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The following figures represent a model of a building, can someone help me, extract the newton equations of motion and determine the degrees of freedom.

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  • $\begingroup$ According to the diagram you have 1 degree of freedom which is phi. For small values of phi, x should equal phi. $\endgroup$
    – Sami Safarini
    Commented Sep 2, 2023 at 11:24
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    $\begingroup$ This reads like a homework question. Please edit into the question details of your attempts to extract the newton equations of motion and determine the degrees of freedom. First step would be to write an expression for the displacement of the tip of the rod with length L in terms of $\phi$ etc. $\endgroup$
    – AJN
    Commented Sep 2, 2023 at 13:44
  • $\begingroup$ The power bond graph technique furnishes a step-by-step cookbook procedure for extracting the equations of motion for practically any dynamical system. The definitive text is by Karnopp and Rosenberg. $\endgroup$
    – niels nielsen
    Commented Sep 2, 2023 at 15:53
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    $\begingroup$ What is theta, half way up the building? You have 2 degrees of freedom, x and phi. $\endgroup$
    – Greg Locock
    Commented Sep 2, 2023 at 21:24
  • $\begingroup$ For what it is worth, assuming the building is modelled as a rigid body, you have a 2dof tuned harmonic damper system. The equations are easy to write but solving them non-numerically is a fairly tedious bit of complex maths. Typically we tune the resonant frequency of the damper/spring/mass system equal to that of the main system, and then change the damper value to maximise the energy absorbed at the resulting two resonances. The first mode has the tip of the building moving in the same direction as m, the second has them in antiphase. $\endgroup$
    – Greg Locock
    Commented Sep 2, 2023 at 22:49

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