Critically damped system(ξ=1): If the damping factor ξ is equal to one, or the damping coefficient c is equal to critical damping coefficient "cc", then the system is said to be a critically damped system.
ξ=1ORccc=1⟹c=cc
Two roots for critically damped system are given by S1 and S2 as below:
S1=[−ξ+ξ2−1]ωnS2=[−ξ−ξ2−1]ωn
For ξ=1; S1=S2=−ωn
Here both the roots are real and equal, so the solution to the differential equation can be given by
x=(A+Bt)e−ωnt...(1)
Now differentiating equation (1) with respect to ‘t’, we get:
x˚=Be−ωnt−ωn(A+Bt)e−ωnt...(2)
Now, let at t=0 : x=X0 t=0 : x˚=0
Substituting these values in equation (1):
X0=A...(3)
Same way, from equation (2), we get
00BB=B−ωn(A+0)=B−ωnA=ωnA=ωnX0...(4)
Now putting the values of A and B in equation (1), we get:
xx=(X0+ωnX0t)e−ωnt=X0(1+ωnt)e−ωnt...(5)
Conclusion
From above equation (5), it is seen that as time t increases, the displacement x decreases exponentially.
The motion of a critically damped system is aperiodic (aperiodic motion motions are those motions in which the motion does not repeat after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.
For critically damped systems, if a system is displaced from its initial position, it will try to reach its mean position in a very short time.
Critically damped systems are generally seen in hydraulic doors closer as it is necessary for the door to come to its initial position in a very short time.
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Logarithmic Decrement (δ)
Over-Damped System (ξ>1)
Under-Damped System (ξ < 1)
Suggested Notes:
Law of Parallelogram of Forces : 5 in 5 MCQs S01-E01
Variation of Tractive Force | Tractive Effort | Effect of Partial Balancing of Locomotives
Balancing of V-Engines
Concept of Direct and Reverse Crank for V-engines & Radial engines
Static and Dynamic Balancing
Multi Cylinder Inline Engine (with firing order) | Numerical
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