Chapter 3 - The Forced Oscillator

Question 1

What does the operation of i have on a vector?

Answer 1

When i precedes or operates on a vector the direction of that vector is turned through a positive angle (anticlockwise). Thus i acting as an operator advances the phase by 90°, and -i retards the phase by 90°.

Question 2

Give the vector form of Ohm’s law, and what are the quantities referred to as the reactances?

Answer 2

1. V = IZ_e

= I[R + i(ωL – 1/ωC)]

where the impedance Z_e = [R + i(ωL – 1/ωC)]

1. The quantities ωL and 1/ωC are called the reactances.
   * Remeber how to calculate the magnitude of Z_e (normal way for vectors)*

Question 3

Define the impedance of a mechanical circuit, and give the equation.

Answer 3

The mechanical impedance is defined as the force required to produce unit velocity in the oscillator, i.e.:

Z_m = F/v or F = vZ_m

The mechanical impedance can be written as:

Z_m = r + i(ωm – s/m) = r + iX_m

where:

Z_m = Z_me^iϕ

and

tanϕ = X_m/r

Question 4

Give the mechanical equation of motion of an oscillator of mass m, stiffness s and resistance r being driven by an alternating force F₀cos(ωt) where F₀ is the amplitude of the force.

Answer 4

mẋ ̇ + rẋ + sx = F₀cos(ωt)

and for the electrical case:

Lq̇ ̇ + Rq̇ +q/C = V₀cos(ωt)

Question 5

Describe the complete solution for x in the equation if motion for a damped driven simple harmonic oscillator.

Answer 5

There are terms:

1. The Transient term

Dies away with time, is the solution to the equation of motion, without the force.

x = Ce^-rt/2me ^{i(a/m-r^2/4m^2)1/2t}

1. Steady state term

The behaviour of the oscillator after the transient term has died away.

x = -iF₀e^i(ωt-ϕ)

     ωZ<sub>m</sub>

where Z_m is the mechanical inductance

Question 6

Explain what insight is gained by the steady state term:

x = -iF₀e^i(ωt-ϕ)

  ωZ<sub>m</sub>

Where:

Z_m = [r² + (ωm – s/ω)²]^1/2

Answer 6

1. That the phase difference ϕ exists between x and the force because of the reactive part (ωm – s/ω) of the mechanical impedance.
2. That an extra difference is introduced by the factor -1 and even if ϕ were zero the displacement x would lag the force F₀cos(ωt) by 90°.
3. That the meximum amplitude of the displacement x is F₀/ωZ_m. We see that this is dimensionally correct because the velocity x/t has dimensions F₀/Z_m.

Question 7

Give the value of x after taking the real part of the solution of the equation of motion from F₀cos(ωt).

Also give the velocity, and the acceleration

Answer 7

x = F₀/ωZ_m­ sin(ωt – ϕ)

x^’ = F₀/Z_m­ cos(ωt – ϕ)

x’’ = -F₀ω/Z_m­ sin(ωt – ϕ)

where

tanϕ = (ωm – s/ω)/r

Question 8

Describe what the frequency of velocity resonance is and how to derive it.

Answer 8

The frequency of velocity resonance is the frequency where the velocity (F₀/Z_m) has its maximum value of v = F₀/r

1. From the velocity amplitude (F₀/Z_m), it is clear that v will have a maximum value when Z_m is a minimum.
2. Z_m = [r² + (ωm – s/ω)²]^1/2
3. Z_m = r when (ωm – s/ω) = 0
4. Thus the frequency of velocity resonance is where (ωm = s/ω)

Question 9

Describe what the frequency of displacement resonance is and derive the equation for it.

Answer 9

The Displacement resonance is the frequency at which the displacement (F₀/ωZ_m­) would have its maximum value. (x_max = F₀/ω’r)

1. Start with the displacement equation: x = F₀/ωZ_m­ sin(ωt – ϕ)
2. Frequency of displacement resonance will be where: ωZ_m is a minimum.
3. Using critical points in differentiation it can be seen:

ω² = s/m – r²/2m² = ω₀² – r²/2m²

• Think about how x_max is derived*
• Note the relationship between the frequency of velocity resonance and the frequency of displacment resonance.*

Question 10

What is the relationship between the frequency of displacement resonance and velocity resonance?

Answer 10

ω² = s/m – r²/2m² = ω₀² – r²/2m²

Where:

ω is the frequency of displacement resonance, and

ω₀ is the frequency of velocity resonance.

Question 11

Derive the instantaneous power by the driving force on an oscillator.

Answer 11

1. p = Fx^’
2. p = F₀cos(ωt) F₀/Z_m cos(ωt – ϕ)
3. p = F₀²/Z_m cos(ωt)cos(ωt – ϕ)

Question 12

What is the average power of a damped driven oscillator?

Answer 12

1. P_av = Total work per oscillation

   Oscillation period

2. P_av = F₀²/2Z_m cosϕ

Question 13

Define the Q-value in terms of Resonance absorption bandwidth.

Answer 13

Q = ω₀(ω₂ - ω₁)

where

(ω₂ - ω₁) are the frequencies at which the power supplied is:

p_av = ½ p_av(maximum)

Question 14

How is the bandwidth defined?

Answer 14

The frequency difference (ω₂-ω₁)

Question 15

Define the Q-value as an amplification factor.

Answer 15

Q = (Maximum displacement at displacement resonance)

Displacement as ω→0

or

Q = A_max/A₀ = A_max/(F₀/s)