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Unformatted text preview: ME 563 Mechanical Vibrations Lecture #13 Multiple Degree of Freedom Modal Coordinate Transformation What Did We Learn? 1 By making the assumption, {x(t)}={A}e st , we observed that the free response was actually a sum of the modes of vibration: If we write this differently, we reveal the most important thing about this course: Coordinate Transformation 2 The solution for the free response is actually a coordinate transformation in disguise: If this is the form of the solution, why not use it to begin with? Generalized coordinates (Physical DOFs) Principal coordinates (Modal DOFs) Modal transformation matrix (Modal vectors) Modal Solution Method 3 Consider the case again with no damping: The equations of motion have been converted to modal equations of motion. Are these equations easier to solve? Orthogonality of the Modes 4 The equations are easier to solve provided the modal vectors are orthogonal with respect to the system matrices: This final equation indicates that two different modes...
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 Fall '09

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