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:
http://farside.ph.utexas.edu/teaching/315/Waves/node12.html
http://ultracold.physics.sunysb.edu/Courses/PHY300-11.Fall/lab/Lab1.pdf
http://isites.harvard.edu/fs/docs/icb.topic945082.files/15C%20Lab%201%20Driven%20Oscill
ator/15c_driven_osc_Sep10.doc
1.3.1 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 N0 directly proportional to the deflection
(Hooke's law):
N o   Do
(1.3.1)
Proportionality constant D0 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  Do  0
(1.3.2)
It is the harmonic oscillator equation of motion, whose general solution is:
 (t )   o cos(o t   )
where the amplitude φ0 and phase δ are determined by the initial conditions, and ω0
defined as:
 2 
Do
 o    
J
 To 
is the frequency of vibration of the pendulum.
(1.3.3)
(1.3.4)
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:
N  D
(1.3.5)
The equation of motion therefore takes the following form:
    o2  0
(1.3.6)
D
J
(1.3.7)
where

is called damping parameter. Below certain value of damping parameter Γ < Γkr, where Γkr=
2ω0, movement of the pendulum is oscillating. For the initial conditions φ(0) = φ0 and (0)
= 0 (pendulum is let go from zero angular velocity of the position of φ0) solution to the
equations of motion (1.3.6) is the function:
(1.3.8)
where
(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 ≥ Γkr the pendulum oscillations do not occur.
For the limiting case of critical damping Γ = Γkr oscillation decay rate is a maximum.
Dependence of the oscillation amplitude of the damped harmonic oscillation with the time are
shown in Figure 1.3.1.
Figure 1.3.1: Deflection φ of damped harmonic oscillator as a function of time. At t = 0 initial
deflection of the oscillator is φ0 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 > 2ω0.
Forced Oscillator
Until we make the described oscillator circuit that forces you have on pendulum additional
torsion torque Nω(t) depends on time. The equation of motion then becomes:
(1.3.11)
where f (t) = Nω(t) / J. In our case, we consider the harmonic force as:
(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).
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) = φ0 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 = 0.25ω0 and the
damping parameter Γ = 0.05ω0.
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 1.3.3 is shown the amplitude of φ0 and stationary phase shift δ forced vibration,
depending on the frequency of exciting force. In the area of a frequency ωr amplitude of the
vibration system significantly increases and reaches a maximum. This phenomenon is called
resonance. Resonant frequency ωr can be found using standard procedure of determining the
extreme of the function φ0(Ω). Maximum function or resonance occurs for values:
(1.3.16)
which is smaller than the vibration frequency ω0. 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 = ω0 are shifted in phase by 90 ◦ and for high frequencies are in anti-phase in to the
exciting force.
1.3.2 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).
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 1.3.5. 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 T0=2π/ω0 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 φm = φm(tm), here tm = 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
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 φ0 = φ0 (Ω), for different values of the damping
parameter Γ, will allow the deletion of the resonance curves similar to curves shown in Figure
1.3.3. 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.
1.3.3 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 ω0 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 φm = φm(tm) 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 φ0 = φ0 (Ω). The curves fit the dependence from equation (3.1.14).
Determine the fit parameters f0, ω0, Γ and the size of ωr. 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: 29th 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
Email: [email protected]