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Consistent Mass Matrices

Natural frequencies and modes: Frequency response ( F(t)=Fo sinwt) Transient



Natural frequencies and modes

Frequency response ( F(t)=Fo sinwt) Transient


response (F(t) arbitrary)


1 Single DOF System

Free Vibration:


f(t) = 0 and no damping (= 0) Eq. (1) becomes

mu ku


(meaning: inertia force + stiffness force = 0) Assume:


This is the circular natural frequency of the single DOF system (rad/s). The cyclic frequency (1/s = Hz) is



2.Multiple DOF System


Equation of Motion                         

Equation of motion for the whole structure is     

Mu    Cu   Ku       f (t) ,  (8)

in which:     u        nodal displacement vector,     


M      mass matrix,


C       damping matrix,


K stiffness matrix, f forcing vector.


Physical meaning of Eq. (8):


Inertia forces + Damping forces + Elastic forces


= Applied forces


Mass Matrices


Lumped mass matrix (1-D bar element):






Study of the dynamic characteristics of a structure: natural frequencies normal modes shapes)


Let f(t) = 0 and C = 0 (ignore damping) in the dynamic equation (8) and obtain


Mu Ku   0


Assume that displacements vary harmonically with time, that is,


where  u  is the vector of nodal displacement amplitudes.


Eq. (12) yields,


This is a generalized eigenvalue problem (EVP).




This is an n-th order polynomial of from which we can find n solutions (roots) or eigenvalues


i (i = 1, 2, , n) are the natural frequencies (or characteristic frequencies) of the structure (the smallest one) is called the fundamental frequency. For each gives one


solution (or eigen) vector



if wi  wj . That is, modes are orthogonal (or independent) to each other with respect to K and





Magnitudes of displacements (modes) or stresses in normal mode analysis have no physical meaning.


For normal mode analysis, no support of the structure is necessary.


i = 0  there are rigid body motions of the whole or a part of the structure. apply this to check


the FEA model (check for mechanism or free elements in the models).


Lower modes are more accurate than higher modes in the FE calculations (less spatial variations in the lower modes fewer elements/wave length are needed).







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