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 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  - FigureFigure  -

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

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

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

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

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)

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

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 Figure

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

 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 (3)

 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. Phase (4)

Barton’s PendulumBarton’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

 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 =  ) Barton’s Pendulum (2) Previous page

Hacksaw Blade Oscillator (1)

 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 Results Hacksaw Blade Oscillator (2)

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

 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 (2)

 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 curvesresonance curves Forced Vibration (3)

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 swingchild’s swing  Destruction of the Tacoma BridgeTacoma Bridge  Sound:  An opera singer shatters a wine glass  Resonance tube Resonance tube  Kundt’s tube Kundt’s tube

 Electricity  Radio tuning  Light  Maximum absorption of infrared waves by a NaCl crystal Resonance (3)

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

 Sonometer 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 Demonstration of Resonance (2)

Oscillation of a metal plate in the magnetic field

Slight Damping

Critical Damping

Heavy Damping

Amplitude

Phase

Barton’s Pendulum

Damped Vibration

Resonance Curves

Swing

Tacoma Bridge Video

Resonance Tube A glass tube has a variable water level and a speaker at its upper end

Kundt’s Tube

Sonometer