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Ever heard about antiresonance? As the name suggests it is opposite of resonance, which is hardly taught in elementary classes. Considered two coupled oscillators where one is forced with say a harmonically driven force and other with without any forcing, just like the building and the TMD system, where the building is forced by say some seismic vibrations. ...

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I assume that your structure is a simple single spring mass damper system? For such a system the assumption of decreasing dampingratio $\zeta$ for increasing mass $m$ is correct. You need more dampingforce to stop a heavier system, therefore resulting in a smaller $\zeta$. In reality though it could be that the additional mass affects the damping behaviour ...

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The equations of motion are two coupled linear second order differential equations. Since the system is linear, any linear combination of solutions is still a solution, hence the set of solutions is a linear space. It can be proved that the solutions of a system of $n$-th order linear differential equations is a linear space of dimension $n$. This means ...

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At equilibrium, the function(potential energy) is minimum( Morse potential curve). we know from mathematics that the first derivative of a function is always zero at minimum. that is why the linear term is missing.

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Using the semi-major axis of $a$, the semi-minor axis of $b$ the polar coordinates of the ellipse are $r(\theta)$ where $\theta$ is the angle the rotating line makes to the horizontal $$r = \frac{a b}{\sqrt{a^2 + (b^2-a^2) \cos^2 \theta }} \tag{1}$$ Point C has $x$-coordinate of $$x_C = r \cos \theta = \frac{a b \cos\theta}{\sqrt{a^2 + (b^2-a^2) \cos^2 \... 1$$\underline {\text {Motion of Point $C$}} $$The equation of ellipse is given by (1) and one can parametrize the curve by (2) .$$\frac {x^2}{a^2} + \frac {y^2}{b^2} =1 \tag {1}  x=a \cos \alpha \;|\; y=b \sin \alpha \tag {2} $$We can determine the equation of motion by using (2)  and (3).$$\tan \theta = \frac {b}{a} \tan \alpha \text { ...

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Pick a constant k to represent how far from a circle the orbit is. Then for any angle $\theta$, $C = k \cos \theta$. That isn't so special. But if instead of constant angular velocity it rotated like an orbit, with the angular rotation varying, then it gets more complicated and I don't have such an easy answer. It's all simple when the angle is the ...

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Well you have actually got the equation of motion of the pendulum. Now all you have to do is understand it It's simple if you can see what is happening in the equation we may write the equation as follows $$\frac{d^2 \theta}{dt^2}=-g\frac{\theta}{l}$$ This equation must be looking very familiar to you if you have solved for the simple harmonic oscillator. ...

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