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MODAL SPACE Back to Basics   by Pete Avitabile

I am still confusedabout how the SDOF

in modal space

is related to thephysical response?

This needssome discussion

 Well - this is a concept that is actually verysimple but does need some explaining tomake sure it is comprehended properly.First let’s start with a few summary

equations that we have presented several times before inprevious articles. Of course the equation of motion in matrixform is the starting point

[M]{x} + [C]{x} +  [K]{x} = {F(t)}

This coupled set of matrix equations is then uncoupledby performing an eigensolution. The modal transformationequation is obtained from the set of modal vectors obtainedfrom the eigensolution. The physical coordinate {x} is related

to the modal coordinate   {p}   using the collection of modal vectors [U]

{x} = [U ]{p} = {u1}p1 + {u2}p2  + {u3}p3 +  . . .

with [U]  =  [{u1}{u2}{u3} . . .]

Substituting this into the physical equation and premulti-plying by the transpose of the projection operator [U] willresult in a very simple diagonal set of equations in modalspace where every equation (modal oscillator) is orthogonaland linearly independent (uncoupled) from each other and isgiven as

  m1

m2

\

p1

p2

...

+

  c1

c2

\

p1

p2

...

+

k1

k2

\

p1

p2

...

=

{u1}T{F}

{u2}T{F}...

The series,   Modal Space—Back to Basics , features a variety of topics relating tomodal analysis and its use for both analytical and experimental work. Some of thematerial presented here is excerpted from the multimedia format  Modal Handbookon CD. Reader’s questions regarding some common problems or misconceptionsrelating to modal analysis are welcome. These questions should be forwarded toExperimental Techniques , Jen Tingets, journals@sem1.com.

 Dr. P. Avitabile (SEM Member) is the director of the Modal Analysis and Controls Laboratory and professor in the Mechanical Engineering Department at the

University of Massachusetts, Lowell, MA.

There are several important things that need to be notedabout this equation. And the mode shape matrix [U] has alot to do with this.

First is that every equation contains only one variable todescribe each equation—the modal displacement for eachparticular mode. Second is that every equation is uncoupledfrom every other equation. Third is that each equation isbasically a very simple single degree of freedom (SDOF)system. Fourth is that the right hand side of the equationidentifiesthe force that is appropriatedto the modal oscillatorfrom the physical force applied to the physical system.Figure 1 shows a schematic of a multiple degree of freedomsystem (MDOF) where coupling between degrees of freedom

exist in the physical model and the resulting equivalent setof SDOF systems representing the modal system in modalspace.

So if we write out any one equation from the modal spaceform and use an ‘‘i’’ subscript the ‘‘ith’’ equation we would get

mipi +  cipi +  kipi  = {ui}T{F} = f i

So with this simple SDOF equation we can calculate theresponse due to any force applied on the equivalent system.Of course we can see that the right hand side of the equationidentifies how much of the force is appropriatedfrom physical

m

k 

1

1 c1

m

p

3

3

c3

m

k 

2

2 c2

f3

k 3

p 2f2

f1p 1

MODE 1 MODE 2 MODE 3

PHYSICALMODEL

COUPLED

MODAL MODEL UNCOUPLED

Fig. 1: MDOF System Schematic and SDOF

Equivalentdoi: 10.1111/j.1747-1567.2010.00648.x© 2010, Copyright the AuthorJournal compilation© 2010, Society for Experimental Mechanics   July/August 2010   EXPERIMENTAL TECHNIQUES   13

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MODAL SPACE Back to Basics

m

k 

1

1c1

f1p

1

MODE 1

)1(n

3

2

1

u

u

u

u

×

1

2 3

Fig. 2: Schematic Response for Mode 1 Contribution

space to the equivalent modal system through the modeshape. With this force, then the response for the equivalentsystem can be identified. This response can be simply foundfrom any Vibrations textbook—usually this is one of thefirst four chapters in most textbooks for free response, forcedsinusoidal response or arbitrary input response. For sake of the discussion here, let’s assume that an impact is applied atone point on the physical system.

Now that physical force will be appropriated to each of themodal DOF in modal space. So if we look at the first modethen we could calculate the impulse response for the SDOFdescribing mode 1. This SDOF response is then distributedover all the physical DOF using the modal transformation

equation; this essentially scales the SDOF response to allthe physical DOF using each value of the first mode shape ateach individual DOF. This is schematically shown in Figure 2(for just a few DOF to illustrate the concept). Now this onlyprovides the part of the response of the physical system thatis related to the contribution that mode 1 makes over allthe physical DOF in the system; the portion of the responserelated to mode 1 is shown in blue.

This is not the entire physical response of the system—itis just the portion of the response that is related to mode1. Now the contribution of the other modes needs to alsobe included. If we look at the second mode then we couldcalculate the impulse response for the SDOF describing mode

2 with the force that is appropriated to mode 2. Again thisSDOF response for mode 2 needs to be distributed over allthe physical DOF using the second mode shape; this onlyrepresents the portion of the response that is related to thesecond mode of the system. This is shown in Figure 3 (for

 just a few DOF to illustrate the concept); the portion of theresponse related to mode 2 is shown in red.

This process is then continued for all the modes thatcontribute to the total response of the physical system.

m

k 

2

2c

2

f2

p2

MODE 2

)2(n

3

2

1

u

u

u

u

×

1

2 3

Fig. 3: Schematic Response for Mode 2 Contribution

{u }

[ M ]{p} + [ C ]{p} + [ K ]{p} = [U] {F(t)}T

m

k 

1

1c1

f1

p1

MODE 1

m

k 

2

2c2

p2

f2

MODE 2

m

p

3

3

c3

f3

k 3

MODE 3

MODAL

PHYSICAL MODEL

[M]{x} + [C]{x} + [K]{x} = {F(t)}

{x} = [U]{p} = [

SPACE 

{x} = [U]{p} =++

{u }p33

{u }p22

{u }p11

{u }3

{u }21

]{p}

.. .

.. .

+

+

Σ=

Fig. 4: Overview of the Modal Space Representation

Of course you have to include all the modes that havea contribution to the overall response otherwise some of the solution is lost. The entire process is best seen inFig. 4. This figure shows the physical equation and themodal transformation equation which allows the coupledphysical system to be written as a set of equivalent SDOFsystems in modal space with the equivalent modal forceapplied on all the modal oscillators in modal space. It is

important to realize that each mode is linearly independentfrom every other mode but that the total response ismade up of the linear combination of the response of all the modes that participate in the response of thesystem.

I hope this explanation helps you to understand how theSDOF response is characterized in physical space from themodal space response. If you have any other questions aboutmodal analysis, just ask me.

14   EXPERIMENTAL TECHNIQUES  July/August 2010