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We have been discussing an application of principles of random vibration to problems
in earth quake engineering. We will continue with discussion on
applications. So, in this lecture, I will consider two more applications - one is in
the area of models for accumulated fatigue damage in randomly vibrating
structures, and the second one is the area of vibration energy flow that is important
in the area of high frequency vibrations of engineering structures.
So, what I intend to do is to be favor of these two applications areas and illustrate
how principles of random vibration analysis are employed here. The objective is
not to describe the problems of fatigue damage nor the problems of energy flow, but to simply
state the basic problem and illustrate how random vibration
principles could be used.
So, we will start with problem of fatigue damage accumulation. So, some, we start with
some empirical background. What is fatigue? So, we can say the fatigue is
loss of mechanical integrity of the structure due to reversal of stresses. So, any structural
material on some scale, on any scale, on, even on very fine scales will
have some imperfections defects. Consequently, the strength of the structural material in
a given volume depends on amount of defect that are present.
Now, when the material is subjected to vibratory loads, there will be reversal of stresses,
and as a consequence of this, the defects and imperfections grow within a
given volume of structural material, and consequently, the strength of the material reduces and this
phenomena is what is known as fatigue. So, fatigue is a
essentially a of phenomena involving of progressive fracture. So, it is a fracture of a structural
member due to repeated cycles of load. Fatigue is the primary mode
of failure for metals subjected to oscillatory loads. So, this is a major source of failures
in aircrafts, railway vehicles, ships, bridges and rotors.
Actually structural components that can carry high constant amplitude loads fail under a
substantially lower magnitude fluctuating load. This is due to, basically
due to fatigue, and during fatigue failure, maximum stress could be well below the tensile
strength of the material but the structure fails after oscillating for a finite
number of cycles, that is, at failure, the response level could be well below the limits
of first passage failure.
So, it is a failure could be quiet catastrophic. So, it is important to obtain a rational description
of how to model these failures. So, the objective of this discussion is
to obtain a probabilistic description of fatigue damage in structures which are driven by random
excitations. For example, the types of questions that would like to
ask are - what is the expected rate at which fatigue damage accumulates? What is the probability
distribution of the life of the structure? What is the influence of
randomness in structural properties? How to characterize the reliability of structure
against fatigue failure? So, we will consider some of these questions and see
how we can progress.
Now, if you look at time histories of a stresses, the various possibilities exist. For example,
in this first one, there is an oscillatory load about a mean level. That
would mean the stress cycles are completely reversed and these are called completely reversed
cyclic stresses. This oscillation could occur about a non-zero mean.
So, the cycling occurs about a non-zero mean which is not 0 or it could be such that there
is no stress reversal, but actually there is a for instance released tension.
Here, the stress never enters the compression zone. It remains tensile but the tensile stress
magnitude keeps oscillating.
Now, here, we introduce certain terms S max S min and S mean are the maximum minimum and
arithmetic mean of this stresses. Delta S is a stress range S max
minus S min and the alternating stress amplitude is given by delta S divided by 2.
The stress amplitude need not be harmonic. The amplitude could vary. For example, here,
there is a stress time history where up to say 1.5 second there is one
amplitude. Beyond that, the amplitude increases. Now, the change in amplitude could also be
accompanied by change in the mean levels about which these oscillations occur. These are
varying amplitude, varying
mean stresses, or all the stress could be a randomly varying in time. It could be multi
frequency. The, here, what we are shown is a narrow band random process.
So, here, there is no, the definition of a cycle here is not immediately evident unless
we clarify what exactly that is.
Just we will illustrate here I am showing certain measure time histories. For example,
in this graph, we are showing the time history of the strain on a steel girder
bridge as a train formation crosses the bridge at a particular velocity. So, this is second
graph is similar result when a longer formation may be free traffic crosses
the bridge. So, the cycle that we see here typically corresponds to the passage of one
locomotive and carriage wagon and this wagon themselves we will have
multiple access and these small oscillations about a higher level represent, you know,
oscillation due to passage of axils of the same wagon.
Now, we use what is known as S N curves to characterize fatigue failure. This S N curve
essentially is a plot of cyclic stress level versus number of cycles to
failure; that means if you take a test coupon and subject it to cyclic loaded into a given
stress value after certain number of cycles of oscillation, this coupon will
fail and the number of oscillation as a function of stress level is known as a S N curve. So,
this is something that is experimentally obtained. This also is known as
Olar curve. The test specimen is typically a cylindrical
specimen subject to uni-axial cyclic stress or a small cantilever beam under bending oscillation.
The stress amplitudes
are kept constant. We could also similar actual real life stress cycles in a fatigue test
but we can, this is not what we are discussing in this lecture.
A simple model for this so called S N curves is of this form N S to the power of b is equal
to c - where b and c are constants, and if you take every them of this, we
get log N plus b log S is equal to log c. That would mean on log scale. S N curve is
a straight line with slope minus 1 by b and intersect log c.
Now, in reality, what happens is - below a particular stress level, the, there would
not be failure due to cyclic oscillation is not possible and this limit of stress is
known as endurance limit, and we are considering stress levels above the endurance limit in
our studies. So, number of cycles from this formula N S to the power b
equal to c. We can, in, for this case, it is N S to c to c into S to the power minus
b for S greater than S not is 0 for S less than or equal to S naught if you taking to
account the effect of endurance limit.
Now, in the S N curve approach, it is an integral approach which actually does not deal with
physical phenomena that would be taking place within the material
which results in loss of mechanical integrity. Actually does not separate the crack initiation
and crack propagation stages, considers only the total life to fracture.
Now, we obtain S N curves in the laboratory conditions and then we want to extrapolate
this information to the field conditions. So, laboratory testing is done on
coupons just like with the way we find Young's Modulus of a, say material using coupons.
In the same sense, S N curves are also obtained in a laboratory
conditions on test coupons, but these results need to be extrapolated to free condition.
When we do that, there are factors like nonzero mean stress, then varying stress amplitudes,
an environmental conditions like temperature, humidity, corrosive
media, etcetera will start influencing the results, and also the size shape and surface
finish of the member under consideration also have significant influence on the
fatigue performance. The frequency of cycling could also important but many times this is
not a very crucial factor.
In the experimental data, actually large scatter is observed is reflects the influence of uncertainties,
and one actually plots, for example, the S N curve, one gets an,
if, in experiment there where green straight line would not emerge. So, at any given point,
you can draw a probability distribution curve, density function, and one
can specify the S N curves in terms of S N and probability curves. This density function
could be modeled as lognormal or Weibull. Endurance limit have already
explained; it is a stress level below which the specimen seems to last in definitely.
Now, we use what is known as Palmgren Miner rule to find out accumulation of a fatigue
damage. Suppose during the life of the structure on x axis, I am plotting
time, and on y axis we are plotting stress amplitudes. Suppose the structure undergoes
N 1 oscillations at stress level S 1 and N 2 oscillations that stress level S 2
and N 3 at N i; N i at stress level S i and N N at stress level S N. Now, if the specimen
is left to oscillate stress level S 1, it will required as a capital N 1 number of
cycles for failure. Similarly, if the specimen is left to oscillate only at S 2, it may required
N 2 number of cycles to failure and these capital N 1 capital N 2 etcetera
are obtain from the S N curve.
Now, according to Palmgren miner hypothesis, what we do is we find out the number of cycles
at different stress levels - S 1 is n 1, S 2 is n 2, S 3 is N 3 and so on
and so forth, and for this stress level from S N curve, we get the number of cycles to
failure. We define in incremental damage as the ratio of n 1 by capital N 1.
Similarly, incremental damage at stress level n 2 by capital level N 2 and so on and so
forth. We define what is known as cumulative damage as the sum of this
incremental damage at any given stage. For example, at the end of application of S 2,
stress level S 2 for N 2 number of cycles, the accumulated damage would be
n 1 by capital N 1 plus n 2 by capital N 2.
Now, according to the palmgren miner hypothesis at the end of m th cycle of oscillation, we
get the accumulated damage as capital delta is summation i equal to 1
N i by capital n i. Now, if I use the formula for the S N curve, I can write this as N i
S i to the power of b by c because capital n i is related to S i and the constant b
and c. According to palmgren miner hypothesis, the condition for failure is that this cumulative
damage should reach a value of 1. So, consequently the safe limit is
i equal to 1 n i S i to the power of b divided by c must be less than or equal to 1. If this
miner sum crosses 1, the specimen deemed to have failed according to this
theory.
A few remarks on the palmgren miner theory could be made. In this theory, the order in
which stresses are applied does not matter; that means the damage is
assume to take place in a linear manner. In reality, actually failure is sensitive to
order of loading. A high stress level followed by low stress level would produce
accumulated damage of a different kind than the other way round where a low stress cycle
is followed by high stress cycle. The method as such does not provide any means
of assessing the effect of variability in constant b and c etcetera, and damage is assume
to accumulate at the same
rate at a given stress level without regard without regard to the past history; that means
it does not take in to account the memory. In actual experiments, the miner
sum at which the failure as occur actually varies over very large range, that is, 0.25
to 4 under harmonic loading, but for random time histories where no ordered
sequence of high and low amplitudes exist, the miner sum is reasonably good 0.6 to 1.6.
Now, the question is we are not so much interested in the limitations and scope of palmgren miner
hypothesis etcetera. What we would like to consider is how can
we extend the palmgren miner hypothesis to random stress time histories. Now, let us
assume that the stress time history is a 0 mean narrow band Gaussian random
process, and consequently if you consider stress to be a time history, there are no
discrete stress levels here, is there not harmonics of different amplitudes
following each other. So how do we proceed?
So, for example, this is a sample of a narrow band time history, time history of a narrow
band random process. The notion of cycle is not very evident here. This is
a narrow band process, but in a broad band process, as we already seen the notion of
a cycle becomes much less apparent.
Here, what we define is function D of T. We define as integral 0 to t chi of T dt - where
chi of t is a rate of accumulation of damage. This is, this we write as N of t
into S to the power of b of t divided by c and this is fashioned after the formula for
accumulation of damage given by n i S i to the power of b divided by c.
Now, the question is - if S of t is a as I said 0 mean narrow band stationary Gaussian
random process, how do interpret these quantities? If the stress time history is
a random process, how do interpret S of t n of t in this expression? So, what we do
is, we take S of t to the peak magnitude and n of t to be the rate of peaks, and
these we have studied already. For a narrow band process, rate of peaks is same as rate
of 0 crossings for narrow band processes; that means every time a 0 is
crossed, there will be a peak as you see here. For example, there is 0 crossing; there is
a peak. So, that is a property of a narrow band process; that is not true for
wideband process but I will to start with will limit or discussion to narrow band process.
Now, our objective is to characterize this function D of capital T, is we are interested
in knowing for example, what is probability that d of capital t is less than or
equal to 1. That is a question that we would like to answer.
Now, the join density a function of S and N is actually not available. So, what we actually
know? We know the peak magnitudes are Rayleigh distributed for
narrow band processes and expected value of n is given by sigma 2 divided by 2 pi by 2
pi into sigma 1 - where sigma 2 is the standard deviation of the derivative
of the process and sigma 1 is the standard deviation of the palmgren process.
Now, we make in Adhoc assumption that S and N are independent. So, we assume that join
density function PSN s, n is product of the marginal density functions.
So, based on that, we find the expected value of chi of t which is the rate at which the
damage is accumulating, and since we know this expression and we know
this probability density function, we can write for this expected value this form. This
is Rayleigh density function. So, consequently this will be the expected value
of S to the power of b, and this is, this quantity is the actual expected value of n
of t, which is again known from study of low level crossings and 0 crossings of
random process.
Now, we recall that the integral of this type 0 to infinity y to the power of 2 nu u minus
1 exponential minus y square dy is actually given by the gamma function.
Now, consequently we can express the expected value of chi of t in terms of the gamma function
and this is what we get. So, at least we are able to proceed and
get the certain Adhoc assumptions the average rate at which fatigue damage is accumulating
in terms of the standard deviation of the parent process, standard
deviation of the derivative process and the material concuss c and b.
Now, we can return to the problem of finding expected value of d of t. Now, if process
is stationary, we see that sigma 2 and sigma 1 are independent of time. So,
they can be pulled out and this 0 to t d t will become this. So, we have this expression
for d of t. That is average of the accumulated damage at time instance capital
T.
Now, we could visualize a T star such that these expected value of D of T becomes 1.
Now, if you do that, that T star is will be 1 by expected value of chi of t. So,
to a first approximation, we can take T star to be the mean fatigue life. So, if you know
the random stress time history and you know its prospect density, you can
derive the variance of the process and its derivative process, and if use this formulary,
you have a handle now that, you can at least, you are at least in a position to
evaluate the mean fatigue life. This is for a stationary narrow band Gaussian random process.
Now, a few remarks are in order. In this particular derivation, we have taken the constant b and
c to be deterministic, but if you happen to know the join
probability density function of b and c, we could incorporate that into the analysis.
So, what the expectation that we have got can be interpreted as a conditional
expectation and the unconditional mean can be evaluated by an integration with respect
to the conditional probability density function to the joint probability
density function of b and c. Is worth considering the equation is T star indeed the expected
fatigue life. Now, we will this needs to be consider carefully. So, for
that purpose, we will consider capital l to be the life time. Life time is a random variable.
So, condition for failure that is a definition of capital l is 0 to l chi of t d t
is 1; that means the expected value of this integral 0 to l chi of t dt is 1.
Now, this is a condition that we actually need to analyze, but what we have done is
we have place this integration, the random quantities in this expression is L as
well as chi of t and there mutually dependent. This integral obviously not equal to this.
This is 0 to mean of L. Then mean of the integrant dt is not the, actually the
condition that we are looking for all those. This is what we seen to have utilized. So,
the conclusion is T star, is not the exact expected fatigue life, but if first
approximation, we could take it as an approximate, an acceptable estimate, and if you want to
really evaluate this integral in exactly, this is the, we do not have
adequate information on the uncertainties here. Therefore, we will not able to proceed
anyway. Now, we have derived now the expression for
expected value of chi of t. With little effect, we can also derive the variance of chi of
t and we can also derive
variance of capital d of t, and we have based our assumption on the fact that where assuming
that the stress time is to the narrow band process. If the process is
broad banded, then the notion of a cycle and amplitude becomes less apparent as I have
already said, but there are algorithms for counting cycles and some of the
algorithms are range per counting range flow counting etcetera. They are available one
could also deal with broad band time histories.
So, what have done in this brief presentation is to illustrate how properties of level crossing
and peak distribution etcetera could be utilized to get a theoretical
estimate of average, the expected life due to when failure is defined with respective
fatigue. Now, we will move on to another application. This is analysis of
vibration in high frequency regime and how we can use random vibration principles for
this. Now, what is high frequency vibration? We consider that a structure is
vibrating in the high frequency regime. If it is response at any frequency, consists
of significant contributions from a large number of modes.
Now, what are the basic problems in such, such, basic difficulties in such problems?
For purpose of illustration, I have shown here the transfer function of a,
frequency response function of a thin walled stiffened cylindrical shell, and as you can
see here, there are several peaks which are closely space. We know that in
frequency response function, peaks occur wherever there is a frequency and their variation is
also governed by value of mode steps at point of driving at point of
measurement. So, if we consider segments like this, it
is very clear that there are large number of natural frequencies that are crowded in
this frequency range. So, if you were to
drive the system harmonically at a single frequency, it is expected that to compute
the response, we need to consider contributions from several neighboring
modes. In that sense, we call this type of oscillations is high frequency oscillations.
This is not true for example in the case of say earthquake response of tall
buildings, but whereas the in acoustic response of missile shells or aircraft wings etcetera,
there will be large number of natural frequencies or in the frequency
range that we are interested. I will clarify the nature of the problem as we go long.
Now, there is one approach to studies such high frequency vibration and that is known
as energy flow models, and one of the techniques are one of the
methodology there is what is known as statistical energy analysis, is a frame work for analyzing
high frequency vibration analysis. Now, the structural behavioral at
high frequencies is very sensitive to minor changes in structural parameters and details
of modeling. A deterministic approach to modeling structural system
parameters is inappropriate; that means if a structure is undergoing high frequency oscillations,
a slight change in say boundary conditions or some of the
parameters of the problem will dramatically alter the behavior of the structure.
So, the essence here is that we need to consider this extreme sensitivity of the response with
respective changes in system parameters. Description of dynamic
behavior of structural joints with increasing frequencies become difficult. This calls for
experimental approaches to characterize structural behavior of joints. So, in
typical structure like automobile vehicle or aircraft structure, there will be several
joints, client joints, point joints, etcetera and their behavior at high frequencies
becomes quiet complicated, and if use methods like finite element method modeling, the flexibility
of the joints becomes a very complicated issue as you go very
higher up in the frequencies. So, the, this statistical energy analysis
procedures essentially take in to account these difficult issues. First thing, it does
is it treats the vibrating system as being
randomly parameter; that means a natural frequency is more shapes of this structure is treated
as being random in nature. The word statistical in statistical energy
analysis refers to uncertainties in specifying the system parameters. The excitations could
be random or could be deterministic. So, the stochastic nature in these
problems come originate due to randomness in system properties.
In such problems, detailed characterization of structural response in terms of spatiotemporal
variations of displacement stress and strain fields become unwieldy.
Macro level description that involves space time frequency averaged response quantities
such as vibration energy content in spatial domains within a structure may
be adequate. So, the method of analysis is aimed at establishing spatial distribution
of vibration energy stored in the structure as a function of frequency.
So, I will just illustrate what, what the, what that means, what I said. Consider a simple
truss here which is carrying a force f of t. Let us assume that the frequency
range of f of t spans several decays of frequencies. It may go up to a few tense of hertz to couple
of thousands of hertz. We could use finite element method to
analyze this problem, but then, if we need to compute the natural frequencies of this
structure over a frequency range of say 2 kilo hertz, there could be several
may be hundreds or couple of hundreds of natural frequencies which could lie in that frequency
range. Now, if we indeed perform, that would mean
that we need to model this structure in greater detail. We need to select element which are
quiet small in size. We
should have large number of such elements that intend requires, that we should be able
to know the property of this structure at smaller levels, smaller spatial
levels and that may not be always possible. Similarly, the question on how does the join
behave as a frequency increases becomes crucial. So, any slight change in system parameters
here can dramatically alter the behavior of the system. So, therefore, if you are interested
in for example displacement
stress or strain at any given point in the structure, we end up doing huge amount of
calculations, but that type of information may not be needed to take decisions
in this types of problems. What do we may like to know is what is the total energy that
resides in this member or how does a total energy gets spatially distributed
in the system. For that purpose, what we do is we divide
the truss into a set of subsystems. Each one is an energy carrying unit, and we would like
to now consider the problem of
this force into the velocity here will be a power input to the system. That power should
be stored at various points in the structure and they, which are they more
energetic members, and if that level of energy is high, how do I reduce it? What are the
vibration paths that take this energy two different parts in the structure?
This type of questions are asked in, in, statistical energy analysis.
So, statistical energy analysis can be viewed as a branch of linear vibration theory with
some of the following characteristic features. Here, the built up structure is
taken to be random in nature. It is divided into a set of subsystems and the subsystem
natural frequencies are taken to be identical and independent random
variables distributed uniformly in the frequency range of interest. That could mean the natural
frequencies of the subsystems are taken to constitute Poisson points
on the frequency axis; that means natural frequencies are like number of natural frequencies
is a Poisson random variables. The external excitation that are often
random in nature are specified in terms of power input, and the governing equation for
system behavior are described in terms of power balance between
subsystems. We do not write the equilibrium equation the way we do in finite element method.
We simply keep track of energy balance. I will come to some
details. The primary objective of the response analysis
is to determine the spatial distribution of total vibration energy residing in the system.
It is not to obtain, for
example, time histories of stresses, strain and displacement at every point on the structure.
Here, it is a macro level description where certain spatial extends are
identified as subsystems, and what happens within that spatial extend in terms of a single
number is what is being sort.
Here are some references - the book by the Lyon and Dejong theory and application of
statistical energy analysis. There is an introductive paper, and a review
paper which provides some background to this subject.
Here, there is another example where there are a stock of three plates. This could be
a typical situation in solar panel arrays in satellite. This e raise to i omega t is
a forcing function that acts on this and this frequency can covers several, you know, it
could run from twenty to say two kilo hertz. What we are interested in
knowing is under this kind of excitations, how does vibration energy reside in different
parts? So, we could model this in terms of subsystems as shown here 1, 6
and 11 correspond to these three plates. This, 1, 6 and 7, 1, 6 and 11 correspond to these
three plates and 2 3 4 5 etcetera, 7 8 9 10 are this coupling members. We
are not interested. It may again emphasis in finding stresses strain and displacement
in various points on the structure.
Now, how do we specify inputs in statistical energy analysis? We do not know time history
of the force but we would like to provide the input power. This input
power is derivable in terms of the input power spectrum and the input power receptance; that
means the force into the velocity, applied force into the velocity in
the direction of the force is a power. Force into displacement is a work done and for that
work done for a unit time is the input power, and in terms of a spectrum if
you want, you have to do certain calculation. Now, we consider inputs to be in the high
frequency range say 20 to 20 k kilo hertz, these excitations could be
random or they could be deterministic also. It is not important that the randomness resides
in the excitation. The randomness is in the system parameters.
How do we model a system? We model a system as collection of energy storing elements called
subsystems. They are not like finite element elements but they are
much more macro level description. They involve much, you know, macro level description of
the system behavior. We focus only on linear and randomly
parametered systems. The behavior is linear but the system is randomly parametered.
This randomness in system parameters could be introduce at the level of definition of
mass stiffness and damping as being random or alternatively we can directly
characterize the natural frequencies and more shapes of the subsystems as being random.
We need not have to specify mass stiffness and damping as random but
since solution is being obtained in model domain. The model parameters like natural
frequency, more shapes participation factor model damping etcetera could be
taken as random.
And what are the response quantities we look for? We look for steady state, time averaged,
total energy stored in each subsystem often averaged over frequency
and ensemble of random realizations; that means we are looking for grossly averaged
features. We are not looking into details. It is average over time, average
over space, average over frequency band and it gives a broad idea about different levels
of energy. They stay at, they decide at different levels.
How do we derive the governing equation? What are the governing equations in SEA? The governing
equation in SEA represents condition of power balance and it
has the form the vector of power input is equal to a matrix multiplied by energy at
different energy residing in different subsystems. This matrix is known as
coupling loss factor matrix. I will talk about this shortly. This is essentially a representation
in the steady state. We do not normally consider transient behavior
using SEA, and the whole analysis is done in frequency domain.
Now, we can ask are there any theoretical foundations for this procedures? Actually
there is no such regress theoretical foundation. There is no, for example, a
variational principle from which we can derive SEA equations nor there is any systematic
proof that as we increase the number of subsystems in SEA formulation.
The answer would converge. So, it is not like a method like finite element method but it
is a heuristic procedure which is found to work well when other methods
do not perform.
Now, the, having said that there is no theoretical basis but there exist an idealize situation
where statistical energy analysis basic results of statistical energy
analysis can be shown to hold good. So, what is that system? This is simple system where
two oscillator which are couples, a coupling spring. The two oscillators
are driven by F 1 of 2, F 1 of t and f 2 of t. These are taken to be independent Gaussian
White Noise Processes. Now, the question that would like to answer is this
F 1 of t will do some work on this displacement. Therefore, there will be a power input here
and there will be a power input here and how does vibration energy
gets shared between this subsystem and this subsystem.
So, what we do is we model the problem using two subsystems - this is one oscillator this
is second oscillator there is a coupling k c, and please note that the
coupling element do not have, does not have damping. It is a conservative coupling. So,
what happens? There is a power input to the first system through F 1 of t
and part of the power is dissipated through damping and part of that flows to the, energy
flows to the second subsystem. In the second system because F 2 of t, there
is some power input, and part of it is dissipated and part of it flows to a subsystem one. So,
now, we set up this simple
equation that pi in 1 is equal to pi 1 2 minus pi 2 1 plus pi diss 1. So, pi in 1 must equal
to this. Similarly, what comes in to second system? Pi in 2 plus pi 1 2, and
what goes out? Pi diss 2 and pi 2 1. So, pi in 2 must equal to pi 2 1 minus pi 1 2 plus
pi diss 2. So, this is, there is no approximation here. This is an exact.
Now, let us try to evaluate the expressions for this input powers and maybe I will take
expected values and see what happens. So, to do that, we will write the
equations of motion. Now, the equation of motion consist this simple to derive the 2
degree approximation, 2 degree freedom model. It guess couple through the
spring k c and this is reasonably straight forward, and we are assuming that F 1 of t
is 0 mean and it is a white noise with strength i 1 and F 2 of t also is a white
noise with strength i 2 is 0 mean, and we assume that the uncorrelated; that means they
are independent because we are assuming Gaussian White Noise models.
The equation can be recast as in this form where we are taking the force acting on the
first system through the coupler on the right hand side.
So, we assume the system since the system is damped, we assume we assume that as c tense
to infinity the system reaches study state. The input stationary; system
is damped. Now, we will digress for a moment. If you consider a single degree freedom system
under a white noise excitation in the steady state, if you want to
find out the cross power spectral density function between displacement and applied
force, this can be derived in terms of this well-known expression, and the
Fourier transform of the response is related to the Fourier transform of the input through
the system transpose function, and we get this S x f of omega to be H of
omega into S F F of omega. Since excitation is white noise, I get this H of omega into
S naught. Now, it is a cross covariants is a Fourier
transform of this. So, I get this expression. Now, a tau equal to 0. I will get R x of f
0 R x f of 0 can be obtained by putting
tau equal to 0 but you can see that the integrant is nothing but Fourier Transform of the impulse
response function. So, R x f of tau is S not H of tau by 2 pi. Now,
in this if we put tau equal to 0, I get R x f of f is 0 which is S not h of 0 by 2 pi.
Now, we know definition of h of 0 is 0. Therefore, r x of 0 is 0. So, that result I will
be using so that this miner digration to make sure that, that is, registers on your mind.
Now, we would also need another result. If X of t and Y of t are jointly stationary,
the expected value of the product of n th derivative and m th derivative of X
and Yy as shown here is given by this. This we have derive when we introduce the basic
motions of mean square derivative for random processes.
Now, we can consider certain expectations F 1 of t into y 1 of t is 0 using the logic
that I just not described. We can also show that sum of these are 0, and since
process is stationary, the process and its derivative at same time are uncorrelated.
So, again, that could mean these expectations are 0, and expected value of y 1
into y 1 double dot using this formula can be shown to be given by this, and we have
several other expectation that we need when work this equations and I leave
it on an exercise to verify each of these statements. These are already covered in the
lectures. So, it is a good time to check you are understanding by showing that
each one of these statements are true.
Now, what is that we are interested in? We are interested in the average power input,
that is, expected value of pi 1 in. What is power input to the first mass that is
applied force F 1 of t into dy 1 would give the work done and into dy 1 by d t would give
the work done for unit time. So, pi 1 in expected value nothing but
expected value of F 1 into y 1 dot. Similarly, pi 2 in will be f 2 into y 2 dot. What is
power dissipated it is c 1 y 1 dot square; pi 2 dissipated is this. What is pi 1 2?
Pi 1 2 is, pi 1 2 is the force in the spring acting on velocity of this. So, that would,
you can show that, that would be given by k c into y 1 into y 2 dot and pi 2 1 will
be k c y 2 into y 1 dot.
Now, if since y one of t and y 2 dot of t is 0 and it not 0, it is equal to minus y
1 dot y 2 t follows that pi 1 2 is equal to minus pi 2 1. Now, we can recast all these
equations in a matrix form. So, on the left side, I have pi one in; here, pi 2 in, and
all other expected values I mean I am writing, I mean you have to recast that set
of equations in this form we get this equation. This is a straight forward representation
of the equation that we consider just now. Now, in this, we will now consider some of
the rows. Suppose if you consider the first two rows, I get c 1 into y one dot square
expected value plus k c into y 1 y 2
dot is equal to F 1 of t into y 1 dot. Similarly, c 2 y 2 dot minus k c into y 2 y 1 dot is
this. These two equations are nothing but the statement of law of conservation
energy, that, is pi one diss plus pi 1 2 is pi 1 in pi 2 diss minus pi 1 2 is pi 2 in.
So, things gel-well, there is no problem.
Now, if you consider rows five and six in that equation and take a look in this equation.
You can show that what they are telling as is the average kinetic energy in
the first system minus plus average potential energy in the first system is given by this.
Similarly, this for the second system; this is the equation. Now, if for
instance, if coupling spring is absent, then we know that in the steady state average kinetic
energy, same as average potential energy and that again this consistent.
Now, if we assume that this coupling spring is much smaller than k 1 or k 2, that means
the coupling is weak. This is one of the assumption that is made in
statistical energy analysis, and if I now consider expected value of y 1 into y 2 dot,
you can derive this expression and you have to carefully look at this
expression. This is expected value of y 1 into y 2 dot is nothing but expected value
of pi 1 2 multiply for the k c. We denote this multiplier.
See, the term inside this bracket is of importance. You can see that this is pi 1 divided by c
1 into expected value of pi 1 in and is m 2 by c 2 into pi 2 in, the second
term. So, what this is telling is that the energy flow from 1 to 2 is proportional to
the, difference in energy at difference in the power input between the first and the
second system some scaled factor. Now, we will manipulate this and see what it tells
us.
Now, again under the assumption of light coupling, we can show that the power input is in steady
state is dissipated by the system. So, that is c 1 y 1 dot square.
So, I can write for pi 1 mean 2 c 1 m 1 into k e 1. Similarly, expected value of pi 2 in
can be given by this expression, and we use the fact that in steady state,
average kinetic energy and average potential energy for the two systems are equal, and
if I now substitute, I get what is known as fundamental statistical energy
analysis result namely that, average energy flow is proportional to the difference between,
the energy between two subsystems, the energy in the two subsystem;
that means just like thermal energy flows from hot spot to cold spot, the vibration
energy also flows from places of high energy concentration to low energy
concentration.
So, this is the essential fundamental result of SEA and this is, this has basis in random
vibration principles. That is what I am trying to discuss. Only that aspect of
SEA is what I am discussing. We have consider two oscillators. Now, what happens if there
are two continuous systems? So, for example, plate is coupled to a
cantilever beam. This plate itself is a multi-model subsystem. It has several natural of frequencies
in its uncoupled state. If I were to perform a model analysis in
the frequency range of interest, there may be several modes; that means this plate will
be represented by a set of single degree freedom systems corresponding to
the normal modes in the independent, generalized coordinates of this plate. Similarly, the
cantilever beam also can be represented as set of oscillators.
Now, that we notionally represent like this. This first picture here is, this is a plate
and these are the oscillators that correspond to generalize coordinates in the
frequency range of interest. Similarly, this is a beam where there are few more oscillators
here which corresponds to the generalize coordinates of the cantilever
beam. This system receives some power pi 1 in and it dissipate some power pi 1 diss.
Similarly, this system receives pi 2 in and dissipates pi 2 in. Now, we have
studied how energy exchange takes place between two oscillators. Now, what we do is we assume
that the energy flow from plate to cantilever consists of energy
exchange between the oscillators in this set with oscillators in this set; that means this
oscillator exchanges energy with this, this and this. Similarly, this oscillator
exchanges energy with all the oscillators here. So, that means all these oscillators
exchange energies.
Now, the average power flow from alpha to beta - that means plate to the cantilever
- is given by the, a constant multiplier which is again dependent on the plate
and cantilever characteristics and in some sense in an energy difference between the
two plates, that is, total energy divided by number of oscillator that are
participated. What is an energy density or the concentration of energy is total energy
divided by number of oscillator? It is like temperature in thermal analogy. So,
it is a hot means total energy divided by number of modes is high. So, here, total energy
divided by number of modes here. So, the flow takes place from hot spot
to the cold spot and that the energy flow is proportional to the energy difference and
its proportionality constant involves the properties of the structure.
So, we have omega is a central frequency; delta omega is a frequency bandwidth and e
expected e alpha is total average energy in the subsystem alpha and n n
alpha is a number of modes per unit frequency interval for the subsystem alpha; that means
omega, that defines this that the way energy is defined this, this, factor
is relevant. So, what this means is - energy flow between conservatively coupled linear
subsystems excited by a broad band random excitation is proportional to
the difference subsystem average model energies. Now, this proportionality constant etcetera
- this is known as coupling loss matrix, coupling loss factor matrix -
the elements all that have to be determined. By, in this particular case, when we discuss
two oscillators, we derived this h 1 2. That is shown here using random
vibration principles.
So, the S e a equation, the equation that you will actually solve when you apply statistical
energy analysis will be in this form. The unknowns are these energy
levels E i and E j. This is E 1 and E 2 in this particular 2 subsystem problem, and the
excitations or the inputs in terms of power pi in 1 pi in j. This is known; these
are unknowns, and this is analogs to are stiffness matrix but this is known as matrix of coupling
loss factors. Typically, this coupling loss factors could be estimated
experimentally where you know the difficulty is associated with joint flexibilities, I
mean randomness etcetera the system properties can be taken in to account or
analytically using wave propagation concepts. I will not get in to that.
The same logic if you applied to a set of n subsystems, we get the unknowns are E 1
E 2 E 3 E n s - where n s are the number of subsystems. Pi in 1 pi in 2 and pi
in n s are the inputs, and this is the matrix of coupling loss factors which has to be determined
as a experimentally or by analytically, and the principles of random
vibration analysis are relevant in this context.
So, this subject of statistical energy analysis quiet worst and I have given few of you suggestions
for further reading. There is a review paper by Fahy's statistical
energy analysis a critical review, and the two papers - one by Langley and another by
Keane which contain quiet useful information. Now, with this, we will
conclude this present lecture. In the next set of four lectures, we will consider certain
problems which basically would help you to learn the subject. So, the
discussion on principles of random vibration analysis and the methods and the basic problems
etcetera now closes in the remaining four lectures of this course. We
will consider a set of problems and tackle them. We conclude this lecture at this stage.