Damped Forced Oscillations Coupled Oscillations 1 The solution for x in the equation of motion of a damped simple harmonic oscillator driven by an external force consists of two terms: a transient term (=temporary) and a steady-state term 2 The transient term dies away with time and is the solution to the equation discussed earlier: Transient Term This contributes the term: x = Ce pt 3 The steady state term describes the behaviour of the oscillator after the transient term has died away. Steady-state Term which decays with time as e -t Solutions Complementary Functions are transients Steady State behaviour is decided by the Particular Integral Both terms contribute to the solution initially, but the ultimate behaviour of the oscillator is described by the steady-state term. It always dies out if there is damping. As a practical matter, it often suffices to know the particular solution. 4 Driven Damped Oscillations: Transient and Steady-state behaviours5 Forced oscillator: Damped case Time dependent function 6 7 Companion equation: General equation Try steady-state solution (Particular solution) Complementary function:Transients 8 Consider a general sinusoidal drive force: Equation of motion becomes: The above equation is the real part of simplest complex differential equation: with x = Re(z) Try steady-state solution (Particular solution) (look at the flow of thought) 9 For now, we write the angular frequency of motion in steady state as s. A general guess would be: Guess a solution based on physical and mathematical intuition. It is not obvious whether the angular frequency of this motion would be that of: - Oscillation without damping or driving ( o ) - Oscillation with damping, but no driving ( ) - Drive frequency ( ) - Or some combination of these? Plug into the equation: And obtain: 10 The left side oscillates at s, while the right side oscillated at . So, if they are to be equal, we must have s = . i.e., In the steady-state, the oscillator moves with the same angular frequency as the drive force. So, our guess now becomes: Now, solving the equation gives: REAL: IMAGINARY: 11 To isolate |z o |, square these 2 equations and add them to give: To isolate , divide the above 2 equations: Our guess is the solution, if |z o | and are given as above. Then, the real part of the solution is: 12 REAL: IMAGINARY: It may appear that the max amplitude appears at = o. However, the term is multiplied by the factor 1/ , which increase as decreases, shifting the peak to a slightly lower value of o. This point of maximum amplitude is the resonance. So that Writing the amplitude in a different form We also see that the phase by which the oscillators response lags behind the drive force also depends on . When = o, then We see that the response amplitude at high frequencies approaches zero. 13 In the opposite limit Amplitude: 14Forced Damped Osc.-2 Show amplitude resonance at: 15 Amplitude Resonance At = 0 16 The low and high frequency behaviour are the same as the situation without damping. The changes due to damping are in the vicinity of = o. Amplitude is finite throughout. The amplitude is maximum at: Amplitude: Phase: 18 So that The phase by which the oscillators response lags behind the drive force also depends on . When = o, then 19 20 For mild damping ( k