Modal Theory of Multiple Degree of Freedom System

Dept. of Mechanical Engineering

Yungpeng Wang

南臺科技大學STUST

Modal Theory of Multiple Degree of Freedom System

1. Theory

2. Solution of Eigenvalue Problem

3. Weighted Orthogonality of Modal Vectors

4. Modal Vector Scaling Example

5. Principal Coordinates - Modal Coordinates

6. Proportional Damping

7. Modal Vectors from the System Matrix

南臺科技大學STUST

Theory

1. Equation of Motion: Multiple degree of freedom system, a set of 2nd order differential equations, matrix form.

2. Eigenvalue Problem

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Solution of Eigenvalue Problem

1. Characteristic Equation

2. Computation of Eigenvalues and Eigenvectors

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Weighted Orthogonality of Modal Vectors

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1. Modal Mass

2. Modal Stiffness

3. Orthonormal Modal Vector

Modal Vector Scaling Example

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1. Example of Two Degrees of Freedom System

Principal Coordinates - Modal Coordinates

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1. principal coordinates, normal coordinates or modal coordinates

2. transformation matrix

3. modal matrix or matrix of modal vectors

4. Modal Mass, Generalized Mass

5. Modal Stiffness, Generalized Stiffness

Proportional Damping

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Structural Damping, Viscous Damping, Coulomb Damping, Hysteretic Damping

Modal Vectors from the System Matrix

南臺科技大學STUST

The evaluation of modal vectors is very useful in relating measured frequency response function data to the system modal vectors.

The residues of a frequency response function for a particular pole ( r ) are directly related to the elements of a modal vector. Also, the symmetry of the adjoint matrix is the justification for not needing to evaluate the complete frequency response function matrix.