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Vibrations in linear 1-dof systems; II. energy considerations for undamped systems

Vibrations in linear 1-dof systems; II. energy considerations for undamped systems (last updated 2011-08-28). Aim. The aim of this lecture is to discuss vibrations in undamped linear 1-dof systems from an energetical point of view.

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Vibrations in linear 1-dof systems; II. energy considerations for undamped systems

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  1. Vibrations in linear 1-dof systems;II. energy considerations forundamped systems (last updated 2011-08-28)

  2. Aim The aim of this lecture is to discuss vibrations in undamped linear1-dof systems from an energetical point of view. For a more comprehensive discussion, see any book in vibration analysisor machine design

  3. Energy considerations Let us as an example consider the lateral motion a of point mass attachedto the end of a cantilever beam, see below By introducing a coordinate x describing the lateral position of the mass we find the following differential equation for the vibration of the system

  4. Energy considerations; cont. Free vibrations By taking the expression for the mass deflection and multiply each term with , where is the velocity, we get We can here identify the kinetic energy Ekin, the potential energy Epot andthe total energy Etot of the mass

  5. Energy considerations; cont. Free vibrations; cont. Furthermore, by using the obtained solution for the free vibration, we find and That is, as expected, the free vibration of the mass-beam systemconsists of a cyclic change between potential energy and kinetic energy.

  6. Energy considerations; cont. Stationary vibrations The work supplied by the force F to the mass-beam system during a period isgiven by (forced vibrations for a non-resonance case) As could be anticipated, the net work is zero for a load cycle. If this wouldhave not been the case, a stationary condition would not have prevailed!

  7. Energy considerations; cont. Stationary vibrations; cont. In the case of resonance, we have a situation in which the motion of the massexposes a phase difference compared to the applied force; the expression forthe motion is proportional to a cosine function. As could be anticipated, this will result in a net work supply to the system ineach load cycle, which is the physical explanation for the ever increasingvibration magnitudes at resonance.

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