Kinematics – Linear, Angular and Rigid Body Motion

Kinematics is the branch of engineering mechanics concerned with the description of motion without considering the forces and moments that cause it. It deals with position, displacement, velocity and acceleration of particles and rigid bodies.

The methods on this page include vector analysis. Background information is provided on the RoyMech Vectors in Engineering Mechanics page.

Contents

• Basic Concepts and Nomenclature
• Particle Kinematics
• Linear Motion
• Angular Motion
• Circular Motion
• Absolute and Relative Motion
• Rotating Unit Vector
• Vector in a Rotating Reference Frame
• Velocity in a Moving Reference Frame
• Acceleration in a Moving Reference Frame
• Rigid-Body Kinematics in a Plane
• Instantaneous Centre of Rotation
• Rigid-Body Kinematics in Three Dimensions
• Finite Rotations
• Euler and Chasles Theorems
• Translation
• Rotation About a Fixed Axis
• Rotation About a Fixed Point
• General Three-Dimensional Motion

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Basic Concepts in Kinematics

• Time – the independent variable used to describe the sequence and duration of events. SI unit: second (s).
• Particle – an idealised body whose dimensions are negligible for the motion being considered.
• Rigid body – an idealised body in which the distance between any two material points remains constant.

Nomenclature

• s = distance or displacement along a defined path (m)
• r = position vector (m)
• u = initial linear velocity (m/s)
• v = linear velocity or final linear velocity (m/s)
• a = linear acceleration (m/s²)
• t = time (s)
• θ = angular displacement (rad)
• ω = angular velocity (rad/s)
• ω₁ = initial angular velocity (rad/s)
• ω₂ = final angular velocity (rad/s)
• α = angular acceleration (rad/s²)
• n = rotational speed (rev/min)

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Particle Kinematics

Particle kinematics describes the position and motion of a particle as functions of time without considering the forces responsible for that motion.

If the position of a particle relative to a selected reference frame is represented by the position vector r, then its velocity and acceleration are obtained by differentiation with respect to time.

$$\mathbf{v}=\frac{d\mathbf{r}}{dt}$$ $$\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^2\mathbf{r}}{d{t}^2}$$

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Linear Motion

Consider a particle moving from position A to position B. Its position relative to the selected coordinate system is described by the position vector r. The displacement between the two positions is:

$$\Delta \mathbf{r}={\mathbf{r}}_B-{\mathbf{r}}_A$$

Distance travelled is a scalar quantity representing the length of the path. Displacement is a vector quantity representing the change in position.

The average velocity over a time interval is:

$${\mathbf{v}}_{\text{avg}}=\frac{\Delta \mathbf{r}}{\Delta t}$$

Taking the limit as the time interval approaches zero gives the instantaneous velocity:

$$\mathbf{v}=\frac{d\mathbf{r}}{dt}$$

The instantaneous velocity vector is tangent to the particle path. Acceleration is the time derivative of velocity:

$$\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^2\mathbf{r}}{d{t}^2}$$

Linear Motion with Constant Acceleration

For one-dimensional motion with constant acceleration a, initial velocity u, final velocity v, displacement s and elapsed time t:

$$v=u+at$$ $$s=ut+\frac{a{t}^2}{2}$$ $$s=\frac{u+v}{2}t$$ $$v^2=u^2+2as$$

These equations apply only when acceleration is constant over the interval being considered.

The original RoyMech diagram below illustrates the integration of constant acceleration to obtain velocity and displacement.

The constants of integration are established from the initial conditions. For an initial velocity u at t = 0, the velocity integration constant is u. If displacement is measured from the initial position, the displacement integration constant is zero.

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Angular Motion

Angular position is described by the angle θ. The instantaneous angular velocity is:

$$\omega =\frac{d\theta }{dt}$$

Angular acceleration is:

$$\alpha =\frac{d\omega }{dt}$$

For a point having position vector r relative to the axis of rotation, its velocity due to rotation is:

$$\mathbf{v}=\mathbf{\omega }\times \mathbf{r}$$

The direction follows from the vector cross product and the right-hand rule. Angular velocity may be represented as a vector; however, finite rotations must not in general be treated as ordinary commutative vectors.

Angular Motion with Constant Angular Acceleration

Rotational speed n in revolutions per minute is related to angular velocity by:

$$\omega =\frac{2\pi n}{60}$$

For constant angular acceleration:

$${\omega }_2={\omega }_1+\alpha t$$ $$\theta =\frac{{\omega }_1+{\omega }_2}{2}t$$ $$\theta ={\omega }_{1}t+\frac{\alpha t^2}{2}$$ $${\omega }_2^2={\omega }_1^2+2\alpha \theta$$

These equations are the angular equivalents of the constant-acceleration relations for linear motion.

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Circular Motion

For a particle moving around a circular path of radius r, tangential speed and angular velocity are related by:

$$v=\omega r$$

Even when the speed is constant, the velocity vector continuously changes direction. The particle therefore has an acceleration directed towards the centre of the circular path.

$$a_n=\frac{v^2}{r}={\omega }^{2}r$$

This is the normal or centripetal acceleration.

For general circular motion, where angular speed may also change, the acceleration has both tangential and normal components:

$$\mathbf{a}=\mathbf{\alpha }\times \mathbf{r}+\mathbf{\omega }\times \left(\mathbf{\omega }\times \mathbf{r}\right)$$

For planar rotation these correspond in magnitude to:

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Absolute and Relative Motion

Motion must always be described relative to a reference frame. In engineering mechanics a reference frame fixed to the Earth is commonly treated as approximately inertial when the effects of the Earth's rotation and orbital motion are negligible for the problem being considered.

The term absolute motion is conventionally used for motion measured relative to the selected fixed or inertial reference frame. Relative motion describes motion measured from another frame that may itself be translating or rotating.

A familiar example is a person walking along a moving railway carriage. The person has a velocity relative to the carriage, while the carriage itself has a velocity relative to the ground.

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Rotating Unit Vector

Consider a unit vector e attached to a reference frame rotating with angular velocity ω.

The time derivative of the rotating unit vector as observed from the fixed frame is:

$$\frac{d\mathbf{e}}{dt}=\mathbf{\omega }\times \mathbf{e}$$

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Vector in a Rotating Reference Frame

Consider a point P having position vector r, expressed using unit vectors e₁ and e₂ attached to a rotating reference frame.

The position vector is:

$$\mathbf{r}=r_1{\mathbf{e}}_1+r_2{\mathbf{e}}_2$$

When differentiating this expression, both the scalar components and the rotating unit vectors may vary with time.

For any vector A, the derivative measured in a fixed frame is related to the derivative measured in a frame rotating with angular velocity ω by:

$${\left(\frac{d\mathbf{A}}{dt}\right)}_{\text{fixed}}={\left(\frac{d\mathbf{A}}{dt}\right)}_{\text{rot}}+\mathbf{\omega }\times \mathbf{A}$$

This transport theorem is fundamental to the analysis of motion in rotating reference frames.

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Velocity of a Point in a Moving Reference Frame

Let O' be the origin of a moving reference frame and let P have position r relative to O'. If r_O' locates O' relative to the fixed frame, then:

$${\mathbf{r}}_P={\mathbf{r}}_{O\prime }+\mathbf{r}$$

If the moving frame translates and rotates with angular velocity ω, the velocity of P is:

$${\mathbf{v}}_P={\mathbf{v}}_{O\prime }+{\mathbf{v}}_{\mathrm{rel}}+\mathbf{\omega }\times \mathbf{r}$$

where v_rel is the velocity of P measured relative to the rotating frame.

If the moving frame does not rotate, ω = 0 and:

$${\mathbf{v}}_P={\mathbf{v}}_{O\prime }+{\mathbf{v}}_{\mathrm{rel}}$$

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Acceleration of a Point in a Moving Reference Frame

Differentiating the velocity relation gives the general acceleration equation:

$${\mathbf{a}}_P={\mathbf{a}}_{O\prime }+{\mathbf{a}}_{\mathrm{rel}}+\mathbf{\alpha }\times \mathbf{r}+\mathbf{\omega }\times \left(\mathbf{\omega }\times \mathbf{r}\right)+2\mathbf{\omega }\times {\mathbf{v}}_{\mathrm{rel}}$$

where:

• a_O' = acceleration of the moving-frame origin;
• a_rel = acceleration measured relative to the rotating frame;
• α × r = tangential acceleration caused by angular acceleration of the frame;
• ω × (ω × r) = centripetal acceleration;
• 2ω × v_rel = Coriolis acceleration.

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Rigid-Body Kinematics in a Plane

A rigid body has fixed distances between all of its material points. A rigid body undergoing general plane motion has three degrees of freedom: two translations and one rotation.

Types of Plane Motion

• Rectilinear translation – all points move along parallel straight paths and the orientation of the body remains unchanged.
• Curvilinear translation – all points move along congruent curved paths while the orientation of the body remains unchanged.
• Rotation about a fixed axis – all points not on the axis move along circular paths centred on that axis.
• General plane motion – a combination of translation and rotation in which the paths of all points lie in planes parallel to a fixed plane.

For two points A and B fixed in the same rigid body:

$${\mathbf{v}}_B={\mathbf{v}}_A+\mathbf{\omega }\times {\mathbf{r}}_{B/A}$$

The corresponding acceleration relation is:

$${\mathbf{a}}_B={\mathbf{a}}_A+\mathbf{\alpha }\times {\mathbf{r}}_{B/A}+\mathbf{\omega }\times \left(\mathbf{\omega }\times {\mathbf{r}}_{B/A}\right)$$

The angular velocity and angular acceleration of a rigid body are properties of the body's motion and do not depend upon which point on the body is selected as the reference point.

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Instantaneous Centre of Rotation

For a rigid body undergoing general plane motion with non-zero angular velocity, there is an instantaneous point having zero velocity. At that instant the velocity distribution of the body can therefore be regarded as a pure rotation about this point. It is called the instantaneous centre of rotation.

The instantaneous centre need not lie within the physical body. For pure translation, where ω = 0, the instantaneous centre may be regarded as lying at infinity.

If the velocity V_A of point A and the angular velocity ω are known, the instantaneous centre lies on the line through A perpendicular to its velocity vector, at a distance:

$$r_A=\frac{V_A}{\left|\omega \right|}$$

If the velocity directions of two points on the body are known, lines drawn through those points perpendicular to their respective velocity vectors intersect at the instantaneous centre, provided the geometry is not degenerate.

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Rigid-Body Kinematics in Three Dimensions

An unconstrained rigid body in three-dimensional space has six degrees of freedom: three translational and three rotational.

Its configuration can therefore be described by the position of a reference point using three coordinates together with three independent parameters describing orientation.

Common modes of rigid-body motion include:

• rectilinear translation;
• curvilinear translation;
• rotation about a fixed axis;
• rotation about a fixed point;
• general spatial motion combining translation and rotation.

Finite Rotations

The order of finite rotations in three dimensions is important. For example, rotating a rigid body through π/2 about the x-axis followed by π/2 about the y-axis does not, in general, produce the same final orientation as performing the rotations in the opposite order.

Finite rotations are therefore not commutative and cannot in general be added as ordinary vectors.

Angular velocity, however, is an instantaneous vector quantity and angular velocity components can be combined vectorially when expressed relative to a common reference frame.

Euler and Chasles Theorems

Euler's Rotation Theorem

Any orientation of a rigid body having one point fixed can be obtained from another orientation by a single rotation about some axis passing through the fixed point.

Chasles' Theorem

A general finite displacement of a rigid body can be represented by a rotation about an axis together with a translation parallel to that axis. This combined motion is commonly described as a screw displacement.

These results concern the representation of rigid-body displacement and do not imply that individual points followed those equivalent paths during the actual motion.

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Translation in Three Dimensions

During pure translation the orientation of the rigid body remains unchanged. Consequently all points have identical velocity and acceleration vectors at a given instant.

For points P and Q:

$${\mathbf{r}}_P={\mathbf{r}}_Q+{\mathbf{r}}_{P/Q}$$

Because r_P/Q remains constant during translation:

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Rotation About a Fixed Axis

For rotation about a fixed axis, points lying on the axis have zero velocity due to the rotation. Other points move on circular paths centred on the axis. The direction of the angular velocity vector is defined by the right-hand rule.

For a point P whose position relative to a point O on the fixed axis is r:

$${\mathbf{v}}_P=\mathbf{\omega }\times \mathbf{r}$$ $${\mathbf{a}}_P=\mathbf{\alpha }\times \mathbf{r}+\mathbf{\omega }\times \left(\mathbf{\omega }\times \mathbf{r}\right)$$

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Rotation About a Fixed Point

If one point of a rigid body remains fixed, the body's instantaneous motion is a rotation about an axis passing through that point. The direction of this instantaneous axis is given by the instantaneous angular velocity vector.

If simultaneous angular velocity components ω₁ and ω₂ are expressed in the same reference frame, the instantaneous angular velocity is their vector sum:

$$\mathbf{\omega }={\mathbf{\omega }}_1+{\mathbf{\omega }}_2$$

The body-cone and space-cone construction provides a geometric interpretation of the changing instantaneous axis during rotation about a fixed point.

For a point having position vector r from the fixed point:

$$\mathbf{v}=\mathbf{\omega }\times \mathbf{r}$$ $$\mathbf{a}=\mathbf{\alpha }\times \mathbf{r}+\mathbf{\omega }\times \left(\mathbf{\omega }\times \mathbf{r}\right)$$

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General Three-Dimensional Motion

General rigid-body motion in three dimensions consists of translation together with rotation. A convenient method of analysis is to select a reference point Q on the body and describe the motion of another point P relative to it.

For two points P and Q fixed in the same rigid body:

$${\mathbf{r}}_P={\mathbf{r}}_Q+{\mathbf{r}}_{P/Q}$$ $${\mathbf{v}}_P={\mathbf{v}}_Q+\mathbf{\omega }\times {\mathbf{r}}_{P/Q}$$ $${\mathbf{a}}_P={\mathbf{a}}_Q+\mathbf{\alpha }\times {\mathbf{r}}_{P/Q}+\mathbf{\omega }\times \left(\mathbf{\omega }\times {\mathbf{r}}_{P/Q}\right)$$

General Rotating Reference Frame

For the more general case, consider a moving reference frame whose origin O' translates and whose axes rotate with angular velocity Ω. A point P may also move relative to those rotating axes.

Its velocity is:

$${\mathbf{v}}_P={\mathbf{v}}_{O\prime }+{\mathbf{v}}_{\mathrm{rel}}+\mathbf{\Omega }\times \mathbf{r}$$

and its acceleration is:

$${\mathbf{a}}_P={\mathbf{a}}_{O\prime }+{\mathbf{a}}_{\mathrm{rel}}+\frac{d\mathbf{\Omega }}{dt}\times \mathbf{r}+\mathbf{\Omega }\times \left(\mathbf{\Omega }\times \mathbf{r}\right)+2\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{rel}}$$

These equations distinguish the motion of the reference frame from the motion of the point measured within that frame. If the point is fixed in the rotating frame, its relative velocity and relative acceleration are zero.

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Related RoyMech Mechanics References

• Vectors
• Statics
• Kinetics and Dynamics
• Linkages
• Simple Machines
• Component Loading
• Basic Strain Theory
• Failure Modes
• Buckling Notes
• Mohr's Circle
• Thick Cylinders
• Rotating Discs and Cylinders
• Torsion
• Stress-Strain Curve
• Loaded Flat Plates

External Kinematics References

• Physics – Kinematics – reference material covering kinematics and motion.
• Basic Kinematics – introductory kinematics reference material.
• Kinematics – Wikipedia – general reference covering kinematics and motion.

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