Rigid body dynamics
Encyclopedia
In physics
, rigid body dynamics is the study of the motion
of rigid bodies. Unlike particles
, which move only in three degrees of freedom (translation
in three directions), rigid bodies occupy space and have geometrical properties, such as a center of mass
, moments of inertia
, etc., that characterize motion in six degrees of freedom (translation in three directions plus rotation
in three directions). Rigid bodies are also characterized as being non-deformable, as opposed to deformable bodies. As such, rigid body dynamics is used heavily in analyses and computer simulation
s of physical systems and machinery where rotational motion is important, but material deformation does not have a significant effect on the motion of the system.
is equal to the sum of all external forces acting on the particle:
where m is the particle's mass, v is the particle's velocity, their product mv is the linear momentum, and fi is one of the N number of forces acting on the particle.
Because the mass is constant, this is equivalent to
To generalize, assume a body of finite mass and size is composed of such particles, each with infinitesimal
mass dm. Each particle has a position vector r. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Since velocity v is the derivative
of position r with respect to time, the derivative of velocity dv/dt is the second derivative of position d2r/dt2, and the linear momentum equation of any given particle is
When the linear momentum equations for all particles are added together, the internal forces sum to zero according to Newton's third law, which states that any such force has opposite magnitudes on the two particles. By accounting for all particles, the left side becomes an integral over the entire body, and the second derivative operator can be moved out of the integral, so
.
Let M be the total mass, which is constant, so the left side can be multiplied and divided by M, so
.
The expression
is the formula for the position of the center of mass
. Denoting this by rcm, the equation reduces to
Thus, linear momentum equations can be extended to rigid bodies by denoting that they describe the motion of the center of mass of the body. This is known as Euler's first law.
where the moment of inertia tensor,
, is given by
Given that Euler's rotation theorem
states that there is always an instantaneous axis of rotation, the angular velocity
,
, can be given by a vector over this axis
where
is a set of mutually perpendicular
unit vectors fixed in a reference frame
.
Rotating a rigid body is equivalent to rotating a Poinsot ellipsoid
.
for a system of particles with linear momenta 
and distances 
from the rotation axis is defined
For a rigid body rotating with angular velocity
about the rotation axis 
(a unit vector), the velocity vector 
may be written as a vector cross product
where
Substituting the formula for
into the definition of 
yields
where we have introduced the special case that the position vectors of all particles are perpendicular to the rotation axis (e.g., a flywheel
):
.
The torque
is defined as the rate of change of the angular momentum 
If I is constant (because the inertia tensor is the identity, because we work in the intrinsecal frame, or because the torque is driving the rotation around the same axis
so that 
is not changing) then we may write
where
is called the angular acceleration (or rotational acceleration) about the rotation axis 
.
Notice that if I is not constant in the external reference frame (i.e. the three main axes of the body are different) then we cannot take the I outside the derivate. In this cases we can have torque-free precession.
s use rigid body dynamics to increase interactivity and realism in video games.
Simulators
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, rigid body dynamics is the study of the motion
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
of rigid bodies. Unlike particles
Point particle
A point particle is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space...
, which move only in three degrees of freedom (translation
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
in three directions), rigid bodies occupy space and have geometrical properties, such as a center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
, moments of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
, etc., that characterize motion in six degrees of freedom (translation in three directions plus rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
in three directions). Rigid bodies are also characterized as being non-deformable, as opposed to deformable bodies. As such, rigid body dynamics is used heavily in analyses and computer simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...
s of physical systems and machinery where rotational motion is important, but material deformation does not have a significant effect on the motion of the system.
Rigid body linear momentum
Newton's Second Law states that the rate of change of the linear momentum of a particle with constant massMass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
is equal to the sum of all external forces acting on the particle:
where m is the particle's mass, v is the particle's velocity, their product mv is the linear momentum, and fi is one of the N number of forces acting on the particle.
Because the mass is constant, this is equivalent to
To generalize, assume a body of finite mass and size is composed of such particles, each with infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
mass dm. Each particle has a position vector r. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Since velocity v is the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of position r with respect to time, the derivative of velocity dv/dt is the second derivative of position d2r/dt2, and the linear momentum equation of any given particle is
When the linear momentum equations for all particles are added together, the internal forces sum to zero according to Newton's third law, which states that any such force has opposite magnitudes on the two particles. By accounting for all particles, the left side becomes an integral over the entire body, and the second derivative operator can be moved out of the integral, so
.Let M be the total mass, which is constant, so the left side can be multiplied and divided by M, so
.The expression
is the formula for the position of the center of massCenter of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
. Denoting this by rcm, the equation reduces to
Thus, linear momentum equations can be extended to rigid bodies by denoting that they describe the motion of the center of mass of the body. This is known as Euler's first law.
Rigid body angular momentum
The most general equation for rotation of a rigid body in three dimensions about an arbitrary origin O with axes x, y, z iswhere the moment of inertia tensor,
, is given byGiven that Euler's rotation theorem
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 point. It also means that the composition of two...
states that there is always an instantaneous axis of rotation, the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
,
, can be given by a vector over this axiswhere
is a set of mutually perpendicularPerpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
unit vectors fixed in a reference frame
Reference frame
Reference frame may refer to:*Frame of reference, in physics*Reference frame , frames of a compressed video that are used to define future frames...
.
Rotating a rigid body is equivalent to rotating a Poinsot ellipsoid
Poinsot's construction
In classical mechanics, Poinsot's construction 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 has four constants: the kinetic energy of the body and the three components...
.
Angular momentum and torque
Similarly, the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
for a system of particles with linear momenta and distances from the rotation axis is definedFor a rigid body rotating with angular velocity
about the rotation axis (a unit vector), the velocity vector may be written as a vector cross productwhere
- angular velocity vector
is the shortest vector from the rotation axis to the point mass.
Substituting the formula for
into the definition of yieldswhere we have introduced the special case that the position vectors of all particles are perpendicular to the rotation axis (e.g., a flywheel
Flywheel
A flywheel is a rotating mechanical device that is used to store rotational energy. Flywheels have a significant moment of inertia, and thus resist changes in rotational speed. The amount of energy stored in a flywheel is proportional to the square of its rotational speed...
):
.The torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
is defined as the rate of change of the angular momentum If I is constant (because the inertia tensor is the identity, because we work in the intrinsecal frame, or because the torque is driving the rotation around the same axis
so that is not changing) then we may writewhere
is called the angular acceleration (or rotational acceleration) about the rotation axis .Notice that if I is not constant in the external reference frame (i.e. the three main axes of the body are different) then we cannot take the I outside the derivate. In this cases we can have torque-free precession.
Applications
Computer physics enginePhysics engine
A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
s use rigid body dynamics to increase interactivity and realism in video games.
See also
Theory- Rigid bodyRigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
- Rigid rotorRigid rotorThe rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
- Soft body dynamicsSoft body dynamicsSoft body dynamics is a field of computer graphics that focuses on visually realistic physical simulations of the motion and properties of deformable objects . The applications are mostly in video games and film. Unlike in simulation of rigid bodies, the shape of soft bodies can change, meaning...
- Multibody dynamics
- PolhodePolhodeThe details of a spinning body may impose restrictions on the motion of its angular velocity vector, ω. The curve produced by the angular velocity vector on the inertia ellipsoid, is known as the polhode, coined from Greek meaning "path of the pole"...
- HerpolhodeHerpolhodeA herpolhode is the curve traced out by the endpoint of the angular velocity vector ω of a rigid rotor, a rotating rigid body. The endpoint of the angular velocity moves in a plane in absolute space, called the invariable plane, that is orthogonal to the angular momentum vector L...
- PrecessionPrecessionPrecession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
- Poinsot's constructionPoinsot's constructionIn classical mechanics, Poinsot's construction 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 has four constants: the kinetic energy of the body and the three components...
Simulators
- Physics enginePhysics engineA physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
- Physics processing unitPhysics processing unitA physics processing unit is a dedicated microprocessor designed to handle the calculations of physics, especially in the physics engine of video games. Examples of calculations involving a PPU might include rigid body dynamics, soft body dynamics, collision detection, fluid dynamics, hair and...
- Physics Abstraction Layer - Unified multibody simulator
- Dynamechs - Rigid body simulator
- RigidChipsRigidChipsRigidChips is a rigid body simulator developed by Takeya Yasuhiko.In RigidChips, various objects and vehicles are constructed by combining parts with a scripting language. The program calculates air and water resistance realistically so that many different models can be made using any basic text...
- Japanese rigid body simulator
External links
- Chris Hecker's Rigid Body Dynamics Information
- Physically Based Modeling: Principles and Practice
- Lectures, Computational Rigid Body Dynamics at University of Wisconsin-Madison
- DigitalRune Knowledge Base contains a master thesis and a collection of resources about rigid body dynamics.