Understanding the concepts on Degrees of Freedom

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 SRSS rule.

INTRODUCTION:

Degrees of freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample. Degrees of freedom is calculated by subtracting one from the number of items within the data sample.

PROCEDURE:

Step 1: M_n = ø_nT_mø_n

Mode 1:

Mode of vibration, T₁ = 2 sec

M₁ = ø₁Tm ø₁

Mode 2:

Mode of vibration, T₂ = 0.6852 sec

M₂ = ø₂T_mø₂

Mode 3:

Mode of vibration, T₃ = 0.4346 sec

M₃ = ø₃Tmø₃

Mode 4:

Mode of vibration, T₄ = 0.3383 sec

M₄= ø₄Tm ø₄

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:

Modal participation factor (Γ) :

_Γ = L/M

_Γ1 =L1/M1 = 1.067

_Γ2 =L2/M2 = -0.335

_Γ3= L3/M3 = 0.178

_Γ4 = L4/M4 = 0.095

_Γ5 = L5/M5 = 0.045

Step 4:

Finding Effective modal mass, (M) :

M = L_n² /M_n

M₁ = L₁² / M₁ = 4.398M

M₂ = L₂² / M₂ = 0.436M

M₃ = L₃² / M₃ = 0.12M

M₄ = L₄² / M₄ = 0.038M

M₅ = L₅² / M₅ = 0.008M

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 = 0.879

Modal mass participating ratio for Mode 2 = M₂ / M_j = 0.0872

Modal mass participating ratio for Mode 3 = M₃ / M_j = 0.0242

Modal mass participating ratio for Mode 4 = M₄ / M_j = 0.01

Modal mass participating ratio for Mode 5 = M₅ / M_j = 0.002

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 Square root of sum of square:

Average base shear is calculated by the method of square of sum of square.

= √(V_b1²+ V_b2²+ V_b3²+ V_b4²+ V_b5²)

= √(2.613)^2+(0.719)^2+(0.272)^2+(0.067)^2+(0.011)^2

= 3.2 kips

• The two modes are to be considered such that a modal mass participating ratio of 90% is obtained.
• M1∗Mj">M1∗MjM1∗Mj + M2∗Mj">M2∗MjM2∗Mj = 88% + 8.6% = 96.6%

RESULT:-

• Finallhy we 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 are obtained.
• As per 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.