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Problem 2.03PP
Write the equations of motion for the double-pendulum system shown in Fig. Assume that the 
displacement angles of the pendulums are small enough to ensure that the spring is always 
horizontal. The pendulum rods are taken to be massless, of length /, and the springs are attached 
three-fourths of the way down.
Step-by-step solution
step 1 of 6
Consider the circuit diagram.
r
m m
Double pendulum system
Step 2 of 6 ^
Re-draw the circuit diagram.
 ̂ / I
^ / /
1
m
Here there are two degrees of freedom
Step 3 of 6 ^
At any instant, let one of them be displayed by 6| and other by 6,.
Step 4 of 6
Writing about point ‘0’ the moment equation is.
Jt(sinO2-sinO,)cos0sm /^d, 
Similarly, write the moment equation for other pendulu 
-mg/sin6, - ^ ^ / j *(sin6,-sine|)cos82 =m/^e.
Step 5 of 6
Assume the angles are small, then sin0| s 0 |, cos9 |S ]and cos02 s l.T he
above equations are modified as.
* (e ,-e ,)= m /= 9 ,
Therefore, the above equation is moment equilibrium about the pivot point of the left pendulum.
Step 6 of 6 ^
Write the equation for moment equilibrium about the pivot point of the right pendulum.
* ( 6 , - 6 , ) - 0
Therefore, the above equation is moment equilibrium about the pivot point of the right pendulum.