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

**CONSISTENT MASS MATRICES**

Natural frequencies and modes

Frequency response ( *F*(*t*)=*F*o sin*wt*) Transient

response (*F*(*t*) arbitrary)

*1 Single DOF System*

** Free Vibration**:

*f*(*t*) = 0 and no damping (*c *= 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):

**VECTOR ITERATION METHODS**

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** i~~s t~~he vector of nodal displacement amplitudes.

Eq. (12) yields,

This is a generalized eigenvalue problem (EVP).

*Solutions*?

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

**M **matrices.

*Note*:

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).

*Example:*

Tags : Mechanical - Finite Element Analysis - Dynamic Analysis Using Element Method

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