Question 3.8: Assuming small oscillations, obtain the differential equatio...

As shown in the figure, let R_{x} and R_{y} be the components of the reaction force at the pin joint. The moments of the externally applied forces about O are

M_{a} = −(kl sin θ)l cos θ − (cl\dot{θ} cos θ)l cos θ − mgl sin θ

For small oscillations, sin θ ≈ θ and cos θ ≈ 1. In this case, M_{a} reduces to
M_{a} = −kl²θ − cl²\dot{θ} − mglθ

One can show that the moment of the inertia (effective) forces about O is given by M_{eff} = ml²\ddot{θ}. Therefore, the second-order differential equation of motion of the free vibration is given by
−kl²θ − cl²\dot{θ} − mglθ = ml²\ddot{θ}

or
ml²\ddot{θ} + cl²\dot{θ} + (kl + mg)lθ = 0
which can be written in the general form of Eq. 3.81 as

m_{e}\ddot{x} + c_{e}\dot{x}+ k_{e}x = 0                                  (3.81)

m_{e}\ddot{θ} + c_{e}\dot{θ}+ k_{e}θ = 0
where
m_{e} = ml²,     c_{e} = cl²,     k_{e} = (kl + mg)l

where the units of m_{e} are kg · m² or, equivalently, N · m · s², the units of the equivalent damping coefficient c_{e} are N · m · s and the units of the equivalent stiffness coefficient k_{e} are N · m. The natural frequency ω is

ω =\sqrt{\frac{k_{e}}{m_{e}}} = \sqrt{\frac{(kl + mg)l}{ml²}} = \sqrt{\frac{kl + mg}{ml}} rad/s

The critical damping coefficient C_{c} is

The damping factor ξ of this system is
ξ = \frac{c_{e}}{C_{c}}= \frac{cl²}{2\sqrt{ml³(kl + mg)}}