2.4.1 Without Complex Amplitudes

To analyze the forced harmonic motion of the damped oscillator, we add a driving force F( t) = F cos( ?t) on the right-hand side of Eq. 2.23. The corresponding steady state expression for the displacement is assumed to be ? = ? cos( ?t ? ). Inserting this into the equation of motion, we get for the first term ? M ? ² ? cos( ?t ? ), for the second, ? R ? ? sin( ?t ? ), and for the third, K ? cos( ?t ? ). Next, we use the trigonometric identities cos( ?t ? ) = cos( ?t) cos + sin( ?t) sin and sin( ?t ? ) = sin( ?t) cos ? cos( ?t) sin and express each of these three terms as a sum of cos( ?t)- and sin( ?t)-terms. Since we have only a cos( ?t)-term on the right-hand side, the sum of the sine terms on the left-hand side has to be zero in order to satisfy the equation at all times and the amplitude of the sum of the cosine terms must equal F. These conditions yield two equations from which ? and can be determined. It is...