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lecture510 - and non-conservative forces, the formula for...

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ME 563 Mechanical Vibrations Lecture #5 Lagrange’s Method for Deriving Equations of Motion
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Derivation 1 Return to Newton’s second law for a particle, i : If we only consider the “active” forces, then we can “project” the equations onto the trajectory of the system to obtain the equation of motion as follows: is called the kinematic variation along the trajectory; we can express it in terms of displacement, velocity, etc.
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Derivation 2 When the variation is substituted into the previous equation, Lagrange’s equations of class II appear (after a lot of calculus). This identity is needed: Proof is available upon request…
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Derivation 3 When the forces are broken up into the active conservative
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Unformatted text preview: and non-conservative forces, the formula for the so-called “non-conservative generalized forces” are found: Non-conservative generalized forces Derivation 4 Lastly, the formula for the kinematic motion is substituted into the equation of motion to obtain the final form: Lagrangian: Derivation 5 If all of the generalized coordinates are independent, then for each of the r th coordinates (degrees of freedom). An alternative form taken from the previous equations is, Lagrange penalty Example Rolling Disc on Incline 6 C F o cos ω o t Eliminate constraints Example Rolling Disc on Incline 7 C F o cos ω o t Calculate generalized forces: Example Rolling Disc on Incline 8...
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lecture510 - and non-conservative forces, the formula for...

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