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Primary Mechanical Vibrators Mass-Spring System Equation Of Primary Mechanical Vibrators Music 206: Mechanical Vibration Tamara Smyth, [email protected] • With a model of a wind instrument body, a model of Department of Music, the mechanical “primary” resonator is needed for a University of California, San Diego (UCSD) complete instrument. November 15, 2019 • Many sounds are produced by coupling the mechanical vibrations of a source to the resonance of an acoustic tube: – In vocal systems, air pressure from the lungs controls the oscillation of a membrane (vocal fold), creating a variable constriction through which air flows. – Similarly, blowing into the mouthpiece of a clarinet will cause the reed to vibrate, narrowing and widening the airflow aperture to the bore. 1 Music 206: Mechanical Vibration 2 Mass-spring System Equation of Motion • Before modeling this mechanical vibrating system, • The force on an object having mass m and let’s review some acoustics, and mechanical vibration. acceleration a may be determined using Newton’s second law of motion • The simplest oscillator is the mass-spring system: – a horizontal spring fixed at one end with a mass F = ma. connected to the other. • There is an elastic force restoring the mass to its equilibrium position, given by Hooke’s law m F = −Kx, K x where K is a constant describing the stiffness of the Figure 1: An ideal mass-spring system. spring. • The motion of an object can be describe in terms of • The spring force is equal and opposite to the force its due to acceleration, yielding the equation of motion: 2 1. displacement x(t) d x m 2 = −Kx. dx dt 2. velocity v(t)= dt dv d2x 3. acceleration a(t)= = dt dt2 Music 206: Mechanical Vibration 3 Music 206: Mechanical Vibration 4 Solution to Equation of Motion Motion of the Mass Spring • The solution to • We can make the following inferences from the d2x oscillatory motion (displacement) of the mass-spring m = −Kx, dt2 system written as: is a function proportional to its second derivative. x(t)= A cos(ω0t) • This condition is met by the sinusoid: – Energy is conserved, the oscillation never decays. – x = A cos(ω0t + φ) At the peaks: dx ∗ the spring is maximally compressed or stretched; = −ω0A sin(ω0t + φ) dt ∗ the mass is momentarily stopped as it is 2 d x 2 changing direction; 2 = −ω0A cos(ω0t + φ) dt – At zero crossings, • Substituting into the equation of motion yields: ∗ the spring is momentarily relaxed (it is neither d2x compressed nor stressed), and thus holds no PE m = −Kx, dt2 ∗ all energy is in the form of KE. ✭✭ ✭✭ ✭✭✭ K ✭✭✭ 2 ✭✭✭✭ ✭✭✭✭ – Since energy is conserved, the KE at zero crossings −ω0✭A✭cos(✭✭ ω0t + φ) = − ✭A✭cos(✭✭ ω0t + φ), m is exactly the amount needed to stretch the spring showing that the natural frequency of vibration is to displacement −A or compress it to +A. K ω0 = . rm Music 206: Mechanical Vibration 5 Music 206: Mechanical Vibration 6 PE and KE in the mass-spring Conservation of Energy oscillator • The total energy of the ideal mass-spring system is • All vibrating systems consist of this interplay between constant: an energy storing component and an energy carrying E = PE + KE (“massy”) component. 1 2 2 2 = KA sin (ω0t + φ)+cos (ω0t + φ) • The potential energy PE of the ideal mass-spring 2 1 system is equal to the work done1 stretching or = KA2 compressing the spring: 2 Potential Energy (PE) 1 1 2 PE = Kx , 0.8 2 1 0.6 2 2 0.4 = KA cos (ω0t + φ). 2 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 • The kinetic energy KE in the system is given by Time (s) Kinetic Energy (KE) the motion of mass: 1 1 0.8 KE = mv2, 2 0.6 0.4 1 ✟✟✯K ✟✟ 2 2 2 = ✟mω0A sin (ω0t + φ) 0.2 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 2 Time (s) = KA sin (ω0t + φ). 2 Figure 2: The interplay between PE and KE. 1work is the product of the average force and the distance moved in the direction of the force Music 206: Mechanical Vibration 7 Music 206: Mechanical Vibration 8 Other Simple Oscillators—Pendulum Other Simple Oscillators—Helmholtz Resonator • Besides the mass-spring, there are other examples of simple harmonic motion. • A Helmholtz Resonator: The mass of air in the • A pendulum: a mass m attached to a string of neck serves as a piston, and the larger volume as the length l vibrates in simple harmonic motion, provided spring. that x ≪ l. S L m l V m x • The resonant frequency of the Helmholtz resonator is Figure 3: A simple pendulum. given by c a f = , 0 2π VL • The frequency of vibration is r where a is the area of the neck, V is the volume, L is 1 g f = , the length of the neck and c is the speed of sound. 2π l r where g is the acceleration due to gravity. • According to this equation, would changing the mass change the frequency? Music 206: Mechanical Vibration 9 Music 206: Mechanical Vibration 10 Damped Vibration Mass-spring-damper system • In a real system, mechanical energy is lost due to • Damping of an oscillating system corresponds to a friction and other mechanisms causing damping. loss of energy or equivalently, a decrease in the • Unless energy is reintroduced into the system, the amplitude of vibration. amplitude of the vibrations with decrease with time. K m • A change in peak amplitude with time is called an envelope, and if the envelope decreases, the vibrating x system is said to be damped. Figure 5: A mass-spring-damper system. 1 • The damper is a mechanical resistance (or viscosity) 0.8 and introduces a drag force Fr typically proportional 0.6 to velocity, 0.4 0.2 Fr = −Rv 0 dx −0.2 = −R , dt −0.4 −0.6 where R is the mechanical resistance. −0.8 −1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4: A damped sinusoid. Music 206: Mechanical Vibration 11 Music 206: Mechanical Vibration 12 Damped Equation of Motion The Solution to the Damped Vibrator • The equation of motion for the damped system is • The solution to the system equation obtained by adding the drag force into the equation of 2 dx dx 2 motion: +2α + ω0x =0 d2x dx dt2 dt m + R + Kx =0, dt2 dt has the form or alternatively x = A(t)cos(ωdt + φ). d2x dx +2α + ω2x =0, where A(t) is the amplitude envelope dt2 dt 0 −t/τ −αt 2 A(t)= e = e , where α = R/2m and ω0 = K/m. • The damping in a system is often measured by the and the natural frequency ωd quantity τ, which is the time for the amplitude to ω = ω2 − α2, decrease to 1/e: d 0 q 1 2m is lower than that of the ideal mass-spring system τ = = . α R K ω0 = . rm • Peak A and φ are determined by the initial displacement and velocity. Music 206: Mechanical Vibration 13 Music 206: Mechanical Vibration 14 Systems with Several Masses System Equations for Two Spring-Coupled Masses • When there is a single mass, its motion has only one a x x degree of freedom and one natural mode of vibration. 1 2 m m • Consider the system having 2 masses and 3 springs: K K K a x1 x2 Figure 7: A short section of a string. m m K K K • The extensions of the left, middle and right springs are x1,x2 − x1, and −x2, respectively. Figure 6: 2-mass 3-spring system. • When a spring is extended by x, the mass attached to • The system will have two “normal” independent the modes of vibration: – left experiences a positive horizontal restoring 1. one in which masses move in the same direction, force Fr = Kx; with frequency – right experiences an equal and opposite force 1 K Fr = −Kx, f1 = where K is the spring constant. 2πrm 2. one in which masses move in different directions • The equation of motion for the displacement of the with frequency first mass: 1 3K mx¨1 = −Kx1 + K(x2 − x1), f2 = and the second mass, 2πr m (assuming equal masses and springs). mx¨2 = −K(x2 − x1)+ K(−x2). Music 206: Mechanical Vibration 15 Music 206: Mechanical Vibration 16 Additional Modes Forced (Driven) Vibration • When modes are independent, the system can vibrate • Getting they system to vibrate at any single mode of in one mode with minimal excitation of another. vibration requires exciting, or driving the system at • Unless constrained to one-dimension, the masses can the desired mode. also move transversely (at right-angles to the springs). • See Dan Russell’s site: The forced harmonic oscillator • An additional mass adds an additional mode of • When a simple harmonic oscillator is driven by an vibration. external force F (t), the equation of motion becomes • An N-mass system has N modes per degree of mx¨ + Rx˙ + Kx = F (t). freedom. • As N gets very large, it becomes convenient to view • The driving force may have harmonic time the system as a continuous string with a uniform dependence, it may be impulsive, or it can be a mass density and tension. random function of time (noise). Figure 8: Increasing the number of masses (Science of Sound).
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