Topic 3 forced oscillator

Topic 1-3 Forced Oscillator
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UEEP1033 Oscillations and Waves
Topic 3:
Forced Oscillator

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Vector: r = a + ib
irsinφ
−irsinφ
r= rejφ
r*= re-jφ
−ib
ib
magnitude or modulus or r:
= complex conjugate of

3

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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(ωL − 1/ωC)
= vector sum of the effective resistances of R, L, and C in the circuit

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magnitude of the impedance:
the vector :
φ = phase difference between the total voltage
across the circuit and the current through it
iωL
Cω
−
1
i






ω
ω=
C
Xe
1
-Lii
=
reactive component of Ze
Electrical impedance: Ze = R + i(ωL − 1/ωC)

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• φ can be positive or negative depending on the relative
value of ωL and 1/ωC
• when ωL > 1/ωC ⇒ φ = positive
but the frequency dependence of the components show that
φ can change both sign and size
• magnitude of Ze is also frequency dependent
when ωL = 1/ωC ⇒ Ze = R
In the vector form of Ohm’s Law: V = I Ze
If V = V0eiωt
and Ze = Zeeiφ
⇒
current amplitude
I lags the V by a phase angle φ
||

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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:

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• A mechanical oscillator of mass m, stiffness s and resistance r
being driven by an alternating force F0cosωt
• Mechanical equation of motion:
Behaviour of a Forced Oscillator
• Voltage equation in electrical case:

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Mechanical forced oscillator with force F0 cosωt
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 with
This term is the
solution to the equation

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(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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×
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 φ exist between x and the force
2. Even if φ = 0, the displacement x would lag the force
F0cos ωt by 900
3. The maximum amplitude of the displacement x is F0/ωZm
The value of x resulting
from F0 cosωt is:

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velocity of the forced oscillation in the steady state:
• The velocity v and the force differ in phase only by φ
• When φ = 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 φ

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Phase angle φ:
Displacement:
Velocity:

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Behaviour of Velocity
in Magnitude and Phase versus Driving Force
• Velocity amplitude:
• Magnitude of velocity vary with frequency ω
• At low frequency, impedance is stiffness controlled
• At high frequency, impedance is mass controlled
• At frequency ω0 where , the impedance has
a minimum Zm = r
• tanφ = 0 at ω0 , the velocity and force being in phase

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Velocity v of Forced Oscillator versus Driving Frequency ω
ω0 = frequency of velocity resonance

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• Velocity
• φ > 0 (i.e. ωm > s/ω)  velocity v lags the force
• ω → ∞ then φ → 90o
: velocity lags the force by 90o
• φ < 0 (i.e. ωm < s/ω)  velocity v ahead of the force
• ω → 0 then φ → -90o
: velocity leads force by 90o
• φ = 0 (i.e. ω0m = s/ω0)  velocity and force are in phase
ω0 = frequency of velocity resonance
ω0
2
= s/m

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Variation of phase angle versus driving frequency, where φ is the phase
angle between the velocity of the forced oscillator and the driving force.
φ = 0 at velocity resonance.
Each curve represents a fixed resistance value.

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Behaviour of Displacement
versus Driving Force Frequency ω
• The phase of the displacement is at all times exactly 90o
behind that of the velocity
• the graph of φ vs ω 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 ω
• At high frequencies the displacement lags the force by
π rad and is exactly out of phase
• the curve showing the phase angle between the
displacement and the force is equivalent to the φ versus ω
curve, displaced by an amount equal to π/2 rad
• at very low frequencies, where φ = −π/2 rad
 velocity leads the force
 displacement and the force are in phase

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Variation of total phase angle between displacement and
driving force versus driving frequency ω
The total phase angle is −φ −π/2 rad

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amplitude of the displacement : x = Fo /ωZm
At low frequencies: Zm = [r2
+ (ωm − s/ω)2
]1/2
→ s/ω
⇒ x ≈ Fo /(ωs/ ω) = Fo /s
At high frequencies: Zm → ωm
⇒ x ≈ Fo /(ω2
m)
At very large frequencies: x → 0

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• The velocity resonance occurs at ωo
2
= s/m , where the
denominator Zm of the velocity amplitude is a minimum
• The displacement resonance will occur when the
denominator ωZm of the displacement amplitude is a
minimum
i.e.

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⇒
⇒
or
⇒ the displacement resonance occurs at a frequency slightly
less than ωo, the frequency of velocity resonance

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Let: displacement
resonance frequency
⇒
Maximum
displacement
mechanical impedance:
The value of mr Zω at rω

26

27

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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:

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Average power vs ω
supplied to an oscillator by the driving force
Q-Value in Terms of the
Resonance Absorption Bandwidth
Bandwidth = ω2 − ω1

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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

Topic 3 forced oscillator

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