Understanding the concepts on Degrees of Freedom
1.
Given:
`T_1`= 2sec, `phi_1` = `((0.334),(0.641),(0.895),(1.078),(1.173))`
`T_2`= 0.6852sec, `phi_2`= `((-0.895),(-1.173),(-0.641),(0.334),(1.078))`
`T_3`= 0.4346sec, `phi_3`= `((1.173),(0.334),(-1.078),(-0.641),(0.895))`
`T_4`= 0.3383sec, `phi_4`= `((-1.078),(0.895),(0.334),(-1.173),(0.641))`
`T_5`=0.2966sec, `phi_5`=`((0.641),(-1.078),(1.173),(-0.895),(0.334))`
a) WKT, `M_n=(phi)_n^T mphi_n`
mode 1:
`M_1`=(0.334,0.641,0.895,1.078,1.173)(m) x `((0.334),(0.641),(0.895),(1.078),(1.173))`
= M x`(0.334^2+0.641^2+0.895^2+1.078^2+1.173^2)`
`M_1`= 3.891 M
mode 2:
`M_2`=(-0.895,-1.173,-0.941,-0.334,1.078)(m) x `((-0.895),(-1.173),(-0.641),(-0.334),(1.078))``
= M x `(-(0.895^2)+(-1.173^2)+(-0.691^2)+(-0.334^2)+(-1.078^2))`
`M_2`= 3.861 M
mode 3:
`M_3`=(1.173,0.334,-1.078,-0.641,0.895)(m) x `((1.173),(0.334),(-1.078),(-0.641),(0.895))`
= M x `((1.173^2)+(0.334^2)+(-1.078^2)+(-0.641^2)+(0.895^2))`
`M_3`= 3.85 M
mode 4:
`M_4`=(-1.078,0.895,0.334,-1.173,0.641)(m) x `((-1.078),(0.895),(0.334),(-1.173),(0.641))`
= M x `((-1.078^2)+(0.895^2)+(0.334^2)+(-1.173^2)+(0.641^2))`
`M_4`= 3.861 M
mode 5:
`M_5`=(0.641,-1.078,1.173,-0.895,0.334)(m) x `((0.641),(-1.078),(1.173),(-0.895),(0.334))`
= M x `((0.641^2)+(-1.078^2)+(1.173^2)+(-0.895^2)+(0.334^2))`
`M_5`= 3.861 M
b) WKT, `L_n = phi_n^T mi`
mode 1:
`L_1`=(0.334,0.641,0.895,1.078,1.173)(m) x `((1),(1),(1),(1),(1))`
=(0.334+0.641+0.895+1.078+1.173)m
`L_1`=4.121 M
mode 2:
`L_2`=(-0.895,-1.173,-0.641,0.334,1.078)(m) x `((1),(1),(1),(1),(1))`
=(-0.895-1.173-0.641+0.331+1.078)m
`L_2`=1.298 M
mode 3:
`L_3`=(1.173,0.334,-1.078,-0.641,0.895)(m) x `((1),(1),(1),(1),(1))`
=(1.173+0.334-1.078-0.641+0.895)m
`L_3`=0.687 M
mode 4:
`L_4`=(-1.078,0.895,0.334,-1.173,0.641)(m) x `((1),(1),(1),(1),(1))`
=(-1.078+0.895+0.334-1.173+0.641)m
`L_4`=-0.381 M
mode 5:
`L_5`=(0.641,-1.078,1.173,-0.895,0.334)(m) x `((1),(1),(1),(1),(1))`
=(0.641-1.078+1.173-0.895+0.334)m
`L_5`= 0.175 M
c) WKT, `|_n=L_n/M_n`
mode 1:
`|_1=L_1/M_1`=4.121M/3.861M=1.067
mode 2:
`|_2=L_2/M_2`=-1.298M/3.861M=-0.33
mode 3:
`|_3=L_3/M_3`=0.687M/3.85M=0.178
mode 4:
`|_4=L_4/M_4`=-0.381M/3.861M=0.09
mode 5:
`|_5=L_5/M_5`=0.175M/3.861M=0.04
d) WKT, `M_n^*=L_n^2/M_n`
mode 1:
`M_1^*=L_1^2/M_1`=(4.121M)^2/3.861M=4.39M
mode 2:
`M_2^*=L_2^2/M_2`=(-0.298M)^2/3.861M=0.43M
mode 3:
`M_3^*=L_3^2/M_3`=(0.687M)^2/3.85M=0.12M
mode 4:
`M_4^*=L_4^2/M_4`=(-0.381M)^2/3.861M=0.03M
mode 5:
`M_5^*=L_5^2/M_5`=(0.175M)^2/3.861M=0.007M
e) Modal mass participating ratio =`M_n^*/M_j`
`M_j`=total mass =5M
mode 1:
`M_1^*/M_j`=4.39M/5M=0.88%
mode 2:
`M_2^*/M_j`=0.43M/5M=8.6%
mode 3:
`M_3^*/M_j`=0.12M/5M=2.4%
mode 4:
`M_4^*/M_j`=0.03M/5M=0.6%
mode 5:
`M_5^*/M_j`=0.007M/5M=0.1%
f) WKT, `V_(bn)=L_n^2/M_nA_n=M_n^*A_n`
here,
`T_1`=2sec, `A_1`=0.27g
`T_2`=0.6352sec, `A_2`=0.76g
`T_3`=0.4346sec, `A_3`=1.03g
`T_4`=0.3383sec, `A_4`=1.03g
`T_5`=0.2966sec, `A_5`=0.76g
mode 1:
`V_(b1)=M_1^*A_1`=1.39M x 0.27g=2.619kips
mode 2:
`V_(b2)=M_2^*A_2`=0.43M x 0.76g=0.719kips
mode 3:
`V_(b3)=M_3^*A_3`=0.12M x 1.03g=0.272kips
mode 4:
`V_(b4)=M_4^* A_4`=0.03M x 1.03g=0.067kips
mode 5:
`V_(b5)=M_5^*A_5`=0.007M x 0.76g=0.011kips
g)
`V_b=sqrt (V_(b1)^2+V_(b2)^2+V_(b3)^2+V_(b4)^2+V_(b5)^2`
`V_b=sqrt(2.613^2+0.719^2+0.272^2+0.067^2+0.011^2)`=3.2kips
h) model mass ration of 90% in obtained by 2 model
`M_1^*/M_j + M_2^*/M_j`=88% +8.6%=96.6%