For the system in the figure derive the following The equati

For the system in the figure derive the following: The equation of motion The natural frequencies and modes for free vibration The initial conditions that would excite only each individual mode of vibration

Solution

Let the displacement of m1 be x1

Eqn of motion of m1 determined by force from K2 and the forces from K1

let the displacment of m2 be x2

m₂ x\"_{2 +} k₁ x₂ = force due to compression of lower spring = k₁( x₁-x₂) (a)

Extension of large spring is (x₁+x₂),

eqn odf motion m₁(x₁+x₂)\" + k₂(x₁+x₂) =0 (b)

Can solve (b) for natural freq : sqrt (k₂/m₁) = Sqrt(2000/5) /sec =20/s

Similarly solve for freq of small block.