Hysteresis math model describing structural deformation of mechanisms for space vehicle docking

Rocket and space engineering


Yaskevich A. V.

e-mail: Andrey.Yaskevich@rsce.ru


Spacecraft mechanical joining during docking is provided by using specialized mechanisms of docking units. These mechanisms may be deformed with hysteresis caused by dissipation of mechanical energy due to internal structural friction. Their deformable elements are complex assemblies composed of a large quantity of parts differing in material and dimensions, or multistage rotation gears with backlashes. The development of a theoretical model of mechanical energy losses in such objects is practically impossible. Simple math models of deformations with hysteresis are considered in this paper. They are based on test data and do not explain the causes of the hysteresis phenomenon. These models are intended for the calculation of reaction forces and moments in mechanism and spacecraft motion equations, i.e. for the estimation of the mechanical energy dissipation effect on docking dynamics. Therefore they have to provide a high computing efficiency for reducing time on numerical integration of differential motion equations.

The hysteresis of mechanism stiffness is of a «double loop» type reflecting mechanical energy losses due to internal structural friction forces. Its math models with constant and variable parameters are considered by the example of a simple «probe-cone» type docking mechanism with independent linear and angular deformations. Hysteresis models for other types of mechanisms may be obtained similarly.

In the hysteresis model with constant parameters, forward and back loop branches for positive and negative deformations are presented with any necessary precision as sets of straight line segments numbered according to a deformation increase from zero to the maximal value of the permissible working range. Transitions between forward and back branches are described by single straight line segments with maximal slope coefficients. The state of deformation model is defined by the sign (positive or negative), branch type (forward, back or transitional) and straight line segment number that corresponds to a current deformation value. Model state changing conditions are described in a table form. Parameters of transitional branches, with a change of deformation velocity sign, are obtained from the solution of second order systems of linear equations.

Hysteresis math models with variable parameters describe deformations of rotation gears with friction clutches. Rotation of the input shaft of such clutch is nonreversible. Therefore arguments of the table description of the stiffness of its input gear are corrected according to the nonreversible rotation angle in case of any change of the rotation rate sign.

Parameters of the piecewise-linear hysteresis model are generally determined by static test data of a mechanism or its elements. Adjustment of the hysteresis model can be made by using dynamic test data in case direct static measurements are difficult or impossible to take.

Piecewise linear models of deformations with hysteresis are used to describe deformations of various docking mechanisms or their elements. They have a high computing efficiency and appropriate engineering accuracy. The paper presents a comparison of math modeling results and dynamic test data obtained at a 6-DOF hybrid facility, for a «probe-cone» spacecraft docking process. Their good coincidence is particularly ensued from taking into account deformations with hysteresis.

The above described hysteresis models based on test data may be used in various engineering applications wherever the effect of this phenomenon on a dynamic process needs to be estimated without explanation of its causes.


structural deformations, hysteresis math model, docking


  1. Syromiatnikov V.S. Stykovochnye ustroistva kosmicheskikh apparatov (Spacecraft docking units), Moscow, Mashinostroenie, 1984, 216 p.

  2. Augusto Visintin. Differential Models of Hysteresis (Applied Mathematical Sciences, vol. 111), Springer, 1994, 407p.

  3. Lapshin R.V. Analytical model for the approximation of hysteresis loop and its application to scanning tunneling microscope. — Review of Scientific Instruments, Vol. 66, No. 9, 1995, pp. 4718-4730.

  4. Mielke T., Roubícek A. Rate-Independent Model for Inelastic Behavior of Shape-Memory Alloys. — Multiscale Modeling & Simulation, Volume 1, Issue 4, 2003, pp. 571–597.

  5. Mayergoyz I.D. Mathematical Models of Hysteresis and their Applications: Second Edition (Electromagnetism). Elsevier, 2003, 499p.

  6. .Smith R. Smart material systems: model development, SIAM, 2005, 501 p.

  7. Lukichev A.A., Iljina V.V. Izvestiya Samarskogo nauchnogo tsentra RAN, 2011, vol. 13, no.4. pp. 39-44.

  8. Tompsett G. A., Krogh L., Griffin D. W., Conner W. C. Hysteresis and Scanning Behavior of Mesoporous Molecular Sieves. — Langmuir, 2005, 21 (18), pp. 8214–8225.

  9. Escolar J.D., Escolar A. Lung hysteresis: a morphological view. — Histology and hystopathalogy, 2004, No. 19, pp. 159-166.

  10. Ju.K. Ponomarev. Izvestiya Samarskogo nauchnogo tsentra RAN, 2011, vol. 13, no. 4(3), pp. 1192-1195.

  11. Popov E.P. Teoriya nelineinykh sistem avtomaticheskogo regulirovaniya i upravleniya (The Theory of Nonlinear Automation Control Systems) Moscow, Nauka, 1988, 256 p.

  12. A.V. Yaskevich. Izvestiya RAN. Kosmicheskie issledovaniya, 2007, vol. 45, no. 4, pp. 325-336.

  13. A Yaskevich. Real time simulation of contact interaction during spacecraft docking and berthing. — Journal of Mechanical Engineering and Automation, Vol.4, No. 1, January 2014, pp. 1-15.


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