THIS LECTURE From free to damped oscillations k In a damped oscillation the mechanical energy is dissipated by frictional forces

Friction k Micrometer scale (~ m) Atomic scale (~ m) Friction arises from electromagnetic forces between atoms at the interface of two objects Empirical expression for the frictional force r = resistance of the medium

Damped oscillators k Light damping Heavy damping Critical damping Dependence of x on t for a mass m under the action of elastic and frictional force x

A shock absorber (or damper) is a mechanical device designed to smooth out or dampen a sudden shock impulse and dissipate kinetic energy. Dampers or shock absorbers Shock absorber in cars Critical damping

Tuned mass dumper Taipei 101Taipei 101 skyscraper (Taiwan) - Contains the world's largest tuned mass damper at 730-tons Dampers or shock absorbers

Damped oscillators k Light damping Heavy damping Critical damping Dependence of x on t for a mass m under the action of elastic and frictional force x Coefficient of damping Dependence of x on t

Which system reaches the equilibrium first? m k m m k (a) No damping (b) With damping (a) reaches the equilibrium first m

Which system reaches the equilibrium first (in the presence of damping)? They reach the equilibrium at the same time m m k k

What is the value of the coefficient of damping,  ? What kind of damping (heavy, light or critical)? What is the value of the natural angular frequency,   ? EXAMPLE

Find the general solution of EXAMPLE A and  are determined by initial conditions Light damping

Sound proofing is any means of reducing the intensity of sound with respect to a specified source and receptor. A sound proof room showing acoustic dampening tiles. Dampers for sound proofing