Vibrations 2

Vibrations 2

1. Forced vibration on SDOF systems

   • Non-homogenous equation
     • Solution
       • Homogenous solution
       • Particular solution
   • Response of an undamped system to harmonic forcing
     • When r = 1 i.e.( Frequency = Natural frequency ), this amplification ratio becomes infinate
       • This condition is called resonance
     • φ = 0 if r < 1
     • φ = π (180 degrees) if r > 1
   • Amplification factor
     • Characteristics
       • Any amount of damping reduces the magnification factor for all values of the forcing frequency.
       • The reduction of the amplification factor in the presence of damping is very significant at or near resonance.
       • The amplitude of forced vibration becomes smaller with increasing values of the forcing frequency
   • Complete solutions

2. Multi-degree of freedom systems

   • Number of degrees in the system
     • Number of (lamped) masses in the system x number of possible types of motion of each mass
   • Coupled second-order differential equations
     • The motion along the D.O.F 1 will influence the motion along the D.O.F 2 and vice versa
   • Equations of motion in matrix form
   • Eigenvalue/characteristic value problem.
     • Omega is the undamped natural frequency of the system.
     • Omega ^2 is known as the eigenvalue or the characteristic value,
     • X is known as the eigenvector, the modal vector or the mode shape.
     • Each undamped natural frequency is associated with one mode shape.
     • Determinant of the coefficient matrix
       • This determinant leads to an n th order polynomial in terms of Omega ^2 . Its roots gives the n eigenvalues, Omega ^2 , whose positive square roots give the n natural frequencies of the system
       • The lowest value ( Omega1 ) is called the fundamental or first natural frequency.
     • For each i th value of Omega ^2 , a vector X i can be calculated. This vector X i is so called the i th mode shape of the system.
   • Uncoupling the equations of motion
     • The equations of motion, which are originally expressed in the physical coordinates , x, are transformed to modal coordinates , q, using the eigenvectors obtained by solving the undamped frequency eigenvalue problem.
       • Advantage ☞ modal coordinates are linearly independent vectors.
   • 1. Consider the equations of motion undamped and unforced to find: – Natural frequencies – Mode shapes ☞ Modal matrix [ X
     • 2. Write the equations of motion in terms of the modal coordinates,
       • 3. Solve each equation of motion for each modal coordinate, q i
         • 4. Express the final solution in terms of the original physical coordinates

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