oscillator diffeq variables

spring constant k

k = F/x [N/m]
**positive

mx” + cx’ + kx = 0

Hooke’s law—why is it negative?

F = -kx
indicates that the force exerted by the spring is in the opposite direction to the displacement (restoring force)

natural frequency w0

sqrt(k/m) [rad/s]

damping frequency wd

sqrt(w0² - p²) where p is the damping coefficient

damping constant

usually given based on properties of spring

mx” + cx’ + kx = 0
if c = 0 → undamped

damping coefficient/ratio p

c/2m where c = damping constant and 2 = mass

“damping coefficient per unit mass”

period T

2pi/w0 [s]

time to complete 1 full oscillation (between zero-crossings)

frequency f

1/T [Hz, cycles/s]

amplitude C

sqrt(A² + B²) where A and B are coefficients

maximum displacement or max of x(t)pol

polar form for undamped

x(t) = C cos (w0t - ϕ)

p² - w0² > 0

overdamped (two distinct real roots—BOTH negative)

no oscillations, will return to equilibrium SLOWER than critically damped

c = 0

physical meaning

undamped

no energy loss in the system as it oscillates → will continue oscillating forever without any decrease in amplitude
exists only in theory (bounded)

p² - w0² = 0

critically damped

no oscillations, will go straight to equilibrium (fastest velocity) (bounded)

p² - w0² < 0

underdamped

damping is too weak to prevent oscillations, so there will be oscillations that exponentially decay (bounded)

what is resonance

unbounded system (exponential growth) because wd = w0 (matches natural frequency)

solve for particular solution and rearrange for C = 1/(w0²-wd²)
** when C is undefined (@ w0 = wd), growth is unbounded

quadrant of phase angle?
A > 0 and B > 0

quadrant I (top right)

phase angle

arctan(B/A)

quadrant of phase angle?
A < 0 and B > 0

quadrant II (top left)

quadrant of phase angle?
A < 0 and B < 0

quadrant III (bottom left)

quadrant of phase angle?
A > 0 and B < 0

quadrant IV (bottom right)

when is resonance possible?

• For unforced systems with damping (overdamped, underdamped, critically damped), unbounded oscillations cannot occur.

• Unbounded oscillations are only possible in a system that is driven by an external force at a frequency matching the natural frequency (ω=ω0​)—resonance—and even then, unbounded behavior is typical only in undamped or weakly damped systems.