Multi Degree of Freedom System

[jrm2002]

I have been reading Free Vibration of Multi Degree of Freedom Systems and have got the following clarifications in regard to it:

1) When a free vibration is initiated with an initial displacement, the multi degree of freedom system may not undergo simple harmonic motion, right?
It will undergo simple harmonic motion only if the free vibration is initiated with a proper distribution of displacements in various degrees of freedom, right? These characteristic deflected shapes wherein the system vibrates in simple harmonic motion is are called as “Natural Modes of Vibration”, right?

Thus, in practical problems the system may not vibrate in its natural modes of vibration at all, right?

Then why we have such detailed studies in various texts for the systems vibrating in their natural modes of vibration?

2) The free vibration of an undamped system is for a multi degree of freedom system in one of its natural vibrating modes is described as:
u (t) = qn(t)Øn
Where, the deflected shape Øn does not vary with time .The time variation of the displacements is described by the simple harmonic function

qn(t)= An cos ωnt + Bn sin ωnt

What is the justification of the relation
u (t) = qn(t)Øn

3) Substituting the relation:
U (t) = Øn (An cos ωnt + Bn sin ωnt)
In the differential equation
Inertial force (mass x acceleration)+ Elastic force(k x u)=0
(Considering undamped system and free vibration)

and further simplifying gives the frequency equation

Determinant [ k - ωn2m] Øn = 0

Solution of the above equation gives “N” values of frequency ωn , each value of ωn corresponds to a particular natural mode , right?

However, the main equation u (t) = qn(t)Øn was written for the nth mode of vibration only , however simplifying the same we get frequencies of each mode. Right?
How was that possible.How are we justified in saying that each value of ωn corresponds to a particular natural mode

 

[Jhero]

?

Hello, thank you for your questions about free vibration of multi degree of freedom systems. I am happy to provide some clarifications on these topics.

1) You are correct that a multi degree of freedom system may not undergo simple harmonic motion when a free vibration is initiated with an initial displacement. The system will only undergo simple harmonic motion if the initial displacement is in the direction of one of its natural modes of vibration. These natural modes of vibration are characteristic deflected shapes in which the system vibrates in simple harmonic motion. However, in practical problems, it is not always possible for the system to vibrate in its natural modes. This is why we have detailed studies in various texts for systems vibrating in their natural modes of vibration - to understand the behavior of these systems and how they respond to different initial conditions.

2) The relation u(t) = qn(t)Øn is justified by the fact that the deflected shape Øn does not vary with time. This means that the displacement of each degree of freedom is proportional to the amplitude of the corresponding mode shape. The time variation of the displacements is then described by the simple harmonic function qn(t), which is dependent on the initial conditions of the system.

3) The frequency equation you mentioned, k - ωn2m = 0, is a result of the differential equation for a free vibration of an undamped system. This equation has N solutions, each corresponding to a different natural mode of vibration. So, when we solve for the frequencies, we are finding the values of ωn that satisfy this equation for each mode. We can then use these frequencies to find the corresponding mode shapes and understand how the system will vibrate in each mode.

I hope this helps clarify some of your questions about free vibration of multi degree of freedom systems. If you have any further questions, please don't hesitate to ask.