Understanding the concepts on Degrees of Freedom

Question 1: A five storey structure with 5 translational DOF is shown here. Each storey has mass ‘m’. The eigenvalue problem has been solved and the 5 periods of vibration T_n and their corresponding mode shapes, φ_n has been derived.

The fundamental period of vibration (T₁ = 2 seconds) and its associated mode shape:

The second mode of vibration (T₂ = 0.6852 seconds) and its associated mode shape:

The third mode of vibration (T₃ = 0.4346 seconds) and its associated mode shape:

The fourth mode of vibration (T₄ = 0.3383 seconds) and its associated mode shape:

The fifth mode of vibration (T₄ = 0.2966 seconds) and its associated mode shape:

. L_n , M_n and Γ_n for each of the five modes are to be computed from Equations below.

. Also, 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.

. Finally, 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.

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

 AIM: To compute L_n, M_n, T_n for 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.  

PROCEDURE:

Step-1: M_n=_n^Tm_n

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

By scientific calculator, we got M₁ as 3.861M M₁=3.861M

Mode-2: The second mode of vibration, T₂=0.6852 seconds

By scientific calculator, we got M₂ as 3.86M

 M₂=3.861M

Mode-3: The third mode of vibration, T₃=0.4346 seconds

By scientific calculator, we got M₃ as 3.85M

M₃=3.85M

Mode-4: The fourth mode of vibration, T₄=0.3383 seconds

By scientific calculator, we got M₄ as 3.861M 

M₄=3.861M

Mode-5: The fifth mode of vibration, T₅=0.2966 seconds

By scientific calculator, we got M₅=3.861M 

M₅=3.861M

Step-2: L_n=_n^Tm

Mode-1: The first mode of vibration, T₁=2 seconds

By scientific calculator, we got L₁ as 4.121M

 L₁=4.121M

Mode-2: The second mode of vibration, T₂=0.6852 seconds

By scientific calculator, we got L₂ as -1.298M

 L₂=-1.298M

Mode-3: The third mode of vibration, T₃=0.4346 seconds

By scientific calculator, we got L_3­ as 0.687M

L₃=0.687M

Mode-4: The fourth mode of vibration, T₄=0.3383 seconds

By scientific calculator, we got L₄ as -0.38M

 L₄=-0.38M

Mode-5: The fifth mode of vibration, T₅=0.2966 seconds

By scientific calculator, we got L₅ as 0.175M

L₅=0.175M

Step-3: Γ_n=L_n/M_n

Mode-1: Γ₁=L₁/M₁ =4.121M/3.861M =1.067

Mode-2: Γ₂=L₂/M₂ =-1.298M/3.861M =0.178

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=L_n²/M_n

Mode-1: M₁^*=L₁²/M₁ =4.39M

Mode-2:  M₂^*=L₂²/M1=0.43M

Mode-3: M₂^*=L₂²/M1 =0.12M

Mode-4: M₃^*=L₃²/M₃ =0.03M

Mode-5: M₅^*=L₅²/M₅ =0.007M

Step-5: Modal mass participating ratio=M_n^*/M_j M_n^*= Effective modal mass

M_j= Total mass= M+M+M+M+M=5M

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: To obtain shear force for each of the 5 modes os vibration.

V_bn=A_nxL_n²/M_n =A_nxM_n^*

From the graph,T₁=2 sec, A₁=0.27g

T₂=0.6852 sec, A₂=0.76g

T₃=0.4346 sec, A₃=1.03g

T₄=0.3383 sec, A₄=1.03g

T₅=2.9666 sec, A₅=0.76g

Mode-1: V_b1=A₁xM₁^*=2.619 kips

Mode-2: V_b2=A₂xM₂^*=0.719 kips

Mode-3: V_b3=A₂xM₂^*=0.272 kips

Mode-4: V_b4=A₄xM₄^*=0.067 kips

Mode-5: V_b5=A₅xM₅^*=0.011 kips

Step-7: Combined base shear, V_b=root(V_b1²+ V_b2²+ V_b3²+ V_b4²+ V_b5²) =root((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,

(M₁^*/M_j)+(M₂^*/M_j)=88%+8.6%=96.6%

Result: L_n, M_n, Γ_n values are calculated for each of the 5 modes. The base shear force for each of the 5 modes are obtained. Effective modal mass and the modal mass participating ratio for 5 modes are obtained.