5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsThe orthogonal properties of the eigenforms constitute the basis of one of the most attractive methods for solving dynamic problems of multi degree-of-freedom systems.The Eigenforms or eigenvectors of the system can be shown to be orthogonal with respect to the mass and stiffness matrices as follows

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsConsider the i th and the j th eigenforms.the i th eigenform is obtained after substituting the i th eigenvalue into eq.[5.14].Therefore: [5.24]whereby: N.B. The 2nd index shows that it is the i th eigenvalue,which is substituted into eq.[5.14]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsSimilarly the j th eigenform is obtained after substituting the j th eigenvalue into eq.[5.14]..

Therefore: [5.25]whereby: N.B. The 2nd index shows that it is the j th eigenvalue,which is substituted into eq.[5.14]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsEquations [5.24] and [5.25] give:

From eq.[5.24]From eq.[5.25][5.26][5.27]Premultiply eq.[5.26] by and eq.[5.27] by To obtain[5.28][5.29]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsSince K and M matrices are symmetrical, the following relationships hold:

[5.30][5.31]Thus, subtracting eq.[5.29] from eq.[5.28] gives:[5.32]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsIf i j eq.[5.32] requires that:[5.33]As a consequence of eq.[5.33] it is evident from eq.[5.28] or eq.[5.29]for i j [5.34]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.6 Orthogonal Properties of the EigenformsEq.[5.33] and eq.[5.34] define the othogonal properties of the eigenforms. That is if i j then:

and

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionWe saw in section 5.2 that the equations of motion for un-damped multi degree of freedom system are given by: See eq.[5.4]We saw in section 5.2 that the equations are coupled.By using the Orthogonal properties of the eigenvectors, The equations of motion can be uncoupled with the help of the following linear transformation:

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of Motion [5.35]Where:X(t) = The Displacement vector in eq.[5.4] = Modal MatrixY(t) = Natural coordinates (also called Normal coordinate or generalized coordinates)In eq.[5.35] it is apparent that the modal matrix serves to transform the geometrical coordinates X to the generalized coordinates Y

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of Motion [5.36]Since the modal matrix is not a function of time ( it contains only the amplitudes) it follows:Substituting eq.[5.36] and eq.[5.35] into eq.[5.4] gives: [5.37]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionPre-multiply eq.[5.37] by the transpose of the modal matrix ( T ) to obtain: [5.38]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of Motion Remember the modal matrix is built as follows:i

j

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionLetTaking into account how the modal matrix is built, it follows from matrix algebra:andandWhere:and

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionFrom the othogonal properties of the eigenforms we saw that if i j (implying i j ) thenandThis implies that for i j : M*(i , j ) = K*( i , j) =0In other words M* and K* are diagonal matrices.Rememberand

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionFrom above it follows that eq.[5.38] has the form:Carrying out the multiplication and addition for the ith eigenform give:[5.39]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionIt can be seen that eq.[5.39] is essentially a single degree of freedom system with mass given by:[5.40]And a spring constant Given by[5.41] and are called generalized mass and generalized stiffness respectively.

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionFrom above it follows that eq.[5.39] has the form:[5.42]Which can be handled as a single degree of freedom system with a natural frequencyi is called modal frequency[5.43]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionHence the solution of eq.[5.42] is given by eq.[2.10] [5.44]Where yi(t)= response of the ith mode yio = initial modal displacement = initial modal velocity i = modal frequency

yio and are derived from the initial displacement vector and the initial velocity vector as shown bellow.

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionPre-multiply eq.[5.45] by To get[5.45]Let the initial displacement vector be Xo and the initial modal displacement vector be Yo: Transforming the geometrical coordinates to the generalized coordinates with the help of the modal matrix gives: [5.46]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionExpanding the right hand side of eq[5.46] and taking into account the orthogonal properties of the eigenforms gives:[5.47]From the preceding sections it follows that Eq.[5.47] can be written in the form:

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of Motion[5.48]From eq.[5.48] it follows :[5.49]Eq.[5.49] gives:[5.50]

5.Free Vibrations of Un-damped Multi Degree of Freedom Systems 5.7 Solution of the Equations of MotionWith similar arguments one can show that the initial modal velocity is given by:[5.51]where is the initial velocity vector.When the response of each mode yi(t) has been determined from eq.[5.44] the displacements expressed in geometrical coordinates are given by the normal-coordinate transformation i.e. by eq.[5.35]See eq.[5.35]