Vibrations of string. Henna Tahvanainen. November 8, ELEC-E5610 Acoustics and the Physics of Sound, Lecture 4

Transcription

1 Vibrations of string EEC-E5610 Acoustics and the Physics of Sound, ecture 4 Henna Tahvanainen Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering November 8, 2015

2 1 Ideal strings Vibrations of string 2/1

3 1 The ideal string An ideal string is a fictitious entity with certain special properties. Namely, it is homogenic perfectly flexible lossless Furthermore, it has three quantities that govern its behaviour: linear mass density µ [ kg m ] tension T 0 [N] length [m] Vibrations of string 3/1

4 In the following, we will consider the movement of string in one plane only. The longitudinal coordinate is x and the transversal displacement is denoted with y. Vibrations of string 4/1

5 1 The Wave Equation If the string displacement is moderate, i.e. y x 1, the movement of the ideal string can be characterized with the 1D wave equation: µÿ = T 0 y, (1) where ÿ is acceleration and y is curvature. The two opposing forces here are inertia (mass acceleration) spring force (tension curvature) If we define a variable c 2 = T 0 µ, Eq. (1) becomes ÿ = c 2 y (2) Vibrations of string 5/1

6 1 Two Solutions Two main solutions for Eq. (1) are d Alembert s solution where the string vibration is seen consisting of two opposite waves y(x, t) = g 1 (ct x) + g 2 (ct + x) (3) (animation: Bernoulli s solution where the vibration is seen as a superposition of standing wave modes ( nπx ) [ ( ) ( )] nπct nπct y(x, t) = sin A n sin + B n cos n=1 (R&F:(2.13)) Vibrations of string 6/1

7 1 Bernoulli s Solution et s take a closer look on Eq. (R&F:(2.13)): y(x, t) = ( nπx ) [ ( nπct sin A n sin n=1 ) + B n cos ( )] nπct String vibration is a sum over mode number n of... spatial sinusoidal terms (modes), multiplied by... temporal sinusoidal terms (vibration in time) A n and B n together define the amplitude of each frequency component Vibrations of string 7/1

8 1 Bernoulli s Solution II... ok, so in addition to the physical string parameters and c, the spectrum of the vibration is defined by A n and B n. How are A n and B n defined? - By excitation, i. e. the initial conditions for the velocity and displacement: A n = 2 nπc B n = ( nπx ẏ(x, 0) sin ( nπx y(x, 0) sin ) dx ) dx For plucked string ẏ(x, 0) = 0 A n = 0, for a struck string y(x, 0) = 0 B n = 0. (R&F:(2.17)) (R&F:(2.18)) Vibrations of string 8/1

9 1 Bernoulli s Solution III Recall that the spatial term in Eq. (R&F:(2.13)): y n = sin ( ) nπx corresponds to different modes... Vibrations of string 9/1

10 1 Bernoulli s Solution IV...furthermore, the temporal term y t = [ A n sin ( ) nπct + Bn cos ( )] nπct vibrates at frequencies f n =? (4) Vibration components evenly spaced in frequency! harmonic spectrum! Vibrations of string 10/1

11 1 Free Vibration Free vibration means that the string is excited with some initial conditions and then left to vibrate on its own ( et s consider striking a string at some location, and see how Bernoulli s solution is affected. add an excitation force to the wave equation: ÿ c 2 y = f (x, t) consider the force as an impulse at some location x 0 : f (x, t) = δ(x x 0 )δ(t), where δ is Dirac s delta function However, instead of force, we would need an initial velocity or displacement to calculate A n and/or B n. How to proceed? Apply Newton s 2nd law (relates force to the time derivative of velocity) Vibrations of string 11/1

12 1 Free Vibration II So, our force impulse becomes initial acceleration ÿ(x, 0) = δ(x x 0)δ(t) µ initial velocity becomes ẏ(x, 0) = ÿ(x, 0)dt = δ(x x 0) µ...while the initial displacement is y(x, 0) =? Insert the initial conditions into Eqs. R&F:(2.17) and R&F:(2.18): A n = 2 ( nπx ) ẏ(x, 0) sin dx nπc 0 B n = 2 ( nπx ) y(x, 0) sin dx 0 Vibrations of string 12/1

13 1 Free Vibration III Next, evaluate the integral (e.g. with δ(x x 0 ) ( nπx sin µ 0 ) dx =? (when x 0 [0, ]). Thus, the string vibration is ( nπx ) [ ( ) ( )] nπct nπct y(x, t) = sin A n sin + B n cos where n=1 A n = 2 ( nπcµ sin nπx0 ), B n = 0 Vibrations of string 13/1

14 1 Free Vibration IV Remember Eq. (4): f n = nc 2 c = 2fn n insert into the vibration equation: y(x, t) = consists of ( nπx sin n=1 spatial terms (eigenmodes) ) [ 2 ( nπµc sin nπx0 ) sin temporal terms (eigenfrequencies) frequency-dependent scaling ( )] nπct (5) Vibrations of string 14/1

15 1 String excited at x 0 After all the math, we obtained a nice equation for the vibration of a string, struck at x 0 : y(x, t) = 1 ( nπx ) ( nπx0 ) 1 sin sin sin (π2f n t) (6) πµ fn n=1 What if x 0 = /2? The middle sine term becomes sin ( ) nπ 2 = 0, when n = 2, 4, 6, 8,... even harmonics absent! Generally: if the string is excited at 1 m of it length, every mth harmonic will be missing if excitation location is moved towards string s end, fewer and fewer harmonics missing sound gets brighter! Vibrations of string 15/1

16 1 String excited at x 0 II An alternative (graphical) way to express the same idea: the closer the excitation is to the antinode of an eigenmode, the better it excites the corresponding eigenfrequency. Vibrations of string 16/1

17 1 String excited at x 0 III Actually used in the piano! The piano hammer typically hits the string at 1 7th of its length the 7th harmonic damped... Table: Eigenfrequencies and closest notes for A 2 note. n f n note f note error (Hz) A A E A C# E G A luckily, since it s so much out of tune! Vibrations of string 17/1

18 1 Forced String Vibration When the excitation is continuous, the string vibration is considered forced. basically, the same mechanisms apply as what discussed above also, the frequency of the excitation force has an effect the excitation must match both the spatial and temporal form of a mode, if that mode is to be excited a continuous excitation at an eigenfrequency exponentially increases the vibration amplitude amplitude would become infinite, if it weren t for the losses Vibrations of string 18/1

19 2 osses, Stiffness, Nonlinearities, Other Polarizations Vibrations of string 19/1

20 2 oss Mechanisms The most important loss mechanisms in a vibrating string are: damping caused by air viscosity internal losses actually, a set of different thermo- and viscoelastic loss mechanisms transfer of mechanical energy through supports depends on the connection impedance between the string and the body a reversible process (in theory) The combined effect of all losses may be expressed as a single force term R(f ) Vibrations of string 20/1

21 2 Effect of osses Strictly speaking, the loss term R(f ) depends not only on the frequency, but also on physical properties of the string string geometry properties of the air etc. difficult to obtain theoretically! In practice, it is measured from the attenuation times at different frequencies. Results in an additional term to the wave equation: ÿ c 2 y + 2R(f )ẏ = f (x, t) (7) Vibrations of string 21/1

22 2 Effect of Stiffness Real strings are never perfectly flexible, but have a nonzero stiffness. This internal stiffness generates another restoring force (in addition to the external tension T 0 ). The amount of the stiffness depends on cross-section area of the string A linear mass density µ Young s modulus E (depends on the material) the radius of gyration κ (depends on string geometry) Stiffness creates yet another force term to the wave equation: ÿ c 2 y + 2R(f )ẏ + EAκ2 µ y = f (x, t) (8) Vibrations of string 22/1

23 2 Effect of Stiffness II Stiffness causes the wave propagation velocity to become frequency-dependent upper harmonics shift higher in frequency the resulting tone no longer (strictly) harmonic! Inharmonicity caused by stiffness has a significant impact on how pianos are tuned. Vibrations of string 23/1

24 2 Other Polarizations Obviously, the string does not only vibrate in a single transversal polarization, but also in the other transversal polarization termination impedance is different in different polarizations different decay times two-stage decay! longitudinal polarization basically the same as transversal vibration, but the propagation velocity is different (see R&F: Sec. 2.14). Typically c c might connect to the instrument body (and become audible) rotational polarization usually negligible, except perhaps with bowed strings The total string vibration is a superposition of all vibrations in different polarizations. Vibrations of string 24/1

25 2 Effect of Nonlinearities Some nonlinear effects in vibrating strings: tension modulation with large amplitudes string tension varies during vibration causes initial pitch glide stick-slip coupling between bow and string hammer nonlinearity (in the piano) hammer seems harder when played with greater intensity timbre changes as a function of key velocity other nonlinear effects in musical instruments discussed in Nonlinear physics of musical instruments by Fletcher. Vibrations of string 25/1

26 3 Membranes Vibrations of string 26/1

27 3 General Membranes can be seen 2-D extensions of ideal strings Real membranes are effected also by losses and nonlinearities We can look at the lossless rectangular and circular membranes briefly Vibrations of string 27/1

28 3 Wave equation for rectangular membrane Similarly to the string, we can write the wave equation for a perfectly homogenous and flexible sheet 2 z t 2 and c 2 = T σ where T = tension applied via edges σ = mass per unit area and the solution... = c2 ( 2 x + 2 )z(x, y) 2 (9) y 2 Vibrations of string 28/1

29 3 Solution to wave equation The solution can be given as z = sin m=1 n=1 ( ) ( ) mπ nπ x sin y [M sin (ω mn t) + N cos (ω mn t) x y which consists of sum over two types of modes spatial modes in x-direction spatial modes in y-direction temporal vibration Both spatial directions have their own modes. (10) Vibrations of string 29/1

30 3 Eigenfrequencies and -modes The eigenfrequencies are given as f mn = 1 2 T σ ( m x ) 2 ( ) 2 n + (11) y where x and y are the dimensions of the membrane and m, n = 1, 2, 3... Vibrations of string 30/1

31 3 Circular membranes For the circular membrane, the displacement z(r, φ) is given in polar coordinates For the fixed boundary condition z(r, φ) = 0, the eigenfrequencies are given by f mn = c 2πR β mn (12) where β mn is the nth zero of the mth Bessel function (of the first kind) Vibrations of string 31/1

32 3 Circular membranes f mn = c 2πR β mn (13) Vibrations of string 32/1

33 3 Circular membranes Vibrations of string 33/1

34 3 Why are drums typically round not rectangular? et s assume that we have a rectangular and a circular membrane of the same material: surface mass σ = 10kgm 2 surface tension T = 100Nm 2 We want to tune the lowest eigenfrequency of the drums to f = 110 Hz. What would be the area of each drum? Calculate also the first five partials of each membrane. Can the membranes be tuned harmonically? Vibrations of string 34/1

35 3 Real membranes Real membranes are effected by finite stiffness and air loading Air loading of the drum typically lowers the eigenfequencies, dominating effect with thin membranes Confined air will raise axisymmetric modes such as (0,1), but decrease (1,1) and (2,1) -> we can tune the lowest eigenfrequencies to near-harmonic relationships (e.g., timpani) Stiffness raises the eigenfrequencies Vibrations of string 35/1