New exact solution of Euler’s equations (rigid body dynamics) in the case of rotation over the fixed point

A new exact solution of Euler’s equations (rigid body dynamics) is presented here. All the components of angular velocity of rigid body for such a solution differ from both the cases of symmetric rigid rotor (which has two equal moments of inertia: Lagrange’s or Kovalevskaya’s case), and from the Euler’s case when all the applied torques are zero, or from other well-known particular cases. The key features are the next: the center of mass of rigid body is assumed to be located at meridional plane along the main principal axis of inertia of rigid body, besides, the principal moments of inertia are assumed to satisfy to a simple algebraic equality. Also, there is a restriction at choosing of initial conditions. Such a solution is also proved to satisfy to Euler–Poinsot equations, including invariants of motion and additional Euler’s invariant (square of the vector of angular momentum is a constant). So, such a solution is a generalization of Euler’s case.     

External stability of resonances in the motion of an asymmetric rigid body with a strong magnet in the geomagnetic field

We consider a precession motion, close to the classical Lagrange case, of an asymmetric rigid body with a strong magnet in an orbit in the geomagnetic field. For the principal moment we take the restoring torque due to the interaction between the planet magnetic fields and the rigid body. The perturbing actions are due to small moments of the rigid body mass-inertial asymmetry and small constant moments. We show that these perturbations result in the realization of secondary resonance effects in the rotational motion of the rigid body caused by the influence of resonance denominators in higher-order approximations of the averaging method. These effects were discovered in the study of rotational motion of a satellite with a magnetic damper in the nearly Euler case. In the present paper, we analyze both the secondary resonance effects themselves and the external stability of resonances. We obtain conditions ensuring a decrease in the angular velocity of the rigid body rotation about its center of mass. We also discover several new laws of influence of resonances on the nonresonance evolution of slow variables, which is related to the appearance of stable resonances.     

The Method of Averaging for Euler's Equations of Rigid Body Motion

We formulate the method of averaging for perturbations of Euler's equations of rotational motion. Euler's equations are three strongly nonlinear coupled differential equations that can be viewed as a three dimensional oscillator. The method of averaging is used to determine the long-term influence of perturbation terms on the motion by averaging about the nominal rigid body motion. The treatment is applicable to a large class of motions including precession with large nutation – it is not restricted to small motions about simple spins or nearly axi-symmetric bodies. Three examples are shown that demonstrate the accuracy of the method's predictions.     

Melnikov's method for rigid bodies subject to small perturbation torques

Summary In this paper, the global motion of rigid bodies subjected to small perturbation torques, either conservative or dissipative, is investigated by means of Melnikov's method. Deprit's variables are introduced to transform the equations of motion into a standard form which is rendered suitable for the application of Melnikov's method. The Melnikov method is used to predict the transversal intersections of stable and unstable manifolds for the pertubed rigid-body motion. The chosen examples are a self-excited rigid body subject to a small periodic torque in a viscous medium, and the heavy rigid body. It is shown in both cases that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.     

Dynamic flight stability of hovering insects

The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the “rigid body” assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects (hoverfly, cranefly, dronefly and hawkmoth) in hovering flight is studied (the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157 Hz). The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the “rigid body” assumption is tested and how differences in size and wing kinematics influence the applicability of the “rigid body” assumption is investigated. The primary findings are: (1) For insects considered in the present study and those with relatively high wingbeat frequency (hoverfly, drone fly and bumblebee), the “rigid body” assumption is reasonable, and for those with relatively low wingbeat frequency (cranefly and howkmoth), the applicability of the “rigid body” assumption is questionable. (2) The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. (3) Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative M _u (pitching moment produced by unit horizontal speed) as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Z _w (vertical force produced by unit vertical speed) as the primary source of the stable slow subsidence mode. The project supported by the National Natural Science Foundation of China (10232010 and 10472008).     

Construction of optimal laws of variation of the angular momentum vector of a rigid body

We consider the problem of constructing optimal preset laws of variation of the angular momentum vector of a rigid body taking the body from an arbitrary initial angular position to the required terminal angular position in a given time. We minimize an integral quadratic performance functional whose integrand is a weighted sum of squared projections of the angular momentum vector of the rigid body. We use the Pontryagin maximum principle to derive necessary optimality conditions. In the case of a spherically symmetric rigid body, the problem has a well-known analytic solution. In the case where the body has a dynamic symmetry axis, the obtained boundary value optimization problem is reduced to a system of two nonlinear algebraic equations. For a rigid body with an arbitrarymass distribution, optimal control laws are obtained in the form of elliptic functions. We discuss the laws of controlled motion and applications of the constructed preset laws in systems of attitude control by external control torques or rotating flywheels.     

漂浮基柔性两杆空间机械臂的关节运动鲁棒控制与柔性振动最优控制Robust joint motion control and vibration optimal control for a free-floating two-flexible-link space manipulator

讨论了载体位置、姿态均不受控情况下,具有有界干扰及有界未知参数的漂浮基柔性两杆空间机械臂的具有鲁棒性的关节运动控制与柔性振动最优控制算法设计问题。首先选择合理的联体坐标系,利用拉格朗日方程并结合动量守恒原理得到漂浮基柔性两杆空间机械臂系统的动力学方程。通过合理选择联体坐标系与利用奇异摄动理论,实现了两个柔性杆柔性振动之间、关节运动与两柔性杆柔性振动的解耦,得到了柔性两杆空间机械臂的慢变子系统与柔性臂快变子系统。针对两个子系统设计相应的控制规律,即增广鲁棒慢变子系统控制律与柔性臂快变子系统最优控制律,这两个相应的子系统控制规律综合到一起构成飘浮基柔性两杆空间机械臂总的关节运动与臂柔性振动控制的组合控制律。系统的数值仿真证实了方法的有效性。该控制方案不需要直接测量漂浮基的位置、移动速度和移动加速度。     

On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector

In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector whose components are ℓ _i (i=1,2,3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter ε (Elfimov in PMM, 42(2):251–258, [1978]). The equations of motion will be studied under certain initial conditions of motion. The Poincaré small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, [1959]; Nayfeh in Perturbation methods, Wiley-Interscience, New York, [1973]) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler’s angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

Dissipation in systems of coupled Lagrange gyroscopes

A system of coupled symmetric solids (Lagrange gyroscopes) moving under the simultaneous action of dissipative and nonconservative positional forces is analyzed for stability. The equations of perturbed motion are transformed so that nonconservative terms are eliminated and the Thomson-Tait-Chetaev theorems can be applied.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 127–135, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Averaging Theory of Arbitrary Order for Piecewise Smooth Differential Systems and Its Application

The averaging theory for studying periodic orbits of smooth differential systems has a long history. Whereas the averaging theory for piecewise smooth differential systems appeared only in recent years, where the unperturbed systems are smooth. When the unperturbed systems are only piecewise smooth, there is not an existing averaging theory to study existence of periodic orbits of their perturbed systems. Here we establish such a theory for one dimensional perturbed piecewise smooth periodic differential equations. Then we show how to transform planar perturbed piecewise smooth differential systems to one dimensional piecewise smooth periodic differential equations when the unperturbed planar piecewise smooth differential systems have a family of periodic orbits. Finally as application of our theory we study limit cycle bifurcation of planar piecewise differential systems which are perturbation of a \(\Sigma \)-center.     

Analytical solutions for dynamics of dual-spin spacecraft and gyrostat-satellites under magnetic attitude control in omega-regimes

The attitude dynamics of a dual-spin spacecraft (a gyrostat with one rotor) with magnetic actuators attitude control is considered in the constant external magnetic field at the presence of the spacecraft’s own magnetic dipole moment, which is created proportionally to the angular velocity components (this motion regime can be called as “the omega-regime” or “the omega-maneuver”). The research of the dual-spin spacecraft angular motion under the action of the magnetic restoring torque is fulfilled in the generalized formulation close to the classical mechanics’ task of the heavy body/gyrostat motion in the Lagrange top. Analytical exact solutions of differential equations of the motion are obtained for all parameters in terms of elliptic integrals and the Jacobi functions. New obtained analytical solutions can be classified as results developing the classical fundamental problem of the rigid body and gyrostat motion around the fixed point. The technical application of the omega-regime to the angular reorientation of the spacecraft longitudinal axis along the angular momentum vector is considered.     

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Δ E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt–Miranker method indeed reflect the behavior of the exact solutions of the O Δ Es. For that reason, a version of the multiple scales perturbation method for O Δ Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Δ Es and to singularly perturbed, linear O Δ Es.     

Some Primitive Concepts in Continuum Mechanics Regarded in Terms of Objective Space-Time Molecular Averaging: The Key Rôle Played by Inertial Observers

In Continuum Mechanics the notions of body, material point, and motion, are primitive. Here these concepts are derived for any (possibly time-dependent) material system via mass and momentum densities whose values are local spacetime averages of molecular quantities. The averaging procedure necessary to ensure molecular-based densities can be agreed upon by all observers (that is, are objective) has implications for constitutive relations. Specifically, such relations should first be expressed in terms of Galilean-invariant functions of the motion relative to an inertial frame. Thereafter such relations can be re-phrased for general observers, thereby yielding general-frame constitutive relations compatible with material frame-indifference. Two postulates concerning observer agreement (which together constitute a statement of material frame-indifference) are shown to imply that any stress response function which is assumed to depend upon the motion in an inertial (general) frame must be Galilean-invariant (invariant under superposed rigid body motions). Accordingly, invariance under superposed rigid body motions is not a fundamental tenet of continuum physics, but rather a consequence of material frame-indifference whenever constitutive dependence upon motion in a general observer frame is postulated.     

The Equations of Motion of a Rigid Body with Constraints Explicitly Depending on Time

The general equations of motion of a body for an observer in S ₀ (an inertial frame) or for an observer in S ₁ (an accelerated frame) are derived. They allow us to determine, in any case, the inertial and gyroscopic forces and to find the difference between them. If the vector that determines the position of the body in S ₀ depends explicitly on time in S ₀, the work-energy principle yields a supplementary condition and these equations can be shown to be equivalent to the Painleve integrals in the Lagrange formulation. Since we are dealing with inertial frames, gyroscopic forces rather than inertial forces are taken into account. If another reference frame is used, we can choose it so that the position vector depends implicitly on time in S ₁ and another set of equations can be obtained for the motion of the body in S ₁. The work-energy principle yields a supplementary condition, but inertial forces should be added. Since an explicit time dependence does not exist in S ₁, gyroscopic forces do not exist as well and instead we have Coriolis forces that behave like gyroscopic forces     

Modeling of the 3D rocking problem

The rocking motion of a rigid rectangular prism on a moving base is a complex three dimensional phenomenon. Although, with very few exceptions, the previous models in the literature make the simplified assumption that this motion is planar, this is usually not true since a body will probably not be aligned with the direction of the ground motion. Thus, even in the case where the body is fully symmetric, the rocking motion involves three dimensional rotations and displacements.In this work, a three dimensional formulation is introduced for the rocking motion of a rigid rectangular prism on a deformable base. Two models are developed: the Concentrated Springs Model and the Winkler Model. Both sliding and uplift are taken into account and the fully non-linear equations of the problem are developed and solved numerically.The models developed are later used to examine the behavior of bodies subjected to general ground excitations. The contribution of phenomena neglected in previous models, such as twist, is stressed.     

Attitude stabilization of a rigid body under the action of a vanishing control torque

The problem of attitude stabilization of a rigid body with the use of restoring and dissipative torques is studied. The possibility of implementing a control system in which the restoring torque tends to zero as time increases, and the only remaining control torque is a linear time-invariant dissipative one, is investigated. Both cases of linear and essentially nonlinear restoring torques are considered. With the aid of the Lyapunov direct method and the comparison method, conditions are derived under which we can guarantee stability or asymptotic stability of an equilibrium position of the body despite the vanishing of the restoring torque. A numerical simulation is provided to demonstrate the effectiveness of analytical results.     

Secular perturbations of quasi-elliptic orbits in the restricted three-body problem with variable masses

In this work we consider the satellite version of the restricted three-body problem when masses of the primary bodies P₀, P₁ vary isotropically with different rates, and their total mass changes according to the joint Meshcherskii law. Equations of motion of the body P₂ of infinitesimal mass are obtained in terms of the osculating elements of the aperiodic quasi-conical motion about the body P₀. Doubly averaging these equations and using the Hill approximation, we have obtained the differential equations determining the secular perturbations of the orbital elements and determined the domains of possible values of the system parameters for which their analytical solutions are expressed in terms of elementary or elliptic functions. The bodies P₀, P₁ mass variation laws for which the corresponding differential equations are integrable, have been found.     

A New Case of an Integrable System with Dissipation on the Tangent Bundle of a Multidimensional Sphere

The equations of motion for a dynamically symmetric n-dimensional fixed rigid body-pendulum situated in a rioricoriservative force field are studied. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of an incident medium. The complete list of transcendental first integrals expressed in terms of finite combinations of elementary functions is found.