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Multi Degrees of
Freedom:
Number of DoF in a System =
Number of Masses in system x
Number of possible types of
motion of each mass
Could be one mass with n directions of motion OR n masses with one type of motion
So 2DOF freedom is either 1 mass with two types of motion or
2 masses with one type of motion
A 2 degree of freedom system refers to the direction the mass can move in a system. It has 2 Natural Modes of Vibration
• –Each Mode has its own natural frequency wn
• –Each Mode has its own initial condition
• Together, these are the ‘Mode Shape’
• Classical Eigenvalue /Eigenvector solution:
• §Produce Free Body Diagrams of the forces acting on each mass
• §Write the equations of motion for each mass
• §Write the equations of motion in matrix format
• §Put values into stiffness matrix i.e. K
• §Put values into M matrix
• §Use det(-lM+K)=0 to solve for l (where l=w2)
• §w=l1/2 for the natural frequencies w1,w2….
• §To solve further for the mode shape use (-ω_1^2 M+K) u_1=0
• Where u_1 is the vector ={■8(X_11@X_21 )} with X1, X2 the magnitude and direction of body 1 & 2’s initial condition in Mode 1
Multi Degrees of Freedom.pptxjjjjjjjjjjjjjjjjjjjjjj
Method calculates eigenvalues,
which is then used to calculate the
natural frequencies.
Now eigenvalues/natural frequencies
have been calculated, another
matrices setup can be used to
calculate the mode shape.

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Multi Degrees of Freedom.pptxjjjjjjjjjjjjjjjjjjjjjj

  • 2. Number of DoF in a System = Number of Masses in system x Number of possible types of motion of each mass Could be one mass with n directions of motion OR n masses with one type of motion So 2DOF freedom is either 1 mass with two types of motion or 2 masses with one type of motion A 2 degree of freedom system refers to the direction the mass can move in a system. It has 2 Natural Modes of Vibration • –Each Mode has its own natural frequency wn • –Each Mode has its own initial condition • Together, these are the ‘Mode Shape’ • Classical Eigenvalue /Eigenvector solution: • §Produce Free Body Diagrams of the forces acting on each mass • §Write the equations of motion for each mass • §Write the equations of motion in matrix format • §Put values into stiffness matrix i.e. K • §Put values into M matrix • §Use det(-lM+K)=0 to solve for l (where l=w2) • §w=l1/2 for the natural frequencies w1,w2…. • §To solve further for the mode shape use (-ω_1^2 M+K) u_1=0 • Where u_1 is the vector ={■8(X_11@X_21 )} with X1, X2 the magnitude and direction of body 1 & 2’s initial condition in Mode 1
  • 4. Method calculates eigenvalues, which is then used to calculate the natural frequencies.
  • 5. Now eigenvalues/natural frequencies have been calculated, another matrices setup can be used to calculate the mode shape.