Kinematics

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Kinematics (Greek κινειν,kinein, to move) is a branch of dynamics which describes the motion of objects without the consideration of the masses or forces that bring out the motion. In contrast, kinetics is concerned with the forces and interactions that produce or affect the motion.

The simplest application of kinematics is to point particle motion (translational kinematics or linear kinematics). The description of rotation (rotational kinematics or angular kinematics) is more complicated. The state of a generic rigid body may be described by combining both translational and rotational kinematics (rigid-body kinematics). A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints. The kinematic description of fluid flow is even more complicated, and not generally thought of in the context of kinematics.

Contents

[edit] Translational motion

Translational or curvilinear kinematics is the description of the motion in space of a point along a trajectory. This path can be linear, or curved as seen with projectile motion. There are three basic concepts that are required for the understanding translational motion:

  1. Displacement is the shortest distance between two points: the origin and the displaced point. The origin is (0,0) on a coordinate system that is defined by the observer. Because displacement has both magnitude (length) and direction, it is a vector whose initial point is the origin and terminal point is the displaced point.
  2. Velocity is the rate of change in displacement with respect to time; that is the displacement of a point changes with time. Velocity is also a vector. For a constant velocity, every unit of time adds the length of the velocity vector (in the same direction) to the displacement of the moving point. Instantaneous velocity (the velocity at an instant of time) is defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \frac {d \vec s}{d t}

, where ds is an infinitesimally small displacement and dt is an infinitesimally small length of time. Average velocity (velocity over a length of time) is defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \frac {\Delta \vec s}{\Delta t} , where Δs is the change in displacement and Δt is the interval of time over which displacement changes.

  1. Acceleration is the rate of change in velocity with respect to time. Acceleration is also a vector. As with velocity if acceleration is constant, for every unit of time the length of the acceleration vector (in the same direction) is added to the velocity. If the change in velocity (a vector) is known, the acceleration is parallel to it. Instantaneous acceleration (the acceleration at an instant of time) is defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a = \frac {d \vec v}{d t}

, where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time. Average acceleration (acceleration over a length of time) is defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a = \frac {\Delta \vec v}{\Delta t} , where Δv is the change in velocity and Δt is the interval of time over which velocity changes.

When acceleration is constant it is said to be undergoing uniformly accelerated motion. If this is the case, there are four equations that can be used to describe the motion of an object.

  1. Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \int \vec a dt = \vec v_0 + \vec a t
Those who are familiar with calculus may recognize this as an initial value problem. Because acceleration (a) is a constant, integrating it with respect to time (t) gives a change in velocity. Adding this to the initial velocity (v0) gives the final velocity (v).
  1. Failed to parse (Missing texvc executable; please see math/README to configure.): \vec s = \int \vec v dt = \int \vec v_0 + \vec at dt = \vec v_0 t + \frac{1}{2} \vec at^2
Using the above formula, we can substitute for v to arrive at this equation, where s is displacement.
  1. Failed to parse (Missing texvc executable; please see math/README to configure.): \vec s = \frac{\vec v+ \vec v_0}{2} t
By using the definition of an average, and the knowledge that average velocity times time equals displacement, we can arrive at this equation.
  1. Failed to parse (Missing texvc executable; please see math/README to configure.): v^2= v_0^2 + 2 a s


[edit] Relative velocity

Main article: Relative velocity

To describe the motion of object A with respect to object O, when we know how each is moving with respect to object B, we use the following equation involving vectors and vector addition:

Failed to parse (Missing texvc executable; please see math/README to configure.): r_{A/O} = r_{B/O} + r_{A/B} \,\!

The above relative motion equation states that the motion of A relative to O is equal to the motion of B relative to O plus the motion of A relative to B.

For example, let Ann move with velocity Failed to parse (Missing texvc executable; please see math/README to configure.): V_{A}

and let Bob move with velocity Failed to parse (Missing texvc executable; please see math/README to configure.): V_{B}

, each velocity given with respect to the ground. 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.): V_{A/B} ), the equation above gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): V_{A} = V_{B} + V_{A/B} \,\! .

To find Failed to parse (Missing texvc executable; please see math/README to configure.): V_{A/B}

we simply rearrange this equation to obtain:
Failed to parse (Missing texvc executable; please see math/README to configure.): V_{A/B} = V_{A} -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.

[edit] Rotational motion

Image:Angular velocity.svg
The angular velocity vector points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule.

Rotational kinematics is the description of the rotation of an object and involves the definition and use of the following three quantities:

Angular position: If a vector is defined as the oriented distance from the axis of rotation to a point on an object, the angular position of that point is the oriented angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta from a reference axis (e.g. the positive x-semiaxis) to that vector. An oriented angle is an angle swept about a known rotation axis and in a known rotation sense. In two-dimensional kinematics (the description of planar motion), the rotation axis is normal to the reference frame and can be represented by a rotation point (or center), and the rotation sense is represented by the sign of the angle (typically, a positive sign means counterclockwise sense). Angular displacement can be regarded as a relative position. It is represented by the oriented angle swept by the above-mentioned point (or vector), from an angular position to another.

Angular velocity: The magnitude of the angular velocity Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

is the rate at which the angular position Failed to parse (Missing texvc executable; please see math/README to configure.):  \theta 
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}

Angular acceleration: The magnitude of the angular acceleration Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

is the rate at which the angular velocity Failed to parse (Missing texvc executable; please see math/README to configure.): \omega
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.): \,\!\theta_f - \theta_i = \omega_i t + \frac{1}{2} \alpha t^2 \qquad \theta_f - \theta_i = \frac{1}{2} (\omega_f + \omega_i)t
Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\omega_f = \omega_i + \alpha t \qquad \alpha = \frac{\omega_f - \omega_i}{t} \qquad \omega_f^2 = \omega_i^2 + 2 \alpha (\theta_f - \theta_i)

.

Here Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\theta_i

and Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\theta_f
are, respectively, the initial and final angular positions, Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\omega_i
and Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\omega_f
are, respectively, the initial and final angular velocities, and Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\alpha
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.

[edit] Coordinate systems

In any given situation, the most useful coordinates may be determined by constraints on the motion, or by the geometrical nature of the force causing or affecting the motion. Thus, to describe the motion of a bead constrained to move along a circular hoop, the most useful coordinate may be its angle on the hoop. Similarly, to describe the motion of a particle acted upon by a central force, the most useful coordinates may be polar coordinates.

[edit] Fixed rectangular coordinates

In this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually i is a unit vector in the x direction, j is a unit vector in the y direction, and k is a unit vector in the z direction.

The position vector, s (or r), the velocity vector, v, and the acceleration vector, a are expressed using rectangular coordinates in the following way:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec s = x \vec i + y \vec j + z \vec k \, \!


Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \dot {s} = \dot {x} \vec {i} + \dot {y} \vec {j} + \dot {z} \vec {k} \, \!


Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a = \ddot {s} = \ddot {x} \vec {i} + \ddot {y} \vec {j} + \ddot {z} \vec {k} \, \!


Note: Failed to parse (Missing texvc executable; please see math/README to configure.): \dot {x} = \frac{\mathrm{d}x}{\mathrm{d}t}

, Failed to parse (Missing texvc executable; please see math/README to configure.):  \ddot {x} = \frac{\mathrm{d}^2x}{\mathrm{d}t^2}


[edit] Two dimensional rotating reference frame

This coordinate system only expresses planar motion.

This system of coordinates is based on three orthogonal unit vectors: the vector i, and the vector j which form a basis for the plane in which the objects we are considering reside, and k about which rotation occurs. Unlike rectangular coordinates, which are measured relative to an origin that is fixed and non-rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied.

[edit] Derivatives of unit vectors

The position, velocity, and acceleration vectors of a given point can be expressed using these coordinate systems, but we have to be a bit more careful than we do with fixed frames of reference. Since the frame of reference is rotating, we must take the derivatives of the unit vectors into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at a rate of Failed to parse (Missing texvc executable; please see math/README to configure.): \omega in the counterclockwise direction (that's Failed to parse (Missing texvc executable; please see math/README to configure.): \omega k using the right hand rule) then the derivatives of the unit vectors are as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{\vec i} = \omega \vec k \times \vec i = \omega \vec j


Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{\vec j} = \omega \vec k \times \vec j = - \omega \vec i


[edit] Position, velocity, and acceleration

Given these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.

[edit] Position

Position is straightforward:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec s = x \vec i + y \vec j


It is just the distance from the origin in the direction of each of the unit vectors.

[edit] Velocity

Velocity is the time derivative of position:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \frac{\mathrm{d}\vec s}{\mathrm{d}t} = \frac{\mathrm{d} (x \vec i)}{\mathrm{d}t} + \frac{\mathrm{d} (y \vec j)}{\mathrm{d}t}


By the product rule, this is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \dot x \vec i + x \dot{\vec i} + \dot y \vec j + y \dot{\vec j}


Which from the identities above we know to be:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = \dot x \vec i + x \omega \vec j + \dot y \vec j - y \omega \vec i = (\dot x - y \omega) \vec i + (\dot y + x \omega) \vec j


or equivalently

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v = (\dot x \vec i + \dot y \vec j) + (y \dot{\vec j} + x \dot{\vec i}) = \vec v_{rel} + \vec \omega \times \vec r


where Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v_{rel}

is the velocity of the particle relative to the coordinate system.

[edit] Acceleration In Three Dimensions

Acceleration is the time derivative of velocity.

We know that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a = \frac{\mathrm{d} \vec v}{\mathrm{d}t} = \frac{\mathrm{d} \vec v_{rel}}{\mathrm{d}t} + \frac{\mathrm{d} (\vec \omega \times \vec r)}{\mathrm{d}t}


Consider the Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d} \vec v_{rel}}{\mathrm{d}t}

part. Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v_{rel}
has two parts we want to find the derivative of: the relative change in velocity (Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a_{rel}

), and the change in the coordinate frame (Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \times \vec v_{rel} ).

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d} \vec v_{rel}}{\mathrm{d}t} = \vec a_{rel} + \omega \times \vec v_{rel}


Next, consider Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d} (\vec \omega \times \vec r)}{\mathrm{d}t} . Using the chain rule:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d} (\vec \omega \times \vec r)}{\mathrm{d}t} = \dot{\vec \omega} \times \vec r + \vec \omega \times \dot{\vec r}


Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{\vec r}=\vec v=\vec v_{rel} + \omega \times \vec r

from above:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d} (\vec \omega \times \vec r)}{\mathrm{d}t} = \dot{\vec \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r) + \vec \omega \times \vec v_{rel}


So all together:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a = \vec a_{rel} + \omega \times \vec v_{rel} + \dot{\vec \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r) + \vec \omega \times \vec v_{rel}


And collecting terms:

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec a = \vec a_{rel} + 2(\omega \times \vec v_{rel}) + \dot{\vec \omega} \times \vec r + \vec \omega \times (\vec \omega \times \vec r)


[edit] Three dimensional rotating coordinate frame

(to be written)

[edit] Kinematic constraints

A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:

[edit] 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, :

Failed to parse (Missing texvc executable; please see math/README to configure.): v_G(t) = \omega \times r_{G/O} \,\!

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

[edit] Inextensible cord

This is the case where bodies are connected by some cord that remains in tension and cannot change length. The constraint is that the sum of all components of the cord, however they are defined, is the total length, and the time derivative of this sum is zero.

[edit] See also

Look up kinematics in
Wiktionary, the free dictionary.

[edit] External links

Kinematics

← Integrate ... Differentiate →
Displacement (Distance) | Velocity (Speed) | Acceleration | Jerk | Snap

ar:علم الحركة

bs:Kinematika ca:Cinemàtica cs:Kinematika da:Kinematik de:Kinematik el:Κινηματική es:Cinemática fa:جنبش‌شناسی fr:Cinématique gl:Cinemática hr:Kinematika id:Kinematika it:Cinematica he:קינמטיקה ka:კინემატიკა lt:Kinematika hu:Kinematika ms:Kinematik nl:Kinematica ja:運動学 pl:Kinematyka pt:Cinemática ru:Кинематика точки sq:Kinematika sk:Kinematika sl:Kinematika fi:Kinematiikka sv:Kinematik ta:அசைவு விபரியல் tr:Kinematik uk:Кінематика ur:جنبشیات yi:קינעמאטיק zh:运动学

Retrieved from "http://en.wikilib.com/wiki/Kinematics"
Personal tools
AD Links