Damped Oscillators SHOs Oscillations Free oscillations because once

Damped Oscillators • SHO’s: Oscillations “Free oscillations” because once begun, they will never stop! • In real physical situations & for real physical systems, of course: – There are retarding forces: These will damp oscillations, which eventually die away to the stop motion. • Usual approximation: Damping force Fr v = x • In what fallows, take Fr = -bv, b = a constant > 0, which depends on the system. • Prototype oscillator: Mass m in 1 d under combined linear restoring force - kx + retarding force -bv.

Prototype Damped Oscillator Newton’s 2 nd Law: F = ma = - bx - kx

• Newton’s 2 nd Law Equation of Motion: F = ma = - bv - kx Or: m(d 2 x/dt 2) = - b(dx/dt) - kx Or: mx + bx + kx = 0 • Definitions: – Damping Parameter: β [b/(2 m)] – Characteristic angular frequency: ω02 (k/m) • Equation of motion becomes: x + 2βx + (ω0)2 x = 0

• Equation of motion: x + 2βx + ω02 x = 0 • A standard, homogeneous, 2 nd order differential equation. • GENERAL SOLUTION (Appendix C!) has the form: where x(t) = e-βt[A 1 eαt + A 2 e-αt] α [β 2 - ω02]½ A 1 , A 2 are determined by initial conditions: (x(0), v(0)). [A 1 eαt + A 2 e-αt] can be oscillatory or exponential depending on the relative sizes of β 2 & ω02.

x(t) = e-βt[A 1 eαt + A 2 e-αt] with α [β 2 - ω02]½ 3 cases of interest (Figure): Underdamping ω02 > β 2 Critical damping ω02 = β 2, Overdamping ω02 < β 2 x(t) (qualitative) in the 3 cases

Underdamped Case x(t) = e-βt[A 1 eαt + A 2 e-αt] with α [β 2 - ω02]½ • Underdamping ω02 > β 2 • Define: ω12 ω02 - β 2 > 0 ω1 “Angular frequency” of the damped oscillator. – Strictly speaking, we can’t define a frequency when we have damping because the motion is NOT periodic! • The oscillator never passes twice through a given point with the same velocity (it is slowing down!) – However we can define: ω1 = (2π)/(2 T 1), T 1 = time between two adjacent crossings of x = 0 – Note: ω1 = [ω02 - β 2]½ < ω0 – If the damping is small, ω1 ω0

x(t) = e-βt[A 1 eαt + A 2 e-αt] with α [β 2 - ω02]½ • Underdamping ω02 > β 2 ω12 ω02 - β 2 > 0 – The solution is an oscillatory function multiplied by an exponential “envelope”. x(t) = e-βt[A 1 exp(iω1 t) + A 2 exp(-iω1 t)] Or, defining 2 new integration constants A & δ: x(t) = A e-βt cos(ω1 t - δ) • The oscillatory part looks just like the undamped solution with ω 02 ω12 ω02 - β 2 • The maximum amplitude of motion decreases with time by a factor e-βt. This function forms an “envelope” function for the oscillatory part: xen = A e-βt

x(t) = A e-βt cos(ω1 t - δ), xen = A e-βt Underdamping! Shown in fig. for δ = 0. Clearly the period is longer (the frequency is shorter) than for the undamped case!

x(t) = A e-βt cos(ω1 t - δ) Underdamping! • Ratio of the amplitudes at 2 successive maxima is of interest. – Define: T Time at which a maximum occurs. τ1 Time between 2 successive maxima. τ1 (2π)/ω1 – The ratio of the amplitudes at 2 successive maxima: D [Aexp(-βT)]/[Aexp(-β{T+ τ1})] or D = exp(βτ1) D The “Decrement” of the motion ln(D) = βτ1 “Logarithmic Decrement” of the motion

x(t) = A e-βt cos(ω1 t - δ) Underdamping! • Consider: v(t) = x(t) = (dx/dt) = - β[A e-βt cos(ω1 t - δ) - ω1[A e-βt sin(ω1 t – δ)] • Consider the Mechanical Energy E: – Due to damping, E is not a constant in time! E IS NOT CONSERVED! – Energy is continually given up to the damping medium. Energy dissipation in terms of heat. E = T + U (U is due to the restoring force -kx only, not due to the retarding force -bv). U = U(t) = (½)k[x(t)]2, T = T(t) = (½)m[v(t)]2. E is clearly mess! Clearly, it decays in time as e-βt

• Energy for underdamped case. See figure. (Also see Prob. 3 -11)

• Energy Loss Rate in the underdamped case. See figure. (d. E/dt) [v(t)]2 (See Prob. 3 -11) (d. E/dt) = is a max when v(t) = max. (near equilibrium position). (d. E/dt) = 0 when v(t) = 0

• Example 3. 2: Phase Diagram - Underdamped case. Use a computer to construct a phase diagram for the underdamped oscillator. A = 1 cm, ω0= 1 rad/s, β = 0. 2 s-1, δ = (½)π. Define: u = ω1 x, w = βx + v. Polar coordinates: ρ = [u 2+w 2]½ φ = ω1 t. Get a logarithmic spiral for the phase trajectory. w vs. u:

• Phase Diagram - Underdamped case. A spiral phase path. The continually decreasing magnitude of the radius vector is always an indication of damped oscillations. v vs. x:

x(t) & v(t) for the case just shown