Optimal Design and Physical Modelling of Mallet Percussion Instruments
Luís Henrique1, José Antunes2
1: Instituto Politécnico do Porto, Escola Superior de Música e Artes do Espectáculo IPP/ESMAE, Rua da Alegria, 4000 Porto, Portugal
2: Instituto Tecnológico e Nuclear, Applied Dynamics Laboratory ITN/ADL, Estrada Nacional 10, 2685 Sacavem, Portugal
Summary At the present time, instruments are mostly designed by trial and error procedures, which are inefficient and costly. In the first part of this paper we present an approach, based on finite-element eigen-analysis coupled with optimization procedures, which enable the computation of optimal instrument shapes in order to obtain a target set of modal frequencies. We briefly discuss various optimization approaches, deterministic and stochastic, in relation with computational efficiency and effectiveness. A satisfying compromise has been found by describing the shape of the vibrating components in terms of orthogonal shape-functions, and then optimizing their amplitude coefficients using a deterministic optimization scheme. Beyond enabling a systematic and cost- effective way of improving conventional instrument designs, an obvious advantage of optimization is the possibility of developing non-conventional instruments with new sound qualities. We illustrate the various aspects discussed by optimizing vibraphone or marimba-type bars, for several modal target sets. In the second part of this paper, we turn towards the sound synthesis of percussion bars. Here, the nonlinear physical modelling is based on a modal representation of the unconstrained bar. Such approach addresses the spatial aspects of the problem, being well suited for both non-dispersive and dispersive systems − which is the case of the flexural waves of interest here. Only the vibratory responses will be simulated, without an explicit accounting of sound radiation phenomena or of the coupling between vibrating bars and acoustic resonators. We illustrate the computational method with numerical simulations (sounds and animations) of marimba and vibraphone bars, for both classic and non-orthodox geometries.
PACS no. 02.60.Pn, 05.45.-a, 43.40.Ga, 46.15. Cc
1. Introduction approach, by coupling optimization procedures with adequate modelling A significant number of modes can be techniques. excited in mallet percussion instruments. Significant developments in system These are of paramount importance, as far modelling and optimization, during the last as instrument intonation and timbre quality decades, are the result of a spectacular are concerned, and every effort must be increase in computational hardware and provided to reach a successful design. At the software capabilities − see, for instance, present time, musical instruments are mostly [1,2]. Although mainly applied in the designed by trial and error procedures, context of industrial problems [3,4], there based on empirical knowledge and have been a number of attempts to apply experimentation. In spite of the makers’ optimization techniques in the context of know-how, such methods are often musical instruments [5-11]. Results have inefficient and costly. We think that many been in general fruitful, and efforts in this aspects of instrument design can be direction are still actively pursued by the improved using a more rigorous design authors and others [12,13]. Some aspects of our own work in this area will be presented function, to be minimized, was then here, in order to elucidate how the various computed between the eigenfrequencies bar shapes used in our numerical obtained at each iteration and the modal simulations were obtained. A more detailed target set. discussion and recent results of our work in Many parameters are involved in the the optimization of mallet percussion geometry optimization problem, with two instruments will be found elsewhere [13]. unwanted consequences: Firstly, the In a previous paper [10], we have optimization becomes computationally developed a method for the optimal design intensive, and this is further true as the of percussion instrument bars, such as found number of parameters to optimize Pp in xylophones, marimbas and vibraphones. ( p = 1,2, ) increases. Secondly, the error The aim was to compute optimal bar L shapes, in order to comply with a pre- hyper-surface ε (Pp ) where the global defined target set of modal frequencies − in minimum is searched will display in general other words, to shape the spectral content of many local minima. the instrument response − given a number of In [10] we avoided converging to sub- technological constraints, which establish optimal local minima by using a robust (but the acceptable ranges of the instrument greedy) global optimization technique − physical and geometrical parameters, as well simulated annealing [1,2,15]. In order to as other features such as shape simplicity. improve the computational efficiency, the Typically, the modal frequencies of global optimization algorithm was coupled interest will display harmonic relationships with a deterministic local optimization (however, one might well wish otherwise, technique [1,2], to accelerate the final stage depending on the musical context). For of the convergence procedure. Very instance, typical values of the first and encouraging results have been obtained, second modal frequencies for xylophones demonstrating the feasibility and robustness and marimbas attempt integer relationships of this approach, as well as the potential to of 1: 3 and 1: 4 , respectively, with some address other aspects of musical instrument variations on the third flexural modal design. However, a negative side effect was frequency, most usually between 1: 9 and the need for significant computation times, 1:10 . To obtain adequate intonation, the bar which seem ill suited to the optimization of profile is progressively changed by trial and large-scale systems − such as, for instance, error, using a suitable undercut, until the carillon bells. target frequencies have been reached within More recently, we tried to alleviate this the allowed tolerance [14]. problem, by reducing the dimension of the In [10] we showed that the design of search space where optimization is tuned bars can be greatly improved and performed [11]. This can be achieved in simplified by coupling finite-element eigen- several ways, by describing the geometrical computations with a suitable optimization profiles of the vibrating components in procedure. Indeed, for complex and terms of a limited number of parameters. innovative modal frequency relationships, Here, we chose to develop H (x) in terms of such approach may prove invaluable. Values a set of orthogonal shape functions Ψs (x ) , of the geometrical parameters − the bar optimizing their amplitude coefficients. For profile H (x) , given by the height complex systems, described by finite- H j ≡ H (x j ) of each mesh element − were element meshes with hundreds or thousands sampled in the admissible search space. of elements, this approach may reduce the Then, for each system configuration, an size of the optimization problem by several eigenvalue analysis was performed using the orders of magnitude. Then, we have found finite-element method. A suitable error that, most often, acceptable solutions can be obtained using efficient local optimization algorithms, leading to a further reduction in phenomena or of the coupling between computation times. Most of the examples vibrating bars and acoustic resonators − see presented in this paper have been obtained [21,27-29]. We also assume that all using such approach. We illustrate the nonlinear effects stem from the mallet/bar various aspects discussed by optimizing interaction, with no material or large- vibraphone or marimba-type bars, for displacement nonlinearities − such as several modal target sets. Both classic and unmistakably found in thin-walled plates, non-orthodox geometries are addressed. shells, cymbals and gongs [24,30-33]. From We then turn to the physical modelling our nonlinear numerical simulations, we of the nonlinear responses of percussion have obtained bar-animations and sounds, instruments. Physical modelling of musical for marimba and vibraphone bars, using instruments is one of the most active areas both conventional and innovative designs. in music acoustics, and very significant developments have been achieved in recent years [16-18]. Among these, string and wind 2. Optimization procedures instruments have been addressed in many significant papers. However, in spite of the We will start by briefly describing the relevant work offered by a few authors, computational approach used here to obtain idiophones − which typically display the bar eigenvalues. These are needed in the strongly dispersive waves − have received error function to be minimized, and we much less attention. Rossing [19] has put discuss the structure of such error function. much effort in understanding the fascinating We then recall the deterministic and dynamics of many percussion instruments. stochastic optimization algorithms used in Among authors interested in the physical the present work, as well as the relevant modelling of idiophones, Chaigne and his constraints which must apply. To conclude co-workers have been particularly active in this section, we propose a computationally this field − see, for instance, [20-24]. efficient approach, using orthogonal shape- Our interest in the physical modelling of functions to describe the system geometry, impacted bars was motivated by the need to and develop the modified constraints which assert the timbral qualities of different apply to such formulation. designs, without the need for costly and lengthy prototype machining of each and 4.1 Computational approach every computed optimized bar profile The cross-section dimensions of bars H (x) . Hence, we will address here a sound used in marimba-like instruments are not synthesis problem already approached by usually small compared to their length. Chaigne & Doutaut using a spatial Indeed, as modal frequencies increase, the discretisation in terms of finite differences Bernoulli-Euler slender beam model [20]. However, we will develop a becomes progressively inadequate. completely different approach, based on the Therefore, flexural modes are here modelled modal representation of unconstrained bars, in terms of the Timoshenko thick-beam to address the spatial aspect of the model, which corrects for the effects of dynamical problem. Such approach is well rotary inertia and shear deformation [34]. adapted to nonlinear problems involving We will assume that only bending both non-dispersive and dispersive systems modes are of interest here. Obviously, for − and, as such, suited to the flexural waves eccentric blows, torsion modes will be excited in idiophones, as we have shown excited [14,35]. On the other hand, as shown recently [25,26]. by Bork et al. [35], modal displacements are Only the vibratory aspects will be three-dimensional at higher frequencies and simulated here, without an explicit the beam approximation becomes then accounting of the sound radiation clearly abusive. However, we will only consider beam modes here, for simplicity. are computed in the usual manner, assuming For the same reason, we will neglect the harmonic solutions: influence of material anisotropy and inhomogeneities, which are significant when {Y(t)}= {ϕm }exp(iωm t) (4) dealing with wood bars [ 35,36]. Thus, for small vibratory motions, the and we obtain the classic eigenvalue transverse displacement y(x,t ) and slope formulation: φ(x,t) of the free conservative system are formulated as: 2 [[K]− ωm [M]]{}{}ϕm = 0 (5)
∂ 2 y ∂φ ∂ 2 y from which the modes are computed, for ρ A(x) + k G A(x) − = 0 (1) 2 2 ∂t ∂ x ∂ x each bar geometry of interest.
4.2 Error function ∂ 2φ ∂ 2φ ρ I(x) − E I(x) + In what follows we postulate that the ∂ t 2 ∂ x 2 material properties ρ , E and G are known (2) ∂ y and constant. Hence, we wish to determine + k G A(x) φ − = 0 the optimal bar height profile H * and total ∂ x j length L* that lead to a given set of modal ref where the local bar cross-section area and frequencies of interest ωm . moment of inertia are respectively Let us assume we have a bar with height A(x) = B H (x) and I(x) = B H (x)3 / 12 profile H j and length L , leading to the (with the bar width B ), ρ is the specific modal frequencies ωm (H j , L) . An obvious mass of the bar material, E is the Young term in the error function must concern modulus, G = 2 2 1+ν is the shear []() deviations from the computed eigenvalues modulus and k is a geometric factor for the and the reference target set ω ref : shear energy (equal to 5 / 6 for rectangular m cross-sections). Inertial and stiffness terms ref can be easily recognized in equations (1) ε1(H j , L) = Wm [ωm (H j , L) − ωm ] (6) L and (2). r Finite element discretisation of the preceding formulation leads to a linear where Wm are weighting factors for the system of equations in the classic form: modal errors and Lr is a suitable norm,
computed over the m = 1,2,L, MT modal []M {Y&&}+ []K {}Y = {}0 (3) frequencies of the target set. Specifically, we tested the following norms: where []M and []K are the inertia and M M stiffness operators, respectively (the later T T M 2 T incorporating any boundary conditions), and L1 = ∆m ; L2 = ∆m ; L∞ = MAX ∆m ∑ ∑ m=1 {}Y is the vector of physical displacements. m=1 m=1 In the case of bars with arbitrary height ref profile H (x) , we discretize the system with ∆m = ωm (H j , L) − ωm . In most cases using J elements of identical length we did not find significant differences l = L / J . between results − therefore, the quadratic From (3), the system modal frequencies norm L2 is consistently used in the ωm and corresponding modeshapes {ϕm } following illustrative computations. To the basic error term (6), one may {Ymin }≤ {Y}{≤ Ymax } (9) wish to add other terms, in order to penalize non-desirable effects, for instance Another common form of G (){}Y , most excessively non-smooth (and difficult to c useful in the context of §4.4, is the machine) geometry profiles H . We did so j following matrix inequality − which for the optimizations presented here, by generalises condition (9): introducing an additional penalty term for the profile curvature: [A]{Y}{}≤ B (10)
2 2 ε 2 (H j , L) = ∂ H j ∂ x (7) Basically, local optimization algorithms Lr search for a decrease in the gradient of the so that our final error function reads: error function ∇ε ({Y}) − that is why they are prone to being trapped in local minima. Such basic approach is inefficient, and more ε = (1−α) ε1 (H j , L) + α ε 2 (H j , L) (8) powerful methods rely on additional devices to improve the convergence speed. Newton where 0 ≤ α ≤ 1 is a weighting factor of the and quasi-Newton schemes use (or build up) geometrical complexity penalization − when second-order local information on the error α = 0 only frequency deviations are surface, and use curvature data − the local accounted; if 1 only geometrical α = Hessian matrix − to improve convergence. smoothness would be of interest. Constraints are imposed through penalty terms affecting the active constraints (using 4.2 Deterministic optimization Lagrange multipliers), which force the Basically, the optimization problem may solution to lay on the admissible workspace. be stated as finding the J element heights The necessary conditions for optimality in a * * of H j and the length L leading to a set of constrained problem are given by the Kuhn- M modal frequencies ω * which will Tucker equations [2,37]: T m minimize (8). This is a problem of Active * * * ∇ Y + ∑ λ ∇G Y = 0 constrained optimization, as typically H j ε ({ }){}c c () c (11) and L will be limited by some admissible λ* ≥ 0 values, due to technological or other c reasons. Then, defining a global vector T The main equation states that, at the {}Y = H j , L of the J +1 unknowns, our optimal point, the gradients of the original optimization problem will fit the following error function and of the active constraints general framework: Find the optimal (weighted by Lagrange multipliers) must solution {Y*} which minimises ε ({Y}) cancel. Obviously, the optimality condition for unconstrained problems would be: while complying with a set of constraints G (){}Y ≤ 0 , with ( c = 1,2, ,C ). c K ∇ε ({Y*}) = 0 (12) In our problem, the error function
ε (){}Y defined by (6-8) depends Many nonlinear programming nonlinearly on the geometrical unknowns algorithms have been developed to solve for {}Y , through the eigenvalue computations. * the λc in equations (11). In the present work However, the constraints Gc (){}Y are here we used the so-called Sequential Quadratic simply stated as: Programming (SQP), which is an efficient quasi-Newton scheme. Further details may be found in references [2,38]. 4.3 Stochastic optimization {Yi } will be randomly sampled − may also Many of the available algorithms for decrease with T . global optimization are inspired by the ways In short: the solution space is initially nature works, and are based on stochastic explored at high “temperatures”, when many procedures. Typical examples include error-increasing moves are accepted, genetic methods and simulated annealing allowing the algorithm to find the region (SA) [2-4]. The method used in the present where the global minimum lays. Later, at work, simulated annealing, is a powerful lower “temperatures”, almost only smaller, stochastic approach originally developed by error-decreasing moves are accepted, Metropolis et al. [39] as a Monte Carlo refining the solution. sampling technique for modelling the The general SA implementation evolution of a solid at a given temperature. procedure is as follows: Later, Kirkpatrick at al. [40] generalized this technique, which was then applied as a 1) Start from an initial “temperature” global optimization procedure. We will T and configuration {}Y . review here briefly the main aspects of SA. 0 0 2) From the last accepted configuration, Minimizing the error function ε ({Y}) is generate a new candidate solution seen as being analogous to the decrease of {Y }, randomly sampled within a the energy state of a molten metal during i cooling. At high temperature energy is high, certain vicinity of {}Yi−1 . as particles move freely everywhere. But, as 3) Compute the error ε (){}Yi and temperature decreases, motions are compare with ε (){}Y : progressively restricted and energy lowers. i−1 If the cooling schedule is sufficiently slow, a) If ε ({Yi }){}< ε ()Yi−1 accept the particles will settle into a very ordered state, move; and the system will reach a global (or near- b) If ε ({Yi }){}> ε ()Yi−1 accept (or global) energy minimum. However, if refuse) the move with probability cooling is enforced at a very fast rate, then P − see equation (13). particles will “freeze” in a disordered state Then go to step 2. (quenching), far from the energy minimum. 4) Repeat steps 2 and 3 for a number The SA analogy proceeds as a sequence N of cycles. Then decrease the of solution configurations {}Yi , each one temperature according to a given randomly sampled within a certain vicinity schedule Tt+1 = F()Tt and continue of the last accepted iteration {}Yi−1 , while a from step 2. control parameter T (analogue to the 5) When ε ({Yi }) becomes less than the cooling temperature) decreases at a allowable value ε , stop the sufficiently smooth rate. A very important Tol aspect is that, when iteration moves induce computation. an increase of the error function ε ({Y}), For further information on cooling such iterations may still be accepted − with strategies and other algorithm details, see a probability which depends on T : for instance references [1,2,41-43]. When the global minimum “valley” has P = exp[]− ()ε ({}Yi ) − ε ( {Yi−1 }) T (13) been found, convergence can be somewhat improved by coupling the global This essential feature enables the optimization algorithm with an efficient algorithm not being trapped in local minima. determinist approach, such as described Additionally, the radius of the hyper-sphere before. As stated before, this has been done centred in {}Yi−1 − where the next iteration in the present computations.
4.4 Shape-function approach H min ≤ H j ≤ H max (15) In order to reduce significantly the size L ≤ L ≤ L (16) of the optimization problem, we suggest min max developing the height profile in terms of a set of orthogonal functions (plus a mean and, from (14) and (15), we obtain: value): S H ≤ A Ψ (x ) ≤ H (17) S min ∑ s s j max H (x) = A + A Ψ (x) (14) s=1 0 ∑ s s s=1 or, in matrix form: Obvious suitable basis functions include Fourier series and Chebyshev polynomials. − [Ψ]{A}≤ −{H min } ; []Ψ {}A ≤ {H max } (18) Here we will illustrate the method using simple trigonometric series. and, assembling conditions (16) and (18), The optimisation is now performed in we obtain the full matrix constraint: terms of S amplitude coefficients As of the shape functions, either simultaneously or in − [Ψ] {0} − {}H min successive approximations, by progressively Ψ 0 [] {}{}A {}H max increasing the number of shape functions. ≤ (19) 0 −1 L − L Note that, for complex geometries, we have min S << J and the dimension of the 0 1 Lmax optimization problem may become orders of magnitude lower than using the physical mesh coordinates, as previously described. 3. Illustrative results Then, the smoothness of the system shape will be mostly governed by the truncation The previous remarks will now be order S of the spatial series. However, if illustrated on marimba-type bars. We will the penalty term (7) is still used, then low- only address here systems symmetric with order spatial terms will be favoured in the respect to their middle point. Therefore, optimization process. optimization is performed only on half of We conjecture that the reduced order of the element heights H j used in the finite T the search space {}Z = As , L may lead to element mesh, so that J = 32 . Concerning a “filtered” error surface ε (){}Z with the shape function approach, we used either greater regularity than the original error C function ε (){}Y . Hence, if deterministic Ψs (x) = cos(2 sπ x / L) S optimization strategies are attempted from or Ψs (x) = sin(2 sπ x / L) scratch, they stand lower chances of being trapped in local minima. Obviously, global in the first half of the bar length, with optimization methods can also be used to symmetry beyond. A maximum value of minimize ε (){}Z . S = 10 was used − however, for the present Using the approach suggested here, we bar optimizations, convergence was often have to adapt the physical search-domain achieved using less orthogonal functions. In constraints (9) in order to accommodate the our first example we kept the bar length L new variables As . This can be easily as a variable to be also optimized, however achieved using the matrix formulation (10), it was enforced as a constant parameter in as follows. The original constraints are: all other computations, for an easy comparison of the optimized height profiles. The following fixed parameters were frequency of 440 Hz (except those in Figure used for the modal computations: 1, which are tuned to 880 Hz). In all the L = 350mm (when imposed), B = 40 mm, optimizations performed we used unit 3 10 2 ρ = 2840 kg/m and E = 6.726 10 N/m . weight factors Wm for the modal errors. The The valid search ranges imposed for the bar value adopted for the penalty coefficient on height and total length are: 5 ≤ H j ≤ 30 mm the non-smoothness of H (x ) was most and 100 ≤ L ≤ 500mm (when “free”). Most often α = 0.2 (except in the examples of of the computed bars have a fundamental Figure 1, where α = 0.3 ).
Alfa = 0.3 10000 0.02 8000 6000 0 (a) 4000 -0.02
Frequency [Hz] Frequency 2000 -0.04 0 0 1 2 3 4 0 0.05 0.1 0.15 0.2 Mode index
10 Cosinus terms used ; Alfa = 0.3 () 10000 0.02 8000 0 6000 (b) 4000 -0.02
Frequency [Hz] Frequency 2000 -0.04 0 0 1 2 3 4 0 0.05 0.1 0.15 0.2 Mode index Figure 1: Optimized vibraphone bars with fundamental frequency 880 Hz and frequency relationships 1: 4 :10 of the tuned modes (bar length not imposed; smoothness parameter α = 0.3): (a) Stochastic optimization using SA, performed on the physical element heights; (b) Deterministic optimization using SQP, performed on the amplitudes of cosinusoidal shape-functions.
In Figure 1 we show two optimized bars orders of magnitude lower than using with frequency relationships 1: 4 :10 of the approach (a). Therefore, the optimization tuned modes (commonly found in concert method described in §4.4 will be used in marimbas). Result (a) was obtained most of the following illustrations (except in performing a stochastic optimization on the Figure 3). physical heights H j , while result (b) Figure 2 shows successive stages of the pertains to a deterministic optimization deterministic optimization approach, as the performed on the shape-function amplitudes number of shape functions increases. Six terms are needed to tune five modes of a A . Both computed shapes are perfectly s bar, on the unusual frequency relationships tuned for the imposed frequency 1: 2 : 4 :8 :16 . Notice that, as the number of relationships. However, beyond 8800 Hz, orthogonal functions is incremented, tuning their modal frequencies will obviously be progresses from the lower modes towards different from each other. the higher modes. This is easily explained, Notice that both optimization schemes as higher frequencies are related to shorter automatically produced designs with central wave-lengths. Tuning of higher-frequency undercuts, as found in commercial marimba modes usually ask for the contribution of C bars. Ten shape functions Ψs (x) were used higher-order shape functions. for the second optimization, but three are We have found that computations enough to converge in such a simple performed by optimizing simultaneously all problem. The important point here is that, the amplitude coefficients of the shape- even for this simple system, the computation function usually lead to results of time using approach (b) is typically two comparable quality. 1 Terms Modal Error: 0.185 0.04 10000 0.02 0 -0.02 5000 -0.04 -0.06 0 2 Terms Modal Error: 0.182 0.04 10000 0.02 0 -0.02 5000 -0.04 -0.06 0 3 Terms Modal Error: 0.151 0.04 10000 0.02 0 -0.02 5000 -0.04 -0.06 0 4 Terms Modal Error: 0.105 0.04 10000 0.02 0 -0.02 5000 -0.04 -0.06 0 5 Terms Modal Error: 0.052 0.04 10000 0.02 0 -0.02 5000 -0.04 -0.06 0 6 Terms Modal Error: 0.001 0.04 10000 0.02 0 -0.02 5000 -0.04 -0.06 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 Length [m] Mode index Figure 2: Deterministic optimization process of a vibraphone bar with fundamental frequency 440 Hz and frequency relationships 1: 2 : 4 :8 :16 of the tuned modes (bar length imposed L = 0.350 m; smoothness parameter α = 0.2 ): Convergence while increasing the number of shape-functions.
The influence of the smoothness 4. Nonlinear dynamical modelling parameter α is shown in Figure 3. Here, stochastic optimization was performed on The computational method used here is the geometrical variables H j . The described next: progressive smoothing effect, when α (1) From the optimized bar profile H (x) increases, is obvious. However, if the and the physical properties of the bar, we smoothness penalty is excessively compute the planar flexural vibration modes emphasised, tuning may become unfeasible of the unconstrained system in the audible − such was the case when α = 0.8 was frequency range. At this stage, as in Section imposed. 2, we use the finite-element method with Figure 4 shows three different examples, computations based on the Timoshenko where deterministic optimization was beam model. The boundary conditions used achieved using either Ψ C (x) or Ψ S (x) as are those of a free-free bar, which imply the s s existence of two rigid-body zero-frequency shape functions. We consistently obtained planar modes (translation and rotation), as adequate results, irrespective of the shape well as the modes with elastic deformation. function set used. Often, both sets produced Again, as in Section 2, we will ignore here rather similar optimal shapes. for simplicity the torsion modes and high- Finally, several examples of optimized frequency 3-D effects. However, the use of bars are shown in Figure 5, corresponding to a more accurate modal basis, computed quite un-orthodox frequency relationships. from an extensive 3-D mesh of massive This illustrates the power of an optimization elements, would not change in any way the approach to generate new instruments, when numerical approach used here for the time- suitable design criteria can be established. domain simulations. Indeed, the use of computations. Note, however, that the modes based on such a refined model would modal model can accommodate most easily further highlight the computational any frequency-dependency of the damping efficiency of the modal approach, when coefficients. The average values used in the compared for instance with time-domain present computations were based on finite-element computations. experimental identifications. Obviously, these values are very different for wood and for aluminium bars. 0.04 0.02 α = 0 0 1:4:9() -0.02 0.04 -0.04 0.02 -0.06 COS 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.02 0.04 -0.04 0.02 α = 0.05 -0.06 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.02 () -0.04 0.04 -0.06 0.02 SIN 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 -0.02 0.04 -0.04 0.02 α = 0.1 -0.06 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.02 -0.04 1:3:10() -0.06 0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 () 0.02 COS 0.04 0 0.02 α = 0.2 -0.02 0 -0.04 -0.02 -0.06 -0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.06 () 0.04 0 0.05 0.1 0.15 ()0.2 0.25 0.3 0.35 0.02 SIN 0.04 0 0.02 α = 0.8 -0.02 0 -0.04 -0.02 -0.06 -0.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1:2:4:8:16() 0.04 Figure 3: Stochastic optimization of a 0.02 COS 0 vibraphone bar with fundamental frequency -0.02 440 Hz and frequency relationships 1: 4 :10 -0.04 (bar length imposed L = 0.350 m): Results as a -0.06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 function of the smoothness parameter α . () 0.04 0.02 SIN 0 (2) When performing the dynamical -0.02 computations, the modal damping values -0.04 used clearly affect the bar dynamics, as well -0.06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 as the subjective perception of the simulated sounds. Due to the nature of the various Figure 4: Deterministic optimization of three dissipative phenomena, damping values are vibraphone bars with fundamental frequency usually frequency-dependent, as discussed 440 Hz (bar length imposed L = 0.350 m; in [22,23,36,44,45]. Although this is a subtle smoothness parameter α = 0.2 ): Results using aspect, we used − again for simplicity − 10 cosinusoidal or sinusoidal shape-functions. constant values for all modes in our
10 Cosinus terms used ; Alfa = 0.2 () 8000 0.04 1:2:4:8:16 6000 0.02 0 4000 -0.02
2000 -0.04 Frequency [Hz] -0.06 0 0 2 4 6 Mode index 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 Cosinus terms used ; Alfa = 0.2 () 5000 0.04 4000 0.02 1:2:5:10 3000 0
2000 -0.02 -0.04 Frequency [Hz] Frequency 1000 -0.06 0 0 1 2 3 4 5 Mode index 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 Cosinus terms used ; Alfa = 0.2 () 4000 0.04 1:3:5:7:9 3000 0.02 0 2000 -0.02
1000 -0.04 Frequency [Hz] Frequency -0.06 0 0 2 4 6 Mode index 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 Cosinus terms used ; Alfa = 0.2 () 6000 0.04 0.02 1:3:6:12 4000 0 -0.02 2000 -0.04 Frequency [Hz] Frequency -0.06 0 0 1 2 3 4 5 Mode index 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 Cosinus terms used ; Alfa = 0.2 () 0.04 6000 0.02 1:5:10:15
4000 0 -0.02 2000 -0.04 Frequency [Hz] Frequency -0.06 0 0 1 2 3 4 5 Mode index 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Figure 5: Deterministic optimization of several non-orthodox vibraphone bars with fundamental frequency 440 Hz (bar length imposed L = 0.350 m; smoothness parameter α = 0.2 ).
(3) The vibrating bars are assumed (4) We assume that the bar is impacted supported at two locations xs1 and xs2 by a point mass M c with initial velocity Vc through flexible dissipative fixtures. The at the nominal contact location xc . As other stiffness constant Ks and damping constant authors, we will use a simple Hertz model,
Cs used in the computations are based on to relate the nonlinear contact force (at the experimental results. nominal contact point) to the relative bar/mallet motion after impact [20]. This L interaction point-model is a very convenient 2 approximation. A more accurate contact mm = m(x)[]ϕm (x) dx ∫0 (22) model might be easily implemented in our 2 computations, by including the elastic cm = 2mmωmζ m ; km = mmωm modes of the mallet as well as by using a time-dependent contact length, as a function where ωm and ζ m stand, obviously, for the of the mallet geometry and of the bar/mallet modal (circular) frequencies and damping elastic interpenetration. Also, dissipative values, and ϕ (x ) are the bar modeshapes. contact phenomena could easily be m The modal masses depend on the incorporated in the model. However, if the modeshapes, as well as on the bar profile parameters used in Hertz-type models are through the mass per unit length well chosen, quite realistic results can be m(x) = ρ A(x) = ρ B H (x) . As stated obtained even with this simple contact model. The mallet physical parameters before, for each bar geometry, all modal (inertia and stiffness) and the impact parameters are computed, once and for all, velocity used in our computations are based using a finite-element model based on in experimental data by Bork [46]. equations (1) and (2). (5) At each time-step n , we compute by The vectors of modal responses and modal superposition the physical vibratory modal forces are given as: responses at locations xs1 , xs2 and xc . This {Q}= q1(t),q2 (t),L,qM (t) enables the computation of the interaction (23) forces between the bar and the two elastic {}F = f1(t),f2 (t),L,fM (t) supports, F (t) and F (t ) , as well as the s1 s2 and, at any location x , the physical contact force between the bar and the mallet, response y(x,t) can be simply computed F (t) . These interaction forces are then c from the modal amplitudes q (t ) through projected on the modal basis, and the m dynamic modal equations are integrated one modal superposition: step ahead using an explicit algorithm. This M scheme is pursued for the full duration T of y(x,t) = ϕ (x) q (t) (24) ∑ m m the simulation. m=1
4.1 Bar dynamics and similarly concerning the velocities and We will now formulate the equations accelerations. used in our computational scheme. Let us The modal forces are obtained through then consider a bar of length L , constant modal projection of all the external forces. width B and variable height profile H (x) . For this system, these are the two support In terms of the chosen modal representation, reactions and the mallet/bar interaction the system dynamics are governed by: force. Hence:
[]M {Q&&}+ []C {Q&}+ []K {}Q = {}F (20) Fs1(t)δ (x − xs1) + L fm (t) = Fs2 (t)δ (x − xs2 ) +ϕm (x)dx Here the modal matrixes are given as: ∫0 Fc (t)δ (x − xc ) (25) = Fs1(t)ϕm (xs1) + []M = Diag()m1,m2 ,L,mM Fs2 (t)ϕm (xs2 ) + []C = Diag()c1,c2 ,L,cM (21) Fc (t)ϕm (xc ) []K = Diag()k1,k2 ,L,kM with: where δ (x) is the Dirac distribution. 4.2 Mallet dynamics (20) − Runge-Kutta, Verlet, among others. In our model, the mallet is modelled We used the Euler-Richardson scheme, simply as a point mass M c with initial which is of the explicit type [1]. In this velocity V , interacting with the bar at algorithm, an estimate of the acceleration c vector is performed at the next half-step location xc . If z(t ) is the motion of the tn+1/ 2 ≡ tn + ∆t / 2 , using the information at impactor, then the mallet dynamics will be time t : governed by: n ∆t {}Q + {}Q& , M c &z& = Fc (t) − M c g (26) n n 2 where g is the acceleration due to gravity. −1 ∆t {}Q&&n+1/ 2 = []M F{}{}Q& n + Q&&n , (29) 2 4.3 Impact force ∆t tn + The Hertz model used here leads to the 2 following contact force: which is then used to update displacements 0 and velocities at time t : n+1 if z(t) − yc (t) ≥ 0 Fc (t) = (27) ∆t {}{}Qn+1 = Qn + {}{Q& n + Q&&n+1/ 2 } ∆t (30) − K []z(t) − y (t) s 2 c c
if z(t) − yc (t) < 0 {Q& n+1}= {Q& n }+ {Q&&n+1/ 2 }∆t (31) where yc (t) ≡ y(xc ,t) and s = 3 / 2 . Note that equation (25) insures that only modes 5. Numerical simulations such that ϕ (x ) ≠ 0 will be excited by the m c mallet. We will present now some sample computations, using several optimized bar 4.4 Support forces geometries computed as shown in Sections 2 The bar supports, modelled as linear and 3. Apart from the “static” plots flexible-dissipative fixtures, lead to the presented here, we also have generated following reaction forces: animations of the bar responses and the corresponding sound files. Parametric Fsj (t) = −K s ysj (t) − Cs y&sj (t) ; j =1,2 (28) computations have been performed, to illustrate the influence of: - the impact location x ; where ysj (t) ≡ y(xsj ,t) . Again from c equation (25), we note that only modes such - the bar material; that ϕ (x ) ≠ 0 will be affected by the - the mallet stiffness; m sj - the bar modal frequency ratios. supports. If − as is usual practice − the Parameters kept constant in our supporting fixtures are located at the nodes computations are the mallet mass M c = 20g of the first mode, then only higher-order and impact velocity V = 1m/s, as well as modes will suffer additional damping from c 4 the supports. the support stiffness K s =10 N/m and
dissipation Cs = 20 Ns/m. For all the 4.5 Time-step integration geometries computed we postulated that There are many integration algorithms supports are located at the nodes of the first that might be used to integrate equations elastic mode of the unconstrained bar. The to highlight both the initial transient and the first modal frequency is 440 Hz, for all the subsequent decay. Sound files have been computed bars, irrespectively of their shape generated at the usual sampling rate of or material. As stated before, besides the f s = 44.1kHz. Computational speed was first tuned modes of the optimized bars, all not an important issue for the numerical modes in the range 0 ~ 20kHz were used in simulations presented here, therefore a very the numerical simulations. comfortable time-step of ∆t = 2.3 10−6 s The numerical simulations presented was used. here extend for 5 seconds − which is enough
0.2
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-10 -10 10 10 [(m/s) [(m/s) r r (c) P.S.D. V P.S.D. V P.S.D.
-20 -20 10 10 0 0.5 1 1.5 2 0 0.5 1 1.5 2 4 4 Frequency [Hz] x 10 Frequency [Hz] x 10
Figure 6: Typical response of a 1: 4 :9 vibraphone bar impacted at location xc = 0 using a mallet 8 3/2 with contact stiffness Kc =10 N/m : (a) Time-history of the velocity response at the bar end x = 0 ; (b) Detail of the velocity response during the initial transient and the final decay; (c) Velocity spectra during the initial transient and the final decay.
5.1 Results 8 3/2 stiffness value of Kc =10 N/m between As a first example, Figure 6 (a-c) shows the mallet tip and the bar − see equation (27) several aspects of the vibratory response of a vibraphone bar, when impacted at location − has been used. The time history trace in Figure 6 (a) x = 0 . For this aluminium bar, which has a c shows the initial transient after the mallet typical undercut, we postulated modal impact, followed by the slow decay of the frequency ratios of 1: 4 :9 for the tuned first elastic mode. The first zoomed trace in modes. A modal damping value of Figure 6 (b) highlights how, during the ζ m = 0.02 % has been assumed for all initial transient, the low-frequency support- modes, as explained before. A contact dependent rigid-body modes are excited, as well as the higher frequency elastic modes. the velocity signal at a bar end for our The second zoomed trace shows that the subjective timbre comparisons) is quite decay response is dominated by the lightly realistic, although the resonator has not been damped first elastic mode. These effects are incorporated yet in the model. confirmed by the corresponding response In Figure 8 we display the relative spectra shown in Figure 6 (c). energies of the system modes (notice that The spectrogram of the bar response at the energy scale is logarithmic). Clearly, location xc = 0 is illustrated in Figure 7. It most of the impact energy goes to the first is clear that the higher frequency modes elastic mode − number 3 in this plot. The only vibrate significantly during a few two low frequency rigid-body modes also tenths of a second − which are however vibrate significantly. The energy of the crucial for the timbre recognition. Overall, higher frequency elastic modes decreases we feel that the perceived “sound” (we used fast, as the modal frequency increases.
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Figure 7: Typical response of a 1: 4 :9 vibraphone bar impacted at the bar end xc = 0 using a mallet 8 3/2 with contact stiffness Kc =10 N/m : (a) Time-history of the velocity response at the impact location; (b) Corresponding spectrogram.
5.2 Impact location location xc , when a mallet with contact We will now discuss, through parametric stiffness K =108 N/m3/2 is used. The computations, several aspects which are c particularly significant for the vibratory corresponding sound files show clear responses of this system. Figure 9 shows the differences in the perceived sounds. The time-responses and corresponding sounds obtained at xc = L / 10 and spectrograms of the same 1: 4 :9 xc = L / 3 are rich and well balanced (the vibraphone bar, as a function of the impact typical “vibraphone-sound” one usually hears). However, because the nodes of the fundamental is concerned. On the other fundamental mode are located near hand, when the mallet strikes the middle of
xc = L / 5 , striking the bar near the supports the bar, only the symmetric modes are will induce a “thin” sound, rich at higher excited. The corresponding sound is frequencies but poor as far as the comparatively dull.
2
0 ) T -2 / E m -4 Log( E -6
-8 1 2 3 4 5 6 7 8 9 10 Mode
Figure 8: Typical response of a 1: 4 :9 vibraphone bar impacted at location xc = 0 using a mallet 8 3/2 with contact stiffness Kc =10 N/m : Relative modal energies.
5.3 Contact stiffness dissipation of wooden bars. Here, we recall
Figure 10 shows the time-responses and that modal damping values of ζ m = 0. 02 % corresponding spectrograms of the same and ζ = 0.5 % were used (for all modes), 1: 4 :9 vibraphone bar, as a function of the m respectively in the case of aluminium and contact stiffness K , when the mallet strikes c wood bars. The consequences of this simple at location xc = L / 10 (the influence of the parameter change, when the subjective bar material is also shown here and will be timbral qualities are concerned, go much discussed in next). The extreme values of beyond the mere decrease in response the contact stiffness used in these duration shown in Figure 10. Indeed, simulations, 107 and 109 N/m3/2, represent simulations using the lower damping value respectively soft and hard mallets. When sound as impacted metal, while those using using soft mallets, the sounds produced are the higher damping value sound as impacted smooth, as most of the energy is excited at wood. lower frequencies. Conversely, the much brighter sounds excited by hard mallets are 5.5 Bar shape: conventional designs attested by the significant excitation of the Figure 11 shows the time-responses and higher frequency modes. corresponding spectrograms of several “conventional” bars, with frequency ratios 5.4 Bar material that might be found in xylophones, Figure 10 also shows how significant the marimbas and vibraphones. Common bar material is, for the instrument timbre. variations include the second partial, at Here, the last two time-plots and either 1: 3 (one octave + one fifth) or 1: 4 spectrograms display the responses of a (two octaves), as well as the third partial, at typical marimba bar, with the same either 1: 9 (three octaves + one minor third) fundamental frequency. Obviously, wood or 1:10 (three octaves + one major third). has a lower density than aluminium (about All these cases are illustrated in the figure, one third). However, the main effect of the based in the optimized shapes obtained as material is, without doubt, the much higher shown in Sections 2 and 3. The corresponding sounds show that influence of can be easily combined in the optimization the second partial frequency is very easily procedure. perceived. However, for the untrained ear, the influence of the third partial is more subtle. 6. Conclusion
5.6 Bar shape: non-orthodox designs Two important aspects of the design of We conclude by illustrating in Figure 12 percussion instruments have been addressed the time-responses and corresponding in this paper: (1) Geometry optimization of spectrograms of the four unconventional the vibrating component, and (2) Sound bars optimized in Section 3. Here, one synthesis through nonlinear physical should bear in mind that the perceived modelling. The feasibility of these two sounds depend, not only on the tuned complementary approaches has been modes, but also on other higher frequency demonstrated. modes which are outside the optimized Beyond enabling a systematic and cost- frequency range (and therefore effective way of improving conventional “uncontrolled”). Here, we chose the modal instrument designs, an obvious advantage of relationships of 1: 2 : 3, 1: 3 : 6 :12 , optimization approaches is the possibility of 1: 2 : 4 : 8 :16 and 1: 5 :10 :15 , not on the developing non-conventional instruments ground of their particular “musical” with new sound qualities. We illustrated the qualities, but only because such modal various aspects discussed by optimizing and tunings lead to very different bar dynamically simulating several vibraphone geometries, as well as response spectra. and marimba bars. Bar/resonator vibro- And, indeed, the sounds obtained are acoustic coupling and radiation effects have different, as can be inferred from the not been yet incorporated in our spectrograms in Figure 12. computational model. Even so, the The case 1: 2 : 3 deserves a short subjective quality of the simulated sounds comment, because it led to a somewhat was found quite satisfactory. unexpected sound. Indeed, from such We intend to extend the present work in frequency combination, one might naively several directions. Concerning optimization, expect a “sweet” sound, but what we these include: obtained sounds harsh, almost as sounds - optimization of resonator geometries, to issued from some gamelan instruments, for control other acoustic modes beyond the instance. What happens here is that the non- first, in connection with the bar optimized forth and upper partials present vibratory modes; frequencies which are relatively low. These - optimization with respect to other are significantly excited and severely aspects beyond modal frequencies, for inharmonic. If we assume that such effect is instance the system modeshapes, in undesirable, there is a lesson to extract from order to better control the radiated this example: Sometimes, one should design spectra; the optimization procedure not only on the - optimization of more complex problems, basis of what is wanted, but also on the basis ranging from soundboard geometries what is not wanted. This can be achieved (and bridge placement) to the design of easily: In the first case, penalty terms in the optimized carillon bells; error-function should tax divergence Turning towards dynamical simulations, between the computed modal frequencies the most urgent aspects to include in our and the target set. In the second case, approach are the bar/resonator vibro- penalties should tax proximity of the acoustic coupling and radiation. Some of computed modal frequencies to any these aspects are currently being addressed. undesirable frequency range. Both aspects []
] 0.1
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r r V -0.1 V -0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Temps [s] Temps [s] 10000 10000
() () 9000 0.04 9000 0.04 0.02 0.02 0 0 8000 -0.02 8000 -0.02 -0.04 -0.04 -0.06 -0.06 7000 7000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 6000 L/10 6000 Aluminium 5000 5000 Kc=1e7 N/m^1.5 4000 4000
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r r V V -0.2 -0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Temps [s] Temps [s] 10000 10000 () () 0.04 9000 0.04 9000 0.02 0.02 0 0 8000 -0.02 8000 -0.02 -0.04 -0.04 -0.06 -0.06 7000 7000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 6000 L/2 6000 Wood 5000 5000
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Figure 9: Vibratory responses of a 1: 4 :9 Figure 10: Vibratory responses of a 1: 4 :9 vibraphone bar as a function of the impact vibraphone bar as a function of the contact 8 3/2 stiffness K and also of the bar material location xc (contact stiffness Kc =10 N/m ). c (impact location xc = L /10 ). []
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[m/s] r r V -0.1 V -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Temps [s] Temps [s] 10000 10000
() 9000 0.04 9000 0.04 0.02 0.02 0 0 8000 -0.02 8000 -0.02 -0.04 -0.04 -0.06 -0.06 7000 7000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 6000 1:4:9 6000 1:2:3 5000 5000
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r r V V -0.2 -0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Temps [s] Temps [s] 10000 10000 () 0.04 () 9000 9000 0.02 0.04 0 0.02 -0.02 8000 0 8000 -0.04 -0.02 -0.06 -0.04 7000 -0.06 7000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
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9000 0.04 9000 0.04 0.02 0.02 0 0 8000 -0.02 8000 -0.02 -0.04 -0.04 -0.06 7000 -0.06 7000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 6000 1:4:10 6000 1:2:4:8:16 5000 5000
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Figure 11: Vibratory responses of four near- Figure 12: Vibratory responses of four non- conventional vibraphone bars (contact stiffness orthodox vibraphone bars (contact stiffness 108 N/m3/2, impact location ). 8 3/ Kc = xc = L /10 Kc =10 N/m 2, impact location xc = L /10 ).
Acknowledgement [9] P. Anglmayer, W. Kausel, G. Widholm: Useful discussions with Octávio Inácio A computer program for optimization (Intituto Politécnico do Porto, IPP/ESMAE, of brass instruments: Part 2 - Portugal) and his valuable assistance while Applications, practical examples. performing preliminary experiments, are Forum Acusticum 1999, Joint 137th gladly acknowledged. Meeting of the Acoustical Society of America and 2nd Meeting of the European Acoustics Association, References Berlin, 15-19 March 1999. [10] L. Henrique, J. Antunes, J. S. Carvalho: [1] W. H. Press, B. P. Flannery, S. A. Design of musical instruments using Teukolky, W. T. Vettering: Numerical global optimization techniques. recipes: The art of scientific International Symposium of Musical computing. Cambridge University Acoustics (ISMA 2001), Perugia, 10-14 Press, Cambridge (1989). September 2001. [2] P. Venkataraman, Applied optimization [11] L. Henrique, J. Antunes: Conception with Matlab programming. 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