lecture1310 - ME 563 Mechanical Vibrations Lecture#13...

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ME 563 Mechanical Vibrations Lecture #13 Multiple Degree of Freedom Modal Coordinate Transformation
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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:
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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)
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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?
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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 r and s are orthogonal with respect to the mass (and stiffness) matrix.
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