Biological Cybernetics

, Volume 97, Issue 1, pp 63–79 | Cite as

High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models

  • Michael Günther
  • Syn Schmitt
  • Veit Wank
Original Paper


High-frequency vibrations e.g., induced by legs impacting with the ground during terrestrial locomotion can provoke damage within tendons even leading to ruptures. So far, macroscopic Hill-type muscle models do not account for the observed high-frequency damping at low-amplitudes. Therefore, former studies proposed that protective damping might be explained by modelling the contractile machinery of the muscles in more detail, i.e., taking the microscopic processes of the actin–myosin coupling into account. In contrast, this study formulates an alternative hypothesis: low but significant damping of the passive material in series to the contractile machinery—e.g., tendons, aponeuroses, titin—may well suffice to damp these hazardous vibrations. Thereto, we measured the contraction dynamics of a piglet muscle–tendon complex (MTC) in three contraction modes at varying loads and muscle–tendon lengths. We simulated all three respective load situations on a computer: a Hill-type muscle model including a contractile element (CE) and each an elastic element in parallel (PEE) and in series (SEE) to the CE pulled on a loading mass. By comparing the model to the measured output of the MTC, we extracted a consistent set of muscle parameters. We varied the model by introducing either linear damping in parallel or in series to the CE leading to accordant re-formulations of the contraction dynamics of the CE. The comparison of the three cases (no additional damping, parallel damping, serial damping) revealed that serial damping at a physiological magnitude suffices to explain damping of high-frequency vibrations of low amplitudes. The simulation demonstrates that any undamped serial structure within the MTC enforces SEE-load eigenoscillations. Consequently, damping must be spread all over the MTC, i.e., rather has to be de-localised than localised within just the active muscle material. Additionally, due to suppressed eigenoscillations Hill-type muscle models taking into account serial damping are numerically more efficient when used in macroscopic biomechanical neuro-musculo-skeletal models.


Isometric Contraction Isometric Force Contractile Element Gastrocnemius Lateralis Pennation Angle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Muscle–tendon complex


Contractile element


Parallel element


Parallel elastic element


Serial element


Serial elastic element


M. soleus


M. flexor digitorum superficialis


M. gastrocnemius medialis


M. gastrocnemius lateralis


normalised muscle activation


minimum value of q


time derivative of q


time constant of rising activation


ratio between τ q and time constant of falling activation


Muscle stimulation


length of CE

\(\dot{l}_{{\rm CE}} = v_{{\rm CE}}\)

contraction velocity of CE


mean anatomical length of MTC


length of model MTC


typical length of model MTC

\(\dot{l}_{{\rm m}}\)

velocity of model MTC


force of model MTC


force of SEE


force of PEE


force of CE


length of SE


rest length of SEE


length of SEE at non-linear-linear transition in F SEE(l SE)


relative stretch at non-linear-linear transition in F SEE(l SE)


force at non-linear-linear transition in F SEE(l SE)


relative stretch in linear part for force increase ΔF SEE,0


stiffness of the linear part of F SEE(l SE)


exponent of F SEE(l SE) in the non-linear part


factor of non-linearity in F SEE(l SE)


normalised isometric force–length relation of CE


optimal fibre length


exponent of F isom(l CE) on either ascending or descending limb


width of F isom(l CE) on either ascending or descending limb


maximum isometric force


coordinate of pole in \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) normalised to current isometric force F max q F isom(l CE)


maximum value of A rel


coordinate of pole in \(F_{{\rm CE}}(\dot{l}_{CE})\) normalised to l CE,opt


maximum value of B rel


concentric contraction velocity at F CE = 0


maximum concentric contraction velocity

\(L_{A_{{\rm rel}}}\)

length dependency of A rel

\(L_{B_{{\rm rel}}}\)

length dependency of B rel

\(Q_{A_{{\rm rel}}}\)

activation dependency of A rel

\(Q_{B_{{\rm rel}}}\)

activation dependency of B rel


rest length of PEE


exponent of F PEE(l CE )


factor of non-linearity in F PEE(l CE)

\({\mathcal{F}}_{{\rm PEE}}\)

force of PEE if l CE is stretched to ΔW limb=des

\({\mathcal{L}}_{{\rm PEE},0}\)

rest length of PEE normalised to l CE,opt


numerical limit for defining zero F isom(l CE)

d V d Fcon

inclination of linear concentric continuation of \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) for F CE < 0


step in inclination of \(F_{{\rm CE}}(\dot{l}_{{\rm CE}} = 0)\) between eccentric and concentric force–velocity relation(s)

\({\mathcal{F}}_{{\rm ecc}}\)

coordinate of pole in \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) normalised to F max q F isom(l CE) for \(\dot{l}_{{\rm CE}} > 0\)


force where linear continuation of eccentric \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) relation starts


velocity where linear continuation of eccentric \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) relation starts

d V d Fecc

inclination of linear eccentric continuation of \(\dot{l}_{{\rm CE}}(F_{{\rm CE}})\) for F CE > F trans


length of PEE ( = l CE)


(constant) damping coefficient of PE


damping coefficient of SE


maximum value in d SE(l CE, q)


minimum value of d SE normalised to d SE,max


dimensionless factor to scale d SE,max

\(\dot{l}_{{\rm SE}} = v_{{\rm SE}}\)

contraction velocity of SE




vector of gravitational acceleration


modelled external damping


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alexander RMcN (1988) Elastic mechanisms in animal movement. Cambrigde University Press, CambridgeGoogle Scholar
  2. Alexander RMcN (2001) Damper for bad vibrations. Nature 414(6866):855–857PubMedCrossRefGoogle Scholar
  3. Biewener AA (1990) Biomechanics of mammalian terrestrial locomotion. Science 250(4984):1097–1103PubMedCrossRefGoogle Scholar
  4. Blickhan R (1989) The spring-mass model for running and hopping. J Biomech 22(11/12):1217–1227PubMedCrossRefGoogle Scholar
  5. Carrion-Vazquez M, Oberhauser AF, Fowler SB, Marszalek PE, Broedel SE, Clarke J, Fernandez JM (1999) Mechanical and chemical unfolding of a single protein: a comparison. Proc Natl Acad Sci USA 96(7):3694–3699PubMedCrossRefGoogle Scholar
  6. Cavagna GA (1970) Elastic bounce of the body. J Appl Physiol 29(3):279–282PubMedGoogle Scholar
  7. Chow JW, Darling WG (1999) The maximum shortening velocity of muscle should be scaled with activation. J Appl Physiol 86(3):1025–1031PubMedGoogle Scholar
  8. Currey JD (2002) Bones: structure and mechanics. Princeton University Press, Princeton, NJGoogle Scholar
  9. Davy DT, Audu ML (1987) A dynamic optimization technique for predicting muscle forces in the swing phase of gait. J Biomech 20(2):187–201PubMedCrossRefGoogle Scholar
  10. Denoth J (1985) The dynamic behaviour of a three link model of the human body during impact with the ground. In: Winter DA, Norman RW, Wells RP, Hayes KC, Patla AE (eds) Biomechanics 9-A, vol 5B of International Series on Biomechanics. Human Kinetics Publishers, Champaign, pp 102–106Google Scholar
  11. Denoth J, Stüssi E, Csucs G, Danuser G (2002) Single muscle fiber contraction is dictated by inter-sarcomere dynamics. J Theor Biol 216(1):101–122PubMedCrossRefGoogle Scholar
  12. Epstein M, Herzog W (2003) Aspects of skeletal muscle modelling. Philos Trans R Soc Lond B 358(1437):1445–1452CrossRefGoogle Scholar
  13. Ettema GJ, Meijer K (2000) Muscle contraction history: modified Hill versus an exponential decay model. Biol Cybern 83(6):491–500PubMedCrossRefGoogle Scholar
  14. Gordon AM, Huxley AF, Julian FJ (1966) The variation in isometric tension with sarcomere length in vertebrate muscle fibers. J Physiol 184:170–192PubMedGoogle Scholar
  15. Granzier HL, Labeit S (2006) The giant muscle protein titin is an adjustable molecular spring. Exerc Sport Sci Rev 34(2):50–53PubMedCrossRefGoogle Scholar
  16. Gruber K, Ruder H, Denoth J, Schneider K (1998) A comparative study of impact dynamics: wobbling mass model versus rigid body models. J Biomech 31(5):439–444PubMedCrossRefGoogle Scholar
  17. Günther M, Ruder H (2003) Synthesis of two-dimensional human walking: a test of the λ-model. Biol Cybern 89(2):89–106PubMedCrossRefGoogle Scholar
  18. Günther M, Sholukha VA, Keßler D, Wank V, Blickhan R (2003) Dealing with skin motion and wobbling masses in inverse dynamics. J Mech Med Biol 3(3/4):309–335CrossRefGoogle Scholar
  19. Hatze H (1977) A myocybernetic control model of skeletal muscle. Biol Cybern 25:103–119PubMedCrossRefGoogle Scholar
  20. Hatze H (1981) Myocybernetic control models of skeletal muscle—characteristics and applications. University of South Africa Press, PretoriaGoogle Scholar
  21. Hill AV (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc Lond B 126:136–195Google Scholar
  22. Julian FJ (1971) The effect of calcium on the force-velocity relation of briefly glycerinated frog muscle fibres. J Physiol 218:117–145PubMedGoogle Scholar
  23. Katz B (1939) The relation between force and speed in muscular contraction. J Physiol 96:45–64PubMedGoogle Scholar
  24. Ker RF (1981) Dynamic tensile properties of the plantaris tendon of sheep (Ovis aries). J Exp Biol 93:283–302PubMedGoogle Scholar
  25. Ker RF, Wang XT, Pike AV (2000) Fatigue quality of mammalian tendons. J Exp Biol 203(Pt 8):1317–1327PubMedGoogle Scholar
  26. Ker RF, Zioupos P (1997) Creep and fatigue damage of mammalian tendon and bone. Comments Theor Biol 4(2-3):151–181Google Scholar
  27. Kistemaker DA, van Soest AJ, Bobbert MF (2006) Is equilibrium point control feasible for fast goal-directed single-joint movements? J Neurophysiol 95(5):2898–2912PubMedCrossRefGoogle Scholar
  28. Krause PC, Choi JS, McMahon TA (1995) The force-velocity curve in passive whole muscle is asymmetric about zero velocity. J Biomech 28(9):1035–1043PubMedCrossRefGoogle Scholar
  29. Krieg M (1992) Simulation und Steuerung biomechanischer Mehrkörpersysteme. Master’s thesis. Eberhard-Karls-Universität, Tübingen, GermanyGoogle Scholar
  30. Lan G, Sun SX (2005) Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation. Biophys J 88(6):4107–4117PubMedCrossRefGoogle Scholar
  31. Lombardi V, Piazzesi G, Ferenczi MA, Thirlwell H, Dobbie I, Irving M (1995) Elastic distortion of myosin heads and repriming of the working stroke in muscle. Nature 374(6522):553–555PubMedCrossRefGoogle Scholar
  32. Lombardi V, Piazzesi G, Reconditi M, Linari M, Lucii L, Stewart A, Sun YB, Boesecke P, Narayanan T, Irving T, Irving M (2004) X-ray diffraction studies of the contractile mechanism in single muscle fibres. Philos Trans R Soc Lond B 359(1452):1883–1893CrossRefGoogle Scholar
  33. Marszalek PE, Lu H, Li H, Carrion-Vazquez M, Oberhauser AF, Schulten K, Fernandez JM (1999) Mechanical unfolding intermediates in titin modules. Nature 402(6757):100–103PubMedCrossRefGoogle Scholar
  34. McMahon TA, Cheng GC (1990) The mechanics of running: how does stiffness couple with speed? J Biomech 23(Suppl. 1):65–78PubMedCrossRefGoogle Scholar
  35. Meijer K, Grootenboer HJ, Koopman HF, van der Linden BJ, Huijing PA (1998) A Hill type model of rat medial gastrocnemius muscle that accounts for shortening history effects. J Biomech 31(6):555–563PubMedCrossRefGoogle Scholar
  36. Minajeva A, Kulke M, Fernandez JM, Linke WA (2001) Unfolding of titin domains explains the viscoelastic behavior of skeletal myofibrils. Biophys J 80(3):1442–1451PubMedGoogle Scholar
  37. Pain MTG, Challis JH (2006) The influence of soft tissue movement on ground reaction forces, joint torques and reaction forces in drop landings. J Biomech 39(1):119–124PubMedCrossRefGoogle Scholar
  38. Petrofsky JS, Phillips CA (1981) The influence of temperature, initial length and electrical activity on force-velocity relationship of the medial gastrocnemius muscle of the cat. J Biomech 14(5):297–306PubMedCrossRefGoogle Scholar
  39. Piazzesi G, Lombardi V (1995) A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle. Biophys J 68(5):1966–1979PubMedCrossRefGoogle Scholar
  40. Piazzesi G, Lombardi V (1996) Simulation of the rapid regeneration of the actin-myosin working stroke with a tight coupling model of muscle contraction. J Muscle Res Cell Motil 17(1):45–53PubMedCrossRefGoogle Scholar
  41. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1994) Numerical recipes in C—the art of scientific computing, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  42. Proske U, Morgan DL (1987) Tendon stiffness: methods of measurement and significance for the control of movement. A review. J Biomech 20(1):75–82CrossRefGoogle Scholar
  43. Reconditi M, Linari M, Lucii L, Stewart A, Sun YB, Boesecke P, Narayanan T, Fischetti RF, Irving T, Piazzesi G, Irving M, Lombardi V (2004) The myosin motor in muscle generates a smaller and slower working stroke at higher load. Nature 428(6982):578–581PubMedCrossRefGoogle Scholar
  44. Reinsch CH (1967) Smoothing by spline functions. Numerische Mathematik 10(3):177–183CrossRefGoogle Scholar
  45. Riemersma DJ, Schamhardt HC (1985) In vitro mechanical properties of equine tendons in relation to cross-sectional area and collagen content. Res Vet Sci 39(3):263–270PubMedGoogle Scholar
  46. Rode C, Siebert T, Herzog W, Blickhan R (2006) The effects of parallel and series elastic components on estimated active muscle force (submitted to the J Biomech)Google Scholar
  47. Schmalz T (1993a) Biomechanische Modellierung menschlicher Bewegung, vol 26 of Wissenschaftliche Schriftenreihe des Deutschen Sportbundes. Karl Hofmann, Schorndorf, GermanyGoogle Scholar
  48. Schmalz T (1993b) Die Nutzung biomechanischer Modelle zur Bestimmung rheologischer Eigenschaften des Muskel-Sehnen-Komplexes. In: Gutewort W, Schmalz T, Weiß T (eds), Symposium Oberhof: Aktuelle Hauptforschungsrichtungen der Biomechanik sportlicher Bewegungen, vol 55, Sankt Augustin, Germany, 1993. Deutsche Vereinigung für Sportwissenschaft (dvs), Academia, pp 102–108Google Scholar
  49. Shadwick RE (1990) Elastic energy storage in tendons: mechanical differences related to function and age. J Appl Physiol 68(3):1033–1040PubMedCrossRefGoogle Scholar
  50. Shampine LF, Gordon MK (1975) Computer solution of ordinary differential equations: the initial value problem. W.H. Freeman & Co., San FranciscoGoogle Scholar
  51. Siebert T, Wagner H, Blickhan R (2003) Not all oscillations are rubbish: forward simulation of quick-release experiments. J Mech Med Biol 3(1):107–122CrossRefGoogle Scholar
  52. Stern JT (1974) Computer modeling of gross muscle dynamics. J Biomech 7:411–428PubMedCrossRefGoogle Scholar
  53. Telley IA, Denoth J, Ranatunga KW (2003) Inter-sarcomere dynamics in muscle fibres. A neglected subject? Adv Exp Med Biol 538:481–500Google Scholar
  54. Tskhovrebova L, Trinick J (2002) Role of titin in vertebrate striated muscle. Philos Trans R Soc Lond B 357(1418):199–206CrossRefGoogle Scholar
  55. van Ingen Schenau GJ (1984) An alternative view to the concept of utilization of elastic energy. Hum Mov Sci 3:301–336CrossRefGoogle Scholar
  56. van Leeuwen JL (1992) Muscle function in locomotion. In: Alexander RMcN (ed) Advances in comparative and environmental physiology, vol 11, chap 7. Springer, Berlin, pp 191–250Google Scholar
  57. van Soest AJ (1992) Jumping from structure to control: a simulation study of explosive movements. PhD thesis, Vrije Universiteit, AmsterdamGoogle Scholar
  58. van Soest AJ, Bobbert MF (1993) The contribution of muscle properties in the control of explosive movements. Biol Cybern 69(3):195–204PubMedCrossRefGoogle Scholar
  59. Wakeling JM, Nigg BM (2001) Soft-tissue vibrations in the quadriceps measured with skin mounted transducers. J Biomech 34(4):539–543PubMedCrossRefGoogle Scholar
  60. Wank V (2000) Aufbau und Anwendung von Muskel-Skelett-Modellen. Habilitationsschrift der Friedrich-Schiller-Universität JenaGoogle Scholar
  61. Wank V, Bauer R, Walter B, Kluge H, Fischer MS, Blickhan R, Zwiener U (2000) Accelerated contractile function and improved fatigue resistance of calf muscles in newborn piglets with IUGR. Am J Physiol Regul Integr Comp Physiol 278(2):R304–R310PubMedGoogle Scholar
  62. Wank V, Fischer MS, Walter B, Bauer R (2006) Muscle growth and fiber type composition in hind limb muscles during postnatal development in pigs. Cells Tissues Organs 182(3-4):171–181PubMedCrossRefGoogle Scholar
  63. Wilson AM, Goodship AE (1994) Exercise-induced hyperthermia as a possible mechanism for tendon degeneration. J Biomech 27(7):899–905PubMedCrossRefGoogle Scholar
  64. Wilson AM, McGuigan MP, Su A, van den Bogert AJ (2001) Horses damp the spring in their step. Nature 414(6866):895–899PubMedCrossRefGoogle Scholar
  65. Zajac FE (1989) Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. In: Bourne JR (ed) CRC critical reviews in biomedical engineering, vol 17. CRC Press, Boca Raton, pp 359–411Google Scholar
  66. Zioupos P, Currey JD, Casinos A (2001) Tensile fatigue in bone: are cycles-, or time to failure, or both, important? J Theor Biol 210(3):389–399PubMedCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut für Astronomie und Astrophysik, Abteilung Theoretische Astrophysik, Biomechanik-GruppeEberhard-Karls-UniversitätTübingenGermany
  2. 2.Orthopädische Klink, BiomechaniklaborEberhard-Karls-UniversitätTübingenGermany
  3. 3.Institut für Sportwissenschaft, Lehrstuhl für BewegungswissenschaftFriedrich-Schiller-UniversitätJenaGermany
  4. 4.Institut für Sportwissenschaft, Arbeitsbereich IIIEberhard-Karls-UniversitätTübingenGermany
  5. 5.Institut für Sport und SportwissenschaftAlbert-Ludwigs-UniversitätFreiburgGermany

Personalised recommendations