Question

The equation of motion of system is given by mx¨+cx˙+kx=0. The damped vibration, ωd can be written as:

• A. (A) ωd=(1−ξ2)ωn
• B. (B) ωd=(ωn)2−(c2m)2
• C. (C) Both A and B
• D. (D) Neither A nor B

Answer 1

Answer: (C)

(C) Both A and B

Answer 2

Explanation:
The general case of damped harmonic motion:Fnet=md2xdt2+cdxdt+kx=0Where k= the stiffness of spring, c= damping coefficient and m= massDamped vibration of a system is defined as:ωd=(ωn)2−(c2m)2⋯⋯(1)Damping ratio (ξ) : It is defined as the ratio of actual damping to the critical damping.ξ= actual damping critical damping =ccc⇒ξ=ccc=c2mωn=c2kmc2m=ξωnSubstitute the value of ξ in equation 1, we get,ωd=(ωn)2−(c2m)2=(ωn)2−(ξωn)2=(1−ξ2)ωnHence the correct option is (C).