Understanding the concepts on Degrees of Freedom

1

AIM:- To compute L_n, M_n, and Γ_n for each of the five modes from the given equations.

To calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.

To obtain the base shear for each of the 5 modes of vibration and the combined base shear as per the SRSS rule.

PROCEDURE:-

Step:-1

Step:-5

• Modal mass participating ratio = Mn∗/Mj

Mn∗Mn∗ = Effective modal mass.

MjMj = Total mass = M+M+M+M+M = 5M

Step:-6

Now we need to obtain the base shear for each of the 5 modes of vibration, as shown below.

(T1) = 2 seconds, A1 = 0.27g

(T2) = 0.6852 seconds, A2 = 0.76g

(T3) = 0.4346 seconds, A3 = 1.03g

(T4) = 0.3383 seconds, A4 = 1.03g

(T5) = 0.2966 seconds, A5 = 0.76g

Mode:-1

• Vb1 = (M1∗x A1) = (4.39M x 0.27g) = 2.619 kips

Mode:-2

• Vb2 = (M2∗x A2) = (0.43M x 0.76g) = 0.719 kips

Mode:-3

• Vb3 = (M3∗x A3) = (0.12M x 1.03g) = 0.272 kips

Mode:-4

• Vb4 = (M4∗x A4) = (0.03M x 1.03g) = 0.067 kips

Mode:-5

• Vb5 = (M5∗x A5) = (0.007M x 0.76g) = 0.011 kips

Step:-7

Combined base shear as per SRSS rule:-

Step:-8

• The two modes are to be considered such that a modal mass participating ratio of 90% is obtained.

RESULT:-

• Calculated L_n, M_n, and Γ_n for each of the five modes from the given equations.
• Calculated the effective modal mass M_n* and the modal mass participating ratio for the 5 modes.
• The base shear for each of the 5 modes of vibration is obtained.
• As per the SRSS rule the combined base shear is obtained.
• The two modes are to be considered such that a modal mass participating ratio of 90% is obtained.