Multi Degrees of Freedom.pptxhhhhhhhhhhhhh

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1. Multi Degrees of Freedom:

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 w n – 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(- l M+K )=0 to solve for l (where l=w 2 ) § w=l 1/2  for the natural frequencies w 1 ,w 2 …. § 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 X 1 , X 2  the magnitude and direction of body 1 & 2’s initial condition in Mode 1