1 / 8

Lecture 1 - Background from 1A

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.

rfears
Download Presentation

Lecture 1 - Background from 1A

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 from:Phase 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 • WARNING:One cannot simply multiply the two complex numbers. • Example (i): To calculate (velocity)2 .Take velocity v = Voeiwt with Vo real.Instantaneous value:Mean value: • Example (ii). Power, (Force . Velocity).Take f = Foei(wt+f) with Fo real. Instantaneous value:Mean value:

  5. The damped oscillator • Equation of motion • Rearranging gives • Two independent solutions of the form x=Aept. 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 motion:Use 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 • Z=applied voltage/current response • Example, series electrical circuit:We can write the mechanical impedance in a similar form: Mechanical impedance

More Related