Hi

Your question is a class room question, but a good class room question that is worth delving on.

The simplest way to undrestand this is to look at the swing, i.e. an item most of us have an in-depth study from when we were kids,

The swing is a pendulum, i.e. its resonance is w = sqrt(g/L), where g is 9.82 m/s2 where I live and L is the pendulum length. This provides the same resonance frequency irrespective of passenger weight.

If you push the swing, you can do so at any freqyency. For simpicity, let us assume that you move the swing slowly at large stroke at a frequency much lower than w. This is not easy when the passenger is heavy as, in essence, you then must lift the passnger. The same applies if you try to push the swing much faster than w, only in this case, it is the passnger inertia that resist fast oscillation.

In both cases, as soon as you stop the forced excitation, the swing will use whatever kinetic and potential is avialble and distribute it into its most favorable setting, which is a split where all kinetic energy can shift to potential energy and vice versa, which can be achieved only at its natural frequency, w.

The reason why it is easier to push the swing at w is that it then stores energy when excited at resonance. Free vibration implies vibration at natural frequency.

If we take a mass spring system. Potential energy is 1/2*K*X^2, where K is spring stiffness and X is displacement. Kinetic energy is 1/2*M*V^2. For harmonic vibration X = A*e^jwt, we find that V = jw*A*e^jwt = jw*X.

Inputting and summing kinetic energy, we find that

1/2*K*X^2 + 1/2*M*(jw)^2 = 0

solving whis equation produces the natural frequency w = sqrt(K/M), which just is another way of stating the energy summation listed above . 

Multidof really is the same thing with information added as to where you drive and how this matches the mode shape.

However, to better understand things, take a look here

http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_mdof/vibrations_mdof.htm

http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_forced/vibrations_forced.htm

For modal summation, I enclose part of a course I hold now & then on structure vibration which intended for an audience without prior knowledge on vibration. This excerpt treats a lumped 2dof system and explains most of what you are asking for.

Have fun

Claes

Thank you so much, Dr. Claes.

Really, i don't know about vibration insight out. And, i also don't understand your mean in below paragraph, especially, because my English is not good. And, i just have finished my course for master degree in UTT (an university in Viet Nam). And now, i'm trying to study on modal analysis. And, as you know that, it is too difficult study without supervisor. I don't how to study and where start from?

"If you push the swing, you can do so at any freqyency. For simpicity, let us assume that you move the swing slowly at large stroke at a frequency much lower than w. This is not easy when the passenger is heavy as, in essence, you then must lift the passnger. The same applies if you try to push the swing much faster than w, only in this case, it is the passnger inertia that resist fast oscillation.

In both cases, as soon as you stop the forced excitation, the swing will use whatever kinetic and potential is avialble and distribute it into its most favorable setting, which is a split where all kinetic energy can shift to potential energy and vice versa, which can be achieved only at its natural frequency, w."

What is vice versa?

And next part of your comment, i understood and i know that is as energy method to find frequencies and mode shapes.

Of cause, i haven't read the file and links that you attached. But i'm going to try. 

Again, Thank you so much for your attention, sharing and thoroughness.

Tuan Dung Le

Hi

OK, Vice versa is a Latin phrase that means "the other way around".

Here is another tip - use wikipedia and google to search for things.

https://en.wikipedia.org/wiki/Vice_Versa

A vibration mode is a form of energy storage. Orthogonality means that modes do not share energy.

I believe it may be useful for you to read these

http://qringtech.com/learnmore/recommended-reading/

I encourage you to click around on this site as it was designed for the express purpose of self study on vibration.  http://qringtech.com/learnmore/

This is another good site

http://fsdiagnostics.com/

and some of his ramblings

https://issuu.com/fullspec

Have fun - because this is fun.

Claes