23-02-2011, 09:44 AM
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IMPACT VIBRATION CONTROL USING SEMI-ACTIVE DAMPER
Introduction
Dynamic vibration absorbers are used for suppressing the steady state vibration of a machine.
It consists of spring-mass-damper system.
For an impact force, it can damp only the residual part of impact vibration
But maximum amplitude of the vibration is almost remain same that of undamped vibration
DYNAMIC VIBRATION ABSORBER
RESPONSE Of An IMPULSE
FREE BODY DIAGRAM
EQUATION OF MOTION
Control force of dynamic damper can be written as
fd=Mdx”=-c(x-xd)-Kd(x-xd)
By substituting in (1) we will get
Mx”+Kx=fim-fd
RESPONSE CALCULATION
We calculate the response numerically by Runge-Kutta method. For this we take
M=100Kg
K=9.87MN/m
Fo=63.66kNs
Md=10Kg
C=1.055KNs/m
Kd=0.816MN/m
RESPONSE DUE TO IMPACT
TOTAL TRANSMITTED FORCE
Mx”+Kx=fim-fid
If we take Laplace transform of equation Ft(s) =G(s)Fim(s)-G(s)Fd(s)
Where G(s)=K/(Ms2+K)
Ft=KX(s)-transmitted force to the ground.
TOTAL TRANSMITTED FORCE
If we take the inverse Laplace transform
ft= ftm-ftd
ft= Total transmitted force
ftm-transmitted force due to impact force fim
ftd=transmitted force due to control force fd
COMPONENTS OF FORCE
TRANSMITTED FORCE
Transmitted force is sum of transmitted force due to impact and due to damping force.
Both these forces are just opposite and cancel each other.
But in the first few waves ftd lacks sufficient magnitude
CONCLUSION FROM GRAPH
Two methods we can adopt to suppress the force vibration of impact
1. Increase the amplitude of first wave of ftd by increasing certain action for driving damper mass
2. Utilise the large portion such as the second or third wave.
As a solution to check we will give an initial displacement to damper say xd(0)
INITITAL DISPLACEMENT
Take the Laplace transform of given equation(1) with initial condition xd(0)
Ft(s)= (s2+2ζ ωds+ωd) ωt2fo+Kd ωt2s xd(0)
Gd(s)
Where
Gd(s)=s4+2(1+µ) ζωd S3+[(1+ µ) ωd 2+ ωt2 ]s2+2 ζ ωd ωt2 s+ ωt2ωd 2
2ζ ωd =C/Md Kd/ Md = ωd 2 K/M= ωt2