Understanding the concepts on Degrees of Freedom

Solution.1)

Aim:

• To calculate L_n , M_n and Γ_n for each of the five modes and the effective modal mass M_n* and the modal mass participating ratio for the 5 modes using the equation given below:

• To 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

Given:

• A five storey structure with 5 translational DOF is shown here. Each storey has mass ‘m’.
• The fundamental period of frequency, 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, T₅ = 0.2966 seconds and its associated mode shape:

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Result: Transformed mass matrix, `L_n`, Modal Participation factor, Effective Modal Mass, Modal Mass Partification Ratio, Modal Base Shear and SSSR Average of Base Shear has been sucessfully calculated.