Here, is the moment due to inertia force at a distance ζ from the roller-supported end and is the moment due to inertia force for the payload at the tip of the manipulator and their expressions are given below:

, .........................................(2.3.4)

and, . .........................(2.3.5)

Considering equivalent viscous damping force due to interaction of the system with the environment and by differentiating Eq. (2.3.3) twice with respect to s , using the Leibniz's rule and applying the binomial expansion, one may obtain the following governing differential equation of motion.

................................................................. (2.3.6)

Example- 2.3.2

Derive the equation motion of a string fixed at one end and attached by a nonlinear spring at the other end.

Solution

.....................................................................................(2.3.7)

....................................... (2.3.8)

Subjected boundary conditions

at and at ..........(2.3.9)