Taylor 5.5-5.7 (today) and 5.7-5.8 (Wednesday).
The following table summarizes relations between the constants used in the damped oscillator results.
| quantity | relations | comments |
|---|---|---|
| ω0 | ω02 = k/m | positive |
| ωd | ωd2 = ω02 - β2 | positive |
| β | β = c/2m | positive |
Four cases to consider: no damping, underdamped, critically damped, and overdamped. These differ in the amount of damping as determined by the relative values of β and ω0.
| damping | β | decay parameter |
|---|---|---|
| none | β=0 | 0 |
| under | β<ω0 | β |
| critical | β=ω0 | β |
| over | β>ω0 | β - sqrt(β² - ω0²) |
Every real oscillator has some friction or energy loss, and so will eventually stop oscillating. It is therefore common to drive an oscillator with a periodic force. We'll investigate the behavior of an oscillator with a driving force F. We now have the solution to the homogeneous differential equation. If an additional driving force is applied, then we need to consider the equation:
which can be rewritten with linear operators as
In general, the force can take on any form, but a particularly interesting case is if the force is time dependent, F = F(t). Then, the differential equation is a non-homogeneous, linear, ODE, and techniques exist to solve it. We will solve the case where the force varies (co)sinusoidally with time: F(t) = F0cos(ωt). Don't confuse the driving frequency ω with the natural frequency of the oscillator ω0. It is for this reason that the notation ω0 for the natural frequency is introduced.
The particular solution is a function xp(t) that satisifies the D.E. with the driving force (the inhomogeneous term).
The equation with zero driving force (no term not linear in x) is called the homogeneous equation. We discussed the solutions to this equation, let's call them xh(t). The homogeneous solution contains arbitrary constants (two for this second order equation) while the particular solution doesn't. Additionally, we saw that in all cases the homogeneous solution eventually decays to zero. While not absolutely required (damping forces may be absent), this is a common feature of these types of problems.
The homogeneous solution can be added to the particular solution, and the result still satisfies the inhomogeneous D.E. A general solution of the inhomogeneous D.E. is the sum of the particular plus the homogeneous solution:
The particular solution has no arbitrary constants. The constants of integration are all contained in the homogeneous solution. Given a sufficiently long time, the homogeneous solution will exponentially approach zero, and we're left with just the particular solution.