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Euler's Equations (rigid Body Dynamics)
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. Their general vector form is : \mathbf \dot + \boldsymbol\omega \times \left( \mathbf \boldsymbol\omega \right) = \mathbf. where ''M'' is the applied torques and ''I'' is the inertia matrix. The vector \boldsymbol\alpha=\dot is the angular acceleration. In orthogonal principal axes of inertia coordinates the equations become : \begin I_1\,\dot_ + (I_3-I_2)\,\omega_2\,\omega_3 &= M_\\ I_2\,\dot_ + (I_1-I_3)\,\omega_3\,\omega_1 &= M_\\ I_3\,\dot_ + (I_2-I_1)\,\omega_1\,\omega_2 &= M_ \end where ''Mk'' are the components of the applied torques, ''Ik'' are the principal moments of inertia and ω''k'' are the components of the angular velocity. Derivation In an inertial frame of reference (subscripted "in"), Euler's second law s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility). The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Joseph-Louis Lagrange, Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation In Three Dimensions
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional object has an infinite number of possible central axes and rotational directions. If the rotation axis passes internally through the body's own center of mass, then the body is said to be ''autorotating'' or ''spinning'', and the surface intersection of the axis can be called a ''pole''. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called ''revolving'' or ''orbiting'', typically when it is produced by gravity, and the ends of the rotation axis can be called the ''orbital poles''. Mathematics Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Bodies Mechanics
Rigid or rigidity may refer to: Mathematics and physics *Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity *Structural rigidity, a mathematical theory of the stiffness of ensembles of rigid objects connected by hinges *Rigidity (electromagnetism), the resistance of a charged particle to deflection by a magnetic field *Rigidity (mathematics), a property of a collection of mathematical objects (for instance sets or functions) *Rigid body, in physics, a simplification of the concept of an object to allow for modelling *Rigid transformation, in mathematics, a rigid transformation preserves distances between every pair of points *Rigidity (chemistry), the tendency of a substance to retain/maintain their shape when subjected to outside force *(Modulus of) rigidity or shear modulus (material science), the tendency of a substance to retain/maintain their shape when subjected to outside force Medicine *Rigidity (neurology), an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Bodies
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass. In the study of special relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light. In quantum mechanics, a rigid body is usually thought of as a collection of point masses. For instance, molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors). Kinematics Linear and angular position The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Course Of Theoretical Physics
The ''Course of Theoretical Physics'' is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s. It is said that Landau composed much of the series in his head while in an NKVD prison in 1938–1939. However, almost all of the actual writing of the early volumes was done by Lifshitz, giving rise to the witticism, "not a word of Landau and not a thought of Lifshitz". The first eight volumes were finished in the 1950s, written in Russian and translated into English in the late 1950s by John Stewart Bell, together with John Bradbury Sykes, M. J. Kearsley, and W. H. Reid. The last two volumes were written in the early 1980s. Vladimir Berestetskii and Lev Pitaevskii also contributed to the series. The series is often referred to as "Landau and Lifshitz", "Landafshitz" (Russian: "Ландафшиц"), or "Lanlifshitz" (Russian: "Ланлифшиц") in informal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Rotor
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the ''linear rotor'' requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water (asymmetric rotor), ammonia (symmetric rotor), or methane (spherical rotor). Linear rotor The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses and the values of the masses are the only characteristics of the rigid model. However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed. Corrections on the rigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is a u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tennis Racket Theorem
The tennis racket theorem or intermediate axis theorem is a result in classical mechanics describing the movement of a rigid body with three distinct principal moments of inertia. It is also dubbed the Dzhanibekov effect, after Soviet cosmonaut Vladimir Dzhanibekov who noticed one of the theorem's logical consequences while in space in 1985, although the effect was already known for at least 150 years before that and was included in a book by Louis Poinsot in 1834. The theorem describes the following effect: rotation of an object around its first and third principal axes is stable, while rotation around its second principal axis (or intermediate axis) is not. This can be demonstrated with the following experiment: hold a tennis racket at its handle, with its face being horizontal, and try to throw it in the air so that it will perform a full rotation around the horizontal axis perpendicular to the handle, and try to catch the handle. In almost all cases, during that rotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general Basis (linear algebra), basis in 3-dimensional linear algebra. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering. Chained rotations equivalence Euler angles can be defined by elemental geometry or by composition of rotations. The geometrical definition demonstrates that three composed ''elemental rotations'' (rotations about the axes of a coordinate system) are always sufficient to reach any target frame. The three elemental rotations may be #Conventions by extrinsic rotations, extrinsic (rotations about the axes ''xyz'' of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poinsot's Construction
In classical mechanics, Poinsot's construction (after Louis Poinsot) is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: the kinetic energy of the body and the three components of the angular momentum, expressed with respect to an inertial laboratory frame. The angular velocity vector \boldsymbol\omega of the rigid rotor is ''not constant'', but satisfies Euler's equations. Without explicitly solving these equations, Louis Poinsot was able to visualize the motion of the endpoint of the angular velocity vector. To this end he used the conservation of kinetic energy and angular momentum as constraints on the motion of the angular velocity vector \boldsymbol\omega. If the rigid rotor is symmetric (has two equal moments of inertia), the vector \boldsymbol\omega describes a cone (and its endpoint a circle). This is the torque-free precession of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left-hand Side And Right-hand Side Of An Equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. definition and example of abbreviation More generally, these terms may apply to an or ; the right-hand side is everyt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
