Abstract
Biological rhythmic movements can be viewed as instances of self-sustained oscillators. Auto-oscillatory phenomena must involve a nonlinear friction function, and usually involve a nonlinear elastic function. With respect to rhythmic movements, the question is: What kinds of nonlinear friction and elastic functions are involved? The nonlinear friction functions of the kind identified by Rayleigh (involving terms such as \(\dot \theta ^3 \)) and van der Pol (involving terms such as \(\theta ^2 \dot \theta \)), and the nonlinear elastic functions identified by Duffing (involving terms such as \(\theta ^3 \)), constitute elementary nonlinear components for the assembling of self-sustained oscillators. Recently, additional elementary nonlinear friction and stiffness functions expressed, respectively, through terms such as \(\theta ^2 \dot \theta ^3 \) and \(\theta \dot \theta ^2 \), and a methodology for evaluating the contribution of the elementary components to any given cyclic activity have been identified. The methodology uses a quantification of the continuous deviation of oscillatory motion from ideal (harmonic) motion. Multiple regression of this quantity on the elementary linear and nonlinear terms reveals the individual contribution of each term to the oscillator's non-harmonic behavior. In the present article the methodology was applied to the data from three experiments in which human subjects produced pendular rhythmic movements under manipulations of rotational inertia (experiment 1), rotational inertia and frequency (experiment 2), and rotational inertia and amplitude (experiment 3). The analysis revealed that the pendular oscillators assembled in the three experiments were compositionally rich, braiding linear and nonlinear friction and elastic functions in a manner that depended on the nature of the task.
Similar content being viewed by others
References
Abraham RH, Shaw CD (1982) Dynamics: the geometry of behavior, part 1: Periodic behavior. Ariel Press, Santa Cruz, Calif.
Beek PJ, Beek WJ (1988) Tools for constructing dynamical models of rhythmic movement. Hum Mov Sci 7:301–342
Beek PJ, Turvey MT, Schmidt RC (1992) Autonomous and nonautonomous dynamics of coordinated rhythmic movements. Ecol Psychol 4:65–95
Beek PJ, Rikkert WEI, Wieringen PCW van (in press) Limit cycle properties of rhythmic forearm movements. J Exp Psychol Hum Percept Perform
Bingham GP, Schmidt RC, Turvey MT, Rosenblum LD (1991) Task dynamics and resource dynamics in the assembly of a coordinated rhythmic activity. J Exp Psychol Hum Percept Perform 17:359–381
Breeden JL, Hübler A (1990) Reconstructing equations of motion from experimental data with unobserved variables. Physiol Rev [A] 42:5817–5826
Cavagna GA (1977) Storage and utilization of elastic energy in skeletal muscle. Exerc Sport Sci Rev 5:89–129
Cremers J, Hübler A (1987) Construction of differential equations from experimental data. Z Naturforsch 42a: 797–802
Eisenhammer T, Hübler A, Packard N, Kelso JAS (1991) Modeling experimental time series with ordinary differential equations. Biol Cybern 65:107–112
Haken H, Kelso JAS, Bunz H (1985) A theoretical model of phase transitions in human hand movements. Biol Cybern 51:347–356
Kadar E, Schmidt RC, Turvey MT (1993) Constants underlying frequency changes in rhythmic movements. Biol Cybern 63:421–430
Kay BA, Kelso JAS, Saltzman EL, Schöner G (1987) Space-time behavior of single and bimanual rhythmical movements: data and limit cycle model. J Exp Psychol Hum Percept Perform 13:178–192
Kay BA, Saltzman ES, Kelso JAS (1991) Steady state and perturbed rhythmical movements: a dynamical analysis. J Exp Psychol Hum Percept Perform 17:183–197
Kelso JAS, Vatikiotis-Bateson E, Saltzman EL, Kay BA (1985) A qualitative dynamic analysis of reiterant speech production: phase portraits and dynamic modeling. J Acoust Soc Am 77:266–280
Kugler PN, Turvey MT (1987) Information, natural law and the selfassembly of rhythmic movement. Erlbaum, Hillsdale, NJ
Kugler PN, Turvey MT, Schmidt RC, Rosenblum LD (1990) Investigating a conservative invariant of motion in coordinated rhythmic movement. Ecol Psychol 2:151–189
Pearson CE (1976) Handbook of applied mathematics. Van NostrandReinhold, New York
Rosenblum LD, Turvey MT (1988) Maintenance tendency in coordinated rhythmic movements: relative fluctuations and phase. Neuroscience 27:289–300
Schmidt RC, Turvey MT (1992) Long term consistencies in assembling coordinated rhythmic movements. Hum Mov Sci 11:349–376
Schmidt RC, Beek PJ, Treffner PJ, Turvey MT (1991) Dynamical substructure of coordinated rhythmic movements. J Exp Psychol Hum Percept Perform 17:635–651
Schmidt RC, Shaw BS, Turvey MT (1993) Coupling dynamics in interlimb coordination. J Exp Psychol Hum Percept Perform 19:397–415
Schneider K, Zernicke RF, Schmidt RA, Hart TJ (1989) Changes in limb dynamics during the practice of rapid arm movements. J Biomech 22:805–817
Sternad D, Turvey MT, Schmidt RC (1992) Average phase difference theory and 1∶1 phase entrainment in interlimb coordination. Biol Cybern 67:223–231
Thompson JMT, Stewart HB (1987) Nonlinear dynamics and chaos. Wiley, New York
Turvey MT, Schmidt RC, Rosenblum LD, Kugler PN (1988) On the ime allometry of coordinated rhythmic movements. J Theor Biol 130:285–325
Vatikiotis-Bateson E, Kelso JAS (1993) Rhythm type and articulatory dynamics in English, French and Japanese. J Phon 21:231–265
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beek, P.J., Schmidt, R.C., Morris, A.W. et al. Linear and nonlinear stiffness and friction in biological rhythmic movements. Biol. Cybern. 73, 499–507 (1995). https://doi.org/10.1007/BF00199542
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00199542