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1.
In the 1950s and 1960s, Charles F. Taylor designed and tested various prototype one-wheeled vehicles. These machines were
stabilized and steered using gyroscopes. In this paper, a simple model of a one-wheeled vehicle is presented and analyzed.
This analysis explains the ability of these machines to exhibit stable steady motions.
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2.
M. Batista 《Regular and Chaotic Dynamics》2008,13(4):344-354
In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick
rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The
solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between
a disk and a plane and the study of disk colliding motion.
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3.
S. Benenti 《Regular and Chaotic Dynamics》2008,13(4):283-315
The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing
the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation
of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential
equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers
is avoided). Various examples are illustrated.
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S. A. Chaplygin 《Regular and Chaotic Dynamics》2008,13(4):369-376
This classical paper by S.A. Chaplygin presents a part of his research in non-holonomic mechanics. In this paper, Chaplygin
suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the
“reducing-multiplier method”. The method is illustrated on two concrete problems from non-holonomic mechanics. This paper
produced a considerable effect on the further development of the Russian non-holonomic community. With the help of Chaplygin’s
reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as
Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated
at its true worth by the English-speaking world.
This publication contributes to the series of RCD translations of Chaplygin’s scientific heritage. In 2002 we published two
of his works (both cited in this one) in the special issue dedicated to non-holonomic mechanics (RCD, Vol. 7, no. 2). These
translations along with translations of his other two papers on hydrodynamics (RCD, Vol. 12, nos. 1,2) aroused considerable
interest and are broadly cited by modern researches.
Originally published in: Matematicheskiĭ sbornik (Mathematical Collection), 1911, vol. 28, issue 1. The content of §§ 2 and 3 of this study was presented at the session of
the Moscow Mathematical Society on December 11, 1906. 相似文献
8.
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the
surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady
rotations of the ball at the upmost, downmost and saddle point.
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9.
Rubber rolling over a sphere 总被引:2,自引:2,他引:0
“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied
by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures
of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration
space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for
rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G
2). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling
of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S
2 and base S
2, that can be reduced to an almost Hamiltonian system in T*S
2 with a non-closed 2-form ωNH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of
radius b (unequal moments of inertia I
j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ωNH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular
there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for
p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates
separates the Hamiltonian in this case. No other integrable cases with different I
j are known.
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10.
研究非完整系统动力学的一类逆问题·给出非完整系统的运动方程及其显式,考虑一类仅受齐次非完整约束的力学系统的Szebehely问题,研究已知一类第一积分的一般非完整系统的情形·最后举例说明其应用· 相似文献
11.
We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without
slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for
one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits
a separation of variables and reduction to Abel-Jacobi quadratures. The separating variables that we found appear to be a
non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable
problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.
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12.
非完整非保守力学系统在相空间的Lie对称性与守恒量 总被引:2,自引:2,他引:0
在相空间引入无限小变换,研究非完整非保守力学系统运动微分方程的不变性和守恒量。建立Lie对称确定方程,得到Lie对称的结构方程和守恒量形式,并举例说明结果的应用。 相似文献
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Summary Stokesian swimming is a geometric exercise, a collective game. In Part I, we review Shapere and Wilczek's gauge-theoretical
approach for a single organism. We estimate the speeds of organisms moving by propagating small amplitude waves, and we make
a conjecture regarding a new inequality for the Stokes' curvature. In Part II, we extend the gauge theory to collective motions.
We advocate the influx of nonlinear control theory and subriemannian geometry. Computationally, parallel algorithms are natural,
each microorganism representing a separate processor. In the final section, open questions motivated by biology are presented.
Dedicated to the memory of Juan C. Simo, a pioneer in the use of geometry to produce better analytical and numerical methods
in mechanics
This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo. 相似文献
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二阶非Четаев型非完整系统的守恒律 总被引:5,自引:2,他引:5
研究二阶非ЧeTaeB型完整系统的守恒律,引入Jourdain生成元,给出无限小变换下Jour-dain原理的不变性条件,在一定条件下得到系统的守恒律,并举例说明结果的应用。 相似文献
18.
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the
contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero.
This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated
and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure
for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a
point on a sphere in some potential field. The integrable cases are found.
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19.
本文给出高阶非完整系统运动方程的一类积分及其存在条件,包括1阶积分(广义能量积分),2阶积分和p(p>2)阶积分,所有这些积分都可按系统的Lagrange函数来构造.举例说明本文方法的应用. 相似文献
20.
讨论了非完整性多体编队运动问题.首先利用动态反馈将单个体的动力学模型线性化为2个三阶输入输出积分链的形式;然后提出一种带有单个体间阻尼注入的非连续分布式控制律,并利用Liapunov方法证明了在该控制律作用下闭环系统的渐进稳定性;最后通过一个平面机器人的编队运动仿真验证了所提方法的有效性. 相似文献