Topic 1-3 Forced Oscillator
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UEEP1033 Oscillations and Waves
Topic 3:Forced Oscillator
Topic 1-3 Forced Oscillator
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Vector: r = a + ib
irsin
irsin
r= rej
r*= re-jib
ib
magnitude or modulus or r:
= complex conjugate of
Topic 1-3 Forced Oscillator
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Topic 1-3 Forced Oscillator
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Vector form of Ohm’s Law
inductance L and capacitance C, will introduce a phase difference between voltage and current
Ohm’s Law takes the vector form:
Ze = R + i(L 1/C)
= vector sum of the effective resistances of R, L, and C in the circuit
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UEEP1033 Oscillations and Waves
magnitude of the impedance:
the vector :
= phase difference between the total voltage across the circuit and the current through it
iL
C
1i
CX e
1-Lii
=
reactive component of Ze
Electrical impedance: Ze = R + i(L 1/C)
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• can be positive or negative depending on the relative value of L and 1/C
• when L > 1/C = positive but the frequency dependence of the components show that
can change both sign and size
• magnitude of Ze is also frequency dependent when L = 1/C Ze = R
In the vector form of Ohm’s Law: V = I Ze
If V = V0eit and Ze = Zeei
current amplitude
I lags the V by a phase angle
||
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UEEP1033 Oscillations and Waves
Impedance of a Mechanical Circuit
• Mechanical circuit has mass, stiffness and resistance
• Mechanical impedance = the force required to produce unit velocity in the oscillator i.e. Zm = F/v
• Mechanical impedance :
• Magnitude of Zm :
Topic 1-3 Forced Oscillator
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UEEP1033 Oscillations and Waves
• A mechanical oscillator of mass m, stiffness s and resistance r being driven by an alternating force F0cost
• Mechanical equation of motion:
Behaviour of a Forced Oscillator
• Voltage equation in electrical case:
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UEEP1033 Oscillations and Waves
Mechanical forced oscillator with force F0 cost applied to damped mechanical circuit
The complete solution for x in the equation of motion consists of two terms:
(1) A ‘transient’ term which dies away with time
x decays withThis term is the
solution to the equation
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UEEP1033 Oscillations and Waves
(2) The ‘steady state’ term – describes the behaviour of the oscillator after the transient term has died away, i.e describes the ultimate
behaviour of the oscillator
Force equation in vector form:
Solution:
Velocity:
Acceleration:
Equation of motion:
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UEEP1033 Oscillations and Waves
This vector form of the steady state behaviour of x gives three pieces of information and completely defines the magnitude of the displacement x and its phase with respect to the driving force after the transient term dies away
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1. Phase difference exist between x and the force
2. Even if = 0, the displacement x would lag the force F0 cos t by 900
3. The maximum amplitude of the displacement x is F0/Zm
The value of x resulting from F0 cost is:
Topic 1-3 Forced Oscillator
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velocity of the forced oscillation in the steady state:
• The velocity v and the force differ in phase only by • When = 0 the velocity and force are in phase• Amplitude of the velocity = F0/Zm
real part of the vector expression for the velocity:
• velocity is 90o ahead of the displacement in phase
• velocity differs from the force only by a phase angle
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Phase angle :
Displacement:
Velocity:
Topic 1-3 Forced Oscillator
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Behaviour of Velocity in Magnitude and Phase versus Driving Force
• Velocity amplitude:
• Magnitude of velocity vary with frequency
• At low frequency, impedance is stiffness controlled• At high frequency, impedance is mass controlled
• At frequency 0 where , the impedance has a minimum Zm = r
• tan = 0 at 0 , the velocity and force being in phase
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Velocity v of Forced Oscillator versus Driving Frequency
0 = frequency of velocity resonance
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• Velocity
• > 0 (i.e. m > s/) velocity v lags the force
• then 90o : velocity lags the force by 90o
• < 0 (i.e. m < s/) velocity v ahead of the force
• 0 then -90o : velocity leads force by 90o
• = 0 (i.e. 0m = s/0) velocity and force are in phase
0 = frequency of velocity resonance
02
= s/m
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UEEP1033 Oscillations and Waves
Variation of phase angle versus driving frequency, where is the phase angle between the velocity of the forced oscillator and the driving force. = 0 at velocity resonance. Each curve represents a fixed resistance value.
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UEEP1033 Oscillations and Waves
Behaviour of Displacement versus Driving Force Frequency
• The phase of the displacement is at all times exactly 90o behind that of the velocity
• the graph of vs remains the same, the total phase difference between the displacement and the force involves the extra 90o retardation introduced by the j operator
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Behaviour of Displacement versus Driving Force Frequency
• At high frequencies the displacement lags the force by rad and is exactly out of phase
• the curve showing the phase angle between the displacement and the force is equivalent to the versus curve, displaced by an amount equal to /2 rad
• at very low frequencies, where = /2 rad velocity leads the force displacement and the force are in phase
Topic 1-3 Forced Oscillator
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Variation of total phase angle between displacement and driving force versus driving frequency
The total phase angle is /2 rad
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amplitude of the displacement : x = Fo /Zm
At low frequencies: Zm = [r2 + (m s/)2]1/2 s/
x Fo /(s/ ) = Fo /s
At high frequencies: Zm m
x Fo /(2m)
At very large frequencies: x 0
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UEEP1033 Oscillations and Waves
• The velocity resonance occurs at o2 = s/m , where the
denominator Zm of the velocity amplitude is a minimum
• The displacement resonance will occur when the denominator Zm of the displacement amplitude is a minimum
i.e.
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UEEP1033 Oscillations and Waves
or
the displacement resonance occurs at a frequency slightly less than o, the frequency of velocity resonance
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Let: displacement resonance frequency
Maximum
displacement
mechanical impedance:
The value of mr Z at r
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Power Supplied to Oscillator by the Driving Force
• Driving force must replace the energy lost
• In steady state:
Instantaneous Power P = Instantaneous Driving Force Instantaneous Velocity
Average power supplied by the driving force = Energy dissipated by the frictional force
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Since:
Topic 1-3 Forced Oscillator
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Average power vs supplied to an oscillator by the driving force
Q-Value in Terms of the Resonance Absorption Bandwidth
Bandwidth = 2 1
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The Q-Value as an Amplification Factor
the value of the displacement at resonance is given by:
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the displacement at low frequencies is amplified by a factor of Q at displacement resonance
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Curves of Figure 3.7 (in slide-26) now given in terms of the quality factor Q of the system, where Q is amplification at resonance of low frequency response x = F0/s