MODAL SPACE - IN OUR OWN LITTLE WORLD by Pete Avitabile

I showed some mode They asked if the

structural design What should
shapes to someone
I tell them?
was OK

Illustration by Mike Avitabile

i showed some mode shapes to someone

they asked me if the structural design was ok
what should i tell them ?

If I could have a dollar for every time I have heard that Notice that there is a force on the right hand side of this
question, I'd be rich! The basic answer is you just don't have equation. When we solve for the characteristic equation of the
enough information to answer that question. People who ask system, we assume that there is no force on the right hand side.
that question have no idea what they are asking about. You This is how we obtain the dynamic characteristics of the
have to be very diplomatic in telling them that the question is a system. One way to look at it is that the modes of the system
silly one to ask. are nothing more than a very elaborate set of filters which have
the ability to amplify and attenuate an input signal on a
One of the reasons why they are apt to ask the question is frequency basis. If we just look at the filters themselves, can
because you probably showed them an animation and their we make any assessment whether the filters are good or bad for
impression is that the structure is deforming (since they see the a particular application? Of course not! All we can say is that
deflections on the computer screen). Of course, you know that the filters have some characteristics which relate to a center
this is only a characteristic shape that the structure will frequency, rolloff and some gain settings as seen in Figure 1.
undergo when subjected to a force that excites that mode.
Sometimes I've been known to say "well let's increase the
amplitude of the animation and see if we get the structure to
G1 R1

break on the screen". (Of course, this is ridiculous!!! This RO1 D1

can't happen.) I use this statement to start to explain what

R2

shapes are all about. Animation is only a mechanism to G2

RO2 D2

R3

understand how the structure may deform if that mode is G3

D3

excited by the forcing function. CF2

RO3

CF1 CF3

F1 F2 F3
One of the key points here is that we need to know the applied
Figure 1
force. For some reason, people forget that we need a force
applied to the system to get a response. The physical equation
Well ... the dynamic characteristics of a structural system are
of motion is
[M] {&&x} + [C] {x& } + [K] {x} = {F( t )}
quite the same. We can identify each mode (each filter) as
having a natural frequency (center frequency), damping
and the equivalent modal space representation is (rolloff) and residue/mode shape (gain). We need to very
\ \ \ clearly understand that the mode shapes are only characteristics
M &&p + C p& + K p = U T F
{ } { } { } [ ] { }
and we cannot determine the goodness or badness of a mode
unless we know the forcing function - that is, the right hand
\ \ \ side of the equation.

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I showed someone some mode shapes ... Copyright 2000
SEM Experimental Techniques - February 2000 Page 1 Pete Avitabile
As another example, let's say we wanted to determine the order to get the output of the system. That output could then be
stiffness of a cantilever beam. Well, we could go out in the lab transformed back to the time domain if desired. Well, the
and apply a force to the tip of the cantilever beam and measure important point to make here is that the FRF is multiplied by
the resulting displacement. We know that we could determine the force spectrum.
the stiffness as K = F / X. Now this stiffness is an important
parameter or characteristic of the beam. But once I determine That means that the input force spectrum is amplified and
the stiffness, do I know if the beam will fail or not? Of course attenuated by this multiplication. The FRF controls how this
not! I would need to know the actual force that was applied to force is amplified and attenuated on a frequency basis. In the
the beam - wouldn't I? You see, in the test lab we applied an figure above, the FRF appears to have contribution for all four
arbitrary force and measured the displacement due to that force modes shown. That assumes that the applied force and
in order to determine the character of the beam. Someone response location exists at a point where there is participation
needs to identify the actual real world force before I can of each of the four modes of the system.
compute the actual displacement. And then I need to have
some specification defined as to how to assess the acceptability But what if the force was applied at a location of a node of a
of the structure due to the design or real world forces - which mode. Let's say that the force was applied along the symmetry
brings me to another important point. line along the length of the plate. Then, the applied force
would not excite any of the torsional modes from that location;
One thing that people often forget is that once the mode shapes then we say that those modes don't participate in the response
are obtained and a dynamic design force is specified, the of the plate due to that force. The same is true for the response
response can be computed, but someone needs to identify a location. So we can see that both the input and output locations
specification defining what is acceptable and unacceptable for will have an effect on the response of the system. (In fact, the
the response. This, at times, can be one of the most frustrating mode shape amplitudes have a strong influence on how much a
parts of the structural dynamic response modeling process. The particular mode contributes to the overall response.)
responses can be computed but no one has defined what the
level of acceptance is. Many times this very important detail is While we could say that certain modes may not participate in
overlooked in the process of extracting pretty animated mode the response of the system, that does not imply that those
shapes. Then everyone asks... how much deflection is modes don't exist - they just are not needed to compute the
acceptable, how long will the component life be, does it "feel" response of the system. But the modes still exist - they define
good, is the response too noisy, etc. the dynamic characteristics of the system. Depending on the
location of the applied force and the point where response
needs to be measured (as well as the frequency content of the
signal), will determine how the structure responds. Some
modes may be more dominant in the level of response and
others may be less dominant in the response - again depending
INPUT TIME FORCE
OUTPUT TIME RESPONSE on the particular input-output location selected. But all the
f(t) y(t)
modes exist - they just may not all be activated on a uniform
FFT IFT basis.

So what we need to remember is that a modal test only defines

the character of the system. We apply an arbitrary force which
is measured along with the response of the system due to the
INPUT SPECTRUM FREQUENCY RESPONSE FUNCTION OUTPUT SPECTRUM
applied force. This enables us to determine the dynamic
f(j ) h(j ) y(j ) characteristics of the system - the frequency, damping and
Figure 2 mode shapes. These are only characteristics of the system. We
display the mode shape (animate them) to better understand
So now that we have discussed a few of these things, let's go how the structure may deform if a force is applied to the system
back to our plate example that we have discussed before that excites one or more modes of the system. Remember,
concerning different aspects of modal analysis. Figure 2 shows modal analysis doesn't use the force on the right hand side of
a schematic of a typical forced vibration problem. There is the equation - the mode shapes are independent of the force.
some force which is applied in the time domain. Well, this
time signal is very confusing so it helps to identify some Now I hope you understand why you can't answer the question
important characteristics of this force if it is transformed to the that you asked. If you have any other questions about modal
frequency domain using the FFT process. Now I know that this analysis, just ask me.
force is multiplied times the frequency response function in

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I showed someone some mode shapes ... Copyright 2000
SEM Experimental Techniques - February 2000 Page 2 Pete Avitabile