Question 4.7.1: A simple pendulum is pivoted at point O as shown in Figure 4......

a. With reference to the FBD in Figure 4E1b and summing moments about an axis through O(assume that moments are positive counter-clockwise and that the amplitude of oscillation is small such that sin θ ≈ θ),

(mL\ddot{θ})  L =  \sum{M_{0}   =   –   L_{1} (k   L_{1} θ)  –  c  ( L_{2} \dot{θ} )L_{2}  –  mg (L θ)}.

Re-arranging terms, the equation of motion for the pendulum becomes

(mL²) \ddot{θ}  +  (c L_{2}²) \dot{θ}  +  (k L_{1}²  +   mg L) θ  = 0 .

b. By definition, the undamped natural frequency of the pendulum is given by

ω_{n}  =  \sqrt{ \frac{k L_{1}²  +   mg L}{mL²} } .                                    (i)

By analogy to the translational oscillation, one has

2ζ ω_{n} =  \frac{c L_{2}²}{mL²} ,   or  ζ  =  \frac{c L_{2}²}{2 mL²} \sqrt{ \frac{mL²}{k L_{1}²  +   mg L} }  .                    (ii)

If ζ < 1, the damped natural frequency is defined by Equation (4.4) as

s_{1}  , s_{2}   =   − α  ±  i ω_{d}                                                       (4.4)

ω_{d}  =  ω_{n}  \sqrt{1  –  ζ²}       or

ω_{d}  = \sqrt{\frac{k L_{1}²  +   mg L}{mL²}  [1  –   (\frac{c L_{2}²}{2  mL²} )²  (\frac{mL²}{k L_{1}²  +   mg L} )]}   .

Simplifying, it becomes

ω_{d}  = \sqrt{(\frac{k L_{1}²  +   mg L}{mL²})  –  (\frac{c L_{2}²}{2  mL²}) ²}   .                                      (iii)

This is the damped natural frequency of the given pendulum. Note that the second term inside the square root in Equation (iii) was incorrectly given as ( \frac{cL_{1} L_{2}² }{2  mL²} )² in the answer to Supplementary Problem 63 in page 29 of [2]. The term  ( \frac{cL_{1} L_{2}² }{2  mL²} )²  is dimensionally incorrect and therefore is inconsistent with the first term inside the square root.

c. When ω_{d}  = 0, from Equation (iii) one obtains

( \frac{cL_{2}² }{2  mL²} )²   =   (\frac{k L_{1}²  +   mg L}{mL²})  .

Therefore:

c  =  \frac{2  mL²}{L_{2}²}  \sqrt{ \frac{k L_{1}²  +   mg L}{mL²} } .                                (iv)

This is the required damping coefficient.
Equation (iv) can be written as

\frac{cL_{2}² }{mL²}=  2  ω_{n} .

But from Equation (ii), the rhs is equal to 2ζω_{n} or ζ = 1. In other words, when the damping is critical, the damped natural frequency of the pendulum becomes zero.