Piecewise Exact Method

[willowman]

Hi,

Looking for some clarification regarding the piecewise exact method when applied to SDOF non-harmonic systems.

I understand that the equation of motion that describes the behaviour of the system is as below.

https://files.engineering.com/getfi...c&file=Screen_Shot_2020-04-12_at_12.38.02.png

I understand that the u (displacement) and u'(velocity) terms are given by some fairly lengthy integration, but none the less they are below attached.

https://files.engineering.com/getfi...d&file=Screen_Shot_2020-04-12_at_12.39.14.png

My question is why do we not need to define a term for u'' as this appears in the original equation describing the motion. Is it because if the step size is sufficiently small that acceleration is assumed to be zero and therefore u'' = zero?

If i'm wrong, is anyone aware of the term for u'' in terms of A, B, C and D before I attempt a length differentiation?

 

[Blackstar123]

My question is why do we not need to define a term for u'' as this appears in the original equation describing the motion.

Because you don't require u" as input, to determine the dynamic response. If the final solution doesn't contain a u" term, it doesn't necessarily mean that the acceleration is negligible or wasn't used in the derivations of the final solution.

In your case, the total dynamic response is the sum of the following three individual cases.
1. Displacement due to Free vibration
2. Displacement due to Constant force forced vibration
3. Displacement due to Ramp force forced vibration
The solution to these case are obtained by solving the 2nd Degree Differential equation.
Lets talk about the general steps involved in finding the solution for a homogeneous 2nd order DE (solution of first case, which is also the simplest to find). We proceed as follows.
1. We assume a general solution u(t) of the DE,
2. We find the derivatives u'(t) and u"(t) of the solution of DE assumed in step No.1.
3. We plug these derivatives in the original 2nd order DE i.e. the motion equation.
4. After simplification, we obtain a characteristics quadratic equation.
5. We find the roots of this equation.
6. We put these roots back in step no. 1 and find the most simplest form a general solution.
As I said earlier,

If the final solution doesn't contain a u" term, it doesn't necessarily mean that, the acceleration is negligible or wasn't used in the derivations of the final solution.

 

[willowman]

Thanks for response.

I end up with an expression to describe each SDOF in modal coordinates of the form (attached)

https://files.engineering.com/getfi...5&file=Screen_Shot_2020-04-12_at_18.29.10.png

It seems like I will need a u'' term in order to solve my equation at any given time-step t?

 

[r13]

It looks like the units do not check out, as your equation indicated ---> acceleration + displacement = F (frequency or force)?

 

[willowman]

Hi,

Sorry I should have been clearer.

The y’’ is multiplied by mass matrix and the y is multiplied by stiffness matrix so hopefully units check out then.

 

[r13]

I see. It is in the form - m*y"(t) + k*y(t) = F(t) for SDOF system without damping, correct? You might want to review the linked paper for more helps. Link

 

[Blackstar123]

Why are you ending up with three force components in motion equation for the equivalent SDOF system? There should be a single force that varies with time and acting on the modal mass.
Refer to the following attached figure, which might explain things more clearly than following explanation.

From your equation, it seems like you're determining the response for the case of an undamped forced vibration, since you don't have a velocity term in the attached equation. It doesn't mean if a motion equation doesn't have a velocity term, than the structure will have zero velocity. It just signifies that, the resisting force due to damping will be zero. Damping helps to reduce the displacement due to forced or free vibration.

Assuming, this post is related to your previous post, you have a MDOF system of degree 3.
To make sure we are on the same page, the steps to determine the dynamic response using modal analysis are as follows.
(Correct me if I don't recall the procedure correctly, since it's quite some time when I studied the structural dynamics course)
1. First step will be to find modes shapes and respective frequencies.
2. Then determine modal masses, stiffness and forces (which should be a single value, for each mode)
3. Convert your MDOF into several SDOF system (3 in your case).
4. Find the dynamic response for each system
5. Multiply each response with respective mode shape to find floor displacement for that mode.
6. Add floor displacement for each mode shape to get the total floor displacement.

The method which you are using to find the dynamic response (step no. 4) will become more simple since you don't have to deal with the damping related terms.

https://files.engineering.com/getfi...986fba426&file=Screenshot_20200413_024544.jpg

https://files.engineering.com/getfi...2ed394322&file=Screenshot_20200413_024502.jpg

 

[willowman]

Hi Blackstar123,

I will check out the attachments you have posted now. I end up with three force components due to the fact that I transferred the force vector into modal coordinates. I followed the logic presented in Chapter 6 of Structural Dynamics and vibration in practice (Douglas Thorby). I get the following:

https://files.engineering.com/getfi...&file=Screen_Shot_2020-04-13_at_05.44.48.png.

Regarding forces F1, F2 and F3. F1 is always zero as I don't have force acting on the 1st floor, and forces F2 and F3 are obtained from force-time history data.

 

[Blackstar123]

Oh! I get it now. Your modal force are Q1, Q2 & Q3, and not F1, F2 and F3.