Two identical pendulums, each with mass \(m\) and length \(l\), are connected by a spring of stiffness \(k\) at a distance \(d\) from the fixed end, as shown in Fig. 5.35.

a. Derive the equations of motion of the two masses.

b. Find the natural frequencies and mode shapes of the system.

c. Find the free-vibration response of the system for the initial conditions \(\theta_{1}(0)=a\), \(\theta_{2}(0)=0, \dot{\theta}_{1}(0)=0\), and \(\dot{\theta}_{2}(0)=0\).

d. Determine the condition(s) under which the system exhibits a beating phenomenon.

P 81(t) m k m 82(1) FIGURE 5.35 Two pendulums connected by a spring.