Question

The Figure below shows a homogeneous cylinder of radius R and mass m. Assuming that the cylinder rolls on a rough surface without sliding, derive the equations of motion using the energy conservation approach, and determine the natural frequency. 179

Answer 1

E.O.M and Natural frequency: problem Dx mass-m Radius = R. Angular displacemen K K No. of Springs - 3 Stiffness of =k each spring Three Springs in Parallel connections: Equivalent stiffness = K, +K ₂ + K3 - ККК Ke=Bk "Here there is no Couple". No Elastic energy relates Couple to springs in our case So, the problem will be 4 A No couple * 3k 4

By Energy Approach, Elastic strain Energy of the spring is Here Beg= 3k SE =} (3K) x+) = (36)(RD) Rotational kinetic Energy of the disk KE = 1/2 2 1 6 h I disc - ImR²; Here x=R.O. > x=RO 0. + 1 = R6 Linear Kinetic Energy of the Disc is K. E = 1 mv2 = 1 m(G) 2 = { mCRO)? = 1 mR²2 According to Conservation of Energy Principle Total Energy is constant d (T. E) = 0 & 2 (T. E) = 0. . -

x R.E Total Energy = S.E + Ki Ep + KĘ = || (36) * + 7 (mp" 30 + mor (1 : x = 2 (36) R2 + 7 (mp 8 + 1 mR" 2 d (T.E)=0 doc" = nx? > 0 = { (3R)R(20)6 +(mR") Ge)ő +mR126) 8 = [3kRo + mk ő + mR8 å Since oto - (mR" + m2) + (3KR²0 = 0 (3 mR!) 8 + (3% R4 = 0 Comparing with Equations of simple Harmonil Motion ģtur o=o. Here won = ( 3kR² ) = (2x . Natural frequency = Wr - 12 h ( had see) | | n 5 (H2) ЭК m MR 2K η Hertz