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# Equations of motion of a damped 2-dof system

## Equations of motion of a damped 2-dof system

(OP)
I am required to find the equations of motion for a damped free vibrating 2 dof system. The equations are needed for each mass individually depending on time i.e x1(t) and x2(t). I have found many examples of these equations with no damping but i can't seem to find any explanation for the damped free vibration case. The initial displacement may be varied but the initial velocity is always zero. Would anyone know the equations for each mass for the system? I have attached a diagram of the system
Replies continue below

### RE: Equations of motion of a damped 2-dof system

What textbook do you use? I'd expect any undergraduate level engineering text to have an outline of how to solve this type of problem (2-dof systems) and inherent limitations of certain types of solutions.

### RE: Equations of motion of a damped 2-dof system

Let's see your solution for m1. It'll still refer to x2. A neat trick is to ignore parallel springs and dampers, just use springs, and then at the end substitute (k+jwc) for each spring.

Cheers

Greg Locock

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### RE: Equations of motion of a damped 2-dof system

mmullan,

This looks like an FEA 101 question, complete with the linear algebra. You need to write simultaneous equations for all your degrees of freedom.

--
JHG

### RE: Equations of motion of a damped 2-dof system

I'd call it dynamics or vibrations, not FEA. The actual solution in the frequency domain is not pretty, but if you use Mathcad or Maxima or Mathematica then they will do the heavy lifting, provided you set the equations up correctly. In the time domain it is much simpler, and can be solved for arbitrary x1 and x2 at t=0 in a spreadsheet or MATLAB or Octave, using a small time step, to give a time history of x1 and x2.

However this is irrelevant, OP is stumbling over writing the equations in the first place. Work out the force in each spring as a function of the motion of its ends. Work out the acceleration of each mass as a function of the spring forces. Then substitute (c.D +k) for each spring value to get the force for the spring and damper combination. D is the differential operator d/dt, hence 1/D is integration.

There, I've done your homework for you.

Cheers

Greg Locock

New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?

### RE: Equations of motion of a damped 2-dof system

Longer ago than I care to remember I went through exactly this exercise when developing a spreadsheet to solve this problem.  Initially I did not have the two "ground" points, only one, but a few years later I extended my workings to accommodate this.  I attach a PDF of my calculation notes from the time.

If you want more precise enunciation of the method, you can download the spreadsheet I developed from my website (http://rmniall.com), then trace the input data through to the development of the required equations.

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