Shivaprasad.PShivaprasad.P

080922004

MTech CAMDA

1st semester

Topics to discuss� Introduction

-Types of Damping

� Hysteresis Damping

Free Vibration with Hysteretic Damper� Free Vibration with Hysteretic Damper

� Forced Vibration with Hysteretic Damper

� Examples

Introduction

� Damped Vibration :

When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibration is said to be damped.vibration is said to be damped.

Types of Damping models

� Viscous damping models

� Hysteretic damping model

Hysteretic Damper� The damping caused by the friction between the

internal planes that slip or slide as the material deforms is called hysteresis (or solid or structural)

damping.damping.

Free Vibration with Hysteretic Damper

� Consider the spring-viscous damper arrangement

� For this system the force needed to cause displacement x(t)

� For a harmonic motion of frequency ω and amplitude X,

� x(t)=X sin ωt

F(t) = k X sin ωt+ c X ω cos ωt

=

� When F versus x is plotted

represents a closed loop.

� The area of the loop denotes the

energy dissipated by the damperenergy dissipated by the damper

in a cycle of motion and is given by

� The energy loss in one loading and unloading cycle is equal to the area enclosed by the hysteresis loop.

� The similarity between the

hysteresis loop and Force vs

displacement of spring mass displacement of spring mass

damper system can be used to

define a hysteresis damping

constant.

� It was found experimentally

that the energy loss per cycle due to internal friction is independent of the frequency but approximately proportional to the square of the amplitude

� The damping coefficient e is assumed to be inversely

proportional to the frequency as

� where h is called the hysteresis damping constant.

� The energy dissipated by the damper in a cycle of motion becomes

� Complex Stiffness

� The spring and the damper

are connected in parallel

The force-displacement relation

can be expressed ascan be expressed as

Where

is called the complex stiffness of the system and is a constant indicating dimensionless measurement of damping.

Response of the system� In terms of β, the energy loss per cycle can be

expressed as

� Under hysteresis damping ,the motion can be nearly � Under hysteresis damping ,the motion can be nearly considered as harmonic and the decrease in the amplitude per cycle can be determined using energy balancing .

� Consider

the energies at points P and Q

--------(a)

Similarly, the energies at points Q and R give

------------(b)

Multiplying equation (a) and (b) we have

� The hysteresis logarithmic decrement can be defined as

� The equivalent viscous damping ratio is given by � The equivalent viscous damping ratio is given by

� The equivalent damping constant Ceq is given by

Forced Vibration with Hysteresis

Damping� Consider a single degree

system with hysteresis

damper.

The system is subjected to� The system is subjected to

Harmonic force

F(t)= F0 sin ωt

� The equation of motion can be derived as

� Where denotes the damping force.

� The steady-state solution of equation of motion can be assumed as

� By substituting we have

� The amplitude ratio attains its maximum

value of at the resonant frequency in the case

of hysteresis damping, while it occurs at a frequency of hysteresis damping, while it occurs at a frequency below resonance in the case of viscous damping.

� The phase angle φ has a value of at ω=0 in the case of hysteresis damping . This indicates that the response can never be in phase with the forcing function in the case of hysteresis damping.

Application� Hysteresis Dampers are used for controlling seismic

response of Bridges and Structures.

Damper brace system

MCB damper system

� Stockbridge damper is also an hysteresis damper.

� Used to arrest the vortex excitation, which

which tends to produce oscillations of

high frequency ,low amplitude in a high frequency ,low amplitude in a

direction transverse to wind stream which

result in fatigue failures.

Reference� Mechanical Vibration by S.S.Rao 4/e, Pearson

Education Inc 2004.

� Technical Review Vol. 42 No. 1 (Feb. 2005) ,Mitsubishi

Heavy Industries, Ltd. Hysteresis Dampers for Heavy Industries, Ltd. Hysteresis Dampers for Controlling Seismic Response of Bridges and Structures, by MOTOETSU ISHII, SATORU UEHIRA, YASUO OGI, KUNIHIRO MORISHITA.

�Thank

you.you.