Kinematics

Classical mechanics
$\mathbf{F} = m \mathbf{a}$ Newton's Second Law
History of classical mechanics · Timeline of classical mechanics
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 · Velocity · Speed · Mass · Acceleration · Gravity · Force · Impulse · Torque / Moment / Couple · Momentum · Angular momentum · Inertia · Moment of inertia · Reference frame · Energy · Kinetic energy · Potential energy · Mechanical work · Virtual work · D'Alembert's principle
Core topics Rigid body · Rigid body dynamics · Euler's equations (rigid body dynamics) · Motion · Newton's laws of motion · Newton's law of universal gravitation · Euler's laws of motion · Equations of motion · Inertial frame of reference · Non-inertial reference frame · Rotating reference frame · Fictitious force · Linear motion · Mechanics of planar particle motion · Displacement (vector) · Relative velocity · Friction · Simple harmonic motion · Harmonic oscillator · Vibration · Damping · Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Non-uniform circular motion · Centripetal force · Centrifugal force · Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement
Scientists Galileo Galilei · Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut · Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson

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. Ampere's cinématique,^[4] which he constructed from the Greek κινῆματ-, kinemat-, or κινῆμα, 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). Kinematics is also used 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. Further, mathematicians have developed the subject 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 (such as displacement, time, velocity and acceleration) 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.

Rigid transformations

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).

Displacements and motion

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 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=(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

$\textbf{P} = [T(\phi, \mathbf{d})]\textbf{p} = \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix}\begin{Bmatrix}x\\y\\1\end{Bmatrix}.$

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]

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,

$\textbf{P}(t)=[T(0,\textbf{d}(t)]\textbf{p} = \textbf{d}(t) %2B \textbf{p}.$

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

$\textbf{V}_P=\dot{\textbf{P}}(t) = \dot{\textbf{d}}(t)=\textbf{V}_O,\quad \textbf{A}_P=\ddot{\textbf{P}}(t) = \ddot{\textbf{d}}(t) = \textbf{A}_O,$

where the dot denotes the derivative with respect to time and 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.

Particle kinematics

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.

Position and reference frames

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.

Position vector

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

$\mathbf{r}_A = (x_A,y_A,z_A),$

where x_A, y_A, and z_A are the Cartesian coordinates of the point. The magnitude of the position vector |r| gives the distance between the point A and the origin.

$|\mathbf{r}| = \sqrt{x_A^{\ 2} %2B y_A^{\ 2} %2B z_A^{\ 2}}.$

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

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 r_A = (x_A,y_A,z_A) and point B has position r_B = (x_B,y_B,z_B), the displacement r_AB of B from A is given by

$\mathbf{r}_{AB} = \mathbf{r}_B - \mathbf{r}_A = (x_B-x_A,y_B-y_A,z_B-z_A).$

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).

Velocity and speed

Average velocity is defined as

$\overline{\mathbf{v}} = \frac {\Delta \mathbf{r}}{\Delta t} \ ,$

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,

$\mathbf{v} = \lim_{\Delta t\rightarrow0}\frac{\Delta\mathbf{r}}{\Delta t} = \frac {d \mathbf{r}}{d t} \, ,$

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:

$|\mathbf{v}| = \left|\frac {d \mathbf{r}}{d t} \right| = \frac {d s}{d t}$

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.

Acceleration

Average acceleration (acceleration over a length of time) is defined as:

$\overline{\mathbf{a}} = \frac {\Delta \mathbf{v}}{\Delta t} \ ,$

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.

$\mathbf{a} = \lim_{\Delta t \rightarrow 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac {d \mathbf{v}}{d t} \, ,$

where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time.

Kinematics of constant acceleration

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:

$\mathbf{v}(t) = \mathbf{v}_0 %2B \int_0^{t} \mathbf{a} \; dt' = \mathbf{v}_0 %2B \mathbf{a}t.$

$\begin{align} \mathbf{r}(t) &= \mathbf{r}_0 %2B \int_0^t \mathbf{v} \; dt' = \mathbf{r}_0 %2B \int_0^t (\mathbf{v}_0 %2B \mathbf{a} t) \; dt' \\ &= \mathbf{r}_0 %2B \mathbf{v}_0 t %2B \tfrac{1}{2} \mathbf{a} t^2. \end{align}$

Additional relations between displacement, velocity, acceleration, and time can be derived. Since a = (v − v₀)/t,

$\mathbf{r}(t) = \mathbf{r}_0 %2B \left(\frac{\mathbf{v}%2B \mathbf{v_0}}{2}\right) t .$

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,

$(\mathbf{r} - \mathbf{r_0}) \cdot \mathbf{a} t = \left( \mathbf{v} - \mathbf{v}_0 \right) \cdot \frac{\mathbf{v} %2B \mathbf{v}_0}{2} t \ ,$

where · denotes the dot product. Dividing the t on both sides and carrying out the dot-products:

$2(\mathbf{r} - \mathbf{r_0}) \cdot \mathbf{a} = v^2 - v_0^{\ 2}.$

In the case of straight-line motion, (r - r₀) is parallel to a. Then

$v^2= v_0^2 %2B 2 a(r-r_0).$

This relation is useful when time is not known explicitly.

Relative velocity

Main article: Relative velocity

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 r_A, r_B, and r_O. Then the position of A relative to the reference object O is

$\mathbf{r}_{A/O} = \mathbf{r}_{A} - \mathbf{r}_{O} \,\!$

Consequently, the position of A relative to B is

$\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B = \mathbf{r}_A - \mathbf{r}_O - \left(\mathbf{r}_B-\mathbf{r}_O\right) = \mathbf{r}_{A/O}-\mathbf{r}_{B/O} \ .$

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:

$\mathbf{r}_{A/O} = \mathbf{r}_{A/B} %2B \mathbf{r}_{B/O} \ ,$

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 $\mathbf{V}_{A}$ relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity $\mathbf{V}_{B}$ , each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity $\mathbf{V}_{A/B}$ ), the equation above gives:

$\mathbf{V}_{A} = \mathbf{V}_{B} %2B \mathbf{V}_{A/B} \,\! .$

To find $\mathbf{V}_{A/B}$ we simply rearrange this equation to obtain:

$\mathbf{V}_{A/B} = \mathbf{V}_{A} -\mathbf{V}_{B} \,\! .$

At velocities comparable to the speed of light, these equations are not valid. They are replaced by equations derived from Einstein's theory of special relativity.

Example: Rectilinear (1D) motion

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

$x_{\mathrm{f}} - x_{\mathrm{i}} = v_{\mathrm{i}} t %2B \tfrac{1}{2} at^2.$

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.

$0 = v_{\mathrm{i}} t %2B \tfrac{1}{2} at^2 = t\left(v_{\mathrm{i}} %2B \tfrac{1}{2} at \right)$

There are two solutions: the first, t = 0, is trivial. The solution of interest is

$t = -\frac{2v_{\mathrm{i}}}{a} = \rm -\frac{2(50\ m\ s^{-1})}{-9.81\ m\ s^{-2}} = 10.2\ s\,.$

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:

$v_{\mathrm{f}}^2 = v_{\mathrm{i}}^2 %2B 2 a (x_{\mathrm{f}} - x_{\mathrm{i}})$

If the origin of our coordinate system is at the ground, then x_i is zero. Then we solve for x_f and substitute known values:

$x_{\mathrm{f}} = \frac{v_{\mathrm{f}}^2 - v_{\mathrm{i}}^2}{2 a} %2B x_{\mathrm{i}} = \rm \frac{(0\ m\ s^{-1})^2-(50\ m\ s^{-1})^2}{2(-9.81\ m\ s^{-2})} %2B 0\ m = 127.55\ m\,.$

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 x_i.

$v_{\mathrm{f}} = \sqrt{v_{\mathrm{i}}^2 %2B 2 a (x_{\mathrm{f}} - x_{\mathrm{i}})} = \rm \sqrt{(0\ m\ s^{-1})^2 %2B 2(-9.81\ m\ s^{-2})(0\ m - 127.55\ m)} = 50\ m\ s^{-1}\,.$

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

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

$\Delta x = x_{\mathrm{f}} - x_{\mathrm{i}} = v_{\mathrm{i}} \cos \theta \; t %2B \tfrac{1}{2} at^2 = v_{\mathrm{i}} \cos \theta \; t$

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.

$0 = v_{\mathrm{i}} \sin \theta \; t %2B \tfrac{1}{2} at^2 = t\left(v_{\mathrm{i}} \sin \theta %2B \tfrac{1}{2} at \right)$

We now solve for t and substitute this expression into the original expression for horizontal displacement.

$\Delta x = v_{\mathrm{i}} \cos \theta \left(\frac{-2 v_{\mathrm{i}} \sin \theta}{a}\right) = -\frac{v_{\mathrm{i}}^2 \sin 2\theta}{a} = 220.93 \ m\,$

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.

Coordinates for particle trajectories

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

$\textbf{P}(t) = X(t)\vec{i} %2B Y(t)\vec{j} %2B Z(t)\vec{k},$

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.

Cylindrical coordinates

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,

$\textbf{P}(t) = R\cos\theta(t)\vec{i} %2B R\sin\theta(t)\vec{j} %2B Z(t)\vec{k}.$

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

$\textbf{e}_r = \cos\theta(t)\vec{i} %2B \sin\theta(t)\vec{j}, \quad \textbf{e}_t = -\sin\theta(t)\vec{i} %2B \cos\theta(t)\vec{j}.$

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

$\textbf{P}(t) = R\textbf{e}_r %2B Z(t)\vec{k},$

where R is constant.

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

$\textbf{V}_P = \frac{d}{dt}(R\textbf{e}_r %2B Z(t)\vec{k}) = R\dot{\theta}\textbf{e}_t %2B \dot{Z}(t)\vec{k},$

where

$\frac{d}{dt}\textbf{e}_r = \dot{\theta}\textbf{e}_t.$

If the trajectory P(t) is not constrained to lie on a circular cylinder, then the radius R varies with time, so we have

$\textbf{P}(t) = R(t)\textbf{e}_r %2B Z(t)\vec{k},$

and

$\textbf{V}_P = \dot{R}\textbf{e}_r %2B R\dot{\theta}\textbf{e}_t %2B \dot{Z}(t)\vec{k}.$

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

$\textbf{A}_P = \frac{d}{dt}(\dot{R}\textbf{e}_r %2B R\dot{\theta}\textbf{e}_t %2B \dot{Z}(t)\vec{k}) = (\ddot{R} - R\dot{\theta}^2)\textbf{e}_r %2B (R\ddot{\theta} %2B 2\dot{R}\dot{\theta})\textbf{e}_t %2B \ddot{Z}(t)\vec{k}.$

Planar circular trajectories

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

$\textbf{P}(t) = R(t)\textbf{e}_r %2B Z_0\vec{k},$

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

$\textbf{V}_P = \frac{d}{dt}(R\textbf{e}_r %2B Z_0\vec{k}) = R\dot{\theta}\textbf{e}_t.$

The acceleration A_P of the particle P, is now given by

$\textbf{A}_P = \frac{d}{dt}(R\dot{\theta}\textbf{e}_t) = - R\dot{\theta}^2\textbf{e}_r %2B R\ddot{\theta}\textbf{e}_t.$

The components

$a_r = - R\dot{\theta}^2, \quad a_t = R\ddot{\theta},$

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

Rotation of a body around a fixed axis

Main article: Circular motion

Rotational or angular kinematics is the description of the rotation of an object.^[12] 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:

$\mathbf{\omega} = \frac {\mathrm{d}\theta}{\mathrm{d}t}$

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:

$\mathbf{\alpha} = \frac {\mathrm{d}\mathbf{\omega}}{\mathrm{d}t}$

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

$\omega_{\mathrm{f}} = \omega_{\mathrm{i}} %2B \alpha t\!$

$\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \omega_{\mathrm{i}} t %2B \tfrac{1}{2} \alpha t^2$

$\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \tfrac{1}{2} (\omega_{\mathrm{f}} %2B \omega_{\mathrm{i}})t$

$\omega_{\mathrm{f}}^2 = \omega_{\mathrm{i}}^2 %2B 2 \alpha (\theta_{\mathrm{f}} - \theta_{\mathrm{i}}).$

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.

Trajectories of points in a moving body

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.

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, 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,

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

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

$[A(t)]^T[A(t)]=I.\!$

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

$\textbf{V}_P = [\dot{T}(t)][T(t)]^{-1}\textbf{P}(t) = \begin{Bmatrix} \textbf{V}_P \\ 0\end{Bmatrix} = \begin{bmatrix} \dot{A}A^T & -\dot{A}A^T\textbf{d} %2B \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix} \begin{Bmatrix} \textbf{P}(t) \\ 1\end{Bmatrix}=[S]\textbf{P}.$

The matrix [S] is given by

$[S] = \begin{bmatrix} \Omega & -\Omega\textbf{d} %2B \dot{\textbf{d}} \\ 0 & 0 \end{bmatrix}$

where

$[\Omega] = \dot{A}A^T,$

is the angular velocity matrix.

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

$\textbf{V}_P = [\Omega](\textbf{P}-\textbf{d}) %2B \dot{\textbf{d}} = \omega\times \textbf{R}_{P/O} %2B \textbf{V}_O,$

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

$\textbf{R}_{P/O}=\textbf{P}-\textbf{d},$

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

$\textbf{V}_O=\dot{\textbf{d}},$

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,

$\textbf{A}_P = \frac{d}{dt}\textbf{V}_P = \frac{d}{dt}\big([S]\textbf{P}\big)=[\dot{S}]\textbf{P} %2B [S]\dot{\textbf{P}} = [\dot{S}]\textbf{P} %2B [S][S]\textbf{P} .$

This equation can be expanded by first computing

$[\dot{S}] = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\textbf{d} -\Omega\dot{\textbf{d}} %2B \ddot{\textbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\textbf{d} -\Omega\textbf{V}_O %2B \textbf{A}_O \\ 0 & 0 \end{bmatrix}$

and

$[S]^2 = \begin{bmatrix} \Omega & -\Omega\textbf{d} %2B \textbf{V}_O \\ 0 & 0 \end{bmatrix}^2 = \begin{bmatrix} \Omega^2 & -\Omega^2\textbf{d} %2B \Omega\textbf{V}_O \\ 0 & 0 \end{bmatrix}.$

The formula for the acceleration A_P can now be obtained as

$\textbf{A}_P = \dot{\Omega}(\textbf{P} - \textbf{d}) %2B \textbf{A}_O %2B \Omega^2(\textbf{P}-\textbf{d}),$

$\textbf{A}_P = \alpha\times\textbf{R}_{P/O} %2B \omega\times\omega\times\textbf{R}_{P/O} %2B \textbf{A}_O,$

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

$\textbf{R}_{P/O}=\textbf{P}-\textbf{d},$

is the relative position vector, and

$\textbf{A}_O = \ddot{\textbf{d}}$

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

Kinematic constraints

A 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:

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,

$\boldsymbol{ v}_G(t) = \boldsymbol{\Omega} \times \boldsymbol{ r}_{G/O}$ .

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

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. See Kelvin and Tait^[13]^[14] and Fogiel.^[15] 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.^[16] An equilibrium problem (not kinematic) of this type is the catenary.^[17]

Kinematic pairs

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^[18] 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

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;^[19]
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^[20] 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.

Notes

^ 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.

References

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 9781402055980.

^ Edmund Taylor Whittaker (1904). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press. Chapter 1. ISBN 0521358833. 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 0891163557. 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.
^ A. M. Ampere, 1834, Essai sur la Pilosophie des Sciences
^ John T. Merz, 1903, A History of European Thought in the Nineteenth Century, Blackwood, London
^ O. Bottema & B. Roth (1990). Theoretical Kinematics. Dover Publications. reface. ISBN 0486663469. 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 0136077919. 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 0521544114. 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 1402054416. 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.
^ Merriam-Webster on-line dictionary, Geometry: the study of properties of given elements that remain invariant under specified transformations
^ Richard Paul, 1981, Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators, MIT Press, Cambridge, MA
^ R. Douglas Gregory (2006). Chapter 16. Cambridge: Cambridge University. ISBN 0521826780. 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 1573929840. 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 0878915192. 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 1110365276. 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 0195061365. http://books.google.com/books?id=aO-v3gvY-I8C&pg=PA472&dq=%22inextensible+cord%22&lr=&as_brr=0&as_pt=ALLTYPES.
^ J. Phillips, Freedom in Machinery, Cambridge University Press, 2006
^ L. W. Tsai, Mechanism design: enumeration of kinematic structures according to function, CRC Press, 2000
^ 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

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.

Kinematics

Contents

Rigid transformations

Displacements and motion

Matrix representation

Pure translation

Particle kinematics

Position and reference frames

Position vector

Displacement

Velocity and speed

Acceleration

Kinematics of constant acceleration

Relative velocity

Coordinates for particle trajectories

Cylindrical coordinates

Planar circular trajectories

Rotation of a body around a fixed axis

Trajectories of points in a moving body

Position

Velocity

Acceleration

Kinematic constraints

Rolling without slipping

Inextensible cord

Kinematic pairs

Kinematic chains

See also

Notes

References

External links