# kinematics

Structuring the mechanics from the
point of view of the forces involved
 mechanics Kinematics Laws of motionwithout forces Dynamics effect offorces Statics Forces in equilibrium ofresting bodies Kinetic forces change the state ofmotion
The kinematics in technical mechanics
 Technical mechanics Statics dynamics Strength theory kinematics kinetics

The kinematics ( altgriech. Κίνημα kinema , movement ') is an area of the mechanism , the movement of bodies purely geometrically describes with the sizes of time , location , speed and acceleration . The force , the mass of the body and all quantities derived from it such as momentum or energy are not taken into account . The kinematics thus only describes how a body moves and is therefore also referred to as movement theory. Why a body moves is left to other areas of mechanics: in physics of dynamics and in technical mechanics of kinetics .

The classification of the kinematics in the mechanics is different. The movement of bodies under the action of forces is the subject of dynamics in physics , which also includes the special case of rest ( statics ). Kinematics and dynamics are direct sub-areas of mechanics there. In technical mechanics , on the other hand, kinematics is viewed together with kinetics as a sub-area of ​​dynamics, which is also directly subordinate to technical mechanics and is therefore on the same level as statics and strength theory .

The term kinematics was coined by André-Marie Ampère in 1834 .

## Reference systems and coordinate systems

Reference systems form the physical framework in which a movement is described. Coordinate systems are mathematical instruments used to describe them; but they are also used outside of physics. In mechanics, solving specific problems always begins with the definition of a reference and coordinate system.

### Reference systems

The variables of location, speed and acceleration depend on the choice of the reference system .

• An observer on a platform perceives an arriving train as moving. For a passenger on the train, however, the train is at rest.
• Observed from the earth, the sun seems to revolve around the still earth. When viewed from space, the sun rests and the earth moves .

The description of movements is basically possible in all reference systems, but the description differs depending on the reference system. The planetary motion is much easier to describe, for example, with a resting sun.

A distinction is made between rest systems , moving and accelerated reference systems , with the accelerated being a special case of moving reference systems. The inertial systems are of particular importance . These are reference systems that either rest or move in a straight line with constant speed (no rotation and no acceleration), because Newton's first law applies in inertial systems: A body free of force then moves at constant speed or remains at rest. In contrast, apparent forces occur in accelerated reference systems . The earth rotates on its own axis and around the sun; so it does not form an inertial system. For most practical questions, however, the earth can be considered to be at rest to a good approximation.

In the context of classical mechanics it is assumed that every body can be assigned a location at any time. This is no longer possible in the context of quantum mechanics . Only probabilities of stay can be specified there. In addition, it is assumed in classical mechanics that bodies can reach any high speed and that time passes at the same speed in any place, regardless of movement. Both are not fulfilled in the theory of relativity .

### Coordinate systems

Polar coordinates

Coordinate systems are used for the mathematical description of the reference systems. Usually a Cartesian coordinate system is used, which consists of axes that are perpendicular to one another. It is particularly suitable for describing linear movements. Polar coordinates are well suited for rotational movements in a plane, especially if the origin is the center of the rotational movement. Cylinder coordinates or spherical coordinates are used in three-dimensional space . If the movement of a vehicle is to be described from the driver's point of view, the accompanying tripod (natural coordinates) is used. The different coordinate systems can be converted with the coordinate transformation . A certain reference system can therefore be described by different coordinate systems.

## Location, speed, acceleration and jerk

Location , speed and acceleration are the three central parameters of kinematics. They are linked to one another over time: a change in location over time is speed and a change in speed over time is acceleration. The terms speed and acceleration always refer to a straight direction, but this direction can change constantly. For rotary movements, there are instead the angle of rotation , the angular velocity and the angular acceleration . All of these quantities are vectors . You not only have an amount, but also a direction.

### place

Numerous notations are used for the location of a point-shaped body : The following is generally used for the location vector . This points from the coordinate origin to the point in the coordinate system at which the body is located. With Cartesian coordinates it is also common, sometimes only stands for the X component of the position vector. If the trajectory of the point is known, then the location is also indicated by the distance covered along the trajectory. For generalized coordinates is common. Since the location of a point changes over time , or is also used. ${\ displaystyle {\ vec {r}}}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle s}$${\ displaystyle q}$${\ displaystyle t}$${\ displaystyle {\ vec {r}} (t), x (t), s (t)}$${\ displaystyle {\ vec {q}} (t)}$

The function that assigns a place to each point in time is the law of distance and time . This can be represented in Cartesian coordinates by the scalar functions , and , which form the components of the position vector: ${\ displaystyle x (t)}$${\ displaystyle y (t)}$${\ displaystyle z (t)}$

${\ displaystyle t \ mapsto {\ vec {r}} (t) = \ left ({\ begin {array} {c} x (t) \\ y (t) \\ z (t) \ end {array} } \ right) = x (t) \ cdot {\ vec {e}} _ {x} + y (t) \ cdot {\ vec {e}} _ {y} + z (t) \ cdot {\ vec {e}} _ {z}}$

where the unit vectors represent the basis (vector space) of the Cartesian coordinate system. ${\ displaystyle \ {{\ vec {e}} _ {x}, {\ vec {e}} _ {y}, {\ vec {e}} _ {z} \}}$

### speed

The change in location over time is the speed . If the position of a body changes during a period of time around the path , then it has the mean speed during this period ${\ displaystyle v}$${\ displaystyle \ Delta t}$${\ displaystyle \ Delta s}$

${\ displaystyle v = {\ frac {\ Delta s} {\ Delta t}}}$.

The speed at any point in time, the instantaneous speed , results from the infinitesimally small change in the position vector during the infinitesimally small period of time : ${\ displaystyle {\ vec {v}} (t)}$${\ displaystyle \ mathrm {d} {\ vec {r}}}$${\ displaystyle \ mathrm {d} t}$

${\ displaystyle {\ vec {v}} (t) = \ lim _ {\ Delta t \ to 0} {\ frac {{\ vec {r}} (t + \ Delta t) - {\ vec {r}} (t)} {\ Delta t}} = {\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm {d} t}} = {\ dot {\ vec {r}}} ( t)}$.

The speed is therefore the derivative of the location with respect to time and is marked with a point above the location vector. In Cartesian coordinates, the velocity vector has the components , and , which each represent the time derivative of the spatial coordinates , and : ${\ displaystyle v_ {x}}$${\ displaystyle v_ {y}}$${\ displaystyle v_ {z}}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$

${\ displaystyle {\ vec {v}} (t) = \ left ({\ begin {array} {c} v_ {x} (t) \\ v_ {y} (t) \\ v_ {z} (t ) \ end {array}} \ right) = \ left ({\ begin {array} {c} {\ frac {\ mathrm {d} x} {\ mathrm {d} t}} (t) \\ {\ frac {\ mathrm {d} y} {\ mathrm {d} t}} (t) \\ {\ frac {\ mathrm {d} z} {\ mathrm {d} t}} (t) \ end {array }} \ right) = \ left ({\ begin {array} {c} {\ dot {x}} (t) \\ {\ dot {y}} (t) \\ {\ dot {z}} ( t) \ end {array}} \ right)}$

The speed vector is made up of its amount and the normalized direction vector . This direction vector represents a momentary tangential vector to the trajectory of the particle. ${\ displaystyle {\ vec {v}} = | {\ vec {v}} | \ cdot {\ vec {e}} _ {\ parallel}}$${\ displaystyle | {\ vec {v}} | = {\ sqrt {v_ {x} ^ {2} + v_ {y} ^ {2} + v_ {z} ^ {2}}}}$${\ displaystyle {\ vec {e}} _ {\ parallel}}$${\ displaystyle {\ vec {r}} (t)}$

### acceleration

The change in speed over time is the acceleration . If the speed of a body changes by this value over a period of time , then it has the mean acceleration ${\ displaystyle a}$${\ displaystyle \ Delta t}$${\ displaystyle \ Delta v}$

${\ displaystyle a = {\ frac {\ Delta v} {\ Delta t}}}$

The acceleration at any given point in time results from the infinitesimally small change in the velocity vector during the infinitesimally small period of time : ${\ displaystyle {\ vec {a}} (t)}$${\ displaystyle \ mathrm {d} {\ vec {v}}}$${\ displaystyle \ mathrm {d} t}$

${\ displaystyle {\ vec {a}} (t) = \ lim _ {\ Delta t \ to 0} {\ frac {{\ vec {v}} (t + \ Delta t) - {\ vec {v}} (t)} {\ Delta t}} = {\ frac {\ mathrm {d} {\ vec {v}}} {\ mathrm {d} t}} = {\ dot {\ vec {v}}} ( t) = {\ ddot {\ vec {r}}} (t)}$.

The acceleration is therefore the first derivative of the speed according to time and is marked with a point above the speed vector, as well as the second derivative of the location according to time and is marked with two points above the location vector. In Cartesian coordinates, the acceleration is represented by its components , and , which result as the second time derivative of the spatial components , and : ${\ displaystyle a_ {x} (t)}$${\ displaystyle a_ {y} (t)}$${\ displaystyle a_ {z} (t)}$${\ displaystyle x (t)}$${\ displaystyle y (t)}$${\ displaystyle z (t)}$

${\ displaystyle {\ vec {a}} (t) = \ left ({\ begin {array} {c} a_ {x} (t) \\ a_ {y} (t) \\ a_ {z} (t ) \ end {array}} \ right) = \ left ({\ begin {array} {c} {\ frac {\ mathrm {d} ^ {2} x} {\ mathrm {d} t ^ {2}} } (t) \\ {\ frac {\ mathrm {d} ^ {2} y} {\ mathrm {d} t ^ {2}}} (t) \\ {\ frac {\ mathrm {d} ^ { 2} z} {\ mathrm {d} t ^ {2}}} (t) \ end {array}} \ right) = \ left ({\ begin {array} {c} {\ ddot {x}} ( t) \\ {\ ddot {y}} (t) \\ {\ ddot {z}} (t) \ end {array}} \ right)}$

The acceleration vector can be split into two components, each of which is tangential and normal to the trajectory. The tangential acceleration describes the change in speed over time and forms a tangent to the trajectory: ${\ displaystyle {\ vec {a}} = {\ vec {a}} _ {\ parallel} + {\ vec {a}} _ {\ perp}}$ ${\ displaystyle {\ vec {a}} _ {\ parallel}}$${\ displaystyle | {\ vec {v}} |}$

${\ displaystyle {\ vec {a}} _ {\ parallel} = {\ frac {\ mathrm {d} | {\ vec {v}} |} {\ mathrm {d} t}} \ cdot {\ vec { e}} _ {\ parallel}}$

The normal acceleration, on the other hand, describes the change in the direction of speed over time and provides a measure of the curvature of the trajectory: ${\ displaystyle {\ vec {a}} _ {\ perp}}$${\ displaystyle {\ vec {e}} _ {\ parallel}}$

${\ displaystyle {\ vec {a}} _ {\ perp} = | {\ vec {v}} | \ cdot {\ frac {\ mathrm {d} {\ vec {e}} _ {\ parallel}} { \ mathrm {d} t}} = {\ frac {| {\ vec {v}} | ^ {2}} {R}} \ cdot {\ vec {e}} _ {\ perp}}$

where is a normalized normal vector of the trajectory and denotes the radius of curvature of the trajectory. ${\ displaystyle {\ vec {e}} _ {\ perp}}$${\ displaystyle R}$

### shock

The change in acceleration over time is the jerk . If the acceleration of a point-like body changes by the value over a period of time , then it has the mean jerk ${\ displaystyle j}$${\ displaystyle \ Delta t}$${\ displaystyle \ Delta a}$

${\ displaystyle j = {\ frac {\ Delta a} {\ Delta t}}}$

The jerk at any given point in time results from the infinitesimally small change in the acceleration vector during the infinitesimally small period of time : ${\ displaystyle {\ vec {j}} (t)}$${\ displaystyle \ mathrm {d} {\ vec {a}}}$${\ displaystyle \ mathrm {d} t}$

${\ displaystyle {\ vec {j}} (t) = \ lim _ {\ Delta t \ to 0} {\ frac {{\ vec {a}} (t + \ Delta t) - {\ vec {a}} (t)} {\ Delta t}} = {\ frac {\ mathrm {d} {\ vec {a}}} {\ mathrm {d} t}} = {\ dot {\ vec {a}}} ( t) = {\ ddot {\ vec {v}}} (t) = {\ overset {...} {\ vec {r}}} (t)}$.

The jerk is therefore the first derivative of the acceleration according to time and is marked with two points above the speed vector, as well as the third derivative of the location according to time and is marked with three points above the position vector. In Cartesian coordinates, the jerk is represented by the components , and : ${\ displaystyle j_ {x} (t)}$${\ displaystyle j_ {y} (t)}$${\ displaystyle j_ {z} (t)}$

${\ displaystyle {\ vec {j}} (t) = \ left ({\ begin {array} {c} j_ {x} (t) \\ j_ {y} (t) \\ j_ {z} (t ) \ end {array}} \ right) = \ left ({\ begin {array} {c} {\ dot {a}} _ {x} (t) \\ {\ dot {a}} _ {y} (t) \\ {\ dot {a}} _ {z} (t) \ end {array}} \ right)}$

According to this definition, which is mainly used in physics, a uniform circular motion would be a motion with constant jerk. In general usage and in technical applications, however, this is a jerk-free movement. The acceleration vector is therefore transformed into a body-fixed coordinate system and the derivation is carried out in this system. One obtains for the jerk in the body-fixed system:

${\ displaystyle {\ vec {j}} '(t) = {\ frac {\ mathrm {d}' {\ vec {a}} '} {\ mathrm {d} t}}}$,

with and the transformation matrix from the body-fixed system to the inertial system. ${\ displaystyle {\ vec {a}} '= (T_ {K'} ^ {I}) ^ {- 1} \ cdot {\ vec {a}}}$${\ displaystyle T_ {K '} ^ {I}}$

In this definition, e.g. B. the lateral jolt , which plays a major role in rail vehicles , proportional to the change in curvature . With the routing elements used , this is given analytically as a function of the path and can be converted into the lateral jolt for a specific speed.

## Types of movement, degree of freedom and constraints

Movements can be classified according to numerous criteria. A special case of movement is the state of rest with zero speed. The subdivision into the

According to the acceleration, a distinction is made between

• Uniform motion (also uniform rotary motion) with an acceleration and a constant speed. When rotating, the magnitude of the velocity is constant and the vector of the angular velocity maintains its direction while the direction changes constantly.${\ displaystyle a = 0}$
Evenly accelerated movement with initial speed and zero initial path. In the diagram on the left, the path is plotted vertically, in the middle diagram the speed and in the right diagram the acceleration; each as a function of time.
• The Uniformly Accelerated Movement with a Constant Acceleration The speed increases at a constant rate. It decreases with negative acceleration. This includes the free fall in which the acceleration due to gravity acts constantly . The oblique throw is a combination of uniform and uniformly accelerated movement: The acceleration due to gravity acts constantly in the vertical direction, while no acceleration acts in the horizontal direction (if the air resistance is disregarded).${\ displaystyle a = const.}$

Depending on whether the observed body can reach any place or not, a distinction is made between

• Free movement in which the body is not restricted and can move freely, like an airplane and the
• Bound movement in which the body is restricted by so-called constraints . A train can only move along the tracks.

The movement possibilities of a body are called the degree of freedom . A point-like body that can move freely in three-dimensional space has three degrees of freedom. If he moves in one plane, he has two degrees of freedom. And only one when moving along a curve or straight line. The restricted degrees of freedom are called binding . An extended, rigid body can also rotate around the body's own axes without changing its center of gravity . He has three more degrees of freedom, since rotation is possible in every dimension. Deformable bodies such as flexible beams, liquids and gases have an infinite number of degrees of freedom.

## Relative movement

The movement of points is often described in accelerated frames of reference that are themselves accelerated compared to another system.

In order to differentiate between the sizes of an object (location, speed, acceleration) in two reference systems, the normal notation is used for the sizes in the base system and the same letter with an apostrophe ( prime ) is used for the accelerated reference system . The latter is then also referred to as the “deleted reference system”, and all quantities related to it are given the addition “relative-” for linguistic differentiation.

meaning
${\ displaystyle {\ vec {r}}}$ Position of the object in S (basic system).
${\ displaystyle {\ vec {r}} {\; '}}$ Relative position of the object in S '.
${\ displaystyle {\ vec {v}} = {\ dot {\ vec {r}}}}$ Speed ​​of the object in S
${\ displaystyle {\ vec {v}} {\; '}}$ Relative speed of the object in S '
${\ displaystyle {\ vec {a}} = {\ dot {\ vec {v}}}}$ Acceleration of the object in S
${\ displaystyle {\ vec {a}} {\; '}}$ Relative acceleration of the object in S '
${\ displaystyle {\ vec {r}} _ {O}}$ Position of the origin of S 'in S
${\ displaystyle {\ vec {v}} _ {O} = {\ dot {\ vec {r}}} _ {O}}$ Velocity of the origin of S 'in S
${\ displaystyle {\ vec {a}} _ {O} = {\ dot {\ vec {v}}} _ {O}}$ Accelerating the origin of S 'in S
${\ displaystyle {\ vec {\ omega}}}$ Angular velocity of the system S 'in S
${\ displaystyle {\ vec {\ alpha}} = {\ dot {\ vec {\ omega}}}}$ Angular acceleration of the system S 'in S

When deriving a vector that is given in a rotating reference system, the angular velocity and the angular acceleration of the reference system must be taken into account. The kinematic relationships are: ${\ displaystyle {\ vec {\ omega}}}$${\ displaystyle {\ dot {\ vec {\ omega}}}}$

kinematic quantities in S
position ${\ displaystyle {\ vec {r}} = {\ vec {r}} _ {O} + {\ vec {r}} {\; '}}$
speed ${\ displaystyle {\ vec {v}} = {\ frac {d {\ vec {r}}} {dt}} = {\ vec {v}} _ {O} + {\ vec {\ omega}} \ times {\ vec {r}} {\; '} + {\ vec {v}} {\;'}}$
acceleration ${\ displaystyle {\ vec {a}} = {\ frac {d {\ vec {v}}} {dt}} = {\ vec {a}} _ {O} + {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} {\; '}) + {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} {\; '} +2 \, {\ vec {\ omega}} \ times {\ vec {v}} {\;'} + {\ vec {a}} {\; '}}$

If S is an inertial system , the absolute acceleration can be inserted into Newton's equation of motion:

${\ displaystyle m {\ vec {a}} = {\ vec {F}}}$

Solved for the term with the relative acceleration , the equation of motion for the relative motion is obtained. ${\ displaystyle m {\ vec {a}} {\; '}}$

## Rigid Body Kinematics

The vector to point P of a rigid body is constant in a body-fixed reference system. The movement of this point in a basic system is calculated as follows: ${\ displaystyle r_ {P}}$

kinematic quantities in S
position ${\ displaystyle {\ vec {r}} = {\ vec {r}} _ {O} + {\ vec {r}} _ {P}}$
speed ${\ displaystyle {\ vec {v}} = {\ frac {d {\ vec {r}}} {dt}} = {\ vec {v}} _ {O} + {\ vec {\ omega}} \ times {\ vec {r}} _ {P}}$
acceleration ${\ displaystyle {\ vec {a}} = {\ frac {d {\ vec {v}}} {dt}} = {\ vec {a}} _ {O} + {\ vec {\ omega}} \ times ({\ vec {\ omega}} \ times {\ vec {r}} _ {P}) + {\ dot {\ vec {\ omega}}} \ times {\ vec {r}} _ {P} }$

## Absolute kinematics

The movement of rigid bodies connected by joints is the basis for analyzing multibody systems. For this purpose, the position, speed and acceleration of the rigid body j relative to the body i are considered. The relative movement can be expressed by the joint coordinates ( generalized coordinates ) and their derivatives. The motion quantities of the body i in the inertial system are assumed to be known.

${\ displaystyle {\ vec {r}} _ {j} = {\ vec {r}} _ {i} + {\ vec {r}} _ {i, j}}$
${\ displaystyle {\ vec {v}} _ {j} = {\ vec {v}} _ {i} + {\ vec {\ omega}} _ {i} \ times {{\ vec {r}} _ {i, j}} + {\ vec {v}} _ {i, j}}$
${\ displaystyle {\ vec {a}} _ {j} = {\ vec {a}} _ {i} + {\ vec {\ omega}} _ {i} \ times ({\ vec {\ omega}} _ {i} \ times {\ vec {r}} _ {i, j}) + 2 {\ vec {\ omega}} _ {i} \ times {\ vec {v}} _ {i, j} + {\ vec {\ alpha}} _ {i} \ times {\ vec {r}} _ {i, j} + {\ vec {a}} _ {i, j}}$
${\ displaystyle {\ vec {\ omega}} _ {j} = {\ vec {\ omega}} _ {i} + {\ vec {\ omega}} _ {i, j}}$
${\ displaystyle {\ vec {\ alpha}} _ {j} = {\ vec {\ alpha}} _ {i} + {\ vec {\ omega}} _ {i} \ times {\ vec {\ omega} } _ {i, j} + {\ vec {\ alpha}} _ {i, j}}$

With:

${\ displaystyle {\ vec {r}} _ {i}, {\ vec {r}} _ {j}}$: Position vectors for the bodies i, j
${\ displaystyle {\ vec {r}} _ {i, j}}$: Vector from body i to body j
${\ displaystyle {\ vec {v}} _ {i}, {\ vec {v}} _ {j}}$: Absolute velocities of the bodies i, j
${\ displaystyle {\ vec {a}} _ {i}, {\ vec {a}} _ {j}}$: Absolute accelerations of the body i, j
${\ displaystyle {\ vec {v}} _ {i, j}}$: Speed ​​of the body j relative to the body i
${\ displaystyle {\ vec {\ omega}} _ {i}, {\ vec {\ omega}} _ {j}}$: absolute angular velocities of bodies i, j
${\ displaystyle {\ vec {\ omega}} _ {i, j}}$: Angular velocity of the body j relative to the body i
${\ displaystyle {\ vec {\ alpha}} _ {i}, {\ vec {\ alpha}} _ {j}}$: absolute angular accelerations of the bodies i, j
${\ displaystyle {\ vec {\ alpha}} _ {i, j}}$: Angular acceleration of body j relative to body i

## Applications

In multi-body systems , the investigation of spatial mechanisms is the subject of kinematics. These mechanisms are often made up of joints and connections. Examples are robots , kinematic chains and wheel suspensions in the automotive industry. With kinematic methods (in robotics, see direct kinematics ) the number of degrees of freedom is determined and the position, speed and acceleration of all bodies are calculated.

## Individual evidence

1. ^ André-Marie Ampère : Essai sur la philosophie des sciences, ou Exposition analytique d'une classification naturelle de toutes les connaissances humaines . Chez Bachelier, Paris 1834 ( limited preview in Google Book Search [accessed December 14, 2017]).
2. Torsten Fließbach: Mechanics - Textbook on Theoretical Physics I , Springer, 7th edition, 2015, pp. 2–8.
3. Wolfgang Nolting: Basic Course Theoretical Mechanics 1 - Classical Mechanics , Springer, 10th edition, 2013, p. 163.
4. Wolfgang Nolting: Basic Course Theoretical Mechanics 2 - Analytical Mechanics , Springer, 9th edition, 2014, p. 3 f.
5. Klaus-Peter Schnelle: Simulation models for the driving dynamics of passenger cars taking into account the non-linear chassis kinematics. VDI-Verlag, Düsseldorf 1990, ISBN 3-18-144612-2 . (Progress reports VDI No. 146)

## literature

• Jens Wittenburg: Kinematics - Theory and Application , Springer, 2016.