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1.
The similarity transform and the properties of quaternion matrices are used to derive invariant expressions for the inertia
matrix and kinetic energy of an asymmetric rigid body undergoing spatial motion around a fixed point in body-fixed and inertial
frames of reference. The elements of the quaternion matrixes are Rodrigues–Hamilton parameters and quasi-velocities 相似文献
2.
This paper proposes a singularity-free beam element with Euler–Bernoulli assumption, i.e., the cross section remains rigid
and perpendicular to the tangent of the centerline during deformation. Each node of this two-nodal beam element has eight
nodal coordinates, including three global positions and one normal strain to describe the rigid translation and flexible deformation
of the centerline, respectively, four Euler parameters or quaternion to represent the attitude of cross section. Adopting
quaternion instead of Eulerian angles as nodal variables avoids the traditionally encountered singularity problem. The rigid
cross section assumption is automatically satisfied. To guarantee the perpendicularity of cross section to the deformed neutral
axes, the position and orientation coordinates are coupled interpolated by a special method developed here. The proposed beam
element allows arbitrary spatial rigid motion, and large bending, extension, and torsion deformation. The resulting governing
equations include normalization constraint equations for each quaternion of the beam nodes, and can be directly solved by
the available differential algebraic equation (DAE) solvers. Finally, several numerical examples are presented to verify the
large deformation, natural frequencies and dynamic behavior of the proposed beam element. 相似文献
3.
David C. P. Ellis François Gay-Balmaz Darryl D. Holm Vakhtang Putkaradze Tudor S. Ratiu 《Archive for Rational Mechanics and Analysis》2010,197(3):811-902
The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible
continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are
nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal
dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that
illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods.
In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion
equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry
group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids.
On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic
structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant
formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation
of the nonlocal equations of molecular strand motion is transformed into quaternionic form. 相似文献
4.
Derivation of the Zakharov Equations 总被引:1,自引:0,他引:1
Benjamin Texier 《Archive for Rational Mechanics and Analysis》2007,184(1):121-183
This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence.
We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data.
We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency
limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions,
the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell
equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances
between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this
work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, MéTIVIER and RAUCH
[12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus. 相似文献
5.
We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric
rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final
angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes
the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes
the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation.
The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved
by using Pontryagin’s maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically
symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the
rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection
on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end
of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation
in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead
of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations
for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process
of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations. 相似文献
6.
Using Hamilton’s principle the coupled nonlinear partial differential motion equations of a flying 3D Euler–Bernoulli beam
are derived. Stress is treated three dimensionally regardless of in-plane and out-of-plane warpings of cross-section. Tension,
compression, twisting, and spatial deflections are nonlinearly coupled to each other. The flying support of the beam has three
translational and three rotational degrees of freedom. The beam is made of a linearly elastic isotropic material and is dynamically
modeled much more accurately than a nonlinear 3D Euler–Bernoulli beam. The accuracy is caused by two new elastic terms that
are lost in the conventional nonlinear 3D Euler–Bernoulli beam theory by differentiation from the approximated strain field
regarding negligible elastic orientation of cross-sectional frame. In this paper, the exact strain field concerning considerable
elastic orientation of cross-sectional frame is used as a source in differentiations although the orientation of cross-section
is negligible. 相似文献
7.
This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear
unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing
flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore,
a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli
beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used
for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials
aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved
by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations
related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem.
The obtained results for a test case are compared with those of some other works and show a good agreement between results. 相似文献
8.
A SYMPLECTIC ALGORITHM FOR DYNAMICS OF RIGID BODY 总被引:1,自引:0,他引:1
For the dynamics of a rigid body with a fixed point based on the quaternion and the corresponding generalized momenta,a displacement-based symplectic integration scheme for differential-algebraic equations is proposed and applied to the Lagrange’s equa- tions based on dependent generalized momenta.Numerical experiments show that the algorithm possesses such characters as high precision and preserving system invariants. More importantly,the generalized momenta based Lagrange’s equations show unique ad- vantages over the traditional Lagrange’s equations in symplectic integrations. 相似文献
9.
A. Babin A. Mahalov B. Nicolaenko 《Theoretical and Computational Fluid Dynamics》1998,11(3-4):215-235
Three-dimensional nonhydrostatic Euler–Boussinesq equations are studied for Bu=O(1) flows as well as in the asymptotic regime of strong stratification and weak rotation. Reduced prognostic equations for
ageostrophic components (divergent velocity potential and geostrophic departure/thermal wind imbalance) are analyzed. We describe
classes of nonlinear anisotropic ageostrophic baroclinic waves which are generated by the strong nonlinear interactions between
the quasi-geostrophic modes and inertio-gravity waves. In the asymptotic regime of strong stratification and weak rotation
we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in the
horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak
rotation. Vertical slanting of these fronts is proportional to μ−1/2 where μ is the ratio of the Coriolis and Brunt–V?is?l? parameters. These fronts select slow baroclinic waves through nonlinear
adjustment of the horizontal scale to the vertical scale by weak rotation, and are the envelope of inertio-gravity waves.
Mathematically, this is generated by asymptotic hyperbolic systems describing the strong nonlinear interactions between waves
and potential vorticity dynamics. This frontogenesis yields vertical “gluing” of pancake dynamics, in contrast to the independent
dynamics of horizontal layers in strongly stratified turbulence without rotation.
Received 8 April 1997 and accepted 29 March 1998 相似文献
10.
We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson
(Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be
formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having
less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability
of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove
finite time stability of solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in
time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations. 相似文献
11.
Banavara N. Shashikanth Artan Sheshmani Scott David Kelly Jerrold E. Marsden 《Theoretical and Computational Fluid Dynamics》2008,22(1):37-64
We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid
body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled
without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart
law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving
rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity
and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings
model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure
when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the
Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian
function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure
is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization,
provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency
conditions.
相似文献
12.
N. P. Plakhtienko 《International Applied Mechanics》2010,46(4):461-466
A refined mathematical dynamic model of a rigid body with a gravity–friction seismic damper moving inside a movable hemispherical
rough depression is set up. This model describe the frequency and decrement properties of a rigid body undergoing translational
vibrations on a gravity–friction seismic-isolation mechanism. The mechanism consists of two platforms. The lower (supporting)
platform has spherical supports that contact with the upper (supported) platform within its hemispherical depressions. The
equations of motion incorporate the nonideality of the unilateral constraints and the shock interaction of a material point
with the sphere. It is shown how to join the models describing the free motion and the sliding motion of the point over a
spherical surface 相似文献
13.
Introducing internal damping in multibody system simulations is important as real-life systems usually exhibit this type of
energy dissipation mechanism. When using an inertial coordinate method such as the absolute nodal coordinate formulation,
damping forces must be carefully formulated in order not to damp rigid body motion, as both this and deformation are described
by the same set of absolute nodal coordinates. This paper presents an internal damping model based on linear viscoelasticity
for the absolute nodal coordinate formulation. A practical procedure for estimating the parameters that govern the dissipation
of energy is proposed. The absence of energy dissipation under rigid body motion is demonstrated both analytically and numerically.
Geometric nonlinearity is accounted for as deformations and deformation rates are evaluated by using the Green–Lagrange strain–displacement
relationship. In addition, the resulting damping forces are functions of some constant matrices that can be calculated in
advance, thereby avoiding the integration over the element volume each time the damping force vector is evaluated. 相似文献
14.
Experimental and numerical investigations of impacting cantilever beams part 1: first mode response 总被引:1,自引:0,他引:1
In this paper, the dynamic behavior of a cantilever beam impacting two flexible stops as well as rigid stops is studied both
experimentally and numerically. The effect of contact stiffness, clearance, and contacting materials is studied in detail.
For the numerical study of the system, a finite element model is created and the resulting differential equations are solved
using a Time Variational Method (TVM). To achieve higher computational efficiency, the Newton–Krylov method is used along
with TVM. Experimental results validate the contact model proposed for predicting the first mode system dynamics. A new nonlinear
force estimation function has been proposed based on measured accelerations, which enables the understanding of the impact
dynamics. 相似文献
15.
多刚体系统分离策略及释放动力学研究 总被引:1,自引:0,他引:1
紧密连接的多刚体系统可在脱离运载航天器后在轨自主分离,无需多次利用航天器发射装置或在航天器中安装多个发射装置进行分离释放,从而有效提高运载航天器空间利用率,简化分离释放操作和降低碰撞风险.本文针对多刚体系统的在轨分离释放问题,研究在轨分离策略及释放过程动力学.首先,考虑刚体相对运动及姿态变化,基于虚功原理及自然坐标方法建立单个刚体的动力学模型.考虑多刚体系统在轨分离释放阶段的轨道运动和连接约束变化,计入分离时刚体间的相互作用,利用拉格朗日乘子法获得含连接约束的非线性动力学模型.考虑到实际工程应用,在多刚体系统分离释放阶段,通过安装在刚体间每个接触表面4个角上的弹射装置实现自主分离.其次,为保证分离过程中刚体之间无碰撞发生,规划了多刚体系统的分离时序,并基于不同弹射方向及分离顺序设计了两种分离释放方案.最后,通过算例研究分析了在轨分离释放过程中刚体的非线性动力学行为,验证了分离释放方案的有效性. 相似文献
16.
The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed.
The differential equations of dynamics of the constrained rod are established with Euler’s angles as variables describing
the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special
configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively.
Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and
relationship between Lyapunov’s and Euler’s stability concepts are discussed. The free frequency of flexural vibration of
the helical rod with cylinder constraint is obtained in analytical form.
The project supported by the National Natural Science Foundation of China (10472067). The English text was polished by Yunming
Chen. 相似文献
17.
N. P. Plakhtienko 《International Applied Mechanics》2009,45(7):786-796
A design model for a chain system of N elastically linked rigid bodies with a spheroidal gravity-friction damper is proposed. The Lagrange–Painlevé equations of
the first kind are used to construct nonlinear dynamical models of a mechanical system undergoing translational vibrations
about the equilibrium position. The conditions under which the system moves in one plane are established. The double nonstationary
phase–frequency resonance of a system with N = 2 is analyze. After the numerical integration of the systems of differential equations, the phase–frequency surfaces are
plotted and examined for several combinations of system parameters under two-frequency loading 相似文献
18.
Eugen Magyari 《Transport in Porous Media》2010,85(2):415-435
The Vadasz–Olek model of chaotic convection in a porous layer is revisited in this article. The first-order differential equations
of this Lorenz-type model are transformed in the governing equations of a damped nonlinear oscillator, modulated by a linear
degenerated overdamped oscillator (relaxator) which in turn is coupled to former one by a nonlinear cross force. The benefit of this mechanical analogy is an intuitive
picture of the regular and chaotic dynamics described by the Vadasz–Olek model. Thus, there turns out that the “eyes” of the
chaotic attractor correspond to the minima of the potential energy of the modulated nonlinear oscillator having a double-well
shape. Several new aspects of the subcritical and supercritical dynamic regimes are discussed in some detail. 相似文献
19.
四元数的一种新的代数结构 总被引:15,自引:0,他引:15
在四元数力学的数学方法研究中,首次引入了友向量的概念,借助四元数的复表示方法,进一步研究了四元数力学中的系列数值计算问题,给出了相应问题的简单计算和论证方法,建立起了一套四元数力学的简单数学方法。 相似文献
20.
L. I. Shkutin 《Journal of Applied Mechanics and Technical Physics》1999,40(5):957-961
A nonlinear deformation model for a shell with rigid transverse fibers is proposed. A complete system of incremental equations,
a variational equation equivalent to this system, and a particular equation of virtual work are formulated. Numerical analysis
of the nonlinear deformation of a spherical dome is performed using the complete equation.
Institute of Computer Modeling, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya
Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 202–207, September–October, 1999. 相似文献