vibrat damp

FORCED VIBRATION & DAMPING

Dampinga 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 VibrationVibrate in the absence of damping and external forceCharacteristics:the system oscillates with constant frequency and amplitudethe system oscillates with its natural frequencythe 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 wateroscillation of a metal plate in the magnetic field

Damped Vibration (2)Total energy of the oscillator decreases with timeThe rate of loss of energy depends on the instantaneous velocityResistive force instantaneous velocityi.e. F = -bv where b = damping coefficientFrequency 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 dampingNo real oscillationTime taken for the displacement to become effective zero is a minimumFigure

Types of Damped Oscillations (3)Heavy damping (Overdamping)Resistive forces exceed those of critical dampingThe system returns very slowly to the equilibrium positionFigure 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 frameThe opposing couple setting up causes the coil to come to rest quickly

Forced OscillationThe system is made to oscillate by periodic impulses from an external driving agentExperimental setup:

Characteristics of Forced Oscillation (1)Same frequency as the driver systemConstant amplitudeTransient 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

EnergyAmplitude of vibration is fixed for a specific driving frequencyDriving force does work on the system at the same rate as the system loses energy by doing work against dissipative forcesPower of the driver is controlled by damping

AmplitudeAmplitude of vibration depends onthe relative values of the natural frequency of free oscillationthe frequency of the driving forcethe extent to which the system is dampedFigure

Effects of DampingDriving frequency for maximum amplitude becomes slightly less than the natural frequencyReduces the response of the forced systemFigure

Phase (1)The forced vibration takes on the frequency of the driving force with its phase lagging behindIf F = F0 cos t, then x = A cos (t - )where is the phase lag of x behind F

Phase (2)Figure1. As f 0, 02. As f , 3. As f f0, /2ExplanationWhen 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 forceAt the maximum displacement, the driving force becomes zero so that the oscillator is not pushed any furtherThereafter, 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 directionHence, 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.

Bartons Pendulum (1)The paper cones vibrate with nearly the same frequency which is the same as that of the driving bobCones vibrate with different amplitudes

Bartons Pendulum (2)Cone 3 shows the greatest amplitude of swing because its natural frequency is the same as that of the driving bobCone 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 vibrationThe card is positioned in such a way as to produce maximum dampingThe blade is then bent to one side. The initial position of the pointer is read from the attached scaleThe blade is then released and the amplitude of the successive oscillation is notedRepeat the experiment several timesResults

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

Forced Vibration (2)In steady state, measure the amplitude of forced vibrationMeasure the time taken for the blade to perform 10 free oscillationsAdjust 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 oscillatorChange the magnitude of damping by rotating the card through different anglesPlot 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 systemThe amplitude of reaches a maximumThe energy of the system becomes a maximumThe phase of the displacement of the driver leads that of the oscillator by 90

Resonance (2)ExamplesMechanics:Oscillations of a childs swingDestruction of the Tacoma BridgeSound:An opera singer shatters a wine glassResonance tubeKundts tube

Resonance (3)ElectricityRadio tuningLightMaximum absorption of infrared waves by a NaCl crystal

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

Resonance: undesirableThe body of an aircraft should not resonate with the propellerThe springs supporting the body of a car should not resonate with the engine

Demonstration of Resonance (1)Resonance tubePlace a vibrating tuning fork above the mouth of the measuring cylinderVary the length of the air column by pouring water into the cylinder until a loud sound is heardThe resonant frequency of the air column is then equal to the frequency of the tuning fork

Demonstration of Resonance (2)SonometerPress the stem of a vibrating tuning fork against the bridge of a sonometer wireAdjust the length of the wire until a strong vibration is set up in itThe vibration is great enough to throw off paper riders mounted along its length

Oscillation of a metal plate in the magnetic field

Slight Damping

Critical Damping

Heavy Damping

Amplitude

Phase

Bartons Pendulum

Damped Vibration

Resonance Curves

Swing

Tacoma BridgeVideo

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

Kundts Tube

Sonometer