A preliminary discrete model of a rope is considered both as a scleronomic and a rheonomic system. Numerical experiments are performed and advantages of the applied algorithm are discussed on the basis of energy conservation. The problem of discretization of the rope is presented in terms of efficient computational simulations. A wave-like effect is discussed with regard to energy transfer and velocity of the model tip. The next directions of the model development are outlined.
A discrete model of a rope with extensible members, involving a simple spring-mass conception, is considered. Lagrange's equations of motion are presented and their complexity is discussed from the computational point of view. Numerical experiments are performed for a system with both scleronomic and rheonomic constraints. Simulated behaviour of the model is analyzed mostly in terms of consequences of the extensible character of the system. Results validity is examined with use of basic energy principles.
A discrete model of a rope is developed and used to simulate the plane motion of the rope fixed at one end. Actually, two systems are presented, whose members are rigid but non-ideal joints involve elasticity or dissipation. The dissipation is reflected simply by viscous damping model, whereas the bending stiffness conception is based on the classical curvature-bending moment relationship for beams and simple geometrical formulas. Equations of motion are derived and their complexity is discussed from the computational point of view. Since modified extended backward differentiation formulas (MEBDF) of Cash are implemented to solve the resulting initial value problems, the technique scheme is outlined. Numerical experiments are performed and influences of the elasticity and damping on behaviour of the model are analyzed. Basic energy principles are used to verify the obtained results.
A discrete model of a rope with spiral springs in joints is considered, the aim being to include transverse elasticity of the rope. Elastic characteristic of the springs is derived on the basis of simple geometrical formulas and the classical curvature-bending moment relationship for beams. Lagrange's equations of motion are presented and their complexity is discussed from the computational point of view. Numerical experiments are performed for a system with both scleronomic and rheonomic constraints. The influence of the elasticity on behaviour of the model is analyzed. Results validity is examined in terms of basic energy principles.
In this work, planar free vibrations of a single physical pendulum are investigated both experimentally and numerically. The laboratory experiments are performed with pendula of different lengths, for a wide range of initial configurations, beyond the small angle regime. In order to approximate the air resistance, three models of damping are consideredinvolving the three components of the resistive force: linear (proportional to velocity), quadratic (velocitysquared) and acceleration-dependent (proportional to acceleration). A series of numerical experiments is discussed, in which the damping coefficients are estimated by means of several computational methods. Based on the observed efficiency, a gradient method for optimization is treated as the main tool for determination of a single set of damping parameters, independent of the system's initial position. In the model of resistive force, the term proportional to acceleration, associated with the empirical Morison equation, seems to be indispensable for the successful approximation of the real pendulum motion.
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