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 = ![\frac{1}{n} In(\frac{B_1}{B_{n+1}} )----(1)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bn%7D%20In%28%5Cfrac%7BB_1%7D%7BB_%7Bn%2B1%7D%7D%20%29----%281%29)
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
%--------------(3)
where
n is the number of cycle
Re arrange eqn (3)
![\frac{B_{n+1}}{B_1} = \frac{100}{5} \\\\\frac{B_{n+1}}{B_1} = 5-----(4)](https://tex.z-dn.net/?f=%5Cfrac%7BB_%7Bn%2B1%7D%7D%7BB_1%7D%20%3D%20%5Cfrac%7B100%7D%7B5%7D%20%5C%5C%5C%5C%5Cfrac%7BB_%7Bn%2B1%7D%7D%7BB_1%7D%20%3D%205-----%284%29)
The given value for n is 30
substitute 30 for n and eqn (4) in eqn(1)
δ = ![\frac{1}{30} In5\\](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B30%7D%20In5%5C%5C)
![= 0.0536](https://tex.z-dn.net/?f=%3D%200.0536)
substitute 0.0536 for δ in eqn (2)
![C = \frac{0.0536}{\sqrt{39.4784+0.00287} } \\C = 0.0085----(5)](https://tex.z-dn.net/?f=C%20%3D%20%5Cfrac%7B0.0536%7D%7B%5Csqrt%7B39.4784%2B0.00287%7D%20%7D%20%5C%5CC%20%3D%200.0085----%285%29)
Consider the formula for damping ratio (C)
C = ![\frac{c}{2\sqrt{mk} } ----(6)](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7B2%5Csqrt%7Bmk%7D%20%7D%20----%286%29)
where
c is the damping constant
m is the mass
substitute 100 for m in eqn(6)
![C = \frac{c}{2\sqrt{100k} } \\\\C = \frac{c}{20\sqrt{k} } ----(7)](https://tex.z-dn.net/?f=C%20%3D%20%5Cfrac%7Bc%7D%7B2%5Csqrt%7B100k%7D%20%7D%20%5C%5C%5C%5CC%20%3D%20%5Cfrac%7Bc%7D%7B20%5Csqrt%7Bk%7D%20%7D%20----%287%29)
compute eqn(5) and eqn(7)
![\frac{c}{20\sqrt{k} } = 0.0085\\\\c = 0.0085(20\sqrt{k} )\\\\c= 0.17\sqrt{k} ----(8)](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7B20%5Csqrt%7Bk%7D%20%7D%20%3D%200.0085%5C%5C%5C%5Cc%20%3D%200.0085%2820%5Csqrt%7Bk%7D%20%29%5C%5C%5C%5Cc%3D%200.17%5Csqrt%7Bk%7D%20----%288%29)
The given time to complete 30 cycles in 60s is
period = 60 / 30
= 2s
Formula for spring constant
![k =mw^2_n----(9)](https://tex.z-dn.net/?f=k%20%3Dmw%5E2_n----%289%29)
where
is the undamped natural frequency
consider the formula for damped natural frequency
![w_d=w_n\sqrt{1-C^2}](https://tex.z-dn.net/?f=w_d%3Dw_n%5Csqrt%7B1-C%5E2%7D)
rearrange
![w_n = \frac{w_d}{\sqrt{1-C^2} } ----(10)](https://tex.z-dn.net/?f=w_n%20%3D%20%5Cfrac%7Bw_d%7D%7B%5Csqrt%7B1-C%5E2%7D%20%7D%20----%2810%29)
substitute eqn (10) to eqn (9)
![k = m(\frac{w_d}{\sqrt{1-C^2} } )\\\\k=\frac{mw^2_d}{1-C^2} ---(11)](https://tex.z-dn.net/?f=k%20%3D%20m%28%5Cfrac%7Bw_d%7D%7B%5Csqrt%7B1-C%5E2%7D%20%7D%20%29%5C%5C%5C%5Ck%3D%5Cfrac%7Bmw%5E2_d%7D%7B1-C%5E2%7D%20---%2811%29)
consider the formula for the damped natural frequency
![w_d=\frac{2\pi }{P} ----(12)](https://tex.z-dn.net/?f=w_d%3D%5Cfrac%7B2%5Cpi%20%7D%7BP%7D%20----%2812%29)
consider eqn (12) in eqn (11)
![k = \frac{m(\frac{2\pi }{P} )^2}{1-C^2}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7Bm%28%5Cfrac%7B2%5Cpi%20%7D%7BP%7D%20%29%5E2%7D%7B1-C%5E2%7D)
substitute 100 for m, 2 for P, 0.0085 for C in the eqn above
![k = \frac{(100)(\frac{2\pi }{2})^2 }{1-(0.0085)^2} \\\\k= \frac{986.96}{0.9999} \\\\k=987N/m](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B%28100%29%28%5Cfrac%7B2%5Cpi%20%7D%7B2%7D%29%5E2%20%7D%7B1-%280.0085%29%5E2%7D%20%5C%5C%5C%5Ck%3D%20%5Cfrac%7B986.96%7D%7B0.9999%7D%20%5C%5C%5C%5Ck%3D987N%2Fm)
substitute 987 for k in eqn (8)
![c=0.17\sqrt{987} \\c=0.17(31.4166)\\c=5.3408Ns/m](https://tex.z-dn.net/?f=c%3D0.17%5Csqrt%7B987%7D%20%5C%5Cc%3D0.17%2831.4166%29%5C%5Cc%3D5.3408Ns%2Fm)
Thus, the value for the damping constant is c = 5.3408Ns/m and the spring constant is k is 987N/m