kinematical

• 運動学的な

関

kinesiological

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/06/27 14:26:20」(JST)

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Classical mechanics
History Timeline
Branches Statics Dynamics Kinetics Kinematics Applied mechanics Celestial mechanics Continuum mechanics Statistical mechanics
Formulations Newtonian mechanics (Vectorial mechanics) Analytical mechanics (Lagrangian mechanics    Hamiltonian mechanics)
Fundamental concepts Space Time Mass Inertia Velocity Speed Acceleration Force Momentum Impulse Torque / Moment / Couple Angular momentum Moment of inertia Frame of reference Energy (kinetic potential) Mechanical work Mechanical power Virtual work D'Alembert's principle
Core topics Rigid body (dynamics Euler's equations) Motion (linear) Euler's laws of motion Newton's laws of motion Newton's law of universal gravitation Equations of motion Inertial / Non-inertial reference frame Fictitious force Mechanics of planar particle motion Displacement (vector) Relative velocity Friction Simple harmonic motion Harmonic oscillator Vibration Damping (ratio)
Rotational motion Circular motion (uniform non-uniform) Rotating reference frame Centripetal force Centrifugal force (rotating reference frame reactive) Coriolis force Pendulum Tangential / Rotational speed Angular acceleration Angular velocity Angular frequency Angular displacement
Scientists Galileo Newton Kepler Horrocks Halley Euler d'Alembert Clairaut Lagrange Laplace Hamilton Poisson Daniel / Johann Bernoulli Cauchy

Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion. The term is the English version of A.M. Ampère's cinématique, which he constructed from the Greek κίνημα, kinema (movement, motion), derived from κινεῖν, kinein (to move).

The study of kinematics is often referred to as the geometry of motion. (See analytical dynamics for more detail on usage).

To describe motion, kinematics studies the trajectories of points, lines and other geometric objects and their differential properties such as velocity and acceleration. Kinematics is used in astrophysics to describe the motion of celestial bodies and systems, and in mechanical engineering, robotics and biomechanics to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the skeleton of the human body.

The study of kinematics can be abstracted into purely mathematical expressions. For instance, rotation can be represented by elements of the unit circle in the complex plane. Other planar algebras are used to represent the shear mapping of classical motion in absolute time and space and to represent the Lorentz transformations of relativistic space and time. By using time as a parameter in geometry, mathematicians have developed a science of kinematic geometry.

The use of geometric transformations, also called rigid transformations, to describe the movement of components of a mechanical system simplifies the derivation of its equations of motion, and is central to dynamic analysis.

Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism, and, working in reverse, kinematic synthesis designs a mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system, or mechanism.

Kinematics of a particle trajectory

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

Particle kinematics is the study of the properties of the trajectory of a particle. The position of a particle is defined to be the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is located at your home, such that East is the x-direction and North is the y-direction, then the coordinate vector to the base of the tower is r=(0, -50, 0). If the tower is 50 m high, then the coordinate vector to the top of the tower is r=(0, -50, 50).

Usually a three dimensional coordinate systems is used to define the position of a particle. However if the particle is constrained to lie in a plane or on a sphere, a two dimensional coordinate system can be used. All observations in physics are incomplete without the reference frame being specified.

The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position of point P can be expressed as

where x_P, y_P, and z_P are the Cartesian coordinates and i, j and k are the unit vectors along the x, y, and z coordinate axes, respectively. The magnitude of the position vector |P| gives the distance between the point P and the origin.

The direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the position vector of a particle isn't unique. The position vector of a given particle is different relative to different frames of reference;

The trajectory of a particle is a vector function of time, P(t), which defines the curve traced by the moving particle, given by

where the coordinates x_P, y_P, and z_P are each functions of time.

The distance traveled is always greater than or equal to the displacement.

Velocity and speed

The velocity of a particle is a vector that tells about the direction and magnitude of the rate of change of the position vector, that is, how the position of a point changes with each instant of time. Consider the ratio of the difference of two positions of a particle divided by the time interval, which is called the average velocity over that time interval. This average velocity is defined as

where ΔP is the difference in the position vector over the time interval Δt.

In the limit as the time interval Δt becomes smaller and smaller, the average velocity becomes the time derivative of the position vector,

Thus, velocity is the time rate of change of position, and the dot denotes the derivative with respect to time. Furthermore, the velocity is tangent to the trajectory of the particle.

As a position vector itself is frame dependent, therefore its velocity is also dependent on the reference frame.

The speed of an object is the magnitude |V| of its velocity. It is a scalar quantity:

where s is the arc-length measured along the trajectory of the particle. This arc-length traveled by a particle over time is a non-decreasing quantity. Hence, ds/dt is non-negative, which implies that speed is also non-negative.

Acceleration

The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio

where ΔV is the difference in the velocity vector and Δt is the time interval.

The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative,

Thus, acceleration is the second derivative of the position vector that defines the trajectory of a particle.

Relative position vector

A relative position vector is a vector that defines the position of a particle relative to another particle. It is the difference in position of the two particles.

If point A has position P_A = (x_A,y_A,z_A) and point B has position P_B = (x_B,y_B,z_B), the displacement R_B/A of B from A is given by

Geometrically, the relative position vector R_B/A is the vector from point A to point B. The values of the coordinate vectors of points vary with the choice of coordinate frame, however the relative position vector between a pair of points has the same length no matter what coordinate frame is used and is said to be frame invariant.

To describe the motion of a particle B relative to another particle A, we notice that the position B can be formulated as the position of A plus the position of B relative to A, that is

Relative velocity

Main article: Relative velocity

Relative velocities between two particles in classical mechanics.

The relations between relative positions vectors become relations between relative velocities by computing the time-derivative. The second time derivative yields relations for relative accelerations.

For example, let the particle B move with velocity V_B and particle A move with velocity V_A in a given reference frame. Then the velocity of B relative to A is given by

This can be obtained by computing the time derivative of the relative position vector R_B/A.

This equation provides a formula for the velocity of B in terms of the velocity of A and its relative velocity,

With a large velocity V, where the fraction V/c is significant, c being the speed of light, another scheme of relative velocity called rapidity, that depends on this ratio, is used in special relativity.

Particle trajectories under constant acceleration

Newton's laws state that a constant force acting on a particle generates a constant acceleration. For example, a particle in a parallel gravity field experiences a force acting downwards that is proportional to the constant acceleration of gravity, and no force in the horizontal direction. This is called projectile motion.

If the acceleration vector A of a particle P is constant in magnitude and direction, the particle is said to be undergoing uniformly accelerated motion. In this case, the trajectory P(t) of the particle can be obtained by integrating the acceleration A with respect to time.

The first integral yields the velocity of the particle,

A second integration yields its trajectory,

Additional relations between displacement, velocity, acceleration, and time can be derived. Since A = (V − V₀)/t,

By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.

A relationship without explicit time dependence may also be derived using the relation At = V − V₀,

where · denotes the dot product. Divide both sides by t and expand the dot-products to obtain,

In the case of straight-line motion, where P and P₀ are parallel to A, this equation becomes

This can be simplified using the notation |A|=a, |V|=v, and |P|=r, so

This relation is useful when time is not known explicitly.

Figure 2: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component a_θ that increases the rate of rotation: dω/dt = |a_θ|/R.

Particle trajectories in cylindrical-polar coordinates

See also: Generalized coordinates, Curvilinear coordinates, Orthogonal coordinates, and Frenet-Serret formulas

It is often convenient to formulate the trajectory of a particle P(t) = (X(t), Y(t) and Z(t)) using polar coordinates in the X-Y plane. In this case, its velocity and acceleration take a convenient form.

Recall that the trajectory of a particle P is defined by its coordinate vector P measured in a fixed reference frame F. As the particle moves, its coordinate vector P(t) traces its trajectory, which is a curve in space, given by

where i, j, and k are the unit vectors along the X, Y and Z axes of the reference frame F, respectively.

Consider a particle P that moves on the surface of a circular cylinder, it is possible to align the Z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the X-Y plane can be used to define the trajectory as,

The cylindrical coordinates for P(t) can be simplified by introducing the radial and tangential unit vectors,

Using this notation, P(t) takes the form,

where R is constant.

Now, in general, the trajectory P(t) is not constrained to lie on a circular cylinder, so the radius R varies with time, and the trajectory in cylindrical-polor coordinates becomes

The velocity vector V_P is the time derivative of the trajectory P(t), which yields,

where

In this case, the acceleration A_P, which is the time derivative of the velocity V_P, is given by

If the radius is constant

If the trajectory of the particle is constrained to lie on a cylinder, then the radius R is constant and the velocity and acceleration vectors simplify. The velocity of V_P is the time derivative of the trajectory P(t),

The acceleration vector becomes

Planar circular trajectories

Each particle on the wheel travels in a planar circular trajectory (Kinematics of Machinery, 1876).

A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the Z axis, in which case

where R and Z₀ are constants. In this case, the velocity V_P is given by

where

is the angular velocity of the unit vector e_t around the z axis of the cylinder.

The acceleration A_P of the particle P is now given by

The components

are called the radial and tangential components of acceleration, respectively.

The notation for angular velocity and angular acceleration is often defined as

so the radial and tangential acceleration components for circular trajectories are also written as

Point trajectories in a body moving in the plane

The movement of components of a mechanical system is analyzed by attaching a reference frame to each part and determining how the reference frames move relative to each other. If the structural strength of the parts are sufficient then their deformation can be neglected and rigid transformations used to define this relative movement. This brings geometry into the study of mechanical movement.

Geometry is the study of the properties of figures that remain the same while the space is transformed in various ways---more technically, it is the study of invariants under a set of transformations. Perhaps best known is high school Euclidean geometry where planar triangles are studied under congruent transformations, also called isometries or rigid transformations. These transformations displace the triangle in the plane without changing the angle at each vertex or the distances between vertices. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry.

The coordinates of points in the plane are two dimensional vectors in R², so rigid transformations are those that preserve the distance measured between any two points. The Euclidean distance formula is simply the Pythagorean theorem. The set of rigid transformations in an n-dimensional space is called the special Euclidean group on Rⁿ, and denoted SE(n).

Displacements and motion

The movement of each of the components of the Boulton & Watt Steam Engine (1784) is modeled by a continuous set of rigid displacements.

The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say M, on one that moves relative to a fixed frame, F, on the other. The rigid transformation, or displacement, of M relative to F defines the relative position of the two components. A displacement consists of the combination of a rotation and a translation.

The set of all displacements of M relative to F is called the configuration space of M. A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the motion of M relative to F. The motion of a body consists of a continuous set of rotations and translations.

Matrix representation

The combination of a rotation and translation in the plane R² can be represented by a certain type of 3x3 matrix known as a homogeneous transform. The 3x3 homogenous transform is constructed from a 2x2 rotation matrix A(φ) and the 2x1 translation vector d=(d_x, d_y), as

$[T(\phi, \mathbf{d})] = \begin{bmatrix} A(\phi) & \mathbf{d} \\ 0, 0 & 1\end{bmatrix} = \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix}.$

These homogeneous transforms perform rigid transformations on the points in the plane z=1, that is on points with coordinates p=(x, y, 1).

In particular, let p define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by

Homogeneous transforms represent affine transformations. This formulation is necessary because a translation is not a linear transformation of R². However, using projective geometry, so that R² is considered to be a subset of R³, translations become affine linear transformations.

Pure translation

If a rigid body moves so that its reference frame M does not rotate relative to the fixed frame F, the motion is said to be pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory d(t) of the origin of M, that is,

Thus, for bodies in pure translation the velocity and acceleration of every point P in the body are given by

where the dot denotes the derivative with respect to time and V_O and A_O are the velocity and acceleration, respectively, of the origin of the moving frame M. Recall the coordinate vector p in M is constant, so its derivative is zero.

Rotation of a body around a fixed axis

Main article: Circular motion

Figure 1: The angular velocity vector Ω points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule. Angular position θ(t) changes with time at a rate ω(t) = dθ/dt.

Rotational or angular kinematics is the description of the rotation of an object. The description of rotation requires some method for describing orientation. Common descriptions include Euler angles and the kinematics of turns induced by algebraic products.

In what follows, attention is restricted to simple rotation about an axis of fixed orientation. The z-axis has been chosen for convenience.

Position: This allows the description of a rotation as the angular position of a planar reference frame M relative to a fixed F about this shared z-axis. Coordinates p=(x, y) in M are related to coordinates P=(X, Y) in F by the matrix equation:

where

$[A(t)] = \begin{bmatrix} \cos\theta(t) & -\sin\theta(t) \\ \sin\theta(t) & \cos\theta(t) \end{bmatrix},$

is the rotation matrix that defines the angular position of M relative to F.

Velocity: If the point p does not move in M, then its velocity in F is given by

It is convenient to eliminate the coordinates p and write this as an operation on the trajectory P(t),

where the matrix

$[\Omega] = \begin{bmatrix} 0 & -\omega \\ \omega & 0 \end{bmatrix},$

is known as the angular velocity matrix of M relative to F. The parameter ω is the time derivative of the angle θ, that is

Acceleration: The acceleration of P(t) in F is obtained as the time derivative of the velocity,

which becomes

where

$[\dot{\Omega}] = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix},$

is the angular acceleration matrix of M on F, and

Description of rotation then involves these three quantities:

• Angular position: The oriented distance from a selected origin on the rotational axis to a point of an object is a vector r ( t ) locating the point. The vector r(t) has some projection (or, equivalently, some component) r_⊥(t) on a plane perpendicular to the axis of rotation. Then the angular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r_⊥(t) in a known rotation sense (typically given by the right-hand rule).
• Angular velocity: The angular velocity ω is the rate at which the angular position θ changes with respect to time t:

The angular velocity is represented in Figure 1 by a vector Ω pointing along the axis of rotation with magnitude ω and sense determined by the direction of rotation as given by the right-hand rule.

• Angular acceleration: The magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t:

The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:

Here θ_i and θ_f are, respectively, the initial and final angular positions, ω_i and ω_f are, respectively, the initial and final angular velocities, and α is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.

Point trajectories in body moving in three dimensions

Important formulas in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in three dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either Newton's second law or Lagrange's equations.

Position

In order to define these formulas, the movement of a component B of a mechanical system is defined by the set of rotations [A(t)] and translations d(t) assembled into the homogenous transformation [T(t)]=[A(t), d(t)]. Let p be the coordinates of a point P in B measured in the moving reference frame M, then the trajectory of this point traced in F is given by

$\textbf{P}(t)=[T(t)]\textbf{p} = \begin{Bmatrix} \textbf{P} \\ 1\end{Bmatrix}=\begin{bmatrix} A(t) & \textbf{d}(t) \\ 0 & 1\end{bmatrix} \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.$

This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context.

This equation for the trajectory of P can be inverted to compute the coordinate vector p in M as,

$\textbf{p} = [T(t)]^{-1}\textbf{P}(t) = \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}=\begin{bmatrix} A(t)^T & -A(t)^T\textbf{d}(t) \\ 0 & 1\end{bmatrix} \begin{Bmatrix} \textbf{P}(t) \\ 1\end{Bmatrix}.$

This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is

Velocity

The velocity of the point P along its trajectory P(t) is obtained as the time derivative of this position vector,

$\textbf{V}_P = [\dot{T}(t)]\textbf{p} = \begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}(t) & \dot{\textbf{d}}(t) \\ 0 & 0 \end{bmatrix} \begin{Bmatrix} \textbf{p} \\ 1\end{Bmatrix}.$

The dot denotes the derivative with respect to time, and because p is constant its derivative is zero.

This formula can be modified to obtain the velocity of P by operating on its trajectory P(t) measured in the fixed frame F. Substitute the inverse transform for p into the velocity equation to obtain

$\textbf{V}_P = [\dot{T}(t)][T(t)]^{-1}\textbf{P}(t) = \begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}A^T & -\dot{A}A^T\textbf{d} + \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix} \begin{Bmatrix} \textbf{P}(t) \\ 1\end{Bmatrix}=[S]\textbf{P}.$

The matrix [S] is given by

where

is the angular velocity matrix.

Multiplying by the operator [S], the formula for the velocity V_P takes the form

where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω], the vector

is the position of P relative to the origin O of the moving frame M, and

is the velocity of the origin O.

Acceleration

The acceleration of a point P in a moving body B is obtained as the time derivative of its velocity vector,

This equation can be expanded by first computing

and

The formula for the acceleration A_P can now be obtained as

or

where α is the angular acceleration vector obtained from the derivative of the angular velocity matrix,

is the relative position vector, and

is the acceleration of the origin of the moving frame M.

Kinematic constraints

Kinematic constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called holonomic constraints, and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called non-holonomic constraints. Constraints can also arise from other interactions such as rolling without slipping, is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:

Kinematic coupling

A kinematic coupling exactly constrains all 6 degrees of freedom.

Rolling without slipping

An object that rolls against a surface without slipping obeys the condition that the velocity of its center of mass is equal to the cross product of its angular velocity with a vector from the point of contact to the center of mass,

For the case of an object that does not tip or turn, this reduces to v = R ω.

Inextensible cord

This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero. A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. An equilibrium problem (not kinematic) of this type is the catenary.

Kinematic pairs

Main article: kinematic pair

Reuleaux called the ideal connections between components that form a machine kinematic pairs. He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification.

Lower pair: A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three dimensional) body to a corresponding point line or plane in the fixed solid body. We have the following cases:

• A revolute pair, or hinged joint, requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom, which is pure rotation about the axis of the hinge.
• A prismatic joint, or slider, requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body maintain contact with a similar parallel plan in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom. This degree of freedom is the distance of the slide along the line.
• A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body. It is a combination of a revolute joint and a sliding joint. This joint has two degrees of freedom. The position of the moving body is defined by both the rotation about and slide along the axis.
• A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom.
• A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.

Higher pairs: Generally, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a cam joint. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.

Kinematic chains

Illustration of a four-bar linkage from Kinematics of Machinery, 1876

Rigid bodies, or links, connected by kinematic pairs, or joints, are called kinematic chains. Mechanisms and robots are examples of kinematic chains. The degree of freedom of a kinematic chain is computed from the number of links and the number and type of joints using the mobility formula. This formula can also be used to enumerate the topologies of kinematic chains that have a given degree of freedom, which is known as type synthesis in machine design.

Examples of kinematic chains: The planar one degree-of-freedom linkages assembled from N links and j hinged or sliding joints are:

• N=2, j=1: this is a two-bar linkage known as the lever;
• N=4, j=4: this is the four-bar linkage;
• N=6, j=7: this is a six-bar linkage. A six-bar linkage must have two links that support three joints, called ternary links. There are two distinct topologies that depend on how the two ternary linkages are connected. In the Watt topology, the two ternary links have a common joint. In the Stephenson topology the two ternary links do not have a common joint and are connected by binary links;

• N=8, j=10: the eight-bar linkage has 16 different topologies;
• N=10, j=13: the 10-bar linkage has 230 different topologies,
• N=12, j=16: the 12-bar has 6856 topologies.

See Sunkari and Schmidt for the number of 14- and 16-bar topologies, as well as the number of linkage topologies that have two, three and four degrees-of-freedom.

See also

• Acceleration
• Analytical mechanics
• Applied mechanics
• Celestial mechanics
• Centripetal force
• Chebychev–Grübler–Kutzbach criterion
• Classical mechanics
• Distance
• Dynamics (physics)
• Fictitious force
• Forward kinematics
• Four-bar linkage
• Inverse kinematics
• Jerk (physics)
• Kepler's laws
• Kinematic coupling
• Kinematic diagram
• Kinetics (physics)
• Motion
• Orbital mechanics
• Statics
• Velocity

Notes

References

Further reading

• Koetsier, Teun (1994) §8.3 "Kinematics", pp 994 to 1001 in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, volume 2, edited by Ivor Grattan-Guinness, Routledge ISBN 0-415-09239-6 .
• Moon, Francis C. (2007). The Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century. Springer. ISBN 978-1-4020-5598-0. 

External links

• Java applet of 1D kinematics
• Physclips: Mechanics with animations and video clips from the University of New South Wales
• Kinematic Models for Design Digital Library (KMODDL) Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes an e-book library of classic texts on mechanical design and engineering.

English Journal

• Kinematic, kinetic and EMG analysis of four front crawl flip turn techniques.

• Pereira SM1,2, Ruschel C2, Hubert M2, Machado L1,3, Roesler H2, Fernandes RJ1,3, Vilas-Boas JP1,3.
• Journal of sports sciences.J Sports Sci.2015 Nov;33(19):2006-15. doi: 10.1080/02640414.2015.1026374. Epub 2015 Mar 27.
• This study aimed to analyse the kinematic, kinetic and electromyographic characteristics of four front crawl flip turn technique variants. The variants distinguished from each other by differences in body position (i.e. dorsal, lateral, ventral) during rolling, wall support, pushing and gliding phas
• PMID 25813081

• Assessment of cartilage contact pressure and loading in the hip joint during split posture.

• Assassi L1, Magnenat-Thalmann N2.
• International journal of computer assisted radiology and surgery.Int J Comput Assist Radiol Surg.2015 Oct 8. [Epub ahead of print]
• PURPOSE: Given the crucial role of the mechanical behavior in the degenerative process of the hip joint, analyzing the contact mechanics in the articular layers during physical activities could contribute to understanding the pathology. Indeed, the development process of hip osteoarthritis generally
• PMID 26450106

• 3D shoulder kinematics for static vs dynamic and passive vs active testing conditions.

• Robert-Lachaine X1, Allard P2, Godbout V3, Begon M2.
• Journal of biomechanics.J Biomech.2015 Sep 18;48(12):2976-83. doi: 10.1016/j.jbiomech.2015.07.040. Epub 2015 Aug 12.
• Shoulder motion analysis provides clinicians with references of normal joint rotations. Shoulder joints orientations assessment is often based on series of static positions, while clinicians perform either passive or active tests and exercises mostly in dynamic. These conditions of motion could modi
• PMID 26298491

Japanese Journal

• 『駆動冗長性を有する高速パラレルメカニズムの高精度制御のための運動学的冗長自由度の導入方法』（Vol.33 No.5 掲載）正誤表

• 日本ロボット学会誌 33(6), 432-432, 2015
• NAID 130005092868

• 駆動冗長性を有する高速パラレルメカニズムの高精度制御のための運動学的冗長自由度の導入方法

• 日本ロボット学会誌 33(5), 379-386, 2015
• NAID 130005092855

• 競泳のキックスタートパフォーマンスと等尺性・等速性脚筋力との関係

• 水泳水中運動科学 18(1), 4-9, 2015
• NAID 130005091144

Related Links

• Kinematical | Define Kinematical at Dictionary.com

kinematics / ˌkɪn əˈmæt ɪks, ˌkaɪ nə-/ Show Spelled [kin-uh-mat-iks, kahy-n uh-] Show IPA noun (used with a singular verb) Physics. 1. the branch of mechanics that deals with pure motion, without reference to the masses or forces ...

• kinematical - Definition of kinematical - Dictionary definitions ...

kinematical Variant of kinematics noun the branch of mechanics that deals with motion in the abstract, without reference to the force or mass Origin: Fr cinématique < Gr kinēma (gen. kinēmatos), motion < kinein, to move ...

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★リンクテーブル★

リンク元	「kinesiological」「運動学的」
関連記事	「kinematic」

「kinesiological」

　　[★]

• 運動学的な

関

kinematical

「運動学的」

　　[★]

英

kinematical、kinesiological

「kinematic」

　　[★]

• 運動力学の

関

kinematics