Understanding the concepts on Degrees of Freedom

Aim:

1. To compute L_n, M_n and T_n for each of the five modes from the given equations
2. To calculate the effective modal mass M_n* and the modal mass participating ratio for the modes

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

Procedure:

Step 1:  M_n = ø_nT_mø_n

Mode 1:

Mode of vibration, T₁ = 2 sec

                        M₁ = ø₁T₁ø₁
                   

Mode 2:

Mode of vibration, T₂ = 0.6852 sec

                        M₂ = ø₂T_mø₂
                   

Mode 3:

Mode of vibration, T₃ = 0.4346 sec

                        M₃ = ø₃T_mø₃
                   

Mode 4:

Mode of vibration, T₄ = 0.3383 sec

                        M₄= ø₄T_mø₄
                   

Mode 5:

Mode of vibration, T₅ = 0.2966 sec

                        M₅= ø₅T_mø₅
                   

Step 2: To find L_n

L_{n =}ø_nT_mI

Mode 1:

Mode of vibration, T₁ = 2 sec

                        L_{1 =}ø₁T_mI

                   

Mode 2:

Mode of vibration, T₂ = 0.6852 sec

                        L_{2 =}ø₂T_mI
                   

 

Mode 3:

Mode of vibration, T₃ = 0.4346 sec

                        L_{3 =}ø₃T_mI                  

 

Mode 4:

Mode of vibration, T₄ = 0.3383 sec

                        L₄ = ø₄T_mI
                   

Mode 5:

Mode of vibration, T₅ = 0.2966 sec

                        L_{5 =}ø₅T_mI
                   

                   

Step 3: Finding T_n

T_n = Ln /Mn

T₁ =L1/M1  = 1.067

T₂ =L2/M2  = -0.33

T₃=  L3/M3 = 0.178

T₄ = L4/M4 = -0.09

T₅ = L5/M5  = 0.04

Step 4: Finding Effective modal mass, M_n*

M_n* = L_n² /M_n

M₁* = L₁² /M₁ = 4.39M

M₂* = L₂² /M₂ = 0.43M

M₃* = L₃² /M₃ = 0.12M

M₄* = L₄² /M₄ = 0.03M

M₅* = L₅² /M₅ = 0.007M

Step 5: Finding Modal mass participating ratio

Modal mass participating ratio = M_n^* / M_j

M_j = M+M+M+M+M = 5M

Modal mass participating ratio for Mode 1 = M₁^* / M_j  = 0.88

Modal mass participating ratio for Mode 2 = M₂^* / M_j  = 0.086

Modal mass participating ratio for Mode 3 = M₃^* / M_j  = 0.024

Modal mass participating ratio for Mode 4 = M₄^* / M_j  = 0.006

Modal mass participating ratio for Mode 5 = M₅^* / M_j  = 0.001

Step 6: To obtain the Base shear

V_bn = M_n^* * A_n

For Mode 1

For T₁ = 2 sec, A₁ = 0.28g

V_b1 = M₁^* * A₁ = 2.612

For Mode 2

For T₂ = 0.6852 sec, A₂ = 0.77g

V_b2 = M₂^* * A₂ = 0.719

For Mode 3

For T₃ = 0.4346 sec, A₃ = 1.03g

V_b3 = M₃^* * A₃ = 0.272

For Mode 4

For T₄ = 0.3383 sec, A₄ = 1.03g

V_b4 = M₄^* * A₄ = 0.067

For Mode 5

For T₅ = 0.2966 sec, A₅ = 0.77g

V_b5 = M₅^* * A₅ = 0.011

Step 7: Using SRSS rule

V_b = (V_b1²+ V_b2²+ V_b3²+ V_b4²+ V_b5²)^1/2

        = 3.2 kip

Result: 

• Two modes are considered such that a model mass participating ratio of 90% is obtained. ie;  M₁^* / M_j  + M₂^* / M_j