Answer :
Answer:
the value for the damping constant is c = 5.3408Ns/m and the spring constant is k is 987N/m
Explanation:
Consider the formula for the logarithmic decrement (δ)
δ = [tex]\frac{1}{n} In(\frac{B_1}{B_{n+1}} )----(1)[/tex]
Consider the formula for damping ratio (C)
C =δ / √4π² + ² ----------(2)
The given amplitude of the 30th cycle is 20% amplitude of the 1st cycle
The ratio of last peak to first peak is
[tex]\frac{B_{n+1}}{B_1} =20[/tex]%--------------(3)
where
n is the number of cycle
Re arrange eqn (3)
[tex]\frac{B_{n+1}}{B_1} = \frac{100}{5} \\\\\frac{B_{n+1}}{B_1} = 5-----(4)[/tex]
The given value for n is 30
substitute 30 for n and eqn (4) in eqn(1)
δ = [tex]\frac{1}{30} In5\\[/tex]
[tex]= 0.0536[/tex]
substitute 0.0536 for δ in eqn (2)
[tex]C = \frac{0.0536}{\sqrt{39.4784+0.00287} } \\C = 0.0085----(5)[/tex]
Consider the formula for damping ratio (C)
C = [tex]\frac{c}{2\sqrt{mk} } ----(6)[/tex]
where
c is the damping constant
m is the mass
substitute 100 for m in eqn(6)
[tex]C = \frac{c}{2\sqrt{100k} } \\\\C = \frac{c}{20\sqrt{k} } ----(7)[/tex]
compute eqn(5) and eqn(7)
[tex]\frac{c}{20\sqrt{k} } = 0.0085\\\\c = 0.0085(20\sqrt{k} )\\\\c= 0.17\sqrt{k} ----(8)[/tex]
The given time to complete 30 cycles in 60s is
period = 60 / 30
= 2s
Formula for spring constant
[tex]k =mw^2_n----(9)[/tex]
where
[tex]w_n[/tex] is the undamped natural frequency
consider the formula for damped natural frequency
[tex]w_d=w_n\sqrt{1-C^2}[/tex]
rearrange
[tex]w_n = \frac{w_d}{\sqrt{1-C^2} } ----(10)[/tex]
substitute eqn (10) to eqn (9)
[tex]k = m(\frac{w_d}{\sqrt{1-C^2} } )\\\\k=\frac{mw^2_d}{1-C^2} ---(11)[/tex]
consider the formula for the damped natural frequency
[tex]w_d=\frac{2\pi }{P} ----(12)[/tex]
consider eqn (12) in eqn (11)
[tex]k = \frac{m(\frac{2\pi }{P} )^2}{1-C^2}[/tex]
substitute 100 for m, 2 for P, 0.0085 for C in the eqn above
[tex]k = \frac{(100)(\frac{2\pi }{2})^2 }{1-(0.0085)^2} \\\\k= \frac{986.96}{0.9999} \\\\k=987N/m[/tex]
substitute 987 for k in eqn (8)
[tex]c=0.17\sqrt{987} \\c=0.17(31.4166)\\c=5.3408Ns/m[/tex]
Thus, the value for the damping constant is c = 5.3408Ns/m and the spring constant is k is 987N/m