relationship between natural frequency obtained from continuous and discrete models

[gopi9]

Hi,

I know how to extract the natural frequencies from continuous and discrete model by using eigen-value equations.
This is the continuous model Mx..+Cx.+Kx=F (x. is differentiation of x). The eigenvalue equation for this is (K-1/ω2 M)u=0, 1/ω2 represents frequency.

If we convert that continuous model in to discrete ie., x.=Ax+Bu, Y=Cx+Du. The eigenvalue equation is (A-(abs(λ))I)x=0. here abs(λ) represents natural frequency.

I want to know the relationship(formula) between the natural frequency obtained from continuous model and discrete model.

 

[electricpete]

Mx..+Cx.+Kx=F
and
x.=Ax+Bu, Y=Cx+Du
can be the same relations, just different form.
The first form is preferred by mechanical engineers
The second form is called state space and is stanard in controls.
The dimension of x in the 2nd form (system of 1st order differential equations) will be twice the dimension of x) will be twice the dimension of f in the 2nd equation.
The eigenvalues are the same.

Both represent a spatially discrete system, although the discrete paraemters may be derived to preserve some aspects of a particular continuous sytem.

Both represent a continuous time system.
When set up properly they represent the same system and have the same eigenvalues.

Given all above, it's not at all clear what the question would be.

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(2B)+(2B)' ?

 

[electricpete]

The dimension of x in the 2nd form (system of 1st order differential equations) will be twice the dimension of x) will be twice the dimension of f in the 2nd equation.

should've been

The dimension of x in the 2nd form (system of 1st order differential equations) will be twice the dimension of x in the 1st form (system of 2nd order differential equations).

=====================================
(2B)+(2B)' ?

 

[electricpete]

Actually there is some terminology differences to reconcile comparing the various and asundry mechanical approaches to the one standard unvarying well-defined state space approach.

An undamped system has a pure imaginary eigenvalue s = j*w0 in the state space approach.

You may solve your mechanical equation of the same undamped problem for areal number w0.

If you the state space solution j*w0 different then the mechanical solution w0, that's up to you. They represent the same solution.


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(2B)+(2B)' ?

 

[electricpete]

The state space solutions to a real problem come in complex conjugate pairs, so should have been
s = +/- j*w0

=====================================
(2B)+(2B)' ?

 

[gopi9]

Thank you so much....I already figured the problem myself..

 

[gopi9]

I have one more question...
can we compare damped natural frequency with the undamped natural frequency.Is there any relationship between them.

 

[electricpete]

Look in your vib textbook probably the first chapter about single degree of freedom systems.

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(2B)+(2B)' ?