Answer to Question #249758 in Mechanical Engineering for Zee

Question

A CEO of a well-known television show wants to build a new rectangular shaped acting stage. The width, thickness and length of the stage are 3 m, 0.2 m and 5 m, respectively. Both ends of the stage must be fixed. The stage should carry a musical instrument, and air conditioner of a total mass of 1500 kg and 800 kg at 1 m, and 2 m from the left end support. Such equipment acts as a concentrated loads on the stage that has a distributed load of 500 kg/m. The Young’s modulus of the material is 200 GPa.  

1.1. Use the engineering sketch to illustrate the structure of the stage and calculate the transverse vibration frequency of the stage.  (18)

1.2. Determine the amount of rection forces on the stage supports.  

Answer 1

a)

$f=\frac{1}{2l}\sqrt{\frac{T}{m}}$

l=5

T=15000+8000=23000N

m=500kg/m

$f=\frac{1}{2*5}\sqrt{\frac{23000}{500}}$

=0.6782Hz

b)

$+\uparrow\sum F_y=0$

-15000-8000-25000+y_l+y_r=0

Where y_l=left support and y_r=right support

Taking moments about left support, clockwise positive;

$\sum M_l=0$

(15000*1)+(8000*2)+(25000*2.5)-(5y_r)=0

15000+16000+62500-5y_r=0

5y_r=93500

Right reaction force,y_r=18700N

-15000-8000-25000+yl+yr=0

-15000-8000-25000+18700+y_r=0

y_r=48000-18700

=29300N