Question

10-30. A block of mass m is dropped from a height h above a spring k, as shown in Fig. P10.30. Beginning at the time of impact (t = 0), the position x(t) of the block satisfies the following second-order differential equation and initial conditions:

mẍ(t) + kx(t) = mg x(0) = 0,

ẋ (0) = √2 g h.

(a) Determine the transient solution
xtran (t), and determine the frequency of oscillation.

(b) Determine the steady-state solution, xss(t).

(c) Determine the total solution x(t), subject to the initial conditions.

(d) Mark each of the following statements as true (T) or false (F):
Increasing the stiffness k will increase the frequency
of x(t).
Increasing the height h will increase the frequency of x(t).
Increasing the mass m will decrease the frequency of x(t).
Doubling the height h will double the maximum value
of x(t).
Doubling the mass m will double the maximum value of x(t).

Answer 1

Given that:

where

Since

  

a))

The transient solution :

b))

The steady state solution :

c))

The total solution of x(t), subject to intial conditions :

d))

Following statemets true or false:

increases increases True

it's does not depends on h(false,(true))

if increases decreases

during will no double max

decreases(false)

during m will also not double false

from(4).