## Longitudinal vibration in a bar that is clamped at left end and F(t) applied at the free end

## Longitudinal vibration in a bar that is clamped at left end and F(t) applied at the free end

(OP)

I am performing a very basic FEM dynamic analysis. I am trying to find out how each node in the bar deforms in this analysis.

(1). For example, when I am using just 1 truss element:

Step1: Found the natural frequency (f) of the bar(using frequency analysis)

Step2: Found time period (T) = 1/f.

Step3: Applied Impulse excitation [F(t)] at the free end of the bar.

Amplitude of Impulse excitation: 0,0; 0.2*T, 1; 0.4*T,0; 2T,0

Step4: Observed the deformation at 'node2' of the bar which is in the form of a wave.

Step5: The total time of the simulation is 2T.

(2). Now, when I am using just 2 truss elements for 1 bar:

Step1: Found the natural frequency (f1, f2) of the bar with 2 elements(using frequency analysis)

where f1 < f2

Step2: Found time periods T1 = 1/f1 and T2 = 1/f2.

Step3: Applied Impulse excitation [F(t)] at the free end of the bar.

Amplitude of Impulse excitation: 0,0; 0.2*T2, 1; 0.4*T2,0; 2T1,0

Step4: Observed the deformation at 'node2 and node3' of the bar which is in the form of waves.

Step5: The total time of the simulation is 2T1.

My questions are:

1. What is the right way to apply the Impulse excitation as we increase the number of elements.?

2. When I apply Neumann bc, I expect the stress profile to have the same profile as the applied load amplitude profile is. However, I see a wave profile for stress at node3 of the bar.

Is it correct to expect the above?

3. With increasing mesh size (1,2,5,10,20,50,100,200,500 truss elements), if I apply Impulse excitation [F(t)] at the free end of the bar (shown below),

[***Amplitude of Impulse excitation for all the elements: 0,0; 0.2*T1, 1; 0.4*T1,0; 2T1,0; where T1 is time period based on first frequency***]

I get waves in such a way that all the nodes deforming have the same time period, but the stress profile of the free end has the same profile as the amplitude shown above.

However, the structure resonates here.

What is the correct approach to perform a vibration analysis of a cantilever bar when impulse excitation is applied at the free end?

Please let me know.

(1). For example, when I am using just 1 truss element:

Step1: Found the natural frequency (f) of the bar(using frequency analysis)

Step2: Found time period (T) = 1/f.

Step3: Applied Impulse excitation [F(t)] at the free end of the bar.

Amplitude of Impulse excitation: 0,0; 0.2*T, 1; 0.4*T,0; 2T,0

Step4: Observed the deformation at 'node2' of the bar which is in the form of a wave.

Step5: The total time of the simulation is 2T.

(2). Now, when I am using just 2 truss elements for 1 bar:

Step1: Found the natural frequency (f1, f2) of the bar with 2 elements(using frequency analysis)

where f1 < f2

Step2: Found time periods T1 = 1/f1 and T2 = 1/f2.

Step3: Applied Impulse excitation [F(t)] at the free end of the bar.

Amplitude of Impulse excitation: 0,0; 0.2*T2, 1; 0.4*T2,0; 2T1,0

Step4: Observed the deformation at 'node2 and node3' of the bar which is in the form of waves.

Step5: The total time of the simulation is 2T1.

My questions are:

1. What is the right way to apply the Impulse excitation as we increase the number of elements.?

2. When I apply Neumann bc, I expect the stress profile to have the same profile as the applied load amplitude profile is. However, I see a wave profile for stress at node3 of the bar.

Is it correct to expect the above?

3. With increasing mesh size (1,2,5,10,20,50,100,200,500 truss elements), if I apply Impulse excitation [F(t)] at the free end of the bar (shown below),

[***Amplitude of Impulse excitation for all the elements: 0,0; 0.2*T1, 1; 0.4*T1,0; 2T1,0; where T1 is time period based on first frequency***]

I get waves in such a way that all the nodes deforming have the same time period, but the stress profile of the free end has the same profile as the amplitude shown above.

However, the structure resonates here.

What is the correct approach to perform a vibration analysis of a cantilever bar when impulse excitation is applied at the free end?

Please let me know.

## RE: Longitudinal vibration in a bar that is clamped at left end and F(t) applied at the free end

GregLocock(among others) posts there and he also posts here so he might weigh in.) In the meantime, I will take a stab at a few of the questions myself.That depends. Are to trying to capture longitudinal modes, lateral, or what? If it's longitudinal, a lot will hinge on support conditions (i.e. the spring constant(s) in that direction).

Are you simulating a longitudinal wave? Or Rayleigh? Love waves?

Another question you have to ask yourself is: do you even need to do a wave propagation type analysis? Most (commercially available) software can give you frequencies/deformation without simulating wave propagation.

Just so you know: your pic didn't post. But are you sure you are looking at the free end at the right time? If you think the amplitudes should be higher....sometimes you may be looking at it during a transient period of vibration. (Since the pulse passes and reflects very quickly and the vibration occurs a short time

afterthat.)By the way, if you are going with that many elements.....have you thought about using Spectral Finite Element software? (If it is available to you.) It really cuts down on the need for that.

## RE: Longitudinal vibration in a bar that is clamped at left end and F(t) applied at the free end

Thank you so much for responding. Yes, it is an elastic wave propagation problem and I am looking to simulate longitudinal waves.

The problem is a 1-D bar fixed at the left end and F(t) applied at the right end.

This problem is similar to what the researchers from NDT (Non-destructive testing) field use to determine material properties in a solid.

My major questions here are:

1. How to capture wave propagation in the elastic medium using commercial software like Abaqus.

2. If 'n' number of elements are used, how does the wave propagation take place without resonating.

Regarding the stress at the tip location part:- in differential equation form, when I specified a Neumann BC (u'(x=1,t) = F(t)/AE)it means that the stress field output that I get should have the same pattern similar to amplitude of my F(t). This means I am doing something really wrong here.

I will also post the same question on the Mechanical Acoustics/Vibration engineering forum. Thank you for the suggestion.

## RE: Longitudinal vibration in a bar that is clamped at left end and F(t) applied at the free end

Double posting is generally discouraged....so you may want to wait a while first. I just mentioned that forum in case you don't get satisfactory answers here. (I should have mentioned that in my first post.)

Abaqus has good technical support. I've seen all sorts of waves simulated on it (i.e. love, lamb, etc.). I think they have some vids on YouTube as well.

A big part of that is the forcing frequency of the signal. If it is matching your system (and the signal is repeated/reflected) you could have a resonance issue.

If you are simulating a situation where the signal would not be reflected nearby.....you may want to have (what is called) a "throw-off element" to prevent reflection.

And I want to emphasize again: if you are getting displacement errors....you mentioned that you have a "clamped" end. Longitudinally, you aren't going to get much vibration if that is the case. You'd have to introduce some flexibility.