In the previous video on damped oscillators, we have characterized the free oscillations of a mass-spring-damper system. Two situations have been evidenced. On the top the underdamped case when zeta is smaller than one, and on the bottom the overdamped case, when zeta is larger than one. We will now address the presence of an external forcing. Our goal is to see what happens to the resonance phenomenon when damping is added to the oscillator. Here you can see the damped oscillator equation we have obtained in the last video. The right-hand forcing term in the equation is now kept. This forcing term f is a function of time and in this video we will address only the particular case of an harmonic forcing f_0 = cos(omega_f t). In order to simplify the calculations, we will make a few asumptions. First of all, we will not use the Laplace transform. Although it is possible to obtain analytical solutions in this case using Laplace transfoms, it is more complicated and does not serve the understanding. Secondly, we will consider only a permanent regime of the oscillator, when all transients have been ultimately damped. It is the same as considering that the harmonic forcing has existed from time equals minus infinity. Thirdly, we will solve the problem using complex harmonic functions. The forcing is then of the form real part of e^(i omega_f t), which is strictly equivalent to the initial form of the harmonic forcing we are interested in. The response is also sought in the form of a complex harmonic function. This time, it is considered that the value q_0 can be complex, hence allowing the real solution to be an harmonic function at the same frequency, but a different phase than the forcing. Let us now solve the equation using these solutions. In practice, the only unknown is now q_0. The oscillator equation is rewritten inserting the previous complex harmonic functions for f and q. The equation is then simplified by removing the exponential in factor of all terms. We finally obtain the ratio between q_0 and f_0. This ratio is called the transfer function between the forcing and the displacement response. It is a complex function. In the following we will study this transfer function by looking at its modulus, which quantifies the relative amplitudes of the forcing and the response, and by looking at its argument, which characterizes the relative phases of the forcing and the responses. Let us first analyse the modulus of the transfer function. It is plotted here as function of the forcing frequency omega_f. Three regimes appear on this plot. First of all, at low forcing frequencies, the transfer function tends to the limit one over k. In this very low frequency regime, the stiffness dominates all forces and the amplitude of the response is that of a spring subjected to a force f_0. Next, at very large frequencies, the transfer function tends asymptotically to zero following the law 1/(m omega_f^2) . Here, it is the inertia that dominates the response, which is then predicted by that of a mass m subjected to a force f_0 cos(omega_f t). These two limit regimes are the same as that predicted by the simple oscillator without damping. Let us now focus on the resonant regime. This occurs when the forcing frequency equals the oscillator frequency. In the undamped case, the oscillator response was infinite at omega_f = omega. Now, the maximum value is a finite value. Hence, at resonance the response is mitigated by the damping. And the higher is the damping, the lower is the resonant response. The frequency at which the resonance occurs can be calculated, it is (omega_f)_max = omega sqrt(1 - 2 zeta^2). Hence, damping reduces the resonance frequency and a resonance peak exists until the quantity in the square root vanishes. That is for zeta = 1/sqrt(2). Let us now look at the expression of the peak amplitude. It is 1/[(2 zeta) sqrt(1 - zeta^2)]. It is remarkable that it does only depend on the reduced damping. The larger is the damping, the smaller is the resonance peak. And finally, in the limit case of overdamped systems, the response does not present a peak anymore, but smoothly connects the stiffness response to the inertial response. Let us now look at the argument of the ratio q_0/f_0. Its value determines the phase relationship between the forcing and the response. It is plotted here in a typical case. In the stiffness regime for small forcing frequencies, it vanishes, indicating that the forcing and the response are in phase. Conversely, in the inertial regime, for large frequencies, the argument of the transfer function equals -pi, indicating that the force and the response are in phase opposition. When damping is increased, this is still valid. Although, the higher is the damping, the smoother is the transition between the stiffness and inertial regimes. You can also note that in any case, it passes through the value -pi/2 for omega = omega_f. Thus, the forcing and the response are in phase quadrature at this frequency. Finally, the previous three cases are illustrated here. On the left, you can see the response dominated by the stiffness at low frequencies, which is in phase with the forcing. On the right, there is the response dominated by inertia at large frequencies. It is in phase opposition with the forcing. And in the center, there is the resonant response, which is in phase quadrature with the forcing. At resonance, the amplitude of the response can be very large. The smaller is the reduced damping, the larger is the response. It is time to conclude. We have seen in this lecture what are the effects of an harmonic forcing on a typical damped oscillator. The effect of damping is in particular to reduce the amplitude of the response at resonance. And, the larger is the damping, the smaller is the resonance peak. Looking at the phase of the transfer function, we have seen that when the forcing frequency equals the frequency of the undamped oscillator, we have phase quadrature. The knowledge of this transfer function properties can be very useful for modal identification of mechanical systems, as we will see in future videos.
