Ideally rigid objects establish sustained contact with one another via complete chatter (a.k.a. Zeno behavior), i.e. an infinite sequence of collisions accumulating in finite time. Alternatively, such systems may also exhibit a finite sequence of collisions followed by separation (sometimes called incomplete chatter). Earlier works concerning the chattering of slender rods in two dimensions determined the exact range of model parameters, where complete chatter is possible. We revisit and slightly extend these results. Then the bulk of the paper examines the chattering of threedimensional objects with multiple points hitting an immobile plane almost simultaneously. In contrast to rods, the motion of these systems is complex, nonlinear, and sensitive to initial conditions and model parameters due to the possibility of various impact sequences. These difficulties explain why we model this phenomenon as a nondeterministic discrete dynamical system. We simplify the analysis by assuming linearized kinematics, frictionless interaction, by neglecting the effect of external forces, and by investigating objects with rotational symmetry. Application and extension of the theory of common invariant cones of multiple linear operators enable us to find sufficient conditions of the existence of initial conditions, which give rise to complete chatter. Additional analytical and numerical investigations predict that our sufficient conditions are indeed exact, moreover solving a simple eigenvalue problem appears to be enough to judge the possibility of complete chatter.
The motion of a rigid, spinning disk on a flat surface ends with a dissipation-induced finite-time singularity. The problem of finding the dominant energy absorption mechanism during the last phase of the motion generated a lively debate during the past two decades. Various candidates including air drag and various types of friction have been considered, nevertheless impacts have not been examined until now. We investigate the effect of impacts caused by geometric imperfections of the disk and of the underlying flat surface, through analysing the dynamics of polygonal disks with unilateral point contacts. Similarly to earlier works, we determine the rate of energy absorption under the assumption of a regular pattern of motion analogous to precession-free motion of a rolling disk. In addition, we demonstrate that the asymptotic stability of this motion depends on parameters of the impact model. In the case of instability, the emerging irregular motion is investigated numerically. We conclude that there exists a range of model parameters (small radii of gyration or small restitution coefficients) in which absorption by impacts dominates all preiously investigated mechanisms during the last phase of motion. Nevertheless the parameter values associated with a homogenous disk on a hard surface are typically not in this range, hence the effect of impacts is in that case not dominant.
This paper presents a partial reconstruction of the rotational dynamics of the Philae spacecraft upon landing on comet 67P/Churyumov-Gerasimenko as part of ESA's Rosetta mission. We analyze the motion and the events triggered by the failure to fix the spacecraft to the comet surface at the time of the first touchdown. Dynamic trajectories obtained by numerical simulation of a 7 degreeof-freedom mechanical model of the spacecraft are fitted to directions of incoming solar radiation inferred from in-situ measurements of the electric power provided by the solar panels. The results include a lower bound of the angular velocity of the lander immediately after its first touchdown. Our study also gives insight into the effect of the programmed turn-off of the stabilizing gyroscope after touchdown; the important dynamical consequences of a small collision during Philae's journey; and the probability that a similar landing scenario harms the operability of this type of spacecraft.
Elasticity ellipses or central ellipses have been long used in graphic statics to capture the elastic behaviour of structural elements. The paper gives a generalisation the concept both in dimensions and in the possibility of degenerate conics/quadrics. The effect of projective transformations of these quadrics is also given, such that the entire mechanical system can be transformed preserving equilibrium and compatibility between its elements.
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