(1.3.1) (1.3.2) It is the harmonic oscillator equation of motion, whose general solution is: (1.3.3)

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1 M22 - Study of a damped harmonic oscillator resonance curves The purpose of this exercise is to study the damped oscillations and forced harmonic oscillations. In particular, it must measure the decay damped oscillator and examine the forced oscillator resonance curves for different damping parameters present. Issues to prepare: - Harmonic oscillator - the equation of motion and its solution, the frequency of its own; - Damped oscillator - the equation of motion and its solution, damping parameter, critical damping; - Forced vibration - sinusoidal force, the phenomenon of resonance; - Rigid body dynamics, torsion pendulum Further Readings: ator/15c_driven_osc_sep10.doc Basic concepts and definitions Torsion pendulum In section 1.1 it was discussed physical pendulum whose vibrations are caused by the force of gravity. Torsion pendulum is a different kind of a pendulum whose vibrations arise due to elastic forces. An example of such a pendulum is solid torsion suspended by a torsion spring wire or shield it with cemented a spiral spring (hairspring or tape). For sufficiently small torsion, torsion pendulum behaves like a harmonic oscillator and can be used as the observation of its essential characteristics. The remainder of this chapter presents elementary knowledge of damped oscillations and forced harmonic oscillations. A detailed analysis of these issues, the solution to the equations of motion and their general character can be found in [45] Damped harmonic oscillator We assume that the torsion pendulum, when deflected by angle φ from the equilibrium position, leads to the creation of elastic torque N 0 directly proportional to the deflection (Hooke's law): No D o (1.3.1) Proportionality constant D 0 depends on the parameters used spring and is called spring constant. The equation of motion of the pendulum moment of inertia J can be written as: J D 0 (1.3.2) It is the harmonic oscillator equation of motion, whose general solution is: o ( t) o cos( ot ) (1.3.3) where the amplitude φ 0 and phase δ are determined by the initial conditions, and ω 0 defined as: 2 o T o D J o (1.3.4) is the frequency of vibration of the pendulum.

2 Now consider the damped oscillator torque directly proportional to angular velocity of the pendulum. In this case, the equation of motion is present additional members: The equation of motion therefore takes the following form: N D (1.3.5) 2 0 (1.3.6) o where D (1.3.7) J is called damping parameter. Below certain value of damping parameter Γ < Γ kr, where Γ kr = 2ω 0, movement of the pendulum is oscillating. For the initial conditions φ(0) = φ 0 and (0) = 0 (pendulum is let go from zero angular velocity of the position of φ 0 ) solution to the equations of motion (1.3.6) is the function: where (1.3.8) (1.3.9) frequency of damping vibration. The incidence is less than the vibration frequency. The amplitude of oscillation decreases with time according to the relation. Time constant (1.3.10) called relaxation time. For weak damping, it determines the decay rate of damped oscillation in the system. If the damping is strong, ie for Γ Γ kr the pendulum oscillations do not occur. For the limiting case of critical damping Γ = Γ kr oscillation decay rate is a maximum. Dependence of the oscillation amplitude of the damped harmonic oscillation with the time are shown in Figure

3 Figure 1.3.1: Deflection φ of damped harmonic oscillator as a function of time. At t = 0 initial deflection of the oscillator is φ 0 and zero angular velocity. Continuous curve ( ) corresponds to chance of weak damping Γ < 2ω0. In addition, the thin lines indicate the oscillation boundary. Curve ( ) corresponds to the critical damping Γ = 2ω0, and curve ( ) shows strong attenuation Γ > 2ω 0. Forced Oscillator Until we make the described oscillator circuit that forces you have on pendulum additional torsion torque N ω (t) depends on time. The equation of motion then becomes: where f (t) = N ω (t) / J. In our case, we consider the harmonic force as: (1.3.11) (1.3.12) The described system may be in transient state or in the stationary state. Transients may occur shortly after turning such exciting force or after changing its frequency. The solution of equation (3.1.11) for these cases is more complicated, and therefore we confine ourselves to stationary states, where the oscillation amplitude reaches a constant value independent of time. We expect that for a time (ie, for times much larger than the relaxation time) vibrations system will occur with a frequency of exciting force Ω (Figure 1.3.2).

4 Figure 1.3.2: damped harmonic oscillator response to sinusoidal forcing. (A) Force F as a function of time. (B) Deflection of the oscillator, φ as a function of time. At t = 0 the oscillator is in equilibrium position: φ(0) = φ 0 and (0) = 0. During the first few oscillation of the oscillator is in the transient state. Steady state is reached after the time t greater than the relaxation time. After this time the oscillator performs oscillations with frequency of the applied force regardless of the initial conditions. Frequency curves for Ω = 0.25ω 0 and the damping parameter Γ = 0.05ω 0. By substitution, you can easily check that the function: (1.3.13) satisfies the equation of motion (3.1.11), if we assume the following relationships: (1.3.14) (1.3.15) In Figure is shown the amplitude of φ 0 and stationary phase shift δ forced vibration, depending on the frequency of exciting force. In the area of a frequency ω r amplitude of the vibration system significantly increases and reaches a maximum. This phenomenon is called resonance. Resonant frequency ω r can be found using standard procedure of determining the extreme of the function φ 0 (Ω). Maximum function or resonance occurs for values: (1.3.16)

5 which is smaller than the vibration frequency ω 0. Resonance frequency deviation the frequency of oscillations increases with the damping parameter Γ. Damping factor also affects the width of the resonance curve and its height. The higher the damping factor, the resonance curve is wider and its amplitude smaller. Width of the resonance curve defines a parameter called half-width (width of the curve measured at half its height). For the weak damping in a large attenuation, one can roughly assume that the half-width of the curve resonance is of order. Figure 1.3.3: Dependency of amplitude (resonance curve) and the stationary phase of the harmonic oscillator on the frequency of exciting force. Graphs are shown for several parameters damping Γ. Though stationary forced vibrations always occur with a frequency of exciting force Ω, the phase shift δ between the excitation and deflection of the oscillator strongly depends on Ω. For small frequency, the pendulum oscillations are consistent with the strength of forcing, for Ω = ω 0 are shifted in phase by 90 and for high frequencies are in anti-phase in to the exciting force The experiment Experimental setup The main element of the experimental set is so-called torsion pendulum balance, or a copper alloy plate, to which a spiral spring is attached (Figure 1.3.4).

6 Figure 1.3.4: Simplified drawing of the pendulum balance used during the experiment. These types of pendulum are used for example in mechanical watches. Note the use of such an element such as mechanical watches Affairs and its analogy to the gravitational pendulum wall clocks. Forcing is done by placing two rods instructed on the one hand to the spring pendulum on the other side (eccentric) to the DC motor. Approximation harmonic oscillator is satisfied for all angles of inclination (measured in arbitrary units). Adjusting the frequency of exciting force followed by change the voltage fed to the engine. Wiring diagram of the system is shown Figure Damping in the system is implemented through the brake induction acting on the basis of the formation of eddy currents. The brake induction attenuation is proportional to the angular velocity of the blade. Changing the damping followed by adjusting the current flowing through the windings of the electromagnet, between which is placed near the Balancing in practice by setting target Power to the adapter. Figure 1.3.5: Wiring diagram of experimental system. Calibration of the frequency of exciting force Depending on the measurement Ω = Ω (U), where Ω is the frequency of exciting force and U is the fed voltage of the motor. This part of the exercise will allow for subsequent submission of resonance curves as a function of frequency. Determination of natural frequency pendulum It is simple stopwatch measurement. Letting go of the wheel Balancing the maximum deflection to measure the free vibration period T 0 =2π/ω 0 and suppressed T=2π/ω pendulum for different damping parameters Γ. The study of damped oscillation Measuring the amplitude of oscillation, depending on the time φ m = φ m (t m ), here t m = T / 2, T is the period of damped oscillation and m = 1,2,.... Letting go of the wheel balancing the maximum deflection can observe the disappearance of oscillation amplitude. It should be noted maximum swing of the pendulum, depending on the number of oscillations m (for both positive and negative positions, ie at t = T / 2). The measurements should be performed for

7 different damping parameter Γ. The measurement will determine the envelope of damped oscillation (See Figure 1.3.1). Measurement of the resonance curve Measurement of the resonance curve φ 0 = φ 0 (Ω), for different values of the damping parameter Γ, will allow the deletion of the resonance curves similar to curves shown in Figure Changing and recording the voltage to the motor, please save the maximum deflection of the wheel balance after stabilization of the amplitude vibrations. For precise voltage changes should be applied more sensitive potentiometer. For small Γ parameter when changing the frequency of exciting force, the pendulum bar very slowly reaches steady state. Therefore, we, with any change in voltage, very gently by hand or by resetting power to stop the wheel balancing. You will see a gradual process of achieving a stable movement. It may help to pre-estimate of the relaxation time τ on the basis of knows damped vibration measurement. For small parameter Γ Balancing wheel begins hit the limiter and the resonance curve cannot be measured in close resonance. In combination with the low precision of the measurements, may make it difficult to further analyse the data. When measuring the resonance curves can be done in a qualitative way observation of the phase shift between the oscillations of the pendulum and the exciting force Production of results Calibration of the frequency of exciting force Draw a graph Ω = Ω (U). To the measured dependence of the linear fitting method, fit a straight line. The resulting fitting parameters allow for later conversion voltage at the frequency of resonance curves and presentations on the frequency scale. Determination of natural frequency pendulum Determine and compare the frequency of the pendulum free oscillation ω 0 and damped oscillation ω for different values of Γ. In this part of the exercise should especially pay attention to measurement uncertainty. The study of damped oscillation Draw the graphs of decay in oscillation amplitude φ m = φ m (t m ) for different damping parameters. The charts (by adjusting the exponent, or straight to linearized dependence) determine the parameters Γ and ω for different values of the current flowing induction by the brake. Measurement of the resonance curve Draw the resonance curve φ 0 = φ 0 (Ω). The curves fit the dependence from equation (3.1.14). Determine the fit parameters f 0, ω 0, Γ and the size of ω r. Compare the values with the values obtained in the previous section. Discuss the results. Optionally, you can make a graph of phase between the oscillations of the pendulum and the exciting force. Date: 29 th March, 2012 Translated from Polish to English From The book I Pracownia Fizyczna -by A. Magiera By Mgr Ghanshyam Khatri Int. PhD Studies Jagiellonian University in Krakow

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