M10e “Resonance and Phase Shift in Mechanical Oscillations”

Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum

Tasks

1. Determine the frequency fd of the damped oscillation and the damping constant  of a rotary pendulum for eight different values of the damping. Plot d as a function of  and compare with theory.

2. Measure the resonance curve of a rotary pendulum as well as the phase shift between drive and deflection of the pendulum disk for one value of the damping constant. Plot the resonance curve, compare with theory and determine resonance frequency and damping constant.

Literature

Physikalisches Praktikum, 14. Auflage, Hrsg. W. Schenk, F. Kremer, Mechanik, 2.3 M. Alonso, E.J. Finn, Physics, Addison-Wesley, 10.13, 10.14

Accessories

Pohl’s wheel with angular sensors, computer interface system, laptop

Key Aspects for Preparation - Moment of inertia, moment of inertia of a disk - Angular momentum - Rotary pendulum, equation of motion - Eigenfrequency, damping constant, frequency of the damped oscillation, logarithmic decrement - Resonance curve, resonance frequency, phase shift

Basics

In this experiment damped and forced oscillations are studied using a rotary pendulum (Pohl’s wheel). The rotary pendulum has an eigenfrequency that is determined by the directional moment of the spring and the moment of inertia of its copper disk. The damping can be varied by controlling the magnitude of eddy current damping in the copper disk that oscillates through a region of variable magnetic field. If the damping is increased, the damped oscillations show a stronger amplitude decrease with time and a small shift of the frequency of the damped oscillation. In case of a periodically driven rotary pendulum, a large oscillation amplitude is observed, when the driving frequency is equal to the resonance frequency. This maximum amplitude is a function of the driving torque, the damping and the eigenfrequency. These basics are presented in more detail in the following.

Rotary pendulum without external drive If the pendulum is deflected by an angle  , it executes a damped oscillation around the rest position

0. This oscillation is called damped eigenoscillation of the system. The equation of motion of the system is obtained from a balance of the torques

MMMTFD (1) with MJT   inertia torque

MDF   restroring torque from the spring

MD  damping torque (moment of inertia J, directional moment of the spring D, damping coefficient  ). Putting these expressions into Eq. (1) yields the equation of motion of the pendulum: JD     0 (2a) 2 or 20   0   (2b) 2 22 with /2J  , 0 DJ/ and d0   . Equation (2b) is a homogeneous, linear differential equation of second order. In solving this differential equation three cases are distinguished: 22 1.  0 : creep (strong damping) (3) 22 2.  0 : aperiodic limiting case (critical damping) (4) 22 3.  0 : oscillation (weak damping). (5) The solution is given by tt (t ) e A1 sin(  d t )   A 2 cos(  d t )  Ce cos(  d t   ) (6) with amplitudes A1, A2 and C and phase : 22 AAAAAA1CCCsin  ,  2  cos  ,   1   2 , tan    1 /  2 .

The eigen angular frequency 0 characterizes the undamped pendulum; d is the angular frequency of the damped oscillation. fdd/(2 ) is often called the frequency of free oscillations.

The eigen angular frequency 0 does not depend on the oscillation amplitude. This is a feature of harmonic oscillators that obey linear equations of motion. The solution for the free, but damped oscillation contains an exponentially decreasing term characterized by the decay time  1 . The frequency of the damped oscillation is smaller than that of the undamped oscillation. The damping constant  , resp. the logarithmic decrement  can be experimentally determined from the temporal decrease of the angular deflection (t): ()t Td ln (period Tdd 2/). (7) ()tT d

Rotary pendulum with external drive

If the pendulum is driven by an external torque M0 sin(t) varying periodically in time, one obtains a forced oscillation. After a settling time the angular frequency of the pendulum is identical to the angular frequency  of the external drive. The equation of motion is given by

J    D   M0 sin(  t ) . (8) This is an inhomogeneous, linear differential equation of second order. The general solution to Eq. (8) is a linear combination of the general solution of the homogeneous differential equation (2b) and a particular solution of the inhomogeneous differential equation (8). A particular solution can be obtained with the ansatz

p(t ) A (  )sin(  t  ) . (9) This yields the amplitude MMJ/ A() 00, (10) 2 2 222 2 2 2 2 2 2 J 00         4   where  is the driving torque‘s angular frequency and 0  DJ/ is the eigen angular frequency of the free, undamped system. The phase shift  is frequency dependent: 2  tan ( )     (11) 2 2 2 2 J00      The general solution of Eq. (8) is obtained by adding equations (6) and (9)

 t (t ) A (  )sin(  t   )  Ce cos( d t   ) . (12) This explicit solution clearly shows that the oscillations of the driven and free system superimpose.

After a settling time  the term with angular frequency d becomes very small due to the exponentially decreasing damping factor. The pendulum then oscillates with the angular frequency  of the drive, albeit with a phase shift  relative to the driving torque.

The limiting values of Eq. (10) are:

MM00 AA(0)2  ; (  )  0 . JD 0 The amplitude of the forced oscillation reaches its maximum at the resonance angular frequency. Minimizing the denominator of Eq. (10) yields 2 2 2 ( 0 )  2    0 . Therefore, the resonance angular frequency is given by 22 R  00 2     / 2 . (13)

R 0  0 / 2

The amplitude at the resonance angular frequency is 2 0 AA(R ) (0)  0 / 2 2d . (14)

AA(R ) (0)  0 / 2

For weak damping (and only then) we get the approximate value: D  . R0 J 3

Plotting the amplitude A() as a function of angular frequency  yields the resonance curve (Fig. 1). The graph of the resonance curve is not symmetric with respect to the resonance angular frequency. The half-width  is defined as the difference between the angular frequencies 1 and 2, at which the resonance amplitude has decreased to the value AAA(1 ) (  2 )  R  / 2 [resp. 2 2 2 AAA(1 ) (  2 )  R  /2 , corresponding to the power PPP(1 ) ( 2 ) max /2 ]. The half-width can 11 only be defined for values of the damping that are smaller than 1    0.38268  , 00 22 since otherwise the branch of the resonance curve on the low frequency side is always larger than

A(R )/ 2 . In the general case one has

2 2 2 1,2  RR 2     . (15)

In the weakly damped case 0 one obtains 1,2  R  and a half-width  2  or     2 . (16) Equation (16) expresses the uncertainty between the frequency and the lifetime of a damped linear oscillator. Large values of damping lead to a small lifetime and cause broad resonance curves. Very narrow resonance curves correspond to weakly damped systems with large lifetime.

A() / A( ) R Fig. 1 Resonance curve. 1.0

0.8  1/2

0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0  /  R

The damping expresses the dissipation of energy brought into the system by the external drive. The smaller the damping of the driven oscillator, the larger its oscillation amplitude at resonance. Another important value characterizing the oscillator is the quality factor Q with

Q 00/     /2  .

According to Eq. (11) the phase shift is negative, i.e. the angular deflection always lags behind the driving motion. Explicitly, the phase shift can be written as 2 (  )  arctan22   0 0  . (17) 2 (  )    arctan22   0  0

As a function of angular frequency the phase shift has an inflection point at the eigenfrequency 0 ,  not at the resonance frequency  . At this inflection point the phase is () . R 0 2

0 Fig. 2 Phase shift.

(rad) -1



-2

Phasenwinkel

-3

0.0 0.5 1.0 1.5 2.0 / 0

Hints towards experimentation and analysis

Task 1 Record the free oscillations for various currents of the eddy-current brake in the range between 400 mA and 1200 mA. Determine the angular frequency of the free oscillation d and the damping 2 2 constant  . Estimate the error of these quantities. Plot d versus  and compare with theory.

Task 2 Measure the frequency dependence of resonance curve and phase shift for one value of the current through the eddy-current brake. The phase shift might be determined from the time shift between drive and pendulum oscillation t using t . Plot the data. Determine the resonance frequency R and damping constant by fitting the theory curves to the data.

Rotary pendulum

Fig. 3 Rotary pendulum after Pohl 1 Resonator (copper disk) 2 Extender wheel of the drive 3 Angular sensor for the drive 4 Motor connection (0 – 24 VDC) 5 Eddy-current brake connections (0 - 2 A) 6 Angular sensor for the resonator 7 Analog-to-digital converter with integrated preamplifier

Software for the interface system