show that the ratio of two consecutive local maximum displac

show that the ratio of two consecutive local maximum displacements is a constant

Solution

Under-damped Case: If the damping is so small that b << n or r /2m<< Kim, then, p = (b2- n2) = - (n2 – b2) = (n2- b2) or p = jw, where w = (n2- b2).

Obviously, p is an imaginary quantity, but w is a real quantity.

Then from equation (2) X =           

Since x is a real quantity, therefore (A¹ + A²) and j(A¹ – A²) must also be real quantities. But A₁ and A₂ are the complex quantities, therefore

(A¹ + A²) = a0 sin and j(A¹ – A²) = a₀  cos in above .equation (3)

x = e ^-bt [ a₀ sin cos wt + a₀ cos sin wt]