12.2. Free Vibration of Undamped 2-DOF Systems

Try these first

1. 1.

   What does a two degree of freedom system represent?

   Hint

   Think about how many directions or movements it can describe.

2. 2.

   Write down the general form of the equation of motion for mass m1.

   Hint

   Remember it includes forces and acceleration terms.

3. 3.

   What does 'M' represent in the equation Mx'' + Kx = 0?

   • Load vector
   • Mass matrix
   • Stiffness matrix

   Hint

   It’s where we account for inertia.

4. 4.

   True or False: Coupling stiffness is not important in a 2-DOF system.

   • True
   • False

   Hint

   Think about how one mass affects the other during vibrations.

5. 5.

   Consider an undamped 2-DOF system with m1 = 500 kg, m2 = 700 kg, k1 = 30000 N/m, k2 = 25000 N/m, and k12 = 5000 N/m. Derive the equations of motion for this system.

   Hint

   Use the definitions of mass and spring forces relevant to each mass.

6. 6.

   Using the derived equations for the 2-DOF system, determine the natural frequencies using appropriate methods (assume a linear approach).

   Hint

   Remember the eigenvalue problem relates back to the determinant of (K - ω²M).