3DOF - Modal Superposition

[willowman]

Hoping someone might be able to assist me with a question regarding modal superposition.

I have a 3DOF system (3 storey structure) which I have calculated a modal mass matrix and a modal stiffness matrix for. Both of these matrices are 3x3 and diagonal so I understand I can now progress to use method of modal superposition to define a response for each mode. The missing part for this method is the modal force vector.

I have done examples in the past where the force P is given as say 0 on the 1st floor, 2000N on the second and 3000N on the third. Calculating the modal force vector is simple enough in this case as it's just the tranpose of the mode shapes * the force matrix.

However, I am now looking at an example where I have been given Force vs Time data individually for each of the forces. 1st floor has no applied loading, 2nd floor has load F2 and 3rd floor has load F3. How do I write a modal force vector when I have 10000+ force amplitudes for F2 and F3 spread across a range of times?

For info I have:

The modal mass matrix is:
[[ 1. -0. 0.]
[-0. 1. 0.]
[ 0. 0. 1.]]

The modal stiffness matrix is:
[[ 86.072 -0. -0. ]
[ -0. 298.588 0. ]
[ -0. 0. 1873.671]]

I am assuming I need a 3x3 modal force matrix and that at any given time-step in the response I should be going to the force value in the provided data?

Any help would be massively appreciated!

 

[hardbutmild]

I think you need to give us some more information about this. For example, I don't even understand what you're trying to achieve here. What is your end goal?
I ask this because it seems like you're not actually doing time-history analysis (or am I wrong?). Your stiffness matrix is constant which means that you're doing some sort of linear analysis, right?

I'll guess that you have a realistic earthquake, so why not use something like duhamel's integral (I'm a bit rusty on dynamics so I can't remember if it can be used for MDOF, but I don't see why not) to get acceleration, velocity and displacement of each node at each time step. From there you could get the forces for each step, right?
If you're having trouble with too much data, you could just keep the largest value (since that's important to you I guess) by comparing the value with the previous step.

Again, without a more detailed end goal I guess I can't give you a better answer. What is the point of getting these forces at each time step if you're not actually updating the stiffness matrix?

 

[Blackstar123]

I am assuming I need a 3x3 modal force matrix and that at any given time-step in the response I should be going to the force value in the provided data?

You mean you'll form another table for story forces respective to a mode shape n,
Pn(t) = Trans[phi]n * f(t)
Where, P(t) and f(t) is equal to size of your matrix that is 1x3 or 3x3 (depends on if you do the multiplication for all mode shapes simultaneously)
That's what I would do too.

Do you see a pattern in f(t) in your given data? May be you can form an equation for f in terms of t and then solve the problem with t as variable. I take your end goal is to find the story displacement?

 

[GregLocock]

" I understand I can now progress to use method of modal superposition to define a response for each mode"

That doesn't make any sense to me. Perhaps you mean the response at each frequency due to each mode?

Cheers

Greg Locock


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[willowman]

Thank you all for the replies, I found this a difficult question to put into words.

To answer GregLocock and hardbutmild , my end goal is to plot on python the modal response for each mode for the duration of the force-time history. I'd like to write an expression for each mode in the attached notation (

https://files.engineering.com/getfi...file=Screen_Shot_2020-04-12_at_06.10.33.png).

What I am unsure of is how to describe the modal force vector in this notation? My current thinking is that I would represent the system as the following.

https://files.engineering.com/getfi...3c0-83696a578259&file=IMG_7B4D47C2131D-1.jpeg

Any thoughts?

 

[GregLocock]

Ok so you want a time history of the displacements of each node in each direction. And you want to do it for each mode separately.

That's fine, that's exactly what superposition is about. Unfortunately I've never had much to do with stiffness matrices.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[WARose]

It's a 3 story building and the mass/stiffness matrix is 3x3? Are you approximating a actual building as a "shear" building? (I.e. where there is no rotation of the horizontal section at the floor level. I.e. the building is like a cantilever.)

In any case, if you have Mario Paz's book in Structural Dynamics, he determines the max displacements based on a single (sudden) load per floor for such a approximation.

It's a little long to reproduce here so I hope you have it. In any case, not sure how you'd handle it for "10000+ force amplitudes". (I would assume either a simplifying assumption or a computer solution.)