Lecture 1 - Background from 1A

1
Lecture 1 - Background from 1A

• Revision of key concepts with application to
  driven oscillators
• Aims
• Review of complex numbers
• Addition
• Multiplication.
• Revision of oscillator dynamics
• Free oscillator - damping regimes
• Driven oscillator - resonance.
• Concept of impedance.
• Superposed vibrations.

2
Complex representation

• Complex nos. and the Argand diagram
• Use complex number A, where the real part
  represents the physical quantity.
  Amplitude PhaseAmplitude follows
  fromPhase follows from
• Harmonic oscillation

3
Manipulation of Complex Nos. I

• Addition
• The real part of the sum is the sum of the real
  parts.

4
Manipulation of Complex Nos. II

• Multiplication
• WARNINGOne cannot simply multiply the two
  complex numbers.
• Example (i) To calculate (velocity)2 .Take
  velocity v Voeiwt with Vo real.Instantaneous
  valueMean value
• Example (ii). Power, (Force . Velocity).Take f
  Foei(wtf) with Fo real. Instantaneous
  valueMean value

5
The damped oscillator

• Equation of motion
• Rearranging gives
• Two independent solutions of the form xAept.
  Substitution gives the two values of p, (i.e.
  p1, p2), from roots of quadratic
• General solution to 1.2

Restoring force
Dissipation (damping)
Natural resonant frequency
6
Damping régimes

• Heavy damping
• Sum of decaying exponentials.
• Critical damping
• Swiftest return to equilibrium.
• Light damping
• Damped vibration.

7
Driven Oscillator

• Oscillatory applied force (frequency w)
• Force
• Equation of motionUse complex variable, z, to
  describe displacement i.e.
• Steady state solution MUST be an oscillation at
  frequency w. So
• A gives the magnitude and phase of the
  displacement response. Substitute z into 1.3
  to get
• The velocity response is

8
Impedance

• Mechanical impedance
• Note, the velocity response is proportional to
  the driving force, i.e. Force
  constant(complex) x velocity
• Z force applied / velocity response
• In general it is complex and, evidently,
  frequency dependent.
• Electrical impedance
• Zapplied voltage/current response
• Example, series electrical circuitWe can
  write the mechanical impedance in a similar form

Mechanical impedance