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 forces that cause it.[1][2][3] The term is the English version of A.M. Ampère's cinématique,[4] which he constructed from the Greek κίνημα, kinema (movement, motion), derived from κινεῖν, kinein (to move).[5] [6]
The study of kinematics is often referred to as the geometry of motion.[7] (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[8] 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.[9] In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system, or mechanism.
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.[10] 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).
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.
The combination of a rotation and translation in the plane R2 can be represented by 3x3 matrix matrices, known as homogeneous transforms. The 3x3 homogenous transform is constructed from a 2x2 rotation matrix A(φ) and the 2x1 translation vector d=(dx, dy), as
- Failed to parse (Missing texvc executable; please see math/README to configure.): [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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P} = [T(\phi, \mathbf{d})]\textbf{p} = \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix}\begin{Bmatrix}x\\y\\1\end{Bmatrix}.
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.[11]
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,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P}(t)=[T(0,\textbf{d}(t)]\textbf{p} = \textbf{d}(t) + \textbf{p}.
Thus, for bodies in pure translation the velocity and acceleration of every point P in the body are given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{V}_P=\dot{\textbf{P}}(t) = \dot{\textbf{d}}(t)=\textbf{V}_O,\quad \textbf{A}_P=\ddot{\textbf{P}}(t) = \ddot{\textbf{d}}(t) = \textbf{A}_O,
where the dot denotes the derivative with respect to time and VO and AO 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.
Particle kinematics is the study of the kinematics of a single particle. The results obtained in particle kinematics are used to study the kinematics of collection of particles, dynamics and in many other branches of mechanics.
The position of a point in space is the most fundamental idea in particle kinematics. To specify the position of a point, one must specify three things: the reference point (often called the origin), distance from the reference point and the direction in space of the straight line from the reference point to the particle. Exclusion of any of these three parameters renders the description of position incomplete. Consider for example a tower 50 m south from your home. The reference point is home, the distance 50 m and the direction south. If one only says that the tower is 50 m south, the natural question that arises is "from where?" If one says that the tower is southward from your home, the question that arises is "how far?" If one says the tower is 50 m from your home, the question that arises is "in which direction?" Hence, all these three parameters are crucial to defining uniquely the position of a point in space.
Position is usually described by mathematical quantities that have all these three attributes: the most common are vectors and complex numbers. Usually, only vectors are used. For measurement of distances and directions, usually three dimensional coordinate systems are used with the origin coinciding with the reference point. A three-dimensional coordinate system (whose origin coincides with the reference point) with some provision for time measurement is called a reference frame or frame of reference or simply frame. 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 sense from the origin. In three dimensions, the position of point A can be expressed as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}_A = (x_A,y_A,z_A),
where xA, yA, and zA are the Cartesian coordinates of the point. The magnitude of the position vector |r| gives the distance between the point A and the origin.
- Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathbf{r}| = \sqrt{x_A^{\ 2} + y_A^{\ 2} + z_A^{\ 2}}.
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.
Displacement is a vector describing the difference in position between two points, i.e. it is the change in position the particle undergoes during the time interval. If point A has position rA = (xA,yA,zA) and point B has position rB = (xB,yB,zB), the displacement rAB of B from A is given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}_{AB} = \mathbf{r}_B - \mathbf{r}_A = (x_B-x_A,y_B-y_A,z_B-z_A).
Geometrically, displacement is the shortest distance between the points A and B. Displacement, distinct from position vector, is independent of the reference frame. This can be understood as follows: the positions of points is frame dependent, however, the shortest distance between any pair of points is invariant on translation from one frame to another (barring relativistic cases).
The distance traveled is always greater than or equal to the displacement.
Average velocity is defined as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \overline{\mathbf{v}} = \frac {\Delta \mathbf{r}}{\Delta t} \ ,
where Δr is the change in position and Δt is the interval of time over which the position changes. The direction of v is same as the direction of the change in position Δr as Δt>0.
Velocity is the measure of the rate of change in position with respect to time, that is, how the distance of a point changes with each instant of time. Velocity also is a vector. Instantaneous velocity (the velocity at an instant of time) can be defined as the limiting value of average velocity as the time interval Δt becomes smaller and smaller. Both Δr and Δt approach zero but the ratio v approaches a non-zero limit v. That is,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v} = \lim_{\Delta t\rightarrow0}\frac{\Delta\mathbf{r}}{\Delta t} = \frac {d \mathbf{r}}{d t} \, ,
where dr is an infinitesimally small displacement and dt is an infinitesimally small length of time.[note 1] As per its definition in the derivative form, velocity can be said to be the time rate of change of position. Further, as dr is tangential to the actual path, so is the velocity.
As a position vector itself is frame dependent, 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): |\mathbf{v}| = \left|\frac {d \mathbf{r}}{d t} \right| = \frac {d s}{d t}
The distance 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.
Average acceleration (acceleration over a length of time) is defined as:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \overline{\mathbf{a}} = \frac {\Delta \mathbf{v}}{\Delta t} \ ,
where Δv is the change in velocity and Δt is the interval of time over which velocity changes.
Acceleration is the vector quantity describing the rate of change with time of velocity. Instantaneous acceleration (the acceleration at an instant of time) is defined as the limiting value of average acceleration as Δt becomes smaller and smaller. Under such a limit, a → a.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{a} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac {d \mathbf{v}}{d t} \, ,
where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time.
Many physical situations can be modeled as constant-acceleration processes, such as projectile motion.
Integrating acceleration a with respect to time t gives the change in velocity. When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the integral relations can be simplified:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}(t) = \mathbf{v}_0 + \int_0^{t} \mathbf{a} \; dt' = \mathbf{v}_0 + \mathbf{a}t.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \mathbf{r}(t) &= \mathbf{r}_0 + \int_0^t \mathbf{v} \; dt' = \mathbf{r}_0 + \int_0^t (\mathbf{v}_0 + \mathbf{a} t') \; dt' \\ &= \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac{1}{2} \mathbf{a} t^2. \end{align}
Additional relations between displacement, velocity, acceleration, and time can be derived. Since a = (v − v0)/t,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}(t) = \mathbf{r}_0 + \left(\frac{\mathbf{v}+ \mathbf{v_0}}{2}\right) 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 for one-dimensional motion. Noting that at = v − v0,
- Failed to parse (Missing texvc executable; please see math/README to configure.): (\mathbf{r} - \mathbf{r_0}) \cdot \mathbf{a} t = \left( \mathbf{v} - \mathbf{v}_0 \right) \cdot \frac{\mathbf{v} + \mathbf{v}_0}{2} t \ ,
where · denotes the dot product. Dividing the t on both sides and carrying out the dot-products:
- Failed to parse (Missing texvc executable; please see math/README to configure.): 2(\mathbf{r} - \mathbf{r_0}) \cdot \mathbf{a} = v^2 - v_0^{\ 2}.
In the case of straight-line motion, (r - r0) is parallel to a. Then
- Failed to parse (Missing texvc executable; please see math/README to configure.): v^2= v_0^2 + 2 a(r-r_0).
This relation is useful when time is not known explicitly.
To describe the motion of object A with respect to object B, when we know how each is moving with respect to a reference object O, we can use vector algebra. Choose an origin for reference, and let the positions of objects A, B, and O be denoted by rA, rB, and rO. Then the position of A relative to the reference object O is
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}_{A/O} = \mathbf{r}_{A} - \mathbf{r}_{O} \,\!
Consequently, the position of A relative to B is
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B = \mathbf{r}_A - \mathbf{r}_O - \left(\mathbf{r}_B-\mathbf{r}_O\right) = \mathbf{r}_{A/O}-\mathbf{r}_{B/O} \ .
The above relative equation states that the motion of A relative to B is equal to the motion of A relative to O minus the motion of B relative to O. It may be easier to visualize this result if the terms are re-arranged:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}_{A/O} = \mathbf{r}_{A/B} + \mathbf{r}_{B/O} \ ,
or, in words, the motion of A relative to the reference is that of B plus the relative motion of A with respect to B. These relations between displacements become relations between velocities by simple time-differentiation, and a second differentiation makes them apply to accelerations.
For example, let Ann move with velocity Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{V}_{A}
relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{V}_{B}
, each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{V}_{A/B} ), the equation above gives:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{V}_{A} = \mathbf{V}_{B} + \mathbf{V}_{A/B} \,\! .
To find Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{V}_{A/B}
we simply rearrange this equation to obtain:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{V}_{A/B} = \mathbf{V}_{A} -\mathbf{V}_{B} \,\! .
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.
-
| Example: Rectilinear (1D) motion |
Figure A: An object is fired upwards, reaches its apex, and then begins its descent under a constant acceleration. Note: The equations described here holds for object fired from ground and should not be mistaken with this picture.
Consider an object that is fired directly upwards and falls back to the ground so that its trajectory is contained in a straight line. If we adopt the convention that the upward direction is the positive direction, the object experiences a constant acceleration of approximately −9.81 m s−2. Therefore, its motion can be modeled with the equations governing uniformly accelerated motion.
For the sake of example, assume the object has an initial velocity of +50 m s−1. There are several interesting kinematic questions we can ask about the particle's motion:
- How long will it be airborne?
To answer this question, we apply the formula
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_{\mathrm{f}} - x_{\mathrm{i}} = v_{\mathrm{i}} t + \tfrac{1}{2} at^2.
Since the question asks for the length of time between the object leaving the ground and hitting the ground on its fall, the displacement is zero.
- Failed to parse (Missing texvc executable; please see math/README to configure.): 0 = v_{\mathrm{i}} t + \tfrac{1}{2} at^2 = t\left(v_{\mathrm{i}} + \tfrac{1}{2} at \right)
There are two solutions: the first, t = 0, is trivial. The solution of interest is
- Failed to parse (Missing texvc executable; please see math/README to configure.): t = -\frac{2v_{\mathrm{i}}}{a} = \rm -\frac{2(50\ m\ s^{-1})}{-9.81\ m\ s^{-2}} = 10.2\ s\,.
- What altitude will it reach before it begins to fall?
In this case, we use the fact that the object has a velocity of zero at the apex of its trajectory. Therefore, the applicable equation is:
- Failed to parse (Missing texvc executable; please see math/README to configure.): v_{\mathrm{f}}^2 = v_{\mathrm{i}}^2 + 2 a (x_{\mathrm{f}} - x_{\mathrm{i}})
If the origin of our coordinate system is at the ground, then xi is zero. Then we solve for xf and substitute known values:
- Failed to parse (Missing texvc executable; please see math/README to configure.): x_{\mathrm{f}} = \frac{v_{\mathrm{f}}^2 - v_{\mathrm{i}}^2}{2 a} + x_{\mathrm{i}} = \rm \frac{(0\ m\ s^{-1})^2-(50\ m\ s^{-1})^2}{2(-9.81\ m\ s^{-2})} + 0\ m = 127.55\ m\,.
- What will its final velocity be when it reaches the ground?
To answer this question, we use the fact that the object has an initial velocity of zero at the apex before it begins its descent. We can use the same equation we used for the last question, using the value of 127.55 m for xi.
- Failed to parse (Missing texvc executable; please see math/README to configure.): v_{\mathrm{f}} = \sqrt{v_{\mathrm{i}}^2 + 2 a (x_{\mathrm{f}} - x_{\mathrm{i}})} = \rm \sqrt{(0\ m\ s^{-1})^2 + 2(-9.81\ m\ s^{-2})(0\ m - 127.55\ m)} = 50\ m\ s^{-1}\,.
Assuming this experiment were performed in a vacuum (negating drag effects), we find that the final and initial speeds are equal, a result which agrees with conservation of energy.
|
-
| Example: Projectile (2D) motion |
Figure B: An object fired at an angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta from the ground follows a parabolic trajectory. Note: The equations described here holds for object fired from ground and should not be mistaken with this picture.
Suppose that an object is not fired vertically but is fired at an angle θ from the ground. The object will then follow a parabolic trajectory, and its horizontal motion can be modeled independently of its vertical motion. Assume that the object is fired at an initial velocity of 50 m s−1 and 30° from the horizontal.
- How far will it travel before hitting the ground?
The object experiences an acceleration of −9.81 m s−2 in the vertical direction and no acceleration in the horizontal direction. Therefore, the horizontal displacement is
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta x = x_{\mathrm{f}} - x_{\mathrm{i}} = v_{\mathrm{i}} \cos \theta \; t + \tfrac{1}{2} at^2 = v_{\mathrm{i}} \cos \theta \; t
Solving the equation requires finding t. This can be done by analyzing the motion in the vertical direction. If we impose that the vertical displacement is zero, we can use the same procedure we did for rectilinear motion to find t.
- Failed to parse (Missing texvc executable; please see math/README to configure.): 0 = v_{\mathrm{i}} \sin \theta \; t + \tfrac{1}{2} at^2 = t\left(v_{\mathrm{i}} \sin \theta + \tfrac{1}{2} at \right)
We now solve for t and substitute this expression into the original expression for horizontal displacement.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta x = v_{\mathrm{i}} \cos \theta \left(\frac{-2 v_{\mathrm{i}} \sin \theta}{a}\right) = -\frac{v_{\mathrm{i}}^2 \sin 2\theta}{a} = 220.93 \ m\,
Note the use of the trigonometric identity 2sinθ cosθ = sin 2θ.
|
Kinematics is the study of how things move. Here, we are interested in the motion of normal objects in our world. A normal object is visible, has edges, and has a location that can be expressed with (x, y, z) coordinates. We will not be discussing the motion of atomic particles or black holes or light.
We will create a vocabulary and a group of mathematical methods that will describe this ordinary motion. Understand that we will be developing a language for describing motion only. We won't be concerned with what is causing or changing the motion, or more correctly, the momenta of the objects. In other words, we are not concerned with the action of forces within this topic.
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
aθ that increases the rate of rotation: d
ω/d
t = |
aθ|/
R.
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 a curve in space, given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P}(t) = X(t)\vec{i} + Y(t)\vec{j} + Z(t)\vec{k},
where i, j, and k are the unit vectors along the X, Y and Z axes of the reference frame F, respectively. There are a number of ways to define the functions X(t), Y(t) and Z(t) to match constraints imposed on the trajectory. Here, the particular case of cylindrical coordinates is presented.
If the particle P 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,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P}(t) = R\cos\theta(t)\vec{i} + R\sin\theta(t)\vec{j} + Z(t)\vec{k}.
The cylindrical coordinates for P(t) can be simplified by introducing the radial and tangential unit vectors,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{e}_r = \cos\theta(t)\vec{i} + \sin\theta(t)\vec{j}, \quad \textbf{e}_t = -\sin\theta(t)\vec{i} + \cos\theta(t)\vec{j}.
Using this notation, P(t) takes the form,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P}(t) = R\textbf{e}_r + Z(t)\vec{k},
where R is constant.
The velocity of VP is the time derivative of the trajectory P(t),
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{V}_P = \frac{d}{dt}(R\textbf{e}_r + Z(t)\vec{k}) = R\dot{\theta}\textbf{e}_t + \dot{Z}(t)\vec{k},
where
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{dt}\textbf{e}_r = \dot{\theta}\textbf{e}_t.
If the trajectory P(t) is not constrained to lie on a circular cylinder, then the radius R varies with time, so we have
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P}(t) = R(t)\textbf{e}_r + Z(t)\vec{k},
and
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{V}_P = \dot{R}\textbf{e}_r + R\dot{\theta}\textbf{e}_t + \dot{Z}(t)\vec{k}.
In this case, the acceleration AP, which is the time derivative of the velocity VP, is given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{A}_P = \frac{d}{dt}(\dot{R}\textbf{e}_r + R\dot{\theta}\textbf{e}_t + \dot{Z}(t)\vec{k}) = (\ddot{R} - R\dot{\theta}^2)\textbf{e}_r + (R\ddot{\theta} + 2\dot{R}\dot{\theta})\textbf{e}_t + \ddot{Z}(t)\vec{k}.
Each particle on the wheel travels in a planar circular trajectory (Kinematics of Machinery, 1876).
[12]
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the Z axis, in which case
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{P}(t) = R\textbf{e}_r + Z_0\vec{k},
where R and Z0 are constants. In this case, the velocity VP is given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{V}_P = \frac{d}{dt}(R\textbf{e}_r + Z_0\vec{k}) = R\dot{\theta}\textbf{e}_t.
The acceleration AP of the particle P, is now given by
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{A}_P = \frac{d}{dt}(R\dot{\theta}\textbf{e}_t) = - R\dot{\theta}^2\textbf{e}_r + R\ddot{\theta}\textbf{e}_t.
The components
- Failed to parse (Missing texvc executable; please see math/README to configure.): a_r = - R\dot{\theta}^2, \quad a_t = R\ddot{\theta},
are called the radial and tangential components of acceleration, respectively.
The notation for angular velocity and angular acceleration is often defined as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \omega = \dot{\theta}, \quad \alpha = \ddot{\theta},
so the radial and tangential acceleration components for circular trajectories are also written as
- Failed to parse (Missing texvc executable; please see math/README to configure.): a_r = - R\omega^2, \quad a_t = R\alpha.
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.[13] 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.
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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\omega} = \frac {\mathrm{d}\theta}{\mathrm{d}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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\alpha} = \frac {\mathrm{d}\mathbf{\omega}}{\mathrm{d}t}
The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \omega_{\mathrm{f}} = \omega_{\mathrm{i}} + \alpha t\!
- Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \omega_{\mathrm{i}} t + \tfrac{1}{2} \alpha t^2
- Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \tfrac{1}{2} (\omega_{\mathrm{f}} + \omega_{\mathrm{i}})t
- Failed to parse (Missing texvc executable; please see math/README to configure.): \omega_{\mathrm{f}}^2 = \omega_{\mathrm{i}}^2 + 2 \alpha (\theta_{\mathrm{f}} - \theta_{\mathrm{i}}).
Here θi and θf are, respectively, the initial and final angular positions, ωi and ωf 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.
Important formulas in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in the plane, or 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.
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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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, 1), and P = (X, Y), 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,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): [A(t)]^T[A(t)]=I.\!
The velocity of the point P along its trajectory P(t) is obtained as the time derivative of this position vector,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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
- Failed to parse (Missing texvc executable; please see math/README to configure.): [S] = \begin{bmatrix} \Omega & -\Omega\textbf{d} + \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix}
where
- Failed to parse (Missing texvc executable; please see math/README to configure.): [\Omega] = \dot{A}A^T,
is the angular velocity matrix.
Multiplying by the operator [S], the formula for the velocity VP takes the form
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{V}_P = [\Omega](\textbf{P}-\textbf{d}) + \dot{\textbf{d}} = \omega\times \textbf{R}_{P/O} + \textbf{V}_O,
where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω], the vector
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{R}_{P/O}=\textbf{P}-\textbf{d},
is the position of P relative to the origin O of the moving frame M, and
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{V}_O=\dot{\textbf{d}},
is the velocity of the origin O.
The acceleration of a point P in a moving body B is obtained as the time derivative of its velocity vector,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{A}_P = \frac{d}{dt}\textbf{V}_P = \frac{d}{dt}\big([S]\textbf{P}\big)=[\dot{S}]\textbf{P} + [S]\dot{\textbf{P}} = [\dot{S}]\textbf{P} + [S][S]\textbf{P} .
This equation can be expanded by first computing
- Failed to parse (Missing texvc executable; please see math/README to configure.): [\dot{S}] = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\textbf{d} -\Omega\dot{\textbf{d}} + \ddot{\textbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\textbf{d} -\Omega\textbf{V}_O + \textbf{A}_O \\ 0 & 0 \end{bmatrix}
and
- Failed to parse (Missing texvc executable; please see math/README to configure.): [S]^2 = \begin{bmatrix} \Omega & -\Omega\textbf{d} + \textbf{V}_O \\ 0 & 0 \end{bmatrix}^2 = \begin{bmatrix} \Omega^2 & -\Omega^2\textbf{d} + \Omega\textbf{V}_O \\ 0 & 0 \end{bmatrix}.
The formula for the acceleration AP can now be obtained as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{A}_P = \dot{\Omega}(\textbf{P} - \textbf{d}) + \textbf{A}_O + \Omega^2(\textbf{P}-\textbf{d}),
or
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{A}_P = \alpha\times\textbf{R}_{P/O} + \omega\times\omega\times\textbf{R}_{P/O} + \textbf{A}_O,
where α is the angular acceleration vector obtained from the derivative of the angular velocity matrix,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{R}_{P/O}=\textbf{P}-\textbf{d},
is the relative position vector, and
- Failed to parse (Missing texvc executable; please see math/README to configure.): \textbf{A}_O = \ddot{\textbf{d}}
is the acceleration of the origin of the moving frame M.
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:
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,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol{ v}_G(t) = \boldsymbol{\Omega} \times \boldsymbol{ r}_{G/O}
. For the case of an object that does not tip or turn, this reduces to v = R ω.
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. See Kelvin and Tait[14][15] and Fogiel.[16] 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.[17] An equilibrium problem (not kinematic) of this type is the catenary.[18]
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[19] 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.
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;[20]
- 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[21] 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.
- ^ Because magnitude of dr is necessarily the distance between two infinitesimally spaced points along the trajectory of the point, it is the same as an increment in arc length along the path of the point, customarily denoted ds.
- 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.
- ^ Edmund Taylor Whittaker (1904). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press. Chapter 1. ISBN 0-521-35883-3. http://books.google.com/books?id=epH1hCB7N2MC&printsec=frontcover&dq=inauthor:%22E+T+Whittaker%22&lr=&as_brr=0&sig=SN7_oYmNYM4QRSgjULXBU5jeQrA&source=gbs_book_other_versions_r&cad=0_2#PPA1,M1.
- ^ Joseph Stiles Beggs (1983). Kinematics. Taylor & Francis. p. 1. ISBN 0-89116-355-7. http://books.google.com/books?id=y6iJ1NIYSmgC&printsec=frontcover&dq=kinematics&lr=&as_brr=0&sig=brRJKOjqGTavFsydCzhiB3u_8MA#PPA1,M1.
- ^ Thomas Wallace Wright (1896). Elements of Mechanics Including Kinematics, Kinetics and Statics. E and FN Spon. Chapter 1. http://books.google.com/books?id=-LwLAAAAYAAJ&printsec=frontcover&dq=mechanics+kinetics&lr=&as_brr=0#PPA6,M1.
- ^ Ampère, André-Marie. Essai sur la Pilosophie des Sciences. Chez Bachelier. http://books.google.com/books?id=j4QPAAAAQAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false.
- ^ Merz, John (1903). A History of European Thought in the Nineteenth Century. Blackwood, London. pp. 5. http://books.google.com/books?id=toZJAAAAYAAJ&pg=PA5&lpg=PA5&dq=history+ampere+cinematique&source=bl&ots=NiWfNE6OI5&sig=4p4KItjzVNzntiBj3cvvZElnRxc&hl=en&sa=X&ei=hrj8TuiFD6aqsQL4rNDQAQ&ved=0CCQQ6AEwAA#v=onepage&q=history%20ampere%20cinematique&f=false.
- ^ O. Bottema & B. Roth (1990). Theoretical Kinematics. Dover Publications. reface. ISBN 0-486-66346-9. http://books.google.com/books?id=f8I4yGVi9ocC&printsec=frontcover&dq=kinematics&lr=&as_brr=0&sig=YfoHn9ImufIzAEp5Kl7rEmtYBKc#PPR7,M1.
- ^ See, for example: Russell C. Hibbeler (2009). "Kinematics and kinetics of a particle". Engineering Mechanics: Dynamics (12th ed.). Prentice Hall. p. 298. ISBN 0-13-607791-9. http://books.google.com/books?id=tOFRjXB-XvMC&pg=PA298. , Ahmed A. Shabana (2003). "Reference kinematics". Dynamics of Multibody Systems (2nd ed.). Cambridge University Press. ISBN 978-0-521-54411-5. http://books.google.com/books?id=zxuG-l7J5rgC&pg=PA28. , P. P. Teodorescu (2007). "Kinematics". Mechanical Systems, Classical Models: Particle Mechanics. Springer. p. 287. ISBN 1-4020-5441-6. http://books.google.com/books?id=k4H2AjWh9qQC&pg=PA287.
- ^ A. Biewener (2003). Animal Locomotion. Oxford University Press. ISBN 19850022X. http://books.google.com/books?id=yMaN9pk8QJAC.
- ^ J. M. McCarthy and G. S. Soh, 2010, Geometric Design of Linkages, Springer, New York.
- ^ "definition of geometry". Merriam-Webster on-line dictionary. http://www.merriam-webster.com/dictionary/geometry. Geometry:the study of properties of given elements that remain invariant under specified transformations.
- ^ Paul, Richard (1981). Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators. MIT Press, Cambridge, MA. ISBN 978-0-262-16082-7. http://books.google.com/books?id=UzZ3LAYqvRkC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false.
- ^ Reuleaux, F., 1876 'The Kinematics of Machinery,' (trans. and annotated by A. B. W. Kennedy), reprinted by Dover, New York (1963)
- ^ R. Douglas Gregory (2006). Chapter 16. Cambridge: Cambridge University. ISBN 0-521-82678-0. http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA457,M1.
- ^ William Thomson Kelvin & Peter Guthrie Tait (1894). Elements of Natural Philosophy. Cambridge University Press. p. 4. ISBN 1-57392-984-0. http://books.google.com/books?id=dHASAAAAIAAJ&pg=PA4&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
- ^ William Thomson Kelvin & Peter Guthrie Tait (1894). op. cit.. p. 296. http://books.google.com/books?id=ahtWAAAAMAAJ&pg=PA296&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES#PPA296,M1.
- ^ M. Fogiel (1980). "Problem 17-11". The Mechanics Problem Solver. Research & Education Assoc.. p. 613. ISBN 0-87891-519-2. http://books.google.com/books?id=XVyD9pJpW-cC&pg=PA613&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
- ^ Irving Porter Church (1908). Mechanics of Engineering. Wiley. p. 111. ISBN 1-110-36527-6. http://books.google.com/books?id=7-40AAAAMAAJ&pg=PA111&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
- ^ Morris Kline (1990). Mathematical Thought from Ancient to Modern Times. Oxford University Press. p. 472. ISBN 0-19-506136-5. http://books.google.com/books?id=aO-v3gvY-I8C&pg=PA472&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
- ^ Phillips, Jack (2007). Freedom in Machinery, Volumes 1-2 (reprint ed.). Cambridge University Press. ISBN 978-0-521-67331-0. http://books.google.com/books?id=Q5btdhoawN4C&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false.
- ^ Tsai, Lung-Wen (2001). Mechanism design:enumeration of kinematic structures according to function (llustrated ed.). CRC Press. p. 121. ISBN 978-0-8493-0901-4. http://books.google.com/books?id=X0AHKxwWTsYC&pg=PA107&lpg=PA107&dq=tsai,+enumeration&source=bl&ots=KsaVQo-N-Z&sig=xUPIkQcpC8ldoxjAJeVifSK2MRY&hl=en&ei=j4oSTpO2FcfYiAKu1ejdDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBgQ6AEwAA#v=onepage&q&f=false.
- ^ R. P. Sunkari and L. C. Schmidt, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm," Mechanism and Machine Theory 41 (2006) 1021–1030
