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Lab 3 Vibration(1)

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  1. Introduction Natural oscillation  –bar oscillator Natural oscillation is taken to mean the non-influenced oscillation of an oscillatory system in its natural state. The system is initially deflected out of its equilibrium position and then oscillates about this position until it is brought to rest by any external or internal damping which may be present. 2. Description of Apparatus  3. Theoretical principles Non-damped oscillation Equation of motion Establishment of the equation of motion involves forming the moment equilibrium about the fulcrum point O of the beam ∑  −==  aF  J  M  C OO  α   The spring force F C  results from the deflection x and the spring constant c. For a small angle, the deflection can be formed         accxF  C   ϕ  ==  The MMI of the beam about the fulcrum point is 3 2 mL J  O  =  The equation of motion is thus the following homogeneous differential equation 03 22 =+  ϕ ϕ  mLca   The solution produces harmonic oscillations with the natural angular  0  or the natural frequency f 222220 321,3 mLca f mLca π ω   ==  The periodic time is 22 32 camLT   π  =  As can be seen, the periodic time/natural frequency can easily be set by way of the lever a of the spring  Damped oscillation Equation of motion Establishment of the equation of motion again involves forming the moment equilibrium about the fulcrum point O of the beam. Allowance is additionally made here for a damper force with the lever arm b bF aF  J  M  d C OO  −−== ∑  α   The damper force F d  results from the speed v and the damper constant d. For small angles the speed can be      lever arm b bd dvF  d   ω  ==  The resultant equation of motion is thus the following homogeneous differential equation 0 0202 =++  ϕ ϕ ϕ   J ca J db   The solution produces decaying harmonic oscillations ( )  ............................ 0 d  t  ω ω ϕ   =  with natural angular frequency 02200  1  J caand  D d   =−=  ω ω ω   and degree of damping 00 2  ω   J b D  =  As can be seen, oscillation is no longer possible with D      d  approaches zero.
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