- #1

zenterix

- 688

- 83

- Homework Statement
- Consider the expression

$$A=\frac{f}{\sqrt{(\omega_0^2-\omega_d^2)^2+\gamma^2\omega_d^2}}\tag{1}$$

- Relevant Equations
- For context, ##A## represents the amplitude of a solution to the equation

$$\ddot{y}+\gamma\dot{y}+\omega_0^2y=f\cos{\omega_d t}\tag{2}$$

If ##\omega_0## is given (by the nature of the physical system under consideration, for example the spring constant and mass of a simple pendulum) then ##A## can be thought of as a function of the driving angular frequency ##\omega_d##.

We can differentiate (1) and find the ##\omega_d## that maximizes amplitude.

We find

##\omega_{d,max}=\sqrt{\omega_0^2-\frac{\gamma^2}{2}}\tag{3}##

Pictorially

What if we consider ##\omega_d## as fixed and consider ##A## a function of ##\omega_0##?

As a concrete example, consider a seismograph.

The floor (the Earth) vibrates at a certain angular frequency that we can measure but not control. This drives the mass on the spring.

We want to choose ##\omega_0## such that the amplitude of the mass is the largest.

We have

$$A(\omega_0)=\frac{f}{\sqrt{(\omega_0^2-\omega_d^2)^2+\gamma^2\omega_d^2}}\tag{4}$$

and if we differentiate and equate to zero we find the solutions

$$\omega_0=0\tag{5}$$

$$\omega_0=\pm \omega_d\tag{6}$$

This is not what I expected.

After all, given ##\omega_0## we find that amplitude is maximized at an ##\omega_d## slightly smaller than ##\omega_0##.

Given ##\omega_d##, I would have expected that amplitude is maximized at a slightly larger ##\omega_0##.

What am I missing here?

We can differentiate (1) and find the ##\omega_d## that maximizes amplitude.

We find

##\omega_{d,max}=\sqrt{\omega_0^2-\frac{\gamma^2}{2}}\tag{3}##

Pictorially

What if we consider ##\omega_d## as fixed and consider ##A## a function of ##\omega_0##?

As a concrete example, consider a seismograph.

The floor (the Earth) vibrates at a certain angular frequency that we can measure but not control. This drives the mass on the spring.

We want to choose ##\omega_0## such that the amplitude of the mass is the largest.

We have

$$A(\omega_0)=\frac{f}{\sqrt{(\omega_0^2-\omega_d^2)^2+\gamma^2\omega_d^2}}\tag{4}$$

and if we differentiate and equate to zero we find the solutions

$$\omega_0=0\tag{5}$$

$$\omega_0=\pm \omega_d\tag{6}$$

This is not what I expected.

After all, given ##\omega_0## we find that amplitude is maximized at an ##\omega_d## slightly smaller than ##\omega_0##.

Given ##\omega_d##, I would have expected that amplitude is maximized at a slightly larger ##\omega_0##.

What am I missing here?