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L06 - Lumped mass and Harmonic Motion.pdf

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McMaster University
Mechanical Engineering

Lecture 6 - Lumped Mass Assumptions • Lumped Analysis - Inertia Effect of Stiffness Elements – Generally ignore mass in stiffness elements • This assumption can lead to significant error, especially if the mass of the stiffness element is not negligible compared to system mass – Concept of a lumped mass • Determine the inertia effects of the stiffness elements referred to a point on a lumped mass • Equivalent kinetic energy that can be referred to the same lumped mass point • Example: 1 DOF spring-mass systems • Consider the distribution of mass in the spring – Calculate an equivalent mass value located at the end Harmonic Motion • Periodic Motion – Motion that is repeated after equal intervals of time – Motion can be very complicated but it must repeat • Cycle – Movement of a vibrating object from equilibrium, to one extreme position, back through equilibrium, to the other extreme position, and back to equilibrium Descriptors • Amplitude  A – Maximum displacement of the vibrating body from equilibrium • Phase angle () – Shift in corresponding points of wave shape • Period  τ (s) – Time taken to complete one cycle • Frequency  f (Hz) = 1 / period – Number of cycles per unit time (cycles/sec) – For a rotational system relates to RPM * Be careful with the units used: • Circular frequency   (rad/= 2 f (Hz) - Practical (RPM) • Natural frequency
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