Forced Oscillations PDF

FORCED OSCILLATIONS Occur when an oscillating system is driven by a periodic force that is external to the oscillating system. In such a case, the oscillator is compelled to move at the frequency of the driving force. The physically interesting aspect of a forced oscillator is its response—how muc...

FORCED OSCILLATIONS Occur when an oscillating system is driven by a periodic force that is external to the oscillating system. In such a case, the oscillator is compelled to move at the frequency of the driving force. The physically interesting aspect of a forced oscillator is its response—how much it moves due to the imposed driving force. Forced Oscillation - an overview | ScienceDirect Topics  FORCED OSCILLATIONS Two entities of importance: 1. The oscillator cos Frequency = without damping = 2 forced oscillation spring mass - Google Search 2. The periodic force Frequency = (not the same as frequency with damping) +2 + = cos = cos - An inhomogeneous differential equation. Forced Oscillations: Solution Microsoft Word - Notes-2nd order ODE pt2 (psu.edu) +2 + = cos Solution = + Complementary solution Particular solution - Solution of the corresponding - Necessarily a solution of the homogeneous equation full inhomogeneous equation for +2 + =0 some particular initial conditions - Never a solution of the full inhomogeneous equation Dies out with time Survives at long times (damped oscillations) Solution of interest Insignificant at long times Forced Oscillations: The Particular Solution +2 + = cos For the sake of mathematical simplicity, let us write the equation in its ‘complex’ form as: +2 + = ……….(1) Observe the behavior of the function at RHS: - It doesn’t change on taking first or second derivative, except for the constant coefficients. Then, the solution can be taken as a general form of the function at RHS. Say, = ……….(2) The solution, then, will be: = ( ) = cos ……….(3) Forced Oscillations: The Particular Solution +2 + = ……….(1) = ……….(2) Putting (2) in (1) + + = + + = = cos + sin Equating real and imaginary parts: + = cos and 2 = sin = and = tan + = cos an oscillation with the frequency of the periodic force This is the solution which survives after some time.  Forced Oscillations: The Particular Solution = cos 0 = + = tan Specific points: 1. When 0, = = , =0 2. When , = = = , = - Remember Q = for underdamped oscillations 3. When , 0, = Forced Oscillations: The Particular Solution Finite = + = tan Resonance The damping ensures that the amplitude does not blow up at = and it is finite for all values of. The amplitude is maximum at = 2 Verify yourself. Forced Oscillations: The Full Solution = + ( )= cos + + cos The frequency of the forced oscillator Corresponds to (under)damped oscillation ( ) cos + + cos Beats in the beginning (transients) Oscillations later (steady state) Damped Driven Oscillator (virginia.edu) Forced Oscillations: The Full Solution ( ) cos + + cos Transients Steady state Transients Steady state Here’s a pair of examples: the same driven damped oscillator, started with zero velocity, once from the origin and once from 0.5. Notice that after about 70 seconds, the two curves are the same, both in amplitude and phase. Damped Driven Oscillator (virginia.edu) Forced Oscillations: The Quality Factor (Q-factor) = Remember Q= for underdamped oscillations and +2 + = cos The Q factor is a measure of the ‘quality’ of an oscillator. It is a measure of how many oscillations take place during the time the energy decays by the factor of 1/e. Forced Oscillations: The Amplitude at Resonance = cos = + The amplitude is maximum ( ) at = 2 = +2 + 2 = 1+ 2 Forced Oscillations: The Average Energy = cos = sin 1 1 , = ( ) , = ( ) 2 2 1 1 = sin ( ) = cos ( ) 2 2 1 1 = sin ( ) = cos ( ) 2 2 1 1 = = 4 4 Forced Oscillations: The Average Energy…. 1 = 4 1 = 4 = + 1 = ( + ) 4 + = 4 +4 Forced Oscillations: Average Power put in the system by Driving Force = = sin Work done: =. Power: , = =. = = sin cos = sin cos 2 sin cos = sin + + sin = = Forced Oscillations: Average Power put in the system by Driving Force… = Width at half-peak power = 2 Verify yourself Remember +2 + = cos Forced Oscillations: Examples 1. Electrical resonance When we turn the knob, the capacitance keeps on changing till the resonant frequency becomes equal to the frequency of the channel which we want to hear. 2. Acoustic resonance 3. Pohl’s pendulum 4. Optical resonance 5. Nuclear magnetic resonance …….