Understanding the concepts on Degrees of Freedom

Understanding the concepts on Degrees of Freedom

TASK:

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 frequency, seconds and its associated mode shape: 

The first mode of vibration, `T1=2` seconds and its associated mode shape:

`Phi1`=`[[0.334],[0.641],[0.895],[1.078],[1.173]]`

The second mode of vibration, `T2=0.6852` seconds and its associated mode shape: 

 `Phi2`=`[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`

The third mode of vibration, `T3=0.4346` seconds and its associated mode shape:

`Phi3`=`[[-0.895],[-1.173],[-0.641],[0.334],[1.078]]`

The forth mode of vibration, `T4=0.3383` seconds and its associated mode shape:

`Phi4`=`[[-1.173],[0.334],[-1.078],[0.641],[0.895]]`

The fifth mode of vibration, `T5=0.2966` seconds and its associated mode shape:

`Phi5`=`[[0.641],[-1.078],[1.173],[0.895],[0.334]]`

• L_n, M_n, and Γ_n for each of the five modes are to be computed from the 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 the 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 calculate L_n,M_n">L_n,M_nL_n,M_n and Γ_n">Γ_nΓn for each of the five modes
• To calculate the effective modal mass M_n">M_nMn* and the modal mass participating ratio for the 5 modes.
• To find the base shear for each of the 5 modes of vibration and the combined base shear as per the SRSS rule.

Calculations:

`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 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.