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Kjell Simonsson 1 Vibrations in linear 1-dof systems; II. energy considerations for undamped systems (last updated 2011-08-28)

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Kjell Simonsson 2 Aim The aim of this lecture is to discuss vibrations in undamped linear 1-dof systems from an energetical point of view. For a more comprehensive discussion, see any book in vibration analysis or machine design

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

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Kjell Simonsson 4 and multiply each term with, where is the velocity, we get Free vibrations By taking the expression for the mass deflection Energy considerations; cont. We can here identify the kinetic energy E kin, the potential energy E pot and the total energy E tot of the mass

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

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

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

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