A mass of 30 kg (shown in) is hung from a spring of stiffness 5 k = × 2.5 10 N/m, which is attached to an aluminum beam ( 9 2 E = × 71 10 N/m , 3 3 ρ = × 2.7 10 kg/m ) of moment of inertia 8 4 I m 3.5 10− = × and of length 35 cm. The beam is supported at its free end and by a circular aluminum cable of diameter 1 mm and length 30 cm.

(a) Determine the equivalent stiffness of the assembly.

(b) Write the differential equation governing in the motion of the mass and find its natural frequencies.

 

Aluminum 1 mm diameter -35 cm- Aluminum I= 3.5 x 10-8 m4 30 cm 2.5 10 N/m 30 kg X