VIEWS: 7 PAGES: 40 POSTED ON: 12/21/2011
Fundamentals of Acoustics and Fourier Transforms Reid McCargar Graduate Research Assistant, NW Electromagnetics and Acoustics Research Laboratory (NEAR Lab) 11/18/2010, Reid McCargar 1 Outline Lesson 1 – a heavy dose of math • A review of sinusoidal functions – non propagating functions • Relationship between oscillation and rotation • Radians • Calculus and oscillators • Complex exponentials, the uber-brief version • Real oscillating systems • Resonance and frequency response Lesson 2 – a lighter dose of math, more conceptual focus • Propagating waves • Time and frequency domain representations, frequency response • Adding waves, adding phasors • Conceptual Fourier synthesis and decomposition • Impulse response and transfer functions • Basics of performing transfer function measurements 11/18/2010, Reid McCargar 2 There is a point to this Do you love sound enough that you would want to do it for a living? Without needing a day job? 11/18/2010, Reid McCargar 3 Our stated purpose, before we dive in To design and implement sound systems in which: 1. All frequencies within the audio band have nearly equal magnitude. 2. All frequencies arrive synchronously. 3. Goals 1 & 2 are achieved as well as possible for as many listeners as possible. 11/18/2010, Reid McCargar 4 A review (hopefully) of sinusoidal functions 11/18/2010, Reid McCargar 5 Visualizing oscillation with rotation Here t = θ 11/18/2010, Reid McCargar 6 Radians 11/18/2010, Reid McCargar 7 Oscillation/rotation math and vocab. Period: T – The time it takes to complete a full cycle [sec] Frequency: f = The number cycles in a given length of time [1/sec] Phase: φ – More specifically the initial phase, specifies an offset in angle (θ ≠ 0 at t = 0) Phase: θ – Everything inside the argument of the cosine DC offset: A0 – The average y-value. Usually taken to be zero, even when it’s not. Non-propagating because function depends only on time. There is no spatial dependence. Example – voltage output of a microphone in a fixed location. 11/18/2010, Reid McCargar 8 Important things—If you remember nothing else, remember this 11/18/2010, Reid McCargar 9 Angular frequency…try to remember this, too It’ll make life easier later. 11/18/2010, Reid McCargar 10 Take 5 11/18/2010, Reid McCargar 11 Position, Velocity and Acceleration…and force We want to fully describe the motion, x (t) or y (t), of an object. If an object has constant velocity, the position is just v X t, and the velocity can be found from the change in position over a period of time. If not, we rely on some calculus to describe the instantaneous velocity and position. Although calculus scares most people (probably for good reason), it is very simple for sinusoids. 11/18/2010, Reid McCargar 12 Calculus for the very mathematically unenthused Derivative : The rate of change of a function – pick a point on the function, the slope of the function is the derivative at that point. 2 f (t) 1.5 slope 1 0.5 0 -0.5 -1 -1.5 -2 T 2T 3T 4T t 11/18/2010, Reid McCargar 13 Calculus for the very mathematically unenthused Velocity is the rate of change of position– it’s the derivative of position. Acceleration is the rate of change of velocity – it’s the 2nd derivative of position. One last thing: Force = mass X acceleration We can now fully describe the motion of many vibratory systems. All we need to know is the forces and the initial conditions. 11/18/2010, Reid McCargar 14 Derivatives of sinusoids You now know 30% of the calculus you need, and you can even forget this part because there’s an easier way. Try to hold on to it for the next 10 minutes. 11/18/2010, Reid McCargar 15 Solution for a simple oscillator Block of mass m, attached to a rigidly-anchored spring. The spring exerts a restoring force proportional to the amount it is stretched, y times a spring stiffness constant, k: Although this seems really simplistic, loudspeakers and microphones have a lot in common with this vibratory system. 11/18/2010, Reid McCargar 16 Solution for a simple oscillator First, we use F = ma How do we express this in terms of a derivative of y? 11/18/2010, Reid McCargar 17 Solution for a simple oscillator First, we use F = ma So the equation describing the motion of the object is? 11/18/2010, Reid McCargar 18 Solution for a simple oscillator First, we use F = ma So the equation describing the motion of the object is For now, don’t sweat the m and k. They’re just numbers that don’t change the form of the answer. The important thing is that we have a situation where the 2nd derivative of a function equals a negative constant times the function itself. Can you think of a type of function that satisfies this equation? 11/18/2010, Reid McCargar 19 Solution for a simple oscillator Substituted into our equation of motion We get We call this our natural frequency 11/18/2010, Reid McCargar 20 Solution for a simple oscillator 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 T 2T 3T 4T time t frequency So our spring system’s response Which can be represented in the is a sinusoid of a single frequency frequency domain as a single spike 11/18/2010, Reid McCargar 21 Break Time 11/18/2010, Reid McCargar 22 Real oscillators, imaginary numbers Imaginary numbers—it’s like the Easter bunny, but not Although it may seem ridiculous to use numbers that don’t exist, the mathematics of sound become absurdly complicated without them. The concept of imaginary numbers was opposed for many years in the scientific community because obviously, they don’t make sense. Does π make sense? Perhaps in some twisted way, but you really can’t run from it. Numbers like these are fundamental to nature. 11/18/2010, Reid McCargar 23 Some basics of complex numbers 11/18/2010, Reid McCargar 24 The complex plane A complex number describes a right triangle in the complex plane 11/18/2010, Reid McCargar 25 Life, death, and another nonsense number Most things in nature grow and die in this fashion. To math geeks, it’s about as exciting as π, or Optimus prime. e gets to be more fun for everybody else when you see how much easier it makes calculus and vibrational system analysis. 11/18/2010, Reid McCargar 26 Complex exponentials Proving this requires some acrobatics that aren’t relevant. Just accept it or talk to me later. Ok, but how does this make our lives any easier? 11/18/2010, Reid McCargar 27 Derivatives of complex exponentials So far it looks pretty simple. Anyone remember what the derivative was for cosine? sine? For the 1st derivative, we just multiplied by . For the second derivative, we just did it twice. Is calculus with complex exponentials as simple as multiplying by every time you see a derivative? Actually yes. You’re now 2/3rds done with calculus, and the remaining bit is even easier. 11/18/2010, Reid McCargar 28 Antiderivatives with complex exponentials Quite often we need to do derivatives in reverse. With complex exponentials, we multiply by to differentiate. To undo differentiation (an operation called integration) we … divide by . Alright, that’s all the calculus, and it’s as easy as multiplying or dividing by . The integral over some range a to b is just the area under the curve. 11/18/2010, Reid McCargar 29 Take 5 11/18/2010, Reid McCargar 30 Solution for an unforced oscillator Let’s take a look at a more realistic oscillator—one in which there is also a damping force proportional to the velocity: Let’s take a look at a more realistic oscillator—one in which there is also a resistive damping force proportional to the velocity: 11/18/2010, Reid McCargar 31 Solving the damped oscillator So the 2 forces acting on our object are the spring restoring force, and the resistive damping force. F = ma gives I won’t solve this now so we can get to resonance, but I’ll give you a hint so you can solve it at home. If we assume the solution is a complex exponential, rearrange this and divide everything by m, we get something in the form of a quadratic equation: You can cancel y, then solve for the angular frequency. 11/18/2010, Reid McCargar 32 Damped oscillator solution If you go through the math, you’ll find that Remember what the natural frequency of our undamped oscillator was? So our damped oscillator oscillates at a lower frequency than the undamped system. Using this, we find the complete solution is: What’s this? I know you remember this. 11/18/2010, Reid McCargar 33 Damped oscillator solution Let’s define a decay coefficient: 11/18/2010, Reid McCargar 34 Forced oscillations There are quite a few systems that behave in (somewhat) principle like unforced, damped oscillators: • cymbals & vibrating drumheads • plucked or struck strings • systems in which there is no energy input into the system after being set in motion And many that don’t: • loudspeakers driven with an amplified electric signal • microphones, which are driven by pressure fluctuations • bowed strings • systems with forces that persist after initial excitation 11/18/2010, Reid McCargar 35 Solution for a forced oscillator Our equation of motion is now: If we suppose a forcing function: After a bunch of math, we find that And 11/18/2010, Reid McCargar 36 Mechanical impedance It makes sense to define a quantity called impedance: For this system, the impedance is: The smallest this can ever be is , which happens when This condition is known as resonance. Designing and implementing good systems is largely a game of working with, and working around resonance. 11/18/2010, Reid McCargar 37 Resonance of a simple oscillator A 1 kHz resonance frequency was chosen to make these plots. There is a mathematical definition of Q, which I’ll spare you. Higher Q results from low damping. 11/18/2010, Reid McCargar 38 Summary What just happened? • We reviewed sinusoidal functions & rotation • We learned about radians, phase, frequency, and period • We learned calculus for sinusoids, which we forgot • We learned about an idealized system with a sinusoidal solution • We learned about complex exponentials in order to forget about sinusoids • We learned the calculus of complex exponentials • We dealt with real oscillating systems and resonance In short, you’ve had a lot of difficult concepts thrown at you. Don’t expect to understand or remember it all right away. Think about it the next time you wait for the bus or would otherwise zone out. 11/18/2010, Reid McCargar 39 Questions?? Next time we’ll be doing more of the same, but dealing with functions of time and space. The good news is that we’ve covered the worst of the math. 11/18/2010, Reid McCargar 40