Experimental Random Signals in Industrial Conditions
and Applications by Computer Analysis
Miroslav Kopecky
Professor of Applied Mechanics
Dept. of Physical Material Engineering
Faculty of Industrial Technologies,
University AD of Trencin , SK-020 01 Puchov, Slovak Republic
e-mail: [email protected]
ABSTRACT :
The paper focused on the service loads and cumulative damage analysis. The description of the service load contains:
the examples of statistical analysis of random stress process, stress spectrum and stress program.
This work presents a method of determining of structural parts random loading distribution, [1] , [3].The methods described in
this paper are the ways to reach the solution goals by means of a characteristic curve of strength reliability with the maximum
use of computer technology.
The results of its application would be presented to mobile facility elements.
A motorcycle running along a road is subjected to two vertically imposed displacements, one at each wheel. The
description of the road surface must be complete enough to describe adequately the displacement imposed at each wheel
at least in statistical terms and the correlation between the two displacements.
In some mobile machinery and equipment, or their elements are the problems of fatigue. The problem of fatigue strength and
service-life, [2] , as the most important phenomena of strength reliability under those conditions, is connected more or less
with a certain degree of uncertainty.
The basic elements of the fatigue life evaluation,[3] , [2], are reviewed in Fig
Key words: Dynamic Failure, Fatigue, Non Linear Dynamic Systems, Numerical Method
and Validation.
1. Introduction
The methods described in this paper are the ways to reach the solution goals by means of a characteristic curve of
strength reliability with the maximum use of computer technology.
The results of its application would be presented to mobile facility elements.
A motorcycle running along a road is subjected to two vertically imposed displacements, one at each wheel.
The description of the road surface must be complete enough to describe adequately the displacement imposed at each
wheel at least in statistical terms and the correlation between the two displacements.
In some mobile machinery and equipment, or their elements are the problems of fatigue. The problem of fatigue strength
and service-life,[2], as the most important phenomena of strength reliability under those conditions, is connected more or less
with a certain degree of uncertainty.
The basic elements of the fatigue life evaluation,[3], are reviewed in Fig. 1.
Fig. 1. Basic elements of the fatigue design process
2. Theory
The result of analysis give us the possibility of estimation of complex character service load and enable on this base the
choice of suitable counts cycle method for block-load spectrum determining, mentioned in schematic diagram on Fig. 2.
Fig. 2. System of computer programs for fatigue life computation
The block-spectrum is the base for elaborating of load program with suitable load sequence. This program is equal with
the service load in fatigue life range. It enables a direct application of that program in evaluation and components tests.
Acquisition of the command with two-wheeled road vehicle for laboratory loading system contains identifications of the
dynamic characteristic features in operating state, during random excitation. The usual way to determine the equivalent
loading of a random evolution is displayed in Fig.3.
An identification of dynamic properties in an operational random excitation.
For a random stationary and distribution time series several parameters describing the time evolution of the quantity
can be expressed by the function of the statistical moments of the power spectral density in relation to the zero
frequency axis is defined as follows:
mn = ∫ f n . S(f) df
(1)
An equivalent frequency of the phenomenon can be obtained taking k – level is = 0 :
fs = √ m2 / m0
(2)
In this case one should consider the equivalent amplitude sinusoid of a random loading equal to the mean square
value of the random evolution with zero mean value :
T
σ2amp = 1 / T ∫ σ2 (t) dt
(3)
0
and
σ red = σamp + σs
(4)
where
n
σs
=1/n
Σ σi
(5)
i=1
Fig. 3. Layout of laboratory test
The starting point of theoretical solutions reliability, [1], is model where the dependence between random loads and life,
N f, of components must be completed by a variable, R(Nf), which expressed digital guarantee in the probability form.
A three-parameter distribution may be expressed as
R(Nf) = exp (-(Nf -Nmin) / (Nsig -Nmin))k)
where: Nmin is a minimum of the longevity,
Nsig is a modal value of the longevity,
k
is a parameter of distribution.
(6)
The probability density function related to Eq. (6) is of the form
f(Nf) = k/(Nsig-Nmin)].[(Nf-Nmin)/(Nsig-Nmin)]k-1 x exp(-[(Nf-Nmin)/(Nsig-Nmin)]k)
(7)
The determination of the parameters of this distribution : k, Nsig, Nmin are achieved by the moment of function Eq. (7)
numerically.
With parameters of distribution, we may define the result by the statistical curve of longevity, which in a form of
probability characterized the longevity form Eq.(6)
ln(-ln R(Nf)) = k [ ln(Nf - Nmin) - ln(Nsig - Nmin) ]
(8)
But for the case Nmin ≤ 0, the function of probability of longevity, Eq.(6), will be reduced to two-parameters of term:
R(Nf) = exp (-(Nf / Nsig)k )
(9)
Estimates of parameters : k, Nsig, by characteristic value Ns, SN, may be expressed:
Nsig = NS / C(k)
(10)
NS = SN . D(k)
(11)
and
where:
C(k) = Γ(1+1/k).
The values of functions: A(k), C(k), D(k) and B(k) , are introduced for the practical application in Eq.(6) for variables
1/k.
3. Application and results
The presented system has been applied in many concrete situations and verified within co-operation with manufacturing
enterprises,[1].
The output data expressing one of the statistical characteristics of loading, Eq. (7), in a graphical way can be seen in Fig. 4.
Fig. 4. The graphical function of failure-free probability
The output data expressing one of the probability density distributions of damage, Eq. (8), in a graphical way can be seen
in Fig. 5.
Fig. 5. The graphical function of probability density distribution of damage
System has been applied also completed on construction subassemblies, as they are shown in Fig. 6.
Fig. 6. View of the load-carrying parts of motorcycles
4 Conclusions
Analysis calculation and experimental results the following conclusion can be drawn. Numerical results are consistent with
experimental ones taking into account type of distribution.
The approaching in presented ranges of investigations would be an essential progress in development of mechanical
science in referring to important problems of fatigue crack of structural parts peaking to very serious catastrophes.
Prototype vehicles or components of vehicles can be more closely evaluated.
The applications of this method shorten knowledge of the time to failure of machine components for transportation and
contribute to the safety and economy of mechanical systems.
Acknowledgments
The author expressed his thanks to Slovak Agency for Research and Science for its support of this work (grant 1/2081/05).
References
[1] Kopecky M. , Experimental-numerical method of random analysis. In: 9th Inter. Conf.
on Experimental Mechanics, vol.3, pp. 1006-1012, Copenhagen, Denmark , 1990.
th
[2] Kopecky M. and Vavro J. , A fatigue curve as a random element. In: 5 Polish - Slovak
Scientific Conference on "Computer Simulation in Machine Design", Wierzba, Poland, pp.7982, ISBN 83-912190-3-8, 2000.
[3] Szala J. , Fatigue fracture of parts of structures as a vibration effect . In:16th Symposium
on “Vibration in physical system”, Poznan-Blazejewko, Poland pp. 40- 49, 1996.