We know that the characteristic equation of the damped free vibration system is,
mS2+cS+K=0
This is a quadratic equation having two roots S1 and S2;
S1,2=2m−c±(2m−c)2−mK
In order to convert the whole equation in the form of ξ , we will use two parameters, critical damping coefficient 'cc' and damping factor 'ξ'. So the roots S1 and S2 can be written as follows;
ξ=cccORξ∴2mc=2mω0c(ascc=2mωn)=ξωn
where, ωn = natural frequency of undamped free vibration = mK rad/s
If the damping factor ‘ξ’ is greater than one or the damping coefficient ‘c’ is greater than critical damping coefficient ‘cc’, then the system is said to be over-damped.
ξ>1OrcccOrc>cc
In overdamped system, the roots are given by;
S1And, S2=[−ξ+ξ2−1]ωn=[−ξ−ξ2−1]ωn
For ξ>1 , we get S1 and S2 as real and negative so we get,
Above equation represents the equation of motion for overdamped system.
The motion obtained by above equation is aperiodic (aperiodic motion motions are those motions in which the motion does not repeats after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.
This type of system does not show much damping, so this systems are used very rarely.
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
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Critically Damped System (ξ = 1)
Damped free Vibration - Numerical 1
Damped free Vibration - Numerical 2
Damped free Vibration - Numerical 4
Logarithmic Decrement (δ)
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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