Understanding the concepts on Degrees of Freedom

Aim:

• To compute L_n , M_n and Γ_n for each five modes from the given equation.
• To calculate the effective modal mass M_n* and the modal mass participating ratio for the 5 modes. The mass matrix will be required to evaluate these parameters.

• To obtain obtain the base shear for each of the 5 modes of vibration and the combined base shear as per SRSS rule. The pseudo acceleration design spectrum to obtain the base shear for each mode is provided below.

• To obtain how many modes such that a modal mass participating ratio of 90% is obtained from the below graph

• How many modes are to be considered such that a modal mass participating ratio of 90% is obtained?

Procedure:

Step 1: To find 5 modes for the below equations

Mode1:

M_n = ɸ_n^Tmɸ_n

The first mode of vibration (T₁) = 2 Seconds

M1 =3.861M

Mode 2: The second mode of vibration (T₂) = 0.6852 Seconds 

M2 =3.861M

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Mode 3: The second mode of vibration (T3) = 0.4346 Seconds 

M3 =3.85M

Mode 4: The second mode of vibration (T4) = 0.3383 Seconds 

M4 =3.861M

Mode 5: The second mode of vibration (T5) = 0.2966 Seconds 

M5 =3.861M

                                                         ****************

Step 2:

L_n = ɸ_n^Tmi

Mode 1: The first mode of vibration (T₁) = 2 Seconds

L₁ = 4.121 M

Mode 2: The first mode of vibration (T2) = 0.6852 Seconds

L₂ = -1.298 M

Mode 3: The first mode of vibration (T3) = 0.4346 Seconds

L₃ = 0.687 M

Mode 4: The first mode of vibration (T4) = 0.3383 Seconds

L₄ = -0.381 M

Mode 5: The first mode of vibration (T5) = 0.2966 Seconds

L₅ = 0.175 M

                                                                *****************

Step 3 :

Mode 3:

Γ3 = L3/M3

      = 0.687M / 3.85M

      = 0.178

Mode 4:

Γ4 = L4/M4

      = -0.381M / 3.861M

      = -0.09

Mode 5:

Γ5 = L5/M5

      = 0.175M / 3.861M

      = 0.04

                                                      **************

Step 4:

M_n^* = Ln²/Mn

Mode 1:

M1^* = L1²/M1

        = (4.121M) ²/3.861M

        = 4.398 M

Mode 2:

M2^* = L2²/M2

        = (-1.298M) ²/3.861M

        = 0.436 M

Mode 3:

M3^* = L3²/M3

        = (0.687M) ²/3.85M

        = 0.12 M

Mode 4:

M4^* = L4²/M4

        = (-0.381M) ²/3.861M

        = 0.03 M

Mode 5:

M5^* = L5²/M5

        = (0.175M) ²/3.861M

        = 0.007 M

                                                    ****************

Step 5:

• Modal mass participating ratio = Mn^*/Mn∗Mj</math>">Mj

Mn∗Mj</math>">Mn^* = Effective nodal mass

Mj   = Total mass = M+M+M+M+M = 5M

          Mn∗</math>">

        

Mode 1

• M₁^* /M_j = 0.88 = 88%

Mode 2

• M₂^* /M_j = 0.086 = 8.6%

Mode 3

• M₃^* /M_j = 0.024 = 2.4%

Mode 4

• M₄^* /M_j = 0.006 = 0.6%

Mode 5

• M₅^* /M_j = 0.001 = 0.1%

                                                              ****************

Step 6 :

Now we need to obtain the base shear for each of the 5 modes of vibration.

V_bn = (ln2Mn</math>">ln2 / Mn)X A_n

(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:

•  V_b1= (M₁^* x A₁) = (4.39M x 0.27g) = 2.613 kips

Mode 2:

• V_b2 = (M₂^* x A₂) = (0.43M x 0.76g) = 0.719 kips

Mode 3:

• V_b3 = (M₃^* x A₃) = (0.12M x 1.03g) = 0.272 kips

Mode 4:

• V_b4 = (M₄^* x A₄) = (0.03M x 1.03g) = 0.067 kips

Mode 5:

• V_b5 = (M₅^* x A₅) = (0.007M x 0.76g) = 0.011 kips

                                                                 ************************

Step 7:

Combined base shear as per SRSS rule:-

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

               = 3.20 kips

                                                                       ********************

Step 8:

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

                                                 88% + 8.6% = 96.6%

Result :

• L_n , M_n and Γ_n for each of the five modes from the given equations has been calculated
• The effective modal mass M_n* and the modal mass participating ratio for the 5 modes is calculated
• The base shear for each of the 5 modes of vibration is 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.