The block A and the platform P of a spring scale are at rest when the lab bench to which t...

The block A and the platform P of a spring scale are at rest when the lab bench to which the scale is rigidly attached begins vibrating sinusoidally with a frequency of 15 Hz and amplitude of 5 mm. The block and the platform are coupled to the base B of the scale by a linear elastic spring and a viscous damper that are internal to the scale. The combined mass of the block and platform is mA = 1.5 kg, the spring constant is k = 50 N/m, and the viscous damping coefficient is c = 7.5 N·s/m.

Figure P9.65

Determine and plot for 10 s the vertical motion of the platform and block as a function of time. The base of the scale B is rigidly attached to the lab bench, and the block A does not separate from the platform P during the vibration. Hint: Parts (a)–(c) of Prob. 9.63 will be helpful.

Prob. 9.63

(a) Derive its equation of motion, using x as the independent variable, and explain in what way the resulting equation of motion is not in the form of Eq. (9.65).

(b) Next, let z = x − y and substitute it into the equation of motion found in Part (a). After doing so, show that you obtain an equation of motion in z that is of the same form as Eq. (9.65).

(c) Find the steady-state solution to the equation of motion found in Part (b) and then using that, determine the steady-state solution for x.