FORCED VIBRATION DAMPING Damping a process whereby energy

FORCED VIBRATION & DAMPING

Damping § § a process whereby energy is taken from the vibrating system and is being absorbed by the surroundings. Examples of damping forces: § § internal forces of a spring, viscous force in a fluid, electromagnetic damping in galvanometers, shock absorber in a car.

Free Vibration § Vibrate in the absence of damping and external force § Characteristics: § the system oscillates with constant frequency and amplitude § the system oscillates with its natural frequency § the total energy of the oscillator remains constant

Damped Vibration (1) § The oscillating system is opposed by dissipative forces. § The system does positive work on the surroundings. § Examples: § a mass oscillates under water § oscillation of a metal plate in the magnetic field

Damped Vibration (2) § Total energy of the oscillator decreases with time § The rate of loss of energy depends on the instantaneous velocity § Resistive force instantaneous velocity § i. e. F = -bv where b = damping coefficient § Frequency of damped vibration < Frequency of undamped vibration

Types of Damped Oscillations (1) § Slight damping (underdamping) § Characteristics: § - oscillations with reducing amplitudes § - amplitude decays exponentially with time § - period is slightly longer § - Figure § -

Types of Damped Oscillations (2) § Critical damping § No real oscillation § Time taken for the displacement to become effective zero is a minimum § Figure

Types of Damped Oscillations (3) § Heavy damping (Overdamping) § Resistive forces exceed those of critical damping § The system returns very slowly to the equilibrium position § Figure § Computer simulation

Example: moving coil galvanometer (1) § the deflection of the pointer is critically damped

Example: moving coil galvanometer (2) § Damping is due to induced currents flowing in the metal frame § The opposing couple setting up causes the coil to come to rest quickly

Forced Oscillation § The system is made to oscillate by periodic impulses from an external driving agent § Experimental setup:

Characteristics of Forced Oscillation (1) § Same frequency as the driver system § Constant amplitude § Transient oscillations at the beginning which eventually settle down to vibrate with a constant amplitude (steady state)

Characteristics of Forced Oscillation (2) § In steady state, the system vibrates at the frequency of the driving force

Energy § Amplitude of vibration is fixed for a specific driving frequency § Driving force does work on the system at the same rate as the system loses energy by doing work against dissipative forces § Power of the driver is controlled by damping

Amplitude § Amplitude of vibration depends on § the relative values of the natural frequency of free oscillation § the frequency of the driving force § the extent to which the system is damped § Figure

Effects of Damping § Driving frequency for maximum amplitude becomes slightly less than the natural frequency § Reduces the response of the forced system § Figure

Phase (1) § The forced vibration takes on the frequency of the driving force with its phase lagging behind § If F = F 0 cos t, then § x = A cos ( t - ) § where is the phase lag of x behind F

Phase (2) § Figure § 1. As f 0, 0 § 2. As f , § 3. As f f 0, /2 § Explanation § When x = 0, it has no tendency to move. maximum force should be applied to the oscillator

Phase (3) § When oscillator moves away from the centre, the driving force should be reduced gradually so that the oscillator can decelerate under its own restoring force § At the maximum displacement, the driving force becomes zero so that the oscillator is not pushed any further § Thereafter, F reverses in direction so that the oscillator is pushed back to the centre

Phase (4) § On reaching the centre, F is a maximum in the opposite direction § Hence, if F is applied 1/4 cycle earlier than x, energy is supplied to the oscillator at the ‘correct’ moment. The oscillator then responds with maximum amplitude.

Barton’s Pendulum (1) § The paper cones vibrate with nearly the same frequency which is the same as that of the driving bob § Cones vibrate with different amplitudes

Barton’s Pendulum (2) § Cone 3 shows the greatest amplitude of swing because its natural frequency is the same as that of the driving bob § Cone 3 is almost 1/4 of cycle behind D. (Phase difference = /2 ) § Cone 1 is nearly in phase with D. (Phase difference = 0) § Cone 6 is roughly 1/2 of a cycle behind D. (Phase difference = ) Previous page

Hacksaw Blade Oscillator (1)

Hacksaw Blade Oscillator (2) § Damped vibration § The card is positioned in such a way as to produce maximum damping § The blade is then bent to one side. The initial position of the pointer is read from the attached scale § The blade is then released and the amplitude of the successive oscillation is noted § Repeat the experiment several times § Results

Forced Vibration (1) § Adjust the position of the load on the driving pendulum so that it oscillates exactly at a frequency of 1 Hz § Couple the oscillator to the driving pendulum by the given elastic cord § Set the driving pendulum going and note the response of the blade

Forced Vibration (2) § In steady state, measure the amplitude of forced vibration § Measure the time taken for the blade to perform 10 free oscillations § Adjust the position of the tuning mass to change the natural frequency of free vibration and repeat the experiment

Forced Vibration (3) § Plot a graph of the amplitude of vibration at different natural frequencies of the oscillator § Change the magnitude of damping by rotating the card through different angles § Plot a series of resonance curves

Resonance (1) § Resonance occurs when an oscillator is acted upon by a second driving oscillator whose frequency equals the natural frequency of the system § The amplitude of reaches a maximum § The energy of the system becomes a maximum § The phase of the displacement of the driver leads that of the oscillator by 90

Resonance (2) § Examples § Mechanics: § Oscillations of a child’s swing § Destruction of the Tacoma Bridge § Sound: § An opera singer shatters a wine glass § Resonance tube § Kundt’s tube

Resonance (3) § Electricity § Radio tuning § Light § Maximum absorption of infrared waves by a Na. Cl crystal

Resonant System § There is only one value of the driving frequency for resonance, e. g. spring-mass system § There are several driving frequencies which give resonance, e. g. resonance tube

Resonance: undesirable § The body of an aircraft should not resonate with the propeller § The springs supporting the body of a car should not resonate with the engine

Demonstration of Resonance (1) 1. Resonance tube 1. Place a vibrating tuning fork above the mouth of the measuring cylinder 2. Vary the length of the air column by pouring water into the cylinder until a loud sound is heard 3. The resonant frequency of the air column is then equal to the frequency of the tuning fork

Demonstration of Resonance (2) § Sonometer § Press the stem of a vibrating tuning fork against the bridge of a sonometer wire § Adjust the length of the wire until a strong vibration is set up in it § The vibration is great enough to throw off paper riders mounted along its length