Introduction to Modal Theory Dept. of Mechanical Engineering Yungpeng Wang 南臺科技大學 STUST

Introduction to Modal Theory 1. Terminology 2. Degrees of Freedom 3. Basic Assumptions 南臺科技大學 STUST

Terminology Modal Frequencies, Eigenvalues, System Poles, Characteristic Roots Modal Vectors, Eigenvectors Modal Coefficients, Residues Real Modal Vectors, Complex Modal Vectors Physical Coordinates, Generalized Coordinates Principal Coordinates, Modal coordinates Modal Mass, Damping, and Stiffness Modal A, Modal B 南臺科技大學 STUST

Degrees of Freedom 1. Theoretical Degrees of Freedom: The number of degrees of freedom for a mechanical system is equal to the number of independent coordinates (or minimum number of coordinates) that is required to locate and orient each mass in the mechanical system at any instant in time. 2. Measurement limitations imposed upon this theoretical concept of the number of degrees of freedom of a mechanical system include: Frequency Range Vibration Amplitude Rotational Degrees of Freedom Measurement 南臺科技大學 STUST

Degrees of Freedom 南臺科技大學 STUST 3. Measurement Degrees of Freedom: The number of measurement degrees of freedom (No, Ni) can be defined as the number of physical locations at which measurements are made times the number of measurements made at each physical location. 4. There is no specific relationship between the number of degrees of freedom (N) and the number of measurement degrees of freedom (No, Ni). In general, in order to define N modes of vibration of a mechanical system, No, Ni must be equal to or larger than N.

Basic Assumptions 南臺科技大學 STUST 1. The structure is a linear system. 2. The structure during the test can be considered as time invariant. 3. The structure is observable. 4. The structure obeys Maxwell’s reciprocity theorem. if one measures the frequency response function between points p and q by exciting at p and measuring the response at q, the same frequency response function will be measured by exciting at q and measuring the response at p. Hpq = Hqp