# vibrat damp

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• 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 (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

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