# Can concepts like “critical damping” or “resonant frequency” be applied to more complex systems than just a spring and damper in parallel?

I am trying to do some modeling analysis by representing materials with parallel systems of springs and dampers. In the simplest case with just one spring parallel to a damper, we have the traditional "damped oscillator":

The fundamental resonant angular frequency is given by:

$$ω = \sqrt{\frac{k}{m}}$$

Critical damping is calculated from the equation of motion:

$$mx_{tt} + cx_t + kx = 0$$

$$ms^2 + cs + k = 0$$

$$s= \frac{-c ± \sqrt{c^2-4mk}}{2m}$$

There are then three conditions:

1. $$c^2 <4mk$$ (under damping)
2. $$c^2 >4mk$$ (over damping)
3. $$c^2 =4mk$$ (critical damping)

The damping ratio is then expressed by $$\frac{c}{\sqrt{4mk}}$$.

I am trying to figure out how to apply similar concepts to more complex models such as the following:

The equation of motion where $$σ$$ is stress (force) and $$σ$$ is strain (stretch, analogous to $$x$$) is as follows:

$$σ + (\frac{η_1}{E_1} + \frac{η_2}{E_2})\dot{σ} + \frac{η_1η_2}{E_1E_2}\ddot{σ}= (η_1+η_2)\dot{ϵ} + \frac{η_1η_2(E_1+E_2)}{E_1E_2}\tag{1}$$

I have been thinking about this problem, and my inclination is that in a more complex system, there will not be one single frequency of oscillation, but rather multiple frequencies.

Critical damping is the damping coefficient which leads to the most rapid extinguishing of oscillations. If a complex system has multiple harmonics, perhaps such a simple concept cannot be applied. Perhaps you would need to pick a specific harmonic to address, and then you could calculate the damping coefficient that would extinguish that harmonic fastest (eg. the fundamental).

I presume such a thing should be possible. It seems reasonable.

How would one predict the fundamental frequency or various frequencies/modes of vibration from an equation of motion like (1)? Then let's say I can calculate a given fundamental frequency of $$f$$. How would I then go about establishing the damping coefficient that would most rapidly extinguish it?

I presume this is solvable as springs and dampers are commonly used to model physical objects, but I'm not sure how to go about it.

Thanks.

• Cross posted: physics.stackexchange.com/q/608201 Jan 17, 2021 at 7:15
• applies to any 2nd order system, particularly common in RLC electrical circuits and feedback control. The "Q" concept also has some intuitive value in higher order linear systems, if they are equivalent to a combination of 1st and 2nd order systems that include at least one pair of complex conjugate poles (or eigenvalues, if you like). Jan 24, 2021 at 5:00