Fakultät für Physik und Geowissenschaften

Physikalisches Grundpraktikum

M 17e “Pohl’s wheel (linear and nonlinear oscillations)” Tasks

1. Determine the frequency fd, the damping constant and the logarithmic decrement of the

rotating pendulum after Pohl at eight different attenuations (currents IW of the eddy current brake).

Plot the logarithmic decrement as a function of the square of the current IW and determine the straight line parameters via linear regression.

2. Experimentally, determine the directional torque D and the moment of inertia J of the rotating pendulum. 3. Record the resonance curve of the rotating pendulum at one of the given attenuations. Determine the resonance frequency and the damping constant and compare the values to those obtained in task 1. 4. Experimentally, try to observe for the nonlinear pendulum the first bifurcation and (qualitatively) the transition to chaotic motion. To this end measure the amplitudes close to the resonance of the

forced oscillation for various values of the eddy current brake current IW (to be decreased in small steps) and plot them graphically.

Additional task For the driven pendulum, simulate the first bifurcation and the transition to chaotic oscillations by changing the attenuation. A computer algebra program is available for download in the download area of the Undergraduate Physics Lab website.

Literature

Physics, M. Alonso, E. F. Finn, Chap. 10.10 The Physics of Vibrations and Waves, H. J. Pain, Wiley 1968, Chap. 3 Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Mechanik, 2.3 Java Applet about mechanical resonance: http://www.walter-fendt.de/ph14d/cpendula.htm

Accessories

Pohl’s wheel (ELWE company), CASSY-interface (Leybold Didactic), PC

Keywords for preparation

- torques on a rotating pendulum, moment of inertia of a cylinder - rotating pendulum, homogeneous and inhomogeneous linear differential equations with constant coefficients, eigenfrequency, damping constant, logarithmic decrement - methods for determining attenuation (logarithmic decrement, resonance curve) - principle of operation of an eddy current brake - principles of nonlinear dynamics, transition into chaotic states, bifurcations

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Remarks

In this experiment free, damped and forced oscillations of a rotating pendulum after Pohl are

studied. The rotating pendulum has an eigenfrequency, depending on the directional torque of the

spring and its moment of inertia. By increasing the damping, in case of free oscillations one can

observe a stronger attenuation of the amplitude as a function of time as well as a decrease of the

eigenfrequency. In case of a periodically driven pendulum the resonance can be realized by matching

external frequency and eigenfrequency. In case of resonance the maximum amplitude is a function of

the damping. Furthermore the pendulum can be provided with an additional mass leading to an

imbalance. This will cause an additional restoring torque and the relation between deflection angle

of the pendulum and restoring torque becomes nonlinear. This results in a qualitatively different

oscillation behaviour of the pendulum, allowing for different amplitude states in between which the

pendulum deflection fluctuates in response to the initial conditions as well as small perturbations.

The movement of the pendulum appears to be random. The emergence of two different amplitude

states is called bifurcation. A further decrease of the attenuation will lead to bifurcations of higher

order. Accordingly, there are various possibilities for the dynamic evolution of such a system (random

or chaotic behaviour).

The pendulum after Pohl used in the experiment (Fig.4) consists of a ring-shaped copper disk with a homogeneous mass distribution, attached to a rotation axis through the center of mass. The rest position is enforced by the spiral spring and the deflection can be read off an angle scale. For controlling the damping of the oscillation the current in the inductor of the eddy current brake can be adjusted. Additionally the pendulum can be actuated on by a regulated stepping motor (that is connected to the disk via an extender wheel and a rod) with different frequencies. The oscillation recorder and further electronic components allow for a contactless measurement of the copper disk’s motion. Voltages proportional to the measured signals, i.e. the time dependence of both angular deflection and angular velocity, are generated and fed to an ADC (Analog-to-Digital Converting) interface for further processing in the computer. Basic knowledge

Linear systems – rotating pendulum without external torque In the simplest case, i.e. without external drive, the system can be modelled in the following way: if

the pendulum is deflected by an angle and released, it will perform a damped oscillation around

the equilibrium angle 0. This oscillation is called damped natural oscillation (eigenoscillation) of the system. The equation of motion of the system can be derived from the equilibrium of torques

T F DM M M , (1)

with TM J torque of inertia

FM D restoring torque of the spring

DM damping torque

(moment of inertia J, directional torque of the spring D, damping coefficient ).

Inserting into the equation above leads to the differential equation of the damped natural oscillation

of the pendulum: 0J D (2a)

or 2

02 0 . (2b)

( / 2J , 2

0 / D J , 2 2

d 0 )

Equation (2b) is a second order, homogeneous, linear differential equation. In the solution of such a differential equation three different cases have to be considered:

3

1. 2 2

0 : overdamped case (3)

2. 2 2

0 : aperiodic limiting case (4)

3. 2 2

0 : oscillatory case (5)

0 denotes the natural angular frequency (angular eigenfrequency of the undamped oscillator) and

d the free angular resonance frequency of the damped oscillator.

The natural angular frequency 0 does not depend on the oscillation amplitude. This result is an important characteristic of harmonic oscillators, as they are described by linear equations of motion. As a solution for the weakly damped case one obtains an exponentially decreasing oscillation that is

characterized by the relaxation time 1 and the free angular resonance frequency d d / 2f .

The free resonance frequency of an attenuated oscillator is always smaller than the eigenfrequency

of the corresponding undamped system. The damping constant , respectively the logarithmic

decrement can be easily obtained from the temporal decrease of the oscillation amplitude (t) :

d

d

( )ln

( )

tT

t T

(period 2 /d dT ). (6)

An illustrative representation of the oscillation behaviour can be found in phase space

( Diagram) . For a point particle in three-dimensional space the phase space is defined as the

set of six-tuples made up by the three spatial and three momentum coordinates. In case of Pohl’s wheel, space and momentum coordinates are one-dimensional quantities. The phase space is then reduced to two dimensions. In a two-dimensional graph the angular velocity ( )t is plotted against

the angular deflection ( )t of the pendulum, which yields a time independent geometrical curve, also

called trajectory. In case of a weakly damped harmonic oscillator the trajectory is a spiral approaching the zero of the coordinate system (which is a so-called attractor). Rotating pendulum with external torque

Attaching an additional, external, periodic torque M0 sin( t) to the pendulum, using an actuator, one obtains a forced oscillation. After a certain settling time the frequency of the pendulum is equal to that of the external drive. The equation of motion is

0 sin( )J D M t . (7)

This is a second order inhomogeneous, linear differential equation. A general solution of Eq. (7) is a superposition of the general solution of the corresponding homogeneous differential equation (2a) and a particular solution of the inhomogeneous equation. A particular solution can be found, using the ansatz

( ) ( )sin( ) p t A t (8)

For the amplitude one obtains

0 0

2 2 2 2 2 2 2 2 2 2 2

0 0

/( )

( ) ( ) 4

M M JA

J

. (9)

denotes the driving frequency and 0 /D J the eigenfrequency of the undamped system.

The phase shift is given by

2 2 2 2

0 0

2tan ( )

( ) ( )J

(9a)

The general solution to Eq. (7) is found by adding this solution to the one of Eq. (2a):

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( ) ( )sin( ) cos( )t

dt A t Ce . (10)

One can see that the oscillations of the free and the driven system superpose. After a settling time the second term is exponentially damped away. The pendulum is then oscillating with the driving

frequency of the stepping motor, albeit with a phase shift with respect to the drive. In the case of a forced oscillation the trajectory in phase space is an ellipse that does not change after the settling time. The amplitude of the forced oscillation reaches a maximum

0

2 2 2 2 2

0

( )2( ) 4

R

dR R

MMA

JJ (11)

at the resonance angular frequency 2 2

0 2R .

For weak damping the resonance angular frequency is approximated by

0R

D

J .

The other limiting values of Eq. (9) are

0 0

2

0

(0)M M

AJ D

; ( ) 0A .

Plotting the amplitude A versus the angular frequency yields the resonance curve, see Fig. 1. The curve is not symmetric with respect to the resonance angular frequency. As full width at half

maximum (FWHM) the difference between the two angular frequencies 1 and 2 is defined at

which the resonance amplitude decreases to the value 1 2( ) ( ) ( ) / 2RA A A ; at this point the

power has decreased to half of the maximum value. An important relation between the FWHM of the resonance curve and the damping constant is

2 , respectively 2 . (12)

Eq. (12) describes the uncertainty between frequency and life time of a damped linear oscillator. Strong damping leads to a short ‘life time’ of the oscillation and results in broad resonance curves. Narrow resonance curves correspond to systems with a longer ‘life time’ and thus a small damping.

Fig. 1 Resonance curve

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

A() / A(R)

/ R

1/2

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The damping describes the dissipation of energy that is fed into the system by the actuator. The smaller the damping of the driven pendulum the bigger is the amplitude in case of resonance. An important parameter for characterization of the resonance behaviour is the quality factor Q with

0 0/ / 2 Q .

Nonlinear systems An additional mass might be attached to the rotating copper disk to make the mass distribution inhomogeneous. This additional mass causes an extra torque nonlinear in the deflection angle and leads to characteristics of a typical nonlinear system (W potential, superposition of the potentials

due to the restoring spring ( 2 ) and the additional mass ( cosm g R ), see Fig. 2). The

oscillation parameters, especially the frequency, are amplitude dependent in this case. This coupling of amplitude and frequency is a characteristic property of nonlinear systems. Rotating pendulum without actuation Attaching an additional mass modifies Eq. (1) into

T F D GM M M M (13)

with

sin( )GM m g R .

m denotes the additional mass, g the acceleration of gravity and R the radius of the copper disk. The equation of motion is

sin( ) 0 J D mgR . (14)

The rest position is changed and can be easily determined from the condition 0 . As the sine

of the deflection angle appears, Eq. (14) is nonlinear in , i.e. this is a second order, nonlinear,

homogeneous differential equation. Full solutions of equations of this kind are in general not available such that one has to resort to approximations. In contrast to the linear system the nonlinear system can oscillate with several frequencies at the same time. The occurring frequencies depend on the deflection of the pendulum. Rotating pendulum with periodic actuation If the system is driven by an external periodic torque, Eq. (14) is modified to

0sin( ) sin( ) J D mgR M t . (15)

The dependence of the oscillatory properties of this system on the parameters in Eq. (15) can be

studied. In the present experiment the amplitude of the external torque 0M , the angular frequency

of the actuator, the additional mass m and the damping of the pendulum can be varied in

certain ranges. In the specific case considered here the damping is varied systematically by varying

the current of the eddy current brake with all other parameters held constant. Hints for measuring and analysis Task 1 To determine the dependence of the damping on the current IW of the eddy current brake, record several free oscillations as function of time for currents between 0.1 A and 0.5 A, using the CASSY- interface system. Starting at a maximum deflection angle of 90°, count the number of periods till the

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amplitude has decreased to about the half of the starting amplitude. The measurements can also be analysed using the CASSY-system. Prepare a table, that contains the current IW, and in addition the eigenfrequency fd, the damping

constant and the logarithmic decrement . The logarithmic decrement is proportional to the square of the current IW. Determine the parameters of the straight line via linear regression. These parameters are to be required in later simulations. Task 2 Determine the directional torque of the spring D as well as the moment of inertia J, using five different additional masses between 30 g and 35 g (realized by combining several wheel weights).

Measure the static deflection angle st and calculate the static restoring torque sinst stM mgR .

The value of R is given at the work station. Plot the data for Mst versus the angle deflection st (in radians) and calculate the directional torque D from the slope. Calculate the moment of inertia J from D and the measured eigenfrequency. Task 3 For the rotating pendulum without additional mass the maximum actuation angle of the extender wheel, see Fig. 4, is to be set to about 3°. The resonance curve should be measured at one damping value (IW =0.3 or 0.5 A); for this record about 12 amplitude values in steps of 0.015 Hz around the resonance frequency. Use the CASSY- interface system and keep in mind that the amplitudes take up to one minute to stabilise.

Plot the amplitudes against the driving frequency. Determine angular eigenfrequency 0 and

damping constant with a fit of Eq. (9) to the data. Task 4 Attach an additional mass (e.g. 33 g) to the pendulum that causes a noticeable additional torque. At the beginning the current is to be set to a maximum of 0.6 A and the period of the external drive should be set to about 4 s as in the simulation. To determine the bifurcations slowly reduce the current IW until the first bifurcation is seen. Take into account that the pendulum needs some time (up to a few minutes) to stabilize the oscillation. Record the results with the CASSY- interface system in the t and representation and discuss these.

Simulations using a computer algebra system As the search for the transition into chaos turns out to be quite intricate, the experiment might be simulated first on a computer and then later verified in the experiment. The simulation program provided is based on Eq. (15) and requires a series of parameters that are known or were determined in tasks 1-3 :

- eigenfrequency (resonance frequency) without additional mass (task 3) - fitting parameters of the linear regression (task 1) - directional torque of the spring (task 2) - moment of inertia (task 2) - maximum torque (task 2) - maximum actuation angle, about 3° (input in radians)

- period of the driving torque (T0 4 s) The values should be entered into the program header. The parameter to be varied to determine the

transition to chaos is the damping constant or, more specifically, the current WI of the eddy current

brake (variable ID in the program). Starting from a maximum of 0.5 A this should be decreased in steps of about 0.01 A. Take a note of the values, at which you observe bifurcations and discuss the evolution of the trajectories in phase space.

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Fig. 2 The W-potential

Fig. 3 Traces of various oscillation modes as a time trace (left), in a phase-space diagram (center) and as a Fourier transform (right) Fig. 4 Pohl’s Wheel (ELWE company) 1 main switch 2 resonator (copper disc) 3 needle with screw thread for additional masses 4 solenoid for the eddy current brake 5 control for eddy current brake 6 extender wheel and gear shift rod for actuation 7 control for actuation 8 voltage output for excursion, angular velocity and angular acceleration 9 fixation for additional masses

2

eff 0,5 cos .U D mg R

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Addendum to Experiment M 17 “Pohl’s Wheel”

Simulations of oscillations, chaotic oscillations and bifurcations using the computer algebra system ‘Maxima’

Sources

T. J. Hemmer, Bachelor Thesis, 2011, Univ. Leipzig, Bereich Didaktik der Physik, Supervisor: Prof. Dr. W. Oehme,

Physikalisches Praktikum, Hrsg. W. Schenk, F. Kremer, Vieweg+Teubner, 2011

Maxima, computer algebra system

Fig. 5 Pohl’s wheel with imbalance. (a) 1: additional mass, 2: eddy-current brake, 3: motor, 4: spiral spring, 5:

oscillating wheel. (b) Force diagram.

For the derivation of the equation of motion consider Pohl’s wheel as shown in Fig. 5. An additional mass mZM

attached at the rest position of the pendulum leads to an imbalance. The spiral spring exerts the following

torque on the wheel:

SM D, (A1)

where D denotes the directional moment of the spiral spring and φ the deflection of the pendulum from its rest

position. The motion of the oscillating wheel is damped by friction (sliding friction, torque Md,G) and by the

eddy-current brake (torque Md,I). Accordingly, the damping torque is given by

d d,I d,G d,GM M M k

. (A2a)

The friction coefficient might be related by

02 J (A2b)

a) b)

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to the damping coefficient ; its current dependence is given by

2

d,Ik I . (A2c)

J0 denotes the moment of inertia of the wheel, kd,I is the proportional constant between I2 and the damping

caused by the eddy-current brake, I denotes the value of the damping current and kd,G is a proportionality

factor to take the sliding friction into account.

The additional mass (moment of inertia 2

Z ZMJ m R) generates a torque (compare Fig. 1b)

m G m G ZM, sin sinM R F M F R m g R . (A3)

R denotes the distance between the center of mass of the additional mass and the oscillation axis. The drive

supplies a torque

A 0 0sin( )M M t , (A4)

where M0 is the amplitude and ω0 the angular frequency of the driving torque.

One obtains the torque balance

ges S d m AM M M M M , (A5)

ZM ZM

20 d,G 0 0( ) sin sin( )J m R D k m g R M t

. (A6)

This yields the following system of differential equations used in the simulation:

, (A7a)

ZM

ZM

2d,I d,G 0 0

20

sin sin( )D k I k m g R M t

J m R

. (A7b)

The program provided for the experiment calculates the time dependence of the angular deflections from the

rest position after some transient oscillations as well as the phase-space diagram. The initial conditions are

specified in the program, since only the behaviour of the pendulum after the transient period is considered.

The solution of the system of differential equations is calculated using the Runge-Kutta method.

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Tab. 1: Overview of variables and source code symbols in the simulation program

Variable Explanation Symbol in source code

J0 Moment of inertia of the wheel J0

mZM Additional mass M

R Distance between additional mass and rotation

axis

R

D Directional moment D

kd,I Proportionality constant between I2 and the

eddy-current brake damping

kdI

kd,G Proportionality constant of sliding friction kdG

I Value of the damping current I

M0 Amplitude of the driving torque M0

0 Angular frequency of the driving torque omega0

Whereas the values of mZM and R are known, the values of the parameters J0, D, 0 as well as kd,I and kd,G are

experimentally measured. The values of the proportionality constants kd,I and kd,G can be determined by a

measurement of the damping coefficient in dependence on the current I flowing through the electromagnet

of the eddy-current brake. If ( 02 J ) is plotted as a function of I

2, by linear regression kd,I and kd,G are

obtained as the slope and the intercept according to 2

d,Ik I and the approximate equation d,G ( 0)k I

.

If all parameters relevant for the simulation are known, these are input to the program (see examples below).

Example data for the simulation of the wheel’s motion

In the program the following parameters have to be specified in order to enable a simulation close to the

experimental conditions:

T0 (About twice the oscillation period) 3,6 – 4 s 1)

f0 (Eigenfrequency) 0,553 Hz

D (Directional moment of the spiral spring) 0,0254 N m

J0 (Moment of inertia of the wheel) 0,0021 N m s²

R (Distance between additional mass and rotation axis) 0,091 m

mZM (Additional mass) 28 g … 33 g

M0 (Maximum torque due to the chosen additional mass) 0,0018 – 0,0029 N m

Amax (Maximum angular deflection, 3 ° - 4,5 °) 0,05 – 0,08 rad

kd,I (Coefficient from 2d,G d,Ik k I ) 0,0052 N m s A

-2

kd,G (Coefficient from 2d,G d,Ik k I ) 6,25E-5 N m

1) corresponds to an eigenfrequency of about 0,50 Hz – 0,56 Hz

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Examples of simulations with a variation of the damping current

The values of the variables can be modified in the program source code. After starting the calculation with

<StrgR> both the time dependence of the angular deflection as well as the phase-space diagram will be shown

graphically.

1. Example (fundamental oscillation)

J0

(kg m2)

m

(kg)

R

(m)

D

(N m )

kdG

(N m)

kdI

(N m s A-2

)

M0

(N m)

T0

(s)

I

(A )

0.0021 0.030 0.09 0.0254 6.25E-5 0.0052 0.002 3.9 0.50

2. Example (2. bifurcation)

J0 (kg m2)

M

(kg)

R

(m)

D

(N m )

kdG

(N m)

kdI

(N m s A-2

)

M0

(N m)

T0

(s)

I

(A )

0.0021 0.030 0.09 0.0254 6.25E-5 0.0052 0.002 3.9 0.40

3. Example (Chaos)

J0 (kg m2)

m

(kg)

R

(m)

D

(N m )

kdG

(N m)

kdI

(N m s A-2

)

M0

(N m)

T0

(s)

I

(A )

0.0021 0.030 0.09 0.0254 6.25E-5 0.0052 0.002 3.9 0.35

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Perform further simulations in order to recognize the sensitive influence of the damping by variation of the

current I in the range between 0.25 A and 0.6 A.

Additionally, the strong influence of other parameters on the oscillatory behaviour can be studied by an

insignificant modification of their values within the error margins (e.g. directional moment of the spiral

spring D, radius R).