Undamped Forced Oscillations

External Forcing SHO with an additional external force Why this particular type of force ? © SB

For any arbitrary time varying force © SB

Driving force: Where

Equation of motion x=xr+ixi

2nd order linear homogeneous differential equation with constant coefficients General solution : x1 (t) and x2 (t) are linearly independent, i.e x1 (t) NOT proportional to (t) x2

2nd order linear inhomogeneous differential equation with constant coefficients General solution : Complementary function Particular integral: obtained by special methods, solves the equation with f(t)0; without any additional parameters A & B : obtained from initial conditions

Obtaining the particular integral Trial solution Therefore Hence

Amplitude, Relative Phase © SB

At resonance [w = w0, for wt=p/2,3p/2…]

Amplitude and Phase

Low Frequency Response Stiffness Controlled Regime

High Frequency Response Mass Controlled Regime

Summary Undamped forced oscillation (from PI) Stiffness controlled regime (w<w0) Resonance (w=w0) Mass controlled regime (w>w0)

References FEYNMAN LECTURES ON PHYSICS VOL I Author : RICHARD P FEYNMAN, IIT KGP Central Library