Oscillations Lecture 5 Overdamping PDF

Lecture 5: Critical and Overdamping Reading: Principles of Physics, 15-8 Hyperphysics Trial solution x(t) = C ert where r is complex x(t) must b...

Lecture 5: Critical and Overdamping Reading: Principles of Physics, 15-8 Hyperphysics Trial solution x(t) = C ert where r is complex x(t) must be real.  Last time: Underdamping b 𝑏 2 𝑥ሺ 𝑡ሻ = ᇣ𝐴𝑒 ᇧᇤ − ᇧᇥ 𝑡 cos(𝜔1𝑡 + 𝜙) with damped frequency 𝜔1 = ඨ ω20 − ൬ ൰ 2𝑚 𝑥𝑚 (𝑡) 2m 𝑘 𝜔0 = ඨ :undamped angular frequency 𝑚 𝑏:damping constant The amplitude decreases exponentially with time: 𝑏 𝑥𝑚ሺ 𝑡ሻ = 𝐴𝑒 − 2𝑚 𝑡 The angular frequency w1 is slightly smaller then from: Knight, Physics the undamped frequency.  OverDamping 𝑥ሺ 𝑡ሻ = 𝐴𝑒𝑟𝑡 (𝐴and 𝑟 may be complex) b b 2 𝑘 solutions: r1,2 = − ±ඨ ൬ ൰ − ω20 𝜔0 = ඨ :undamped 2m 2m 𝑚 angular frequency overdamping (strong damping): b 𝑏 2 b ⇒ r1,2 = − ±ඨ ൬ ൰ − 𝜔02 > ω0 2m 2𝑚 2m The most general solution is a superposition of the two solutions:  Overdamping 𝑏 ට 𝑏 2 𝑏 ට 𝑏 2 −ቆ − ቀ ቁ −𝜔02ቇ 𝑡 −ቆ + ቀ ቁ −𝜔02ቇ 𝑡 𝑥ሺ 𝑡ሻ = 𝐶1𝑒 2𝑚 2𝑚 + 𝐶2𝑒 2𝑚 2𝑚 The system no longer oscillates, but returns to its equilibrium position in an exponential decay without oscillation when it is displaced and released Note – there are two Exponentials rates from: Taylor, Classical Mechanics Which term in x(t) for overdamping dominates at long times? A. C1 exp ( - (g - (g2 – w02)1/2 t) B. C2 exp ( - (g + (g2 – w02)1/2 t) C. Both terms decay at the same rate g = (b/2m)  Which term in x(t) for overdamping dominates at long times? A. C1 exp ( - (g - (g2 – w02)1/2 t) B. C2 exp ( - (g + (g2 – w02)1/2 t) C. Both terms decay at the same rate g = (b/2m) e answer is B as the A term will decay away quickly just leaving the B te  Overdamping Example Show that the system d2x/dt2 + 4dx/dt + 3x = 0 is overdamped and graph the solution with initial conditions x(0) = 1, dx(0)/dt = 0. Which root dominates at long times?  Overdamping Example Show that the system d2x/dt2 + 4dx/dt + 3x = 0 is overdamped and graph the solution with initial conditions x(0) = 1, dx(0)/dt = 0, assuming mass m = 1. Which root dominates at long times? General solution x(t) = c1 exp(r1t) + c2 exp(r2 r2 + (b/m) r + (k/m) = (i.e. 0 m=1, b=4, k=3) r12 = (- 4 ± (16 – 12)1/2)/2 = -2 ± 1 -> x(t) = c1 exp(-t) + c2 exp(-3t) is the exp(-t) root that controls how fast it returns to equilibr  Overdamped Initial Conditions x(t) = c1 exp(-t) + c2 exp(-3t) -> dx(t)/dt = -c1 exp(-t) - 3c2 exp(-3t) Initial Conditions x(0) = 1, dx(0)/dt = 0 x(t) = 3/2 exp(-t) - 1/2 exp(-3t) exp(-3t) exp(-t c1 + c2 = 1 and - c1 - 3c2 = 0 3/2 exp(-t) > c2 = -1/2 and c1 = 3/2 -1/2 exp(-3t)  Critical damping critical damping b b ⇒ only one solution r = − = ω0 2m 2m For a second order differential equation, there must be two free parameters. Therefore, we need to find (guess) a second 𝑏 𝑥ሺ 𝑡ሻ = 𝐴⋅ 𝑡 ⋅ 𝑒 solution by some other method. − 2𝑚 𝑡 is also a solution of the equation of motion for the case = 𝜔0 (see tutorial problem) 𝑏 2𝑚 As in the overdamped case, the system no longer oscillates, but returns to its equilibrium position in an exponential decay without oscillation.  Critical damping Example Show that the system d2x/dt2 + 4dx/dt + 4x = 0 is critically damped and graph the solution with initial conditions x(0) = 1, dx(0)/dt = 0.  Critical damping Example Show that the system d2x/dt2 + 4dx/dt + 4x = 0 is critically damped for a mass m = 1, and graph the solution with initial conditions x(0) = 1, dx(0)/dt = 0. General solution x(t) = (c1 + c2 t) exp(r t) r2 + (b/m) r + (k/m) = 0 (where m=1, b=4, k=4) r12 = (- 4 ± (16 – 16)1/2)/2 = -2 ± 0 is is critically damped as the expression in the square root = -> x(t) = (c1 + c2t) exp(-2t)  Crtically damped Initial Conditions x(t) = (c1+ c2t) exp(-2t) -> dx(t)/dt = -2(c1+ c2t) exp(-2t) + c2 exp( Initial Conditions x(0) = 1, dx(0)/dt = 0 x(t) = (1+ 2t) exp(-2t) c1 = 1 and - 2c1 + c2 = 0 -> c2 = 2 2t exp(-2t) exp(-2t) Comparing k = 3, k = 4 (with b = 4 and m = 1) x(t) = (1+ 2t) exp(-2t) Critically Damped x(t) = 3/2 exp(-t) - 1/2 exp(-3t Overdamped  Critical damping x(t) = (A + B t) * Exp( - gct) t) has two terms how many terms do the following h i) dx(t)/dta) 1 b) 2. c) 3. d) 4 e) 5 ii) d2x(t)/dta)2 1 b) 2. c) 3. d) 4 e)  Critical damping x(t) = (A + B t) * Exp( - gct) t) has two terms how many terms do the following h i) dx(t)/dta) 1 b) 2. c) 3. d) 4 e) 5 ii) d2x(t)/dta)2 1 b) 2. c) 3. d) 4 e)  Decay parameter for underdamping, overdamping and critical damping The rate at which the motion dies out can be characterized by the coefficient in the exponent which dominates the long-term motion. Damping Decay Parameter b =0 0 2m none b b < ω0 2m 2m under b b critical 2m = ω0 2m b b b 2 > ω0 −ඨ ൬ ൰ − 𝜔02 2m 2m 2m over amount of damping b increases For which case does the motion die out most quickly? For critical damping, the motion dies out most quickly.  Exam Question Example What happens under conditions of underdamping, critical damping and overdamping if you jump in the hole, in our otherwise frictionless asteroid Which condition will get you to the centre of the asteroid first a) Underdamping b) Critical damping c) Overdamping  Exam Question Example What happens under conditions of underdamping, critical damping and overdamping if you jump in the hole, in our otherwise frictionless asteroid  Car Suspension Is a typical passenger car suspension – Underdamped – Critically damped – Overdamped  Car Suspension Is a typical passenger car suspension – Underdamped – Critically damped – Overdamped For optimal passenger comfort, which is likely to be the second worst (simple) shock absorber setting for a typical passenger car A. undamped B. underdamped C. critically damped D. overdamped For optimal passenger comfort, which is likely to be the second worst (simple) shock absorber setting for a typical passenger car A. undamped B. underdamped C. critically damped D. overdamped  Analogy with Electrical Systems We have just talked about mechanical systems but NOTE many of these ideas are more commonly applied to electrical systems I = dq(t)/dt VL = L dI/dt = L d2q(t)/dt2 VR = IR = R dq(t)/dt VC = q(t)/C Note by analogy with spring problems L = m, k = 1/C, R = b, q(t) = x(t), I = v VL + VC + VR = 0 Note Kirchoffs Voltage Law q(t) here is charge across capacit  Electrical Systems -> L d2Q/dt2 + R dQ/dt + Q/C = 0 Given the differential equation for an RLC circuit, which quantity is analagous to the damping term in a mechanical oscillator? A) R, resistance B) L, inductance C) C, capacitance  Electrical Systems -> L d2Q/dt2 + R dQ/dt + Q/C = 0 Given the differential equation for an RLC circuit, which quantity is analagous to the damping term in a mechanical oscillator? A) R, resistance B) L, inductance C) C, capacitance  Analogy with Electrical Systems We have just talked about mechanical systems but NOTE many of these ideas are more commonly applied to electrical systems i.e. we know solution to: md2x/dt2 + b dx/dt + kx = 0 L dI/dt + IR + q/C = 0 -> L d2q/dt2 + R dq/dt + q/C = 0 -> w2 = w02 – (b/2m)2 -> w12 = 1/LC – (R/2L)t 2= L/R Q = w1L/R -> q(t) = q(0) Exp[-(Rt/2L) ± ((R/2L)2 – 1/LC)1/ VL + VC + VR = 0 From Kirchoffs Law Eelectric = ½ q(t)2/C Emagnetic = ½ L I2  What you should know General Solutions for – Overdamped Systems – Critically Damped systems Electrical Analogy to Mechanical System