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Encyclopedia > Rigid body dynamics

In physics, rigid body dynamics is the study of the motion of a rigid object. Rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate. Image File history File links Broom_icon. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... Space has been an interest for philosophers and scientists for much of human history. ... This article is about rotation as a movement of a physical body. ...


Briefly summerised particles have:

  • no rotation
  • linear veloctiy
  • mass

Wheras rigid bodies have:

Rigid bodies can not be deformed, unlike Soft body dynamics. Most game Physics engines simulate forward rigid body dynamics. Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2), is the rotational analog of mass. ... Soft body dynamics is an area of physics simulation software that focuses on accurate simulation of a flexible object. ... A physics engine is a computer program that simulates Newtonian physics models, using variables such as mass, velocity, friction and wind resistance. ...


Note: This article has much overlap with the rigid rotor and rigid body articles. Articles should eventually be merged. The rigid rotor is a mechanical model that is used to explain rotating systems. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...


Equations from particle dynamics can be generalized to rigid body dynamics as follows:

Contents

Rigid body linear momentum

The equation for particle linear momentum is In physics, momentum is a physical quantity related to the velocity and mass of an object. ...

frac{mathrm{d}(m v)}{mathrm{d}t}=sum_{i=1}^N f_i

where:

  • m is the particle's mass.
  • v is the particle's velocity.
  • fi is one of the N forces acting on the particle.

Assuming constant mass, this reduces to

m frac{mathrm{d}v}{mathrm{d}t}=sum_{i=1}^N f_i.

To generalize assume a body of finite mass and size is composed of such particles. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Each particle has:

  • a mass dm.
  • a position vector r.

Thus, the linear momentum equation of any given particle would look like this:

mathrm{d}m frac{mathrm{d}^2r}{mathrm{d}t^2}= sum_{i=1}^M f_{i,text{internal}} + sum_{j=1}^N f_{j,mathrm{external}}.

If the equation for each particle were added together, the internal forces would cancel out, since by Newton's third law, any such force would have opposite magnitudes on the two particles. Also, the left side would become an integral over the entire body, and the second derivative operator could come out of the integral, leaving

 frac{mathrm{d}^2}{mathrm{d}t^2} int r, mathrm{d}m = sum_{j=1}^N f_{j,mathrm{external}}.

Letting M be the total mass, the left side can be multiplied and divided by M without changing the validity:

 M frac{mathrm{d}^2 frac{int r, mathrm{d}m}{M}}{mathrm{d}t^2} = sum_{j=1}^N f_{j,mathrm{external}}

However, frac{int r, mathrm{d}m}{M} is the formula for the position of center of mass. Denoting this by rcm, the equation reduces to In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

 M frac{mathrm{d}^2 r_{cm}}{mathrm{d}t^2} = sum_{j=1}^N f_{j,mathrm{external}}.

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.


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 is

M b_{G/O} times frac{mathrm{d}^2 R_O}{mathrm{d}t^2} + frac{mathrm{d}(mathbf{I}boldsymbol{omega})}{mathrm{d}t} = sum_{j=1}^N tau_{O,j}

where:

 mathbf{I} = begin{pmatrix} int (y^2+z^2), mathrm{d}m & -int xy, mathrm{d}m & -int xz, mathrm{d}m -int xy, mathrm{d}m & int (x^2+z^2), mathrm{d}m & -int yz, mathrm{d}m  -int xz, mathrm{d}m & -int yz, mathrm{d}m & int (x^2+y^2), mathrm{d}m end{pmatrix} quad hbox{and} quad boldsymbol{omega} = begin{pmatrix} omega_x  omega_y  omega_z end{pmatrix}.

Here mathbf{I} is the moment of inertia tensor and boldsymbol{omega} is the angular velocity (a vector). Based on this, a theorem states that any rigid body is equivalent when moving to a Poinsot's ellipsoid. Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2), is the rotational analog of mass. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Poinsots construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body. ...


Further

  • ωq is the angular velocity about axis q.
  • M is the total mass.
  • bG/O is the vector from O to the body's center of mass.
  • RO is the position of O.
  • t is time.
  •  int , mathrm{d}m is an integral over the mass of the body.
  • τO,j is one of the N moments about O.

This equation follows from equation for linear momentum of a particle and kinematics; no additional observations of nature are necessary to arrive at it.


There are many special cases that simplify this equation. The first term goes to zero if any of three conditions are met:

  • O is a fixed point (since its second derivative would be zero).
  • A set of axes is chosen with its origin attached to the body's center of mass (since this would reduce the vector b to zero).
  • The vector b always points in the direction of the acceleration of O (since the cross product of parallel vectors is zero).

Also, if the axes are chosen are the principal axes (i.e., the moments about the xy, xz, and yz planes is zero), the off-diagonal terms of the matrix are zero. This case is further discussed by Euler's equations. In physics, Eulers equations govern the rotation of a rigid body. ...


When learning about angular motion, students are generally first exposed to the case of rotation only in the x-y plane and a fixed axis or axis at the center of mass with constant rotational inertia. That equation is

int left(x^2+y^2right), mathrm{d}m frac{mathrm{d}omega_z}{mathrm{d}t} = sum_{j=1}^N tau_{O,j,z}.

Angular momentum and torque

Similarly, the angular momentum mathbf{L} for a system of particles with linear momenta pi and distances ri from the rotation axis is defined This gyroscope remains upright while spinning due to its angular momentum. ...

 mathbf{L} = sum_{i=1}^{N} mathbf{r}_{i} times mathbf{p}_{i} = sum_{i=1}^{N} m_{i} mathbf{r}_{i} times mathbf{v}_{i}

For a rigid body rotating with angular velocity ω about the rotation axis mathbf{hat{n}} (a unit vector), the velocity vector mathbf{v}_{i} may be written as a vector cross product In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... In mathematics, the cross product is a binary operation on vectors in three dimensions. ...

 mathbf{v}_{i} = omega mathbf{hat{n}} times mathbf{r}_{i}  stackrel{mathrm{def}}{=} boldsymbolomega times mathbf{r}_{i}

where

angular velocity vector boldsymbolomega  stackrel{mathrm{def}}{=} omega mathbf{hat{n}}
mathbf{r}_{i} is the shortest vector from the rotation axis to the point mass.

Substituting the formula for mathbf{v}_{i} into the definition of mathbf{L} yields

 mathbf{L} = sum_{i=1}^{N} m_{i} mathbf{r}_{i} times (boldsymbolomega times mathbf{r}_{i}) = boldsymbolomega sum_{i=1}^{N} m_{i} r_{i}^{2} = I omega mathbf{hat{n}}

where we have introduced the special case that the position vectors of all particles are perpendicular to the rotation axis (e.g., a flywheel): boldsymbolomega cdot mathbf{r}_{i} = 0. Spoked flywheel Flywheel from stationary engine. ...


The torque mathbf{N} is defined as the rate of change of the angular momentum mathbf{L} Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ...

 mathbf{N}  stackrel{mathrm{def}}{=} frac{dmathbf{L}}{dt}

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 mathbf{hat{n}} so that I is not changing) then we may write

 mathbf{N}  stackrel{mathrm{def}}{=} I frac{domega}{dt}mathbf{hat{n}} = I alpha mathbf{hat{n}}

where

α is called the angular acceleration (or rotational acceleration) about the rotation axis mathbf{hat{n}}.

Notice that if I is not constant in the external reference frame (ie. 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. Precession of a gyroscope Precession refers to a change in the direction of the axis of a rotating object. ...


Applications

Computer physics engines use rigid body dynamics to increase interactivity and realism in video games. A physics engine is a computer program that simulates Newtonian physics models, using variables such as mass, velocity, friction and wind resistance. ... This article is about computer and video games. ...


See also

Theory

Simulators In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ... The rigid rotor is a mechanical model that is used to explain rotating systems. ... Soft body dynamics is an area of physics simulation software that focuses on accurate simulation of a flexible object. ... A multibody system is used to model the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements. ... The 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. The surface created by the angular velocity vector... Precession of a gyroscope Precession refers to a change in the direction of the axis of a rotating object. ... Poinsots construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body. ...

A physics engine is a computer program that simulates Newtonian physics models, using variables such as mass, velocity, friction and wind resistance. ... The Physics Abstraction Layer (PAL) is an open source cross platform physics engine API abstraction system. ... RigidChips is a solid body simulator developed by The student. In RigidChips, various objects can be freely produced by combining parts, and setting the script to them. ...

External links

  • Chris Hecker's Rigid Body Dynamics Information
  • Physically Based Modeling: Principles and Practice
  • Gyration - Open source software simulating a block-shaped mass floating in free space
  • DigitalRune Knowledge Base contains a master thesis and a collection of resources about rigid body dynamics.
  • Stability of a rigid body spinning freely in space from Hugh Hunt's Dynamics Movies page

  Results from FactBites:
 
Rigid body dynamics - definition of Rigid body dynamics in Encyclopedia (520 words)
Rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate, which introduces other considerations.
To generalize, assume a body of finite mass and size is composed of such particles.
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.
Trinkle's Rigid Body Dynamics (1008 words)
The field of rigid body dynamic (more generally, multibody dynamics) is all about designing mathematical models and algorithms to predict the motions of bodies and the contact forces, including friction, that arise between them when.
In the early 1980's Lodtstedt published the first paper I know of in which the instantaneous dynamic equations of a system of rigid bodies in unilateral contact were formulated as a complementarity problem.
Here are some papers related to rigid body dynamics simulation and design problems: theory, velocity-base time-stepping, theory and examples with torsional friction, the quasistatic problem, the design and manipulation planning with intermittent contacts.
  More results at FactBites »

 
 

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