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MECHANICAL VIBRATIONSASSIGNMENT 01TOPICS

Types of dampingViscous damping Coulomb dampingStructural dampingNon-linear damping

Differential equations of damped free vibrations.Submitted by-Abhishek negiSG-11902Mechanical 6th semester Submitted by-Abhishek negiSG-11902Mechanical 6th semester 1INTRODUCTIONDamping is the resistance offered by a body to the motion of a vibratory systems. The resistance may be applied by a liquid or solid internally or externally.Example- motion of the car wheel in water is resisted by water itself, because of resistance vibrations die out over a few cycles of motion. for small damping value negligible influence on natural frequency of system.

amplitude of vibration is max. at start which decreases and finally lost with time.

rate of decrement in amplitude depend on amount of damping. for small damping value negligible influence on natural frequency of system. amplitude of vibration is max. at start which decreases and finally lost with time. rate of decrement in amplitude depend on amount of damping.2ADVANTAGE

It control the amplitude of vibration so that failure due to resonance may avoided. It improve the efficiency of the system. It makes some of the machinery user friendly(regarding to comfort).TYPES

Viscous damping Coulomb dampingStructural dampingNon-linear dampingADVANTAGE It control the amplitude of vibration so that failure due to resonance may avoided. It improve the efficiency of the system. It makes some of the machinary user friendly(regarding to comfort).3Viscous DampingWhen the system allowed to vibrate in a viscous medium the damping is called as Viscous damping.

Viscosity is the property of a fluid due to which it offers resistance.

It offers resistance to the motion of one layer over the adjacentIn fig. upper plate move parallel to fix plate with a velocity . Net force F is required to maintain vel. of the plate.F= At

A = area of platet = thickness of the fluid film = coeff. Of absolute viscosity

F= AtA = area of platet = thickness of the fluid film = coeff. Of absolute viscosity 4This force F can also be written as F = c so c = A/tc = viscous damping coefficient Main component of viscous damper cylinder piston viscous fluidMore the clearance more velocity of piston hence it will offer less value of c.Damping resistance depends upon pressure difference on both side of piston.

Fig. shows the example of free vibrations with viscous damping.The equation of motion for the system can be written as m + c + kx = 0Fig. shows the example of free vibrations with viscous damping.The equation of motion for the system can be written as m + c + kx = 05Energy dissipation in viscous damping

for a viscously damped system F = c = dx/dt

Work done dW = F.dx =(c.dx/dt).dxThe rate of change of work per cycle i.e.

Assuming simple harmonic motion for which x = A sint so putting this x we finally get E = cA2

Specific Damping ratio = E/E = 2c / m

Damping ratio = E/E = 2c / m6Coulomb DampingDamping offered by the frictional force between two surfaces due to relative motion of those two surfaces. F = RN

The damping resistance is almost constant and does not depend on the rubbing velocity.

The three possible condition of coulomb damping are shown in figure.

for leftward motion of body m + kx = F For for leftward motion of body m + kx = F 7Where n = (k/m)

Solving for constant using boundary conditions we finally get

This solution hold good for half the cycle. So for t = /n x = (x0 F/k)cos + F/k = (x0 2F/k)

The initial displacement x0 is reduced by 2F/k during half cycle.

In one complete cycle initial displacement reduced by 4F/k.

The natural frequency of the system remains unchanged in coulumb damping. Solution for above equation can be written as d w Solution for above equation can be written asSolution for above equation can be written as

8Structural damping

This type of damping is due to resistance offered by elastic properties from within the body.The magnitude of this damping is very small as compared to other dampings. experimentally for elastic body loading and unloading conditions a loop is formed on stress-strain curve. this is called hysteresis loop. area of loop is amount of energy dissipated in one cycle during vibrations. E is not the function of frequency but approximately proportional to the square of amplitude of vibration and proportional to stiffness E = kA2 where A = amplitude of vibration = dimensionless damping factor E = kA2 where A = amplitude of vibration = dimensionless damping factor9Also E = A2Where = k

If this energy is treated equals to energy dissipation by viscous damping

cA2 = A2 c = / F = c = A/

Amplitude decay is found to be exponential in nature.

When force is plotted against displacement then a close loop is formed.

The area of the loop denotes the energy dissipated by the damper in one cycle of motion.10Non-linear/slip/interfacial DampingMicroscopic slip occurs on the interfaces of machine element which causes dissipation of vibrational energy when the interface of machine element are under fluctuating loads. amount of damping depends upon surface roughness, contact pressure and amplitude of vibration. the energy dissipated per cycle depends upon the coefficient of friction, contact pressure and amplitudes. there is a optimum value of pressure for which the energy dissipated is maximum. this value is different for different amplitudes. Larger the energy dissipation larger is the effective damping in the system.

amount of damping depends upon surface roughness, contact pressure and amplitude of vibration. the energy dissipated per cycle depends upon the coefficient of friction, contact pressure and amplitudes. there is a optimum value of pressure for which the energy dissipated is maximum. this value is different for different amplitudes. Larger the energy dissipation larger is the effective damping in the system.11Differential equations of damped free vibrationLet us take a damped spring and mass system for vibration analysis.We know that viscous damping force is proportional to the velocity across the damper, so we are using viscous damping here.A mass m is attached from one end of the spring k, the other end is fixed.A damper is also provided as shown in figure.

Damping force = cAccelerating force = mSpring force = kx

Thus the equation of motion will bem + c + kx = 0Also called the Characteristics equation.

This is differential equation of 2nd order in x.Free body diagramFree body diagram12Assuming a solution of the form x = eutThen = ueut = u2eutSubstituting these values we get mu2eut + cueut + keut = 0 u2 + (cu/m) + (k/m) = 0Solving for u we get

Now solution can be written as

x = A1eu1t + A2eu2t

Where A1 and A2 are two arbitrary constants and u1 and u2 are two rootsNow s x = A1eu1t + A2eu2tWhere A1 and A2 are two arbitary constants and u1 and u2 are two roots13Critical damping (cc) is defined as the value of damping coefficient c for which the term (c/2m)2 k/m becomes zero. cc/2m = (k/m) cc = 2m Damping ratio = = c/ccWe can write the term c/2m asc/2m = (c/cc)(cc/2m) =

So the characteristic equation now written as

Critical damping (cc) is defined as the value of damping coefficient c for which the term (c/2m)2 k/m becomes zero. cc/2m = (k/m) cc = 2m Damping ratio = 14