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What does the position $x(t)$ looks like in an overdamped system?

For the overdamped case the solutions are just exponential decays. There is no harmonic term. If you use complex represenation, for the underdamped case you have some imaginary exponent which results ...
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Equation of motion of a particle in a sinusoidal well

There is an exact analytic answer to equations with a $sin(x)$ in terms of elliptic functions. See the section on "arbitrary amplitude" in the link above.
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Equation of motion of a particle in a sinusoidal well

In newtonian mechanics, you can do what's sometimes called a partial resolution. You won't get $x(t)$ analytically, but you'll get velocity as a function of position. Multiply the differential ...
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Would it be correct to state that the damping force and the spring force are equal in the case of critical damping?

Would it be correct to state that the damping force and the spring force are equal in the case of critical damping? No. Regardless of whether a damped harmonic oscillator is underdamped, critically ...
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Intuition behind the differential equation for forced oscillations

It's better to write the differential equation for forced oscillations in a different way that makes it more relatable to Newton's second law: $$m\ddot{x}=F_0\sin{(\omega''t)}-b\dot{x}-kx$$ Now we can ...
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How to describe a series of damped harmonic oscilators?

I don’t have a particular reference to propose. However, in general, you won’t be able to fond an effective $k$ and $c$. The simplest way to see this is going to frequency space. I’ll choose the ...
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