Recent Applications of Damping Systems for Wind Response
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Topics to discuss
Introduction -Types -Types of Damping Hysteresis Damping Free Free Vibration with Hysteretic Damper Forced Forced Vibration with Hysteretic Damper Examples
Introduction
Damped Vibration : When the th e energy of a vibrating system is gr gradually adually dissipated by friction and other resistances, resistances, the .
Types of Damping Damping models models Viscous damping models Hysteretic damping model
Hysteretic Hyst eretic Damper Damper
The damping caused by the friction between the internal planes planes that slip or slide as the material deforms is called hysteresis (or solid or structural) .
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) F(t) = k X sin sin ωt+ ωt+ c X ω cos cos ωt =
When F versus x is plotted represents a closed loop. lo op. The area area of the loop loop denotes the energy ss pa e y e amper 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 isp isp acem acemen entt o sprin pring g mas masss damper system can be used to define define a hysteresis hysteresis damping constant. It was found experimentally experim entally that the energy loss per cycle due to to internal internal friction frict ion is independent of the frequency but approximately proportional to the square of the amplitude
The damping coefficient coeff icient e is assumed to be inversely proportional to the frequency frequency as
where h is called the hysteresis damping constant. The energy dissipated by the damper in a cycle cycle of motion becomes
Complex Stiffness The spring and the damper are connected connected in parallel The force-displacement relation
Where is called the complex stiffness of the system and dimensionless measurement of damping.
is a constant indicating
Response of the system
In terms of β, the th e energy loss per cycle cycle can be expressed as n er ysteres s amp ng ,t e mot on can e near y considered as harmonic and the decrease in the amplitude amplitude per cycle can be determined using energy e nergy balancing .
Consider the energies at points P and Q
--------(a) Similarly, Similarly, the energies energie s at points Q and R give ------------(b) Multiplying equation equatio n (a) and (b) we we have
The hysteresis logarithmic logarithm ic decrement can be defined defined as e equ va ent v scous amp ng rat o s g ven y
The equivalent damping constant Ceq is given by
Forced Vibra Forced Vibration tion with Hyster Hysteresis esis Damping
Consider a single degree system with hysteresis damper. 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 solution of equation equation of motion can be assumed as By substituting substituting we have
The amplitude amplitude ratio
attains its maximum
value of at the resonant frequency frequency in the case of hysteresis damping, while while it occurs occurs at a frequency 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 indicates that the response can never be in phase with the forcing function function in the case of hysteresis damping.
Application
Hysteresis Dampers are are used for controlling seismic seism ic 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 whic h which tends to produce oscillations oscillations of high frequency frequency ,low ,low amplitude amplitude in a direction transverse transverse to wind stream stream which result in fatigue failures.
Reference
Mechanical Vibration by S.S.Rao S.S.Rao 4/e, Pearson Pearson Education Inc 2004. Technical echnical Review Vol. Vol. 42 No. 1 (Feb. (Feb. 2005) 2005) ,Mitsubishi ,Mitsubishi eav y n us r es, . ys eres s ampers or
Controlling Seismic Response of Bridges and Structures, by MOTOETSU ISHII, SATORU UEHIRA, YASUO YASUO OGI, OGI, KUNIHIRO KUNIHIRO MORISHIT MORISHITA. A.
