Minimum bow force and the impact of torsion for the bowed string

Transcription

1 Minimum bow force and the impact of torsion for the bowed string 1 / 9 Robert Mores University of Applied Sciences Hamburg 1

2 Minimum bow force - definition Definition: transition between normal Helmholtz motion and vibration regimes with additional slips Parameters: bow force bow velocity relative position of bow string impedance (transl.) FB vb β Zo in N (here in grams) in cm / s < 0.5, with respect to the bridge in kg / s Recall: Schelleng diagramm (for const. vb) 2 2 / 9

3 Minimum bow force - definition Raman s model (1918) reused by Askenfelt (1989), Galuzzo & Woodhouse (2014), Mansour et al. (2017) 2 Z0 v Fmin B2 (1) 2 R S D include rotational admittance Schelleng (1973) reused by Schoonderwaldt et al. (2008) Fmin Z0 2 2 R S ' D 3 / 9 v B z (2) μs static friction μd dynamic friction μ'd asymptotic dynamic friction, μ at relative slip velocity z = vb / β z0 introduced minimum offset for slip velocity z vs slip velocity R restistance at the bridge ξ = Z0 / Zr (rotational) 3

4 Minimum bow force existing measurements Early observations by Raman (1918) and by Kar et al. (1951) for the proportionality - ~1/ß² ~1/ß Schoonderwaldt et al. (2008): bow, monochord, cello/violin strings, motor 4 / extended measurements, vb = 5, 10, 15, 20 cm/s / FB = N plus extended experiments on minimum bow force however: Δμ results from fitting ( ), not from measurement main findings: (i) velocity-independence, (ii) x10, (iii) role of damping Galuzzo & Woodhouse et al. (2014): rod, cello string, motor vb = 5 cm/s / FB = N again: Δμ results from fitting, not from measurement confirmation of existing models? 4 9

5 Method Measuring all components of formula (1) or (2) including the friction coefficients In situ capturing of vibrational classes at the contact point Use precision instrumentation construction principle total arrangement operational range precision Cello strings on a monochord, and a real bow, NOT a motor 5 5 / 9

6 Bowing pendulum Find demonstration videos on youtube (search for cello bowing machine) I II height-compensated height-compensated & weight-compensated III bow force control IV V VI keeping track from single shots to bifurcation between bifurcation regimes 6 / 9 Construction principle: Mores (2015) 6

7 Capturing vibration and classification Pick-up for lateral velocity (non-linear) analog integrator (3Hz 20kHz) delivers displacement (still non-linear) 7 / 9 US B1 Classification: displacements directed opposite to bowing are slipping MATLAB: events = diff (sign(diff(low-pass_filtered_signal))) 7

8 Measuring R Schoonderwaldt (2008) Three different materials Support material at t he... nut... br idge R 2 Z0 T1 (3) 8 / 9 n f0 = 1 / T1 in Hz τ in s R in kg/s felt felt ± ± ± 3.3 MDF MDF + Bary-X ± ± ± 1.6 MDF + Bary-X MDF + Bary-X ± ± ± 4.2 felt felt ± ± ± 13 8

9 Friction coefficients and vs. v S D S for = 1/80 D 0.6 S Hyperbolic fit for μd but without μs μ'd = after numeric optimization, R² = vs in cm/s z0 = m/s (velocity offset) 9 / 9 for = 1/6 D μd is measured on dampened strings, no audible pitch μs : 30 instances of stick to slip transitions (single shots, not Helmholtz) μs = ±

10 Vibrational classification 10 / 9 Cello D string on monochord, Helmholtz motion (HM) and non-helmholtz motion with x extra slips per cycle (nhm-x) 10

11 Populations of raw data related populations for the three strokes 11 / 9 11

12 Method 12 / 9 Schoonderwaldt Populations of HM to nhm-1 transitions (x), and nhm-1 to HM transitions (o) for 500+ strokes, linearily approximated for individual β and R (- -) and across the entire data set ( ), results from Eq. 2 for comparison ( ), instrumentation error in the lower left graph applies for all graphs: maximum worst case error (larger bars), standard deviation for the bow force σ = ± 2.8 grams (little bars) 12

13 Fitting results Mode [1 R vb 1/β vb x 1/R x 1/β² 1/ R x 1/β] [1 vb vb x 1/R x 1/β² 1/ R x 1/β] [vb x 1/R x 1/β² 1/ R x 1/β] R² R vb 1/β Coefficients vb x 1/R x 1/β² 1/ R x 1/β 95% confidence intervals 1 R vb 1/β vb x 1/R x 1/β² 1/ R x 1/β Fmin Z0 2 2 R S ' D c1 v B c2 z (4) c1 = 1.5 Z0 = 0.66 kg/s Larsen Solo Medium cello D-string c2 = / 9

14 Brief validation Applying the same classification extraction fitting on the cello G string R = 939 kg/s Z0 = 0.93 kg/s 14 / 9 Populations for the G string, with linear regression across the entire data set ( ), and with predictions from the D string (- -), D string c1 = 1.5 c2 = 177 R² = 0.91 G string c1 = 1.6 c2 = 160 R² =

15 co-determines Helmholtz motion bridge distance = 80 mm bridge distance = 120 mm on cello steel string (open G, 100Hz) left: torsion vs. ß right: torsion trace of 8 consecutive periods (green), and average (red), and the average of the string displacement from the associated 8 consecutive periods / 9

16 Observations 16 / 9 on cello steel string (open D, 147Hz), numbers indicate the multiple of the fundamental frequency Full dot: torsion with high amplitude and near-harmonic appearance Cross: no Helmholtz motion (suppressed torsion) 16

17 Summary Minimum bow force: working with either formula, (1) or (2), is fine however, findings suggest bias force 100 x larger than assumed so far The bias force is proportional to 1/R ß, Fmi n then grows with vb /ß² strongly co-determines Helmholtz motion and also the minimum bow force, there are spots of very low bow force there are spots impossible to play The findings on min bow force still hold for the typical range of musical performance / 9

18 Thank you References Askenfelt, A. (1989). Measurement of the bowing parameters in violin playing. II: bow-bridge distance, dynamic range, and the limits of bow force, J. Acoust. Soc. Am. 86, Kar, K. C., N. K. Datta and S. K. Ghosh (1951). Investigations on the bowed string with an electrically driven bow, Ind. J. Theor. Phys / 9 Mores, R. (2015). Precise cello bowing pendulum, in Proc. of the Third Vienna Talk on Music Acoustics, 106 ff. Raman, C. V. (1918). On the mechanical theory of vibrations of bowed strings, etc., Indian Assoc. Cult. Sci. Bull. 15, Schelleng, J. (1973). The bowed string and the player, J. Acoust. Soc. Am. 53, Schoonderwaldt, E., Guettler, K., and Askenfelt, A. (2008). An empirical investigation of bowforce limits in the Schelleng diagram, Acta Acustica united with Acustica 94, Woodhouse, J., and Galluzzo, P.M. (2014). High-performance bowing machine tests of bowed-string transients, Acta Acustica united with Acustica 100, Mansour,H., J. Woodhouse, and G. Scavone (2017). On minimum bow force for bowed strings, Acta Acustica united with Acustica 103,

19 Total arrangement S1 S2 S3 M1 M2 M3 M3 position sensor force sensor force sensor cellist s arm potential traction weight compensation 19 / 9 S2 M1 string d S1 M2 c b M1 wheel stationary point string d S2 unit size a S3 h string e ri ra 19 S3

20 Range and operations All string instruments up to 60 kg can be instrumented, including double bass. Straight-line movement (limits the bow size): 90 cm Bow velocity: 0 30 cm/s Bow force: 0 5N Bow force difference up- vs. downstroke: < 0.01 N Lateral displacement of bow: < 0.5 mm (for bow forces below 4 N) Friction (string d): < 0.4 N (to be measured for each individual session) / 9

21 Precision Maximum total error physical property sensor maximum error sb in cm bow position S1 ± 0.18 cm vb in cm/s bow velocity derived from sb ± cm/s Ft in N traction force S2 ± 0.15 N ± 1 % Fb in N bow force S3 ± 0.11 N Dynamic response physical property sensor impulse / step T10/90 sb bow position S1 3 cm step ~6 ms Ft traction force S2 10 N impulse ~0.8 ms Fb bow force S3 10 N impulse ~8 ms / 9