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

irsin

irsin

r= rej

r*= re-jib

ib

magnitude or modulus or r:

= complex conjugate of


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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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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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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 F0cost

• Mechanical equation of motion:

Behaviour of a Forced Oscillator

• Voltage equation in electrical case:


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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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(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 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:


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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:


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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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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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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


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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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• 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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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:


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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

Date post:	11-May-2015
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Topic 3 forced oscillator

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