Week 4

Forced Damped Simple Harmonic Motion

This week we began by reviewing the simple harmonic oscillator including dissipative damping via a viscous force. I encourage you to read last week’s blog posts to see the material associated with this, as well as this weeks mathematica notebook:

https://jhubisz.expressions.syr.edu/phy360/wp-content/uploads/sites/5/2016/09/Week-4.nb

The new material for this week is the forced damped simple harmonic oscillator. For this case, we are now continuously injecting energy into and out of the system via some sort of periodic driving mechanism.

The demonstration associated with this is the mass on spring in water, where the spring is hung from a motor and piston that drives the entire mechanism up and down sinusoidally. Experimentally, we verified that at high frequencies, there is not a great deal of steady-state response (by steady-state, we mean the motion that persists a long time after the driving force has turned on). Similar behavior was noted for low frequencies. If, however, we drove the system at a frequency that is close to the natural frequency (the frequency of oscillation if there were no damping or driving) that the response is significant (the mass oscillates with a large steady state amplitude).

The phenomenon of large response at a given characteristic frequency is called resonance. You are all likely familiar with resonance from using swings as children, and driving your legs at the right frequency leads to a more exciting (larger amplitude) ride.

How does the mathematics work out for this scenario? There is now one additional force, the force that is an external input into the system. This force is what it is, independent of the displacement of the oscillator from equilibrium:

$\vec{F}_\text{drive} = F_0 \cos \omega t$ .

Note that here $\omega$ is not a characteristic of the oscillator itself, but rather of the external input. This force is not dependent on x(t).

The total force then (let’s say the system is a mass on a spring with spring constant k, and there is a viscous damping coefficient b) is given by:

$\vec{F} = F_0 \cos \omega t - k x(t) - b \dot{x}(t)$

Applying Newton’s 2nd law, we have an equation of motion given by

$\vec{F} = F_0 \cos \omega t - k x(t) - b \dot{x}(t) = m \ddot{x}(t)$

We rearrange the equation of motion, dividing by $m$ , and we have:

$\ddot{x}(t) + \gamma \dot{x}(t) + \omega_0^2 x(t) = \frac{F_0}{m} \cos \omega t$

where $\gamma = b/m$ and $\omega_0^2 = k/m$ .

Note the important difference from the equations we have looked at before – there is now a term in the equation that does not depend on x(t) at all. To solve this equation, let us presume that the long-time behavior of the system (a long time after the driving force is turned on) is given by oscillation with the same frequency as the driving force. Let us express this in the complex plane, however, when the equation of motion is expressed as

$\ddot{z}(t) + \gamma \dot{z}(t) + \omega_0^2 z(t) = \frac{F_0}{m} e^{i \omega t}$ .

Then our guess that the response is characterized by the same frequency is equivalent to the mathematical statement

$z(t) = Z_0 e^{i \omega t}$ .

Here $Z_0$ is an as yet undetermined constant. If we plug this guess into the equation of motion, we obtain an equation for $Z_0$ :

$Z_0 = \frac{ F_0/m}{(\omega_0^2-\omega^2) + i \omega \gamma}$

Now this looks like a relatively complicated formula, with both real and complex parts, but let us unpack it a bit. First, let us recall that any complex number can be expressed as an amplitude and a phase:

$Z_0 = A e^{i \alpha}$

We can find $A$ first:

$A = \frac{F_0/m}{\sqrt{\omega_0^2-\omega^2)^2+\omega^2 \gamma^2}}$

You can do some geometry in the complex plane to convince yourself that

$\alpha = \arctan \left[ \frac{ - \gamma \omega}{\omega_0^2-\omega^2} \right]$

Be careful that the signs in the numerator and denominator have meaning, as the arctan function is double valued. You must figure out which quadrant the point is in based on the signs of the numerator and denominator separately.

So what does all this mean in terms of the physics??

We are characterizing a response (the motion of the mass) due to a stimulus (the applied driving force). As we have found a solution, we see that for a given stimulus (an $F_0$ and an $\omega$ , there is a response with amplitude $A$ that leads the stimulus in phase $\alpha$ . To get a sense of what this means, please take a look at this weeks mathematica file, in which you can vary the stimulus frequency and watch the stimulus and response change with respect to each other in the complex plane.

Note that the complete solution can be written as

$z = C_+ e^{\rho_+ t} + C_- e^{\rho_- t} + Z_0 e^{i \omega t}$ ,

where the first two parts of the solution are damped, and thus not relevant for the behavior of the system at times that are large in comparison with the damping times. These are frequently referred to as transients of the system.

Physics 360

Vibrations and Waves

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