Modal Analysis

Orthogonal Properties of the Eigen Vectors:

Stem can be shown to be orthogonal with respect to the mass and stiffness matrices.

In equation (1) x = X sin ωt the equation for the i th mode be
              
KX_i = λ_iMX_i                                                                                   (1)

Premultiplying by the transpose of mode j,
               
X_jKX_i = X_jλ_iMX_i = λ_i(X_jMX_i)                                                                  (2)

Now start with the equation for the jth mode and premultiplying by X_i’ to obtain,
           
X_i’KX_j = λ_i(X_i’MX_j)                                                                              (3)

Since K and M are symmetric matrices
              
X_j’MX_i = X_i’MX_j and X_j’KX_i = X_i’ KX_j                                                         (4)

Subtracting (3) from (2),
                
(λ_i – λ_j)X₂’MX = 0                                                                             (5)
                
As λ₂ ≠ λ_j,X₂’MX_j = 0 and X₂’ KX_j = 0                                                       (6)

Equation (6) shows the orthogonal character of the normal modes.

If i = j,

X₂’MX_i = M_i

X₂KX_i = K_i                                                                                  (7)

M_i and K_i are known as the generalized mass and generalized stiffness of the ith mode.

Modal Participation in Free Vibration

To find how much of each mode will be present in the resulting free vibration, one may write the free vibration
 
u(0) as

u(0) = C₁X₁ + C₂X₂ + .... + C_iX_i + ...                                       (8)

Where X_i is the ith normal mode and the coefficient C_i represent the amount of i th mode present in the free vibration. Premultiplying equation (8) by X₂’M and taking note of the orthogonal property one gets,

X_i’Mu(0) = 0 + 0 + .... + (X_i’MX_i)C_i + 0 +m_iC_i              (9)

So,



Modal matrix P:

The modal matrix for a three d of system can be given by





P’MP and P’KP will be diagonal matrices since the off diagonal terms simply expresses the orthogonality relation and are zero.

For a 2DOF:



Similarly    



Weighted Model Matrix : If each column of the model matrix P is divided by the square root of the generalized mass M_i , the new matrix is called weighted model matrix and designated as .



Here I is the unity matrix and matrix λ is a diagonal matrix with eigenvalues in the diagonal elements.

To develop the uncoupled equation of motion by use of modal matrix P



Substituting

X = PY                                        (17)

one will obtain

Premultiplying P’  in equation (18) yields



As (P’MP) and (P’KP)  are diagonal matrices equation (16) represents a set of uncoupled equation which can be easily solved by using the principles used for single d of system.

It can also be reduced to a simpler form by using weighted modal matrix.

Considering



equation (16) will reduce to



Now premultiplying in equation (21) one will get



But from equation (15) it is known that, , so the above equation reduces to

Services: - Modal Analysis Homework | Modal Analysis Homework Help | Modal Analysis Homework Help Services | Live Modal Analysis Homework Help | Modal Analysis Homework Tutors | Online Modal Analysis Homework Help | Modal Analysis Tutors | Online Modal Analysis Tutors | Modal Analysis Homework Services | Modal Analysis