# About solving state space equation (mass, spring, damper)

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Changwoo Lee on 1 Oct 2022
Hello, I have 4-DOF mass, spring, damper sysmtem.
1) My first question is how to construct the state space equation of the below diagram?
-The first force (F1) is imposed during the time t0~t1.
-Then, second force (F2) is imposed during the time t1~t2.
2) Then, within th time t0~t2, I want to get the each displacement (x1,x2,x3,x4)
In this case, how solve the system?
3) In my case, actually, F1 and F2 are not constants.
F1 is function of x3 (for example, F3=3*sin(x3))
F2 is function of x4 (for example, F4=2*cos(x4)) William Rose on 1 Oct 2022
@Changwoo Lee, this looks and sounds a lot like homework. This is not a homework site. Also, your quesitons are about math and physics, not Matlab. There are better sites for those questions - stack exchange for example.
Here are hints to get you started: for each mass, the equation is F = m*a = m*dv/dt = m*d2x/dt2. You can look at the diagram to determine the all the forces acting on each mass. Each mass has a position and a velocity. Those quantities are enough to describe the system.

Sulaymon Eshkabilov on 1 Oct 2022
(1) Derive the equation of motion that will be 2nd order Differential equations
(2) Transform the derived equations into the Laplace transform i.e., tranform from 't' domain into 's' domain
(3) Write down the transfer functions of the system in 's' domain using tf()
(4) Convert the found trnasfer function into state-space representation using tf2ss()
(5) etc.
In fact, to find the response of your system you can use tf() formulation from step 3 directly.

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