In this lecture, we will address the case where damping is present in the mechanical system. This is a very important case. In fact, damping is always present and can be neglected in only very rare situations. We will hence introduce an additional force in the equilibrium equation of the oscillator taking into account the presence of damping. The equations will be solved in two main practical cases: the free oscillations of the system, and forced oscillations when an harmonic force is exterted. In a previous session, you have studied the dynamics of an oscillator consisting of a mass m and a stiffness k. Free oscillations, without force have been considered. They are characterized by harmonic functions sinus and cosinus, with a frequency omega = sqrt(k/m) and by amplitude which depend respectively on the initial position and velocity of the mass at t = 0. For instance, if we stretch the system and release it at t=0 with no velocity, the system will endlessly oscillate with an harmonic evolution of the form q_0 cos(omega t). But this does not look like what is observed in real life, where there is a decay of the oscillation amplitude. For instance, if you consider a classic tuner, the vibrations occur with an amplitude that decays with time. The same happens with a real mass-spring oscillator. In fact, various physical phenomena induce an energy loss in the system during the oscillations. This can be for instance the solid friction between two solids, or the friction occurring when a solid is moving inside a viscous fluid. Energy loss can also occur when vibration involves the deformation of viscoelastic materials, for instance, the rubber of a tyre. Also a vibrating structure may loose energy because of acoustic radiation, this is the case of musical instruments. In this session, we will consider a simple form of damping, which is linear damping. It is what would be caused for instance by the presence of a viscous fluid around the moving mass, or the addition a viscous damper in parallel with the spring. We will keep the later form of the oscillator in the following. Consider that the oscillator moves at a velocity (q)dot. The force exerted by this damper is proportional to minus the velocity, F = -c (q)dot. c is called the viscous damping coefficient. Inserting this force in the oscillator equation, we obtain a damped oscillator equation. And if there is an external forcing, there is a force on the right-hand side of this equation. And if we divide all the terms of the equation by m, here is what we have. I can now introduce a new dimensionless parameter, zeta, which equals c/(2 m omega). Here, omega still equals sqrt(k/m). The damping parameter zeta is called the reduced damping. In order to simplify the equations, we state that omega = 1. By doing this, we actually do the same non dimentionalization of time as in the previous lectures. We finally obtain on the left-hand side an oscillator equation that depends only on one parameter, the reduced damping zeta. In this session, let us focus on free vibrations. In other words, the oscillator we consider is not forced and the right-hand term of the equation term is hence now cancelled. We can now solve this equation for given initial contitions, q_0 and (q)dot_0 using the Laplace transform, as done for the undamped case in a previous lesson. In order to calculate the solution of this differential equation, the Laplace transform is first calculated. Then a simple algebraic equation is solved. And the inverse Laplace transform is used. We want to solve the oscillator equation, with damping, for given values of the displacement q_0 and velocity (q)dot_0 at t = 0. The Laplace transform of the differential equation is readily calculated using the same method as presented in the previous lectures. Solving the resulting algebraic equation is as easy, and we obtain the solution in the Laplace domain. The first case we consider is zeta lower than one. The solution we obtain after using the inverse Laplace transform takes a slightly more complicated form than the undamped case. It consists of two harmonic functions sinus and cosinus at a frequency omega_d different than in the undamped case, multiplied by an exponential decay in time. The influence of damping is hence twofold. First, it modifies the oscillation frequency. Second, it induces a time decaying amplitude of the oscillations. The larger is the damping, the faster is the decay. The resulting solution is illustrated here for two values of the reduced damping zeta of 0.05 and 0.1. The larger is the damping, the faster is the decay. Let us consider again the solution we have found and consider the particular case of small damping. When the damping is small, zeta is small compared to one, and the frequency of the damped oscillator can be reasonably approximated by omega. The time evolution of the displacement then corresponds to a slightly damped harmonic oscillation at the same frequency as in the undamped case. This is a common practical case. For instance, the oscillations of an instrument tuner fall in the category of slightly damped oscillations. Let us now address the opposite case of large damping, occurring when zeta>1. In this case, the system is referred to as overdamped. A different solution is found here. It consists this time of purely exponentially decaying functions, without any oscillation. The larger is the reduced damping, the faster is the decay. As an example, here is a typical free dynamics of an overdamped oscillator. The door closer is based on this principle. A strong damper is placed in parallel of a spring in order to smoothly close the door. It is time to conclude. We have seen in this lecture what are the effects of linear damping on the free dynamics of an oscillator. In a real structure, dampings may have many physical origins, and things may be more complicated to model. But the solution that is often used is then to calculate the eigenfrequencies and eigenmodes without taking into account any dissipation effect, and to estimate the damping for each mode through dynamical measurements. This is where all the properties of damped oscillators we have seen in this lecture becomes very useful. Next, let us see what happens with a force acting on the oscillator.
