CSBSJU PHYS 370 - Mechanics- Oscillator (6 pages)

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Mechanics- Oscillator



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Mechanics- Oscillator

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Pages:
6
School:
College of Saint Benedict and Saint John's University
Course:
Phys 370 - Advanced Physics Lab
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LD Physics Leaflets Mechanics Oscillations Torsion pendulum P1 5 3 2 Forced rotational oscillations Measuring with a hand held stop clock Objects of the experiment g Measuring the amplitude of forced rotational oscillations as function of exciter frequency for various damping constants g Determining the natural frequency of the oscillator g Investigating the phase shift between the exciter and the oscillator Principles 0 Oscillations and wave phenomena are well known due to their presence everywhere in nature and technique is small The rotary oscillations are a special case among various mechanical oscillator models which allow to investigate the most important phenomena occurring in all types of oscillations In experiment P1 5 3 1 the free damped rotary oscillations have been investigated In this experiment it will be investigated how the oscillator reacts to an external periodic force When applying the periodic torque Mex M0 sin ex t we obtain the following equation of motion for the damped rotary oscillating system compare equation I in leaflet P1 5 3 1 J d2 d k D M0 sin ex t dt dt 2 is large I II R 0 0 J moment of inertia is small D directional quantity restoring torque k damping coefficient coefficient of friction angle of rotation 2 M0 maximum of external torque is large ex 2 frequency of the external torque Bi F 0505 The solution of this inhomogeneous differential equation is the sum of a specific particular solution and the general solution of the corresponding homogeneous differential equation M0 0 The latter however decreases exponentially compare equation V in leaflet P1 5 3 1 and is no longer significant after a sufficiently long period of time 0 Fig 1 Resonance curves top and phase shift between exciter and oscillator bottom for various damping constants LD Didactic GmbH Leyboldstrasse 1 D 50354 Huerth Germany Phone 02233 604 0 Fax 02233 604 222 e mail info ld didactic de by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved P1 5 3 2 LD Physics leaflets 2 Apparatus 1 Torsion pendulum 346 00 1 DC power supply 0 16V 0 5 A 521 545 1 Plug in power supply for torsion pendulum 562 793 1 Ammeter DC I 2 A e g LDanalog 20 531 120 1 Voltmeter DC U 24 V e g LDanalog 20 531 120 1 Connecting lead 100 cm blue 500 442 2 Pair cables red and blue 100 cm 501 46 1 Stop clock 313 07 The frequency at which the amplitude of the oscillation is maximal is called the resonance frequency R amplitude resonance This is the case when the radicand in the denominator is minimal By equating the derivative of the radicand with respect to to zero the following relationship for the resonance frequency is found R 02 k2 02 2 2 2 J2 with 0 D J k 2 J For the specific solution the following relationship can be used t 0 ex sin ex t III Substituting equation III in equation II gives after several trigonometric transformations the amplitude of the forced oscillation M0 J 0 ex k 0 ex ex J IV 2 2 VI natural frequency VII damping constant VIII The lower the damping the less the resonance frequency differs from the natural frequency 0 and the larger is the amplitude In the limit of disappearing damping k 0 the amplitude at the resonance frequency ex 0 would tend towards infinity so called resonance catastrophe From equation IV follows that amplitude of the forced oscillation tends towards zero for very high frequencies For very low frequencies 0 the amplitude tends towards the value M0 J which is not equal zero The resonance curve is not symmetrical with respect to the resonance frequency R Fig 2 Schematic representation wiring diagram of the experimental setup A exciter B eddy current brake 24 V A AC OFF 531 120 V 3 100 1m 10m 100m 10 30 100 300 AC 3 100 3 30 10 1 3 1 V A 1 300 1m 300m 100 100m 100m 10m DC A 521 545 DC NETZGER T 0 16 V 0 5 A DC POWER SUPPLY 0 16 V 0 5 A Ux 24 V 650 mA A B V FINE OFF 531 120 V 3 100 1m 10m 100m 10 30 100 300 POWER AC 3 100 3 30 10 1 3 1 V A 1 300 1m 300m 100 100m 100m 10m DC A LD Didactic GmbH Leyboldstrasse 1 D 50354 Huerth Germany Phone 02233 604 0 Fax 02233 604 222 e mail info ld didactic de by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved LD Physics leaflets Note The energy resonance has to be distinguished from the amplitude resonance considered above It is possible to show that the oscillator possesses a maximum in energy when the frequency of the external torque equals the natural frequency ex 0 energy resonance The energy and amplitude resonances are thus obtained at different excitation frequencies The phase shift between the external excitation and the oscillating system is given by tan 2 ex 20 2ex P1 5 3 2 3 IX From this relation follows Measure the period of the exciter and determine the frequency To determine the period measure the time 10 T for 10 revolutions of the drive wheel Read off the amplitude when the forced oscillation has reached a steady state and the amplitude of successive oscillations are constant Note When measuring the amplitude as function of the frequency of the exciter i e the resonance curve several minutes have to be waited until the amplitude is sufficiently constant and the settling process of the forced oscillation has been completed This holds especially for the case of weak damping The settling process is particularly noticeable as a beat close to the resonance For this reason a medium current has to be chosen as starting value For ex 0 the oscillator and the exciter oscillate almost in phase 0 For ex 0 the oscillator and the exciter oscillate almost in anti phase For ex 0 the oscillator lags behind the exciter exactly by 2 When changing the frequency of the exciter to a new value it might be necessary to readjust the voltage of the exciter after measuring and determining the frequency to have an appropriate frequency value with respect to the previous frequency setting In the region of rapid amplitude increase the frequency has to be changed in small steps Setup The set up of the experiment is shown in Fig 2 schematically The period T of the exciter is measured by the stop clock not shown in Fig 2 Safety notes g The current through the eddy current brake should not exceed 2 A for a long time g Avoid overheating of the coils by measuring too long with large current I 1 A It is recommended to stop the oscillator completely between different exciter frequency settings and start the forced oscillation from scratch Thus the time of the settling process is


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