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Transcript
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 1d under combined linear restoring force - kx + retarding force -bv. Prototype Damped Oscillator Newton’s 2nd Law: F = ma = - bx - kx • Newton’s 2nd Law Equation of Motion: F = ma = - bv - kx Or: m(d2x/dt2) = - b(dx/dt) - kx Or: mx + bx + kx = 0 • Definitions: – Damping Parameter: β [b/(2m)] – Characteristic angular frequency: ω02 (k/m) • Equation of motion becomes: x + 2βx + (ω0)2x = 0 • Equation of motion: x + 2βx + ω02x = 0 • A standard, homogeneous, 2nd order differential equation. • GENERAL SOLUTION (Appendix C!) has the form: where x(t) = e-βt[A1 eαt + A2 e-αt] α [β2 - ω02]½ A1 , A2 are determined by initial conditions: (x(0), v(0)). [A1 eαt + A2 e-αt] can be oscillatory or exponential depending on the relative sizes of β2 & ω02. x(t) = e-βt[A1 eαt + A2 e-αt] with α [β2 - ω02]½ 3 cases of interest (Figure): Underdamping ω02 > β2 Critical damping ω02 = β2, Overdamping ω02 < β2 x(t) (qualitative) in the 3 cases Underdamped Case x(t) = e-βt[A1 eαt + A2 e-αt] with α [β2 - ω02]½ • Underdamping ω02 > β2 • Define: ω12 ω02 - β2 > 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)/(2T1), T1 = time between two adjacent crossings of x = 0 – Note: ω1 = [ω02 - β2]½ < ω0 – If the damping is small, ω1 ω0 x(t) = e-βt[A1 eαt + A2 e-αt] with α [β2 - ω02]½ • Underdamping ω02 > β2 ω12 ω02 - β2 > 0 – The solution is an oscillatory function multiplied by an exponential “envelope”. x(t) = e-βt[A1 exp(iω1t) + A2 exp(-iω1t)] Or, defining 2 new integration constants A & δ: x(t) = A e-βt cos(ω1t - δ) • The oscillatory part looks just like the undamped solution with ω02 ω12 ω02 - β2 • 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(ω1t - 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(ω1t - 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(ω1t - δ) Underdamping! • Consider: v(t) = x(t) = (dx/dt) = - β[A e-βt cos(ω1t - δ) - ω1[A e-βt sin(ω1t – 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. (dE/dt) [v(t)]2 (See Prob. 3-11) (dE/dt) = is a max when v(t) = max. (near equilibrium position). (dE/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 = ω1x, w = βx + v. Polar coordinates: ρ = [u2+w2]½ φ = ω1t. 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