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# State space model of mass spring damper with base excitation

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## State space model of mass spring damper with base excitation

(OP)
Dear,

I want to make a state space model of a simple mass spring damper system with a base excitation as shown in the figure below.

The equation of motion is:
mx'' + bx' + kx = ky + by'

How can I write this in state space (A,B,C,D) format? I have difficulties with finding the B matrix.

y(t) = Y sin (Wt)

### RE: State space model of mass spring damper with base excitation

(OP)
Hi IRstuff,

Thank you for the link.
This is almost what I'm looking for (slide 52). However, I'm struggling to write the harmonic forcing functions (2.63) in state-space format (B matrix). Analytical, they do a superposition of the two individual particular solutions. Have you an idea of how this done in state-space format?

### RE: State space model of mass spring damper with base excitation

Forcing functions are inputs that are external to the state space model. They don't go in the B matrix, but are in the u term, assuming xdot = Ax + Bu; y = Cx + Du.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
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### RE: State space model of mass spring damper with base excitation

(OP)
Hi xnuke,

Thank you for your reply.
So my B matrix is: [0; 1/m]
And u is: bWY cos (Wt) + kY sin (Wt)
If xdot is: [x'; x'']

Right?

Steven

### RE: State space model of mass spring damper with base excitation

Steven, that looks correct for u and B based on your definition of the state vector.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

### RE: State space model of mass spring damper with base excitation

(OP)
Thank you!

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