Answer to Question #165311 in Physics for al-amin

Question

 What are free, damped, and forced vibrations? A particle showing damped harmonic motion is subjected to an external periodic force. Establish the differential equation of motion of the particle, explain each terms and solve it to obtain an expression for the displacement of the oscillator. (diagrams are necessary) 

Answer 1

Free Oscillation

The free oscillation possesses constant amplitude and period without any external force to set the oscillation. Ideally, free oscillation does not undergo damping. But in all-natural systems damping is observed unless and until any constant external force is supplied to overcome damping. In such a system, the amplitude, frequency and energy all remain constant.

Damped Oscillation

The damping is a resistance offered to the oscillation. The oscillation that fades with time is called damped oscillation. Due to damping, the amplitude of oscillation reduces with time. Reduction in amplitude is a result of energy loss from the system in overcoming external forces like friction or air resistance and other resistive forces. Thus, with the decrease in amplitude, the energy of the system also keeps decreasing.

Forced Oscillation

When a body oscillates by being influenced by an external periodic force, it is called forced oscillation. Here, the amplitude of oscillation, experiences damping but remains constant due to the external energy supplied to the system.

For example, when you push someone on a swing, you have to keep periodically pushing them so that the swing doesn’t reduce.

$$F _ d ​ =−bv$$

$$F =ma =−kx−mbv$$

$$\omega^2=\frac{k}{m}-\frac{b^2}{4m^2}$$

$$x=Ae^{-\frac{bt}{2m}}\cos{(\omega t+\phi_0)}$$