1 Waves1 Lecture 1 - Background from 1A Revision of key concepts with application to driven oscillators: D Aims: ëReview of complex numbers: > Addition; > Multiplication. ëRevision of oscillator dynamics: > Free oscillator - damping regimes; > Driven oscillator - resonance. ëConcept of impedance. ëSuperposed vibrations.

2 Waves1 Complex representation D Complex nos. and the Argand diagram: ëUse complex number A, where the real part represents the physical quantity. Amplitude Phase Amplitude follows from: Phase follows from: ëHarmonic oscillation:

3 Waves1 Manipulation of Complex Nos. I D Addition ë ëThe real part of the sum is the sum of the real parts.

4 Waves1 Manipulation of Complex Nos. II D Multiplication ëWARNING: One cannot simply multiply the two complex numbers.  Example (i): To calculate (velocity) 2. Take velocity v = V o e i  t with V o real. Instantaneous value: Mean value:  Example (ii). Power, (Force. Velocity). Take f = F o e i(  t+  ) with F o real. Instantaneous value: Mean value:

5 Waves1 The damped oscillator D Equation of motion ë Rearranging gives  Two independent solutions of the form x=Ae pt. Substitution gives the two values of p, (i.e. p 1, p 2 ), from roots of quadratic: ëGeneral solution to [1.2] Restoring force Dissipation (damping) Natural resonant frequency

6 Waves1 Damping régimes D Heavy damping ë Sum of decaying exponentials. D Critical damping ë Swiftest return to equilibrium. D Light damping ë Damped vibration.

7 Waves1 Driven Oscillator  Oscillatory applied force (frequency  ): ëForce:  Equation of motion: Use complex variable, z, to describe displacement: i.e.  Steady state solution MUST be an oscillation at frequency . So  A gives the magnitude and phase of the “displacement response”. Substitute z into [1.3] to get ëThe “velocity response” is

8 Waves1 Impedance D Mechanical impedance ëNote, the velocity response is proportional to the driving force, i.e. Force =constant(complex) x velocity ë Z = force applied / velocity response ëIn general it is complex and, evidently, frequency dependent. D Electrical impedance ëZ=applied voltage/current response ëExample, series electrical circuit: We can write the mechanical impedance in a similar form: Mechanical impedance