![SteamEngine Boulton&Watt 1784](https://upload.wikimedia.org/wikipedia/commons/f/f2/SteamEngine_Boulton%26Watt_1784.png)
In the
physical science of
dynamics, rigid-body dynamics studies the movement of
systems of interconnected
bodies
Bodies may refer to:
* The plural of body
* ''Bodies'' (2004 TV series), BBC television programme
* Bodies (upcoming TV series), an upcoming British crime thriller limited series
* "Bodies" (''Law & Order''), 2003 episode of ''Law & Order''
* ...
under the action of external
forces. The assumption that the bodies are ''rigid'' (i.e. they do not
deform
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* De ...
under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
s attached to each body. This excludes bodies that display
fluid, highly
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
, and
plastic
Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
behavior.
The dynamics of a rigid body system is described by the laws of
kinematics and by the application of Newton's second law (
kinetics) or their derivative form,
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a
function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of
mechanical systems.
Planar rigid body dynamics
If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement. In this case, Newton's laws (kinetics) for a rigid system of N particles, P, ''i''=1,...,''N'', simplify because there is no movement in the ''k'' direction. Determine the
resultant force
In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant for ...
and torque at a reference point R, to obtain
where r denotes the planar trajectory of each particle.
The
kinematics of a rigid body yields the formula for the acceleration of the particle P in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ''ω'' and angular acceleration vector ''α'' of the rigid system of particles as,
For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors e from the reference point R to a point r and the unit vectors
, so
This yields the resultant force on the system as
and torque as
where
and
is the unit vector perpendicular to the plane for all of the particles P.
Use the
center of mass C as the reference point, so these equations for Newton's laws simplify to become
where is the total mass and ''I'' is the
moment of inertia about an axis perpendicular to the movement of the rigid system and through the center of mass.
Rigid body in three dimensions
Orientation or attitude descriptions
Several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections.
Euler angles
The first attempt to represent an orientation is attributed to
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. He imagined three reference frames that could rotate one around the other, and realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space (using two rotations to fix the vertical axis and another to fix the other two axes). The values of these three rotations are called
Euler angles. Commonly,
is used to denote precession,
nutation, and
intrinsic rotation.
File:Euler.png, Diagram of the Euler angles
File:Euler AxisAngle.png, Intrinsic rotation of a ball about a fixed axis.
File:ToupieEuler.png, Motion of a top in the Euler angles.
Tait–Bryan angles
![taitbrianzyx](https://upload.wikimedia.org/wikipedia/commons/5/53/Taitbrianzyx.svg)
These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a set of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are usually referred to as Euler angles.
Orientation vector
Euler also realized that the composition of two rotations is equivalent to a single rotation about a different fixed axis (
Euler's rotation theorem
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed p ...
). Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed.
Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector.
A similar method, called
axis-angle representation, describes a rotation or orientation using a
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
aligned with the rotation axis, and a separate value to indicate the angle (see figure).
Orientation matrix
With the introduction of matrices the Euler theorems were rewritten. The rotations were described by
orthogonal matrices
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix.
The above-mentioned Euler vector is the
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of a rotation matrix (a rotation matrix has a unique real
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
).
The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe.
The
configuration space of a non-
symmetrical
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
object in ''n''-dimensional space is
SO(''n'') × R''n''. Orientation may be visualized by attaching a basis of
tangent vectors
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
to an object. The direction in which each vector points determines its orientation.
Orientation quaternion
Another way to describe rotations is using
rotation quaternions, also called versors. They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more easily converted to and from matrices. When used to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions.
Newton's second law in three dimensions
To consider rigid body dynamics in three-dimensional space, Newton's second law must be extended to define the relationship between the movement of a rigid body and the system of forces and torques that act on it.
Newton formulated his second law for a particle as, "The change of motion of an object is proportional to the force impressed and is made in the direction of the straight line in which the force is impressed." Because Newton generally referred to mass times velocity as the "motion" of a particle, the phrase "change of motion" refers to the mass times acceleration of the particle, and so this law is usually written as
where F is understood to be the only external force acting on the particle, ''m'' is the mass of the particle, and a is its acceleration vector. The extension of Newton's second law to rigid bodies is achieved by considering a rigid system of particles.
Rigid system of particles
If a system of ''N'' particles, P
i, i=1,...,''N'', are assembled into a rigid body, then Newton's second law can be applied to each of the particles in the body. If F
i is the external force applied to particle P
i with mass ''m''
i, then
where F
ij is the internal force of particle P
j acting on particle P
i that maintains the constant distance between these particles.
![Rigid bodies](https://upload.wikimedia.org/wikipedia/commons/2/29/Rigid_bodies.jpg)
An important simplification to these force equations is obtained by introducing the
resultant force
In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant for ...
and torque that acts on the rigid system. This resultant force and torque is obtained by choosing one of the particles in the system as a reference point, R, where each of the external forces are applied with the addition of an associated torque. The resultant force F and torque T are given by the formulas,
where R
i is the vector that defines the position of particle P
i.
Newton's second law for a particle combines with these formulas for the resultant force and torque to yield,
where the internal forces F
''ij'' cancel in pairs. The
kinematics of a rigid body yields the formula for the acceleration of the particle P
i in terms of the position R and acceleration a of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as,
Mass properties
The mass properties of the rigid body are represented by its
center of mass and
inertia matrix. Choose the reference point R so that it satisfies the condition
then it is known as the center of mass of the system.
The inertia matrix
R">Rof the system relative to the reference point R is defined by
where
is the column vector ;
is its transpose, and
is the 3 by 3 identity matrix.
is the scalar product of
with itself, while
is the tensor product of
with itself.
Force-torque equations
Using the center of mass and inertia matrix, the force and torque equations for a single rigid body take the form
and are known as Newton's second law of motion for a rigid body.
The dynamics of an interconnected system of rigid bodies, , , is formulated by isolating each rigid body and introducing the interaction forces. The resultant of the external and interaction forces on each body, yields the force-torque equations
Newton's formulation yields 6''M'' equations that define the dynamics of a system of ''M'' rigid bodies.
Rotation in three dimensions
A rotating object, whether under the influence of torques or not, may exhibit the behaviours of
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
and
nutation
Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
.
The fundamental equation describing the behavior of a rotating solid body is
Euler's equation of motion
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to ...
:
where the
pseudovectors τ and L are, respectively, the
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
s on the body and its
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, the scalar ''I'' is its
moment of inertia, the vector ω is its angular velocity, the vector α is its angular acceleration, D is the differential in an inertial reference frame and d is the differential in a relative reference frame fixed with the body.
The solution to this equation when there is no applied torque is discussed in the articles
Euler's equation of motion
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to ...
and
Poinsot's ellipsoid.
It follows from Euler's equation that a torque τ applied perpendicular to the axis of rotation, and therefore perpendicular to L, results in a rotation about an axis perpendicular to both τ and L. This motion is called ''precession''. The angular velocity of precession Ω
P is given by the
cross product:
Precession can be demonstrated by placing a spinning top with its axis horizontal and supported loosely (frictionless toward precession) at one end. Instead of falling, as might be expected, the top appears to defy gravity by remaining with its axis horizontal, when the other end of the axis is left unsupported and the free end of the axis slowly describes a circle in a horizontal plane, the resulting precession turning. This effect is explained by the above equations. The torque on the top is supplied by a couple of forces: gravity acting downward on the device's centre of mass, and an equal force acting upward to support one end of the device. The rotation resulting from this torque is not downward, as might be intuitively expected, causing the device to fall, but perpendicular to both the gravitational torque (horizontal and perpendicular to the axis of rotation) and the axis of rotation (horizontal and outwards from the point of support), i.e., about a vertical axis, causing the device to rotate slowly about the supporting point.
Under a constant torque of magnitude ''τ'', the speed of precession ''Ω''
P is inversely proportional to ''L'', the magnitude of its angular momentum:
where ''θ'' is the angle between the vectors Ω
P and L. Thus, if the top's spin slows down (for example, due to friction), its angular momentum decreases and so the rate of precession increases. This continues until the device is unable to rotate fast enough to support its own weight, when it stops precessing and falls off its support, mostly because friction against precession cause another precession that goes to cause the fall.
By convention, these three vectors - torque, spin, and precession - are all oriented with respect to each other according to the
right-hand rule.
Virtual work of forces acting on a rigid body
An alternate formulation of rigid body dynamics that has a number of convenient features is obtained by considering the
virtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for ...
of forces acting on a rigid body.
The virtual work of forces acting at various points on a single rigid body can be calculated using the velocities of their point of application and the
resultant force and torque. To see this, let the forces F
1, F
2 ... F
''n'' act on the points R
1, R
2 ... R
''n'' in a rigid body.
The trajectories of R
''i'', are defined by the movement of the rigid body. The velocity of the points R
''i'' along their trajectories are
where ω is the angular velocity vector of the body.
Virtual work
Work is computed from the dot product of each force with the displacement of its point of contact
If the trajectory of a rigid body is defined by a set of
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
, , then the virtual displacements are given by
The virtual work of this system of forces acting on the body in terms of the generalized coordinates becomes
or collecting the coefficients of
Generalized forces
For simplicity consider a trajectory of a rigid body that is specified by a single generalized coordinate q, such as a rotation angle, then the formula becomes
Introduce the resultant force F and torque T so this equation takes the form
The quantity ''Q'' defined by
is known as the
generalized force Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,..., n, acting on a system that has its configuration defined in terms of generali ...
associated with the virtual displacement δq. This formula generalizes to the movement of a rigid body defined by more than one generalized coordinate, that is
where
It is useful to note that conservative forces such as gravity and spring forces are derivable from a potential function , known as a
potential energy. In this case the generalized forces are given by
D'Alembert's form of the principle of virtual work
The equations of motion for a mechanical system of rigid bodies can be determined using D'Alembert's form of the principle of virtual work. The principle of virtual work is used to study the static equilibrium of a system of rigid bodies, however by introducing acceleration terms in Newton's laws this approach is generalized to define dynamic equilibrium.
Static equilibrium
The static equilibrium of a mechanical system rigid bodies is defined by the condition that the virtual work of the applied forces is zero for any virtual displacement of the system. This is known as the ''principle of virtual work.''
This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is ''Q''
''i''=0.
Let a mechanical system be constructed from rigid bodies, B
''i'', , and let the resultant of the applied forces on each body be the force-torque pairs, and , . Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity and angular velocities , , for each rigid body, are defined by a single generalized coordinate q. Such a system of rigid bodies is said to have one
degree of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
.
The virtual work of the forces and torques, and , applied to this one degree of freedom system is given by
where
is the generalized force acting on this one degree of freedom system.
If the mechanical system is defined by m generalized coordinates, , , then the system has m degrees of freedom and the virtual work is given by,
where
is the generalized force associated with the generalized coordinate . The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is
These equations define the static equilibrium of the system of rigid bodies.
Generalized inertia forces
Consider a single rigid body which moves under the action of a resultant force F and torque T, with one degree of freedom defined by the generalized coordinate ''q''. Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force associated with the generalized coordinate is given by
This inertia force can be computed from the kinetic energy of the rigid body,
by using the formula
A system of rigid bodies with m generalized coordinates has the kinetic energy
which can be used to calculate the m generalized inertia forces
Dynamic equilibrium
D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that
for any set of virtual displacements . This condition yields equations,
which can also be written as
The result is a set of m equations of motion that define the dynamics of the rigid body system.
Lagrange's equations
If the generalized forces Q
''j'' are derivable from a potential energy , then these equations of motion take the form
In this case, introduce the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, , so these equations of motion become
These are known as
Lagrange's equations of motion.
Linear and angular momentum
System of particles
The linear and angular momentum of a rigid system of particles is formulated by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles P
i, be located at the coordinates r
''i'' and velocities v
''i''. Select a reference point R and compute the relative position and velocity vectors,
The total linear and angular momentum vectors relative to the reference point R are
and
If R is chosen as the center of mass these equations simplify to
Rigid system of particles
To specialize these formulas to a rigid body, assume the particles are rigidly connected to each other so P, i=1,...,n are located by the coordinates r and velocities v. Select a reference point R and compute the relative position and velocity vectors,
where ω is the angular velocity of the system.
The
linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
and
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of this rigid system measured relative to the center of mass R is
These equations simplify to become,
where M is the total mass of the system and
is the
moment of inertia matrix defined by
where
''i'' − R">''i'' − Ris the skew-symmetric matrix constructed from the vector r
''i'' − R.
Applications
* For the analysis of robotic systems
* For the biomechanical analysis of animals, humans or humanoid systems
* For the analysis of space objects
* For the understanding of strange motions of rigid bodies.
* For the design and development of dynamics-based sensors, such as gyroscopic sensors.
* For the design and development of various stability enhancement applications in automobiles.
* For improving the graphics of video games which involves rigid bodies
See also
*
Analytical mechanics
*
Analytical dynamics
In classical mechanics, analytical dynamics, also known as classical dynamics or simply dynamics, is concerned with the relationship between Motion (physics), motion of bodies and its causes, namely the force (physics), forces acting on the bodies ...
*
Calculus of variations
*
Classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
*
Dynamics (physics)
Dynamics is the branch of classical mechanics that is concerned with the study of forces and their effects on motion. Isaac Newton was the first to formulate the fundamental physical laws that govern dynamics in classical non-relativistic physi ...
*
History of classical mechanics
This article deals with the history of classical mechanics.
Precursors to classical mechanics
Antiquity
The ancient Greek philosophers, Aristotle in particular, were among the first to propose that abstract principles govern nature. Aris ...
*
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
*
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
*
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
*
Rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
*
Rigid rotor
In rotordynamics, the rigid rotor is a mechanical model of Rotation, rotating systems. An arbitrary rigid rotor is a 3-dimensional Rigid body, rigid object, such as a top. To orient such an object in space requires three angles, known as Euler an ...
*
Soft body dynamics
Soft-body dynamics is a field of computer graphics that focuses on visually realistic physical simulations of the motion and properties of deformable objects (or ''soft bodies''). The applications are mostly in video games and films. Unlike in si ...
*
Multibody dynamics
Multibody system is the study of the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements.
Introduction
The systematic treatment of the dynamic behavior of inter ...
*
Polhode
*
Herpolhode
*
Precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In oth ...
*
Poinsot's construction
In classical mechanics, Poinsot's construction (after Louis Poinsot) is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion ha ...
*
Gyroscope
*
Physics engine
A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics (including collision detection), soft body dynamics, and fluid dynamics, of use in the domains of computer gr ...
*
Physics processing unit
A physics processing unit (PPU) is a dedicated microprocessor designed to handle the calculations of physics, especially in the physics engine of video games. It is an example of hardware acceleration.
Examples of calculations involving a PPU mig ...
*
Physics Abstraction Layer – Unified multibody simulator
*
Dynamechs – Rigid-body simulator
*
RigidChips – Japanese rigid-body simulator
*
Euler's Equation
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
References
Further reading
* E. Leimanis (1965). ''The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point.'' (
Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
, New York).
* W. B. Heard (2006). ''Rigid Body Mechanics: Mathematics, Physics and Applications.'' (
Wiley-VCH
Wiley-VCH is a German publisher owned by John Wiley & Sons. It was founded in 1921 as Verlag Chemie (meaning "Chemistry Press": VCH stands for ''Verlag Chemie'') by two German learned societies. Later, it was merged into the German Chemical Soci ...
).
External links
Chris Hecker's Rigid Body Dynamics InformationDigitalRune Knowledge Base contains a master thesis and a collection of resources about rigid body dynamics.
F. Klein, "Note on the connection between line geometry and the mechanics of rigid bodies"(English translation)
F. Klein, "On Sir Robert Ball's theory of screws"(English translation)
E. Cotton, "Application of Cayley geometry to the geometric study of the displacement of a solid around a fixed point"(English translation)
{{DEFAULTSORT:Rigid Body Dynamics
Rigid bodies
Rigid bodies mechanics
Engineering mechanics
Rotational symmetry