INTELLIGENT WORK FORUMS
FOR ENGINEERING PROFESSIONALS

Log In

Come Join Us!

Are you an
Engineering professional?
Join Eng-Tips Forums!
  • Talk With Other Members
  • Be Notified Of Responses
    To Your Posts
  • Keyword Search
  • One-Click Access To Your
    Favorite Forums
  • Automated Signatures
    On Your Posts
  • Best Of All, It's Free!

*Eng-Tips's functionality depends on members receiving e-mail. By joining you are opting in to receive e-mail.

Posting Guidelines

Promoting, selling, recruiting, coursework and thesis posting is forbidden.

Jobs

longitudinal vibration of continuous system (1D bar)

longitudinal vibration of continuous system (1D bar)

(OP)
Hi,
I have been looking at working out the natural frequency and mode shapes of a 1D bar. I have now understood the standard cases, for instance the fixed-free bar with BCs: u@(x=0) = 0; and du/dx@(x=L) = 0.
I need to tackle a case where the free end is replaced by du/dx@(x=L) = P(t), where P(t) is a pressure force and it comes in the form of a set of numbers.
So I have started by expressing u as:
u(x,t) = G(x)*F(t) = [B1*cos(w*x/c)+B2*sin(w*x/c)]*[A1*cos(w*t)+A2*sin(w*t)]

Applying u@(x=0) = 0 gives B1 = 0
Applying du/dx@(x=L) = P(t) gives u(x,t) = (w*L/c)*[B2*cos(w*L/c)]*[A1*cos(w*t)+A2*sin(w*t)] = P(t)

(Note: B1,B2,A1,A2 are constants, c = sqrt(E/rho))
This is the point I get stuck as the 2nd boundary condition does not help. I would appreciate any hints ASAP. Please let me know if this cannot be solved.
Thank you!

RE: longitudinal vibration of continuous system (1D bar)

Congratulation for this work.

Sometimes you can use B1*exp(jkx) and B2*exp(-jkx) which is more physical instead of sin() and cos() but from mathematics it's completly identical.

The first case is a free response, the second case is a forced response.

The time is given by the external force P(t) = P1*exp(jwt). So I believe that you can forget the temporal part of the solution.

RE: longitudinal vibration of continuous system (1D bar)

(OP)
Hi and thank you.
I will have a think about that and possible come back here with more questions on the same problem.

RE: longitudinal vibration of continuous system (1D bar)

The vibration response is always the overlap of a free response, which is a transitional regime, and a forced response.
There are always the both.

Nevertheless, if you apply an impact (with a hammer for example), the vibration response is just composed of the free response (forced response only exists during the impact which lasts a very short time).

If a forced excitation is applied (with a shaker for example), you have first the both regimes (free and forced), but very quickly, the free regime decreases and becomes completely negligeable. Then the vibration response is only equal to the forced response.

This physical explanation should help you to writte the right équations.

Red Flag This Post

Please let us know here why this post is inappropriate. Reasons such as off-topic, duplicates, flames, illegal, vulgar, or students posting their homework.

Red Flag Submitted

Thank you for helping keep Eng-Tips Forums free from inappropriate posts.
The Eng-Tips staff will check this out and take appropriate action.

Reply To This Thread

Posting in the Eng-Tips forums is a member-only feature.

Click Here to join Eng-Tips and talk with other members!


Resources


Close Box

Join Eng-Tips® Today!

Join your peers on the Internet's largest technical engineering professional community.
It's easy to join and it's free.

Here's Why Members Love Eng-Tips Forums:

Register now while it's still free!

Already a member? Close this window and log in.

Join Us             Close