫
x x p x x x p x x x x p x x x p x
d d
d
∫∫
dx (5.12)σ2=
( )
x2 − ( ) + ( ) =2 x 2 x 2( )
x2 − ( )x 2 (5.13)5.1.6 Gaussian and Rayleigh distributions
The Gaussian and Rayleigh distributions frequently occur in vehicle noise and vibration data. The Gaussian distribution is a bell-shaped curve sym- metric about the mean value:
p x( ) = 1 e−(x ) 2
22 2
σ
σ
π (5.14)
The expected value of the product of four Gaussian random process vari- ables, E[x1, x2, x3, x4], the fourth moment of the Gaussian random inputs, can be reduced to the products of the expected values of two process variables:
E x x x x E x x E x x E x x E x x E x x E x x
1 2 3 4 1 2 3 4 1 3 2 4
1 4 2 3
, , ,
[ ] = [ ] [ ]+ [ ] [ ]+
[ ] [ ]]−2μ μ μ μ1 2 3 4 (5.15) where μi is the mean value of the xi process (i = 1, 2, 3, 4). In most Gaussian random processes, μi= 0 (i = 1, 2, 3, 4).
Random variables restricted to positive values, such as the absolute value A of the amplitude, often tend to follow the Rayleigh distribution, defi ned as:
p A A
A
p A A
( ) = A >
( ) = <
⎧
⎨⎪
⎩⎪
−( )
σ
σ 2
22 2
0
0 0
e (5.16)
The shape is shown in Fig. 5.5. The mean and mean square values for the Rayleigh distribution can be found from the fi rst and second moments:
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A Ap A A A
A A
=∞
∫
( ) =∞∫
−( ) =0
2 2
2 0
2 2
d e d 2
σ σ πσ
(5.17)
A A p A A A
A A
2 2
0
3 2
2 0
2 2 2
( )
=∞∫
( )d =∞∫
σ e−( σ )d =2σ (5.18) The variance associated with the Rayleigh distribution is:σA2 A2 A2 4 σ2
=
( )
−( )
=( )
−2π (5.19)i.e.
σA ≈ 2σ
3 (5.20)
The probability of A exceeding a specifi ed value λσ is:
prob eA A d
A A [ >λσ] = ∞
∫
σ −( )λσ
σ 2
22 2
(5.21) The cumulative probability distribution for a sine wave is shown in the fi rst column of the table in Fig. 5.6, and is written as:
P x x
( ) = +12 1πsin−1
( )
A (5.22)Its probability density is:
p x
A x
x A
p x x A
( ) =
− <
( ) = >
⎧
⎨⎪
⎩⎪
1 0
2 2
π (5.23)
Rayleigh distribution
0 1 2 3 4A
p(A) 0.6 0.5 0.4 0.3 0.2 0.1
5.5 Rayleigh distribution.
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5.6 Three frequently encountered signals and their conversions.
Sine wave
Autocorrelation 2 A2
Autospectrum Autospectrum Autospectrum
Wideband record
A
A x
x
x
x x
0 0 0 x
0
1.0 1.0
–A
A
0 0
0
0 0
–A
Narrowband record
P(x) P(x)
p(x)
R(t)
Sxx(w)
w w
w R(t) = –cos w0t
–w0 w0 4 A2 Area = –
0 –w0 0 w0 –w0 0 w0
Sxx(w) Sxx(w)
4 A2 Area = –
R(t) = ce–k/t
t
p(x) p(x)
P(x) 1.0