LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: [email protected] by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved P1.5.3.2 LD Physics Leaflets Mechanics Oscillations Torsion pendulum 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 Bi / Fö 0505 Principles Oscillations (and wave) phenomena are well known due to their presence everywhere in nature and technique. 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 oscilla-tions. In experiment P1.5.3.1 the free damped rotary oscilla-tions have been investigated. In this experiment it will be investigated how the oscillator reacts to an external periodic force. When applying the periodic torque )tsin(MMex0ex⋅ω⋅= (I) we obtain the following equation of motion for the damped rotary oscillating system (compare equation (I) in leaflet P1.5.3.1): )tsin(MDdtdkdtdJex022⋅ω⋅=ϕ⋅+ϕ+ϕ (II) J: moment of inertia D: directional quantity (restoring torque) k: damping coefficient (coefficient of friction) ϕ: angle of rotation M0: maximum of external torque ωex = 2π⋅ν: frequency of the external torque The solution of this inhomogeneous differential equation is the sum of a specific (particular) solution and the general solution of the corresponding homogeneous differential equa-tion (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. δ is largeδ is smallωRω ϕ0ω0 Fig. 1: Resonance curves (top) and phase shift between exciter and oscillator (bottom) for various damping constants δ. δ is largeδ is smallφππ/20ωω0P1.5.3.2 - 2 - LD Physics leaflets LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: [email protected] by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved For the specific solution the following relationship can be used: )tsin()()t(exex0φ−⋅ω⋅ωϕ=ϕ (III) Substituting equation (III) in equation (II) gives after several trigonometric transformations the amplitude of the forced oscillation: 2ex2ex00ex0Jk)(J/M)(ω+ω−ω=ωϕ (IV) The frequency at which the amplitude of the oscillation is maximal is called the resonance frequency ωR (amplitude resonance). This is the case when the radicand in the de-nominator is minimal. By equating the derivative of the radi-cand with respect to ω to zero the following relationship for the resonance frequency is found: 2202220R2J2kδ−ω=−ω=ω (VI) with JD0=ω (natural frequency) (VII) J2k⋅=δ (damping constant) (VIII) The lower the damping the less the resonance frequency differs from the natural frequency ω0 and the larger is the amplitude. In the limit of disappearing damping (k → 0) the amplitude at the resonance frequency (ωex = ω0) would tend towards infinity (so called resonance catastrophe). From equation (IV) follows that amplitude of the forced oscil-lation tends towards zero for very high frequencies. For very low frequencies (ω → 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 ωR. Fig. 2: Schematic representation (wiring diagram) of the experimen-tal setup: (A) exciter, (B) eddy current brake. 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 Ux24 V–650 mAVVAA31100m10m1m100µ31100m10m1m100µ30010030103300100301031300m100mACDC531 120OFFAC ~24 V(B)(A)VFINEA521 545DC NETZGERÄT 0–16 V / 0–5 ADC POWER SUPPLY 0–16 V / 0–5 APOWERVVAA31100m10m1m100µ31100m10m1m100µ30010030103300100301031300m100mACDC531 120OFFLD Physics leaflets - 3 - P1.5.3.2 LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: [email protected] by LD Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved 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: ωex = ω0 (energy resonance). The energy and amplitude resonances are thus obtained at different excitation frequen-cies. The phase shift φ between the external excitation and the oscillating system is given by: )(2tan2ex20exω−ωωδ=φ (IX) From this relation follows: For ωex << ω0 the oscillator and the exciter oscillate al-most in phase (φ ~ 0). For ωex >> ω0 the oscillator and the exciter oscillate al-most in anti-phase (φ ~ π). For ωex = ω0 the oscillator lags behind the exciter exactly by π/2. 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). Carrying out the experiment a) Determining the amplitude as function of the fre-quency −−−− recording the resonance curve - Set the current for