World Library  
Flag as Inappropriate
Email this Article

Newton–Euler equations

Article Id: WHEBN0003015195
Reproduction Date:

Title: Newton–Euler equations  
Author: World Heritage Encyclopedia
Language: English
Subject: List of things named after Isaac Newton, Early life of Isaac Newton, Rigid bodies, Inverse dynamics, Screw theory
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Newton–Euler equations


In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body.[1][2] [3][4][5]

Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.

Contents

  • Center of mass frame 1
  • Any reference frame 2
  • Applications 3
  • See also 4
  • References 5

Center of mass frame

With respect to a coordinate frame whose origin coincides with the body's center of mass, they can be expressed in matrix form as:

\left(\begin{matrix} {\bold F} \\ {\boldsymbol \tau} \end{matrix}\right) = \left(\begin{matrix} m {\bold I_3} & 0 \\ 0 & {\bold I}_{\rm cm} \end{matrix}\right) \left(\begin{matrix} \bold a_{\rm cm} \\ {\boldsymbol \alpha} \end{matrix}\right) + \left(\begin{matrix} 0 \\ {\boldsymbol \omega} \times {\bold I}_{\rm cm} \, {\boldsymbol \omega} \end{matrix}\right),

where

F = total force acting on the center of mass
m = mass of the body
I3 = the 3×3 identity matrix
acm = acceleration of the center of mass
vcm = velocity of the center of mass
τ = total torque acting about the center of mass
Icm = moment of inertia about the center of mass
ω = angular velocity of the body
α = angular acceleration of the body

Any reference frame

With respect to a coordinate frame located at point P that is fixed in the body and not coincident with the center of mass, the equations assume the more complex form:

\left(\begin{matrix} {\bold F} \\ {\boldsymbol \tau}_{\rm p} \end{matrix}\right) = \left(\begin{matrix} m {\bold I_3} & - m ^{\times}\\ m ^{\times} & {\bold I}_{\rm cm} - m^{\times}^{\times}\end{matrix}\right) \left(\begin{matrix} \bold a_{\rm p} \\ {\boldsymbol \alpha} \end{matrix}\right) + \left(\begin{matrix} m^{\times}^{\times} {\bold c} \\ ^\times ({\bold I}_{\rm cm} - m ^\times^\times)\, {\boldsymbol \omega} \end{matrix}\right),

where c is the location of the center of mass expressed in the body-fixed frame, and

[\mathbf{c}]^{\times} \equiv \left(\begin{matrix} 0 & -c_z & c_y \\ c_z & 0 & -c_x \\ -c_y & c_x & 0 \end{matrix}\right) \qquad \qquad [\mathbf{\boldsymbol{\omega}}]^{\times} \equiv \left(\begin{matrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{matrix}\right)

denote skew-symmetric cross product matrices.

The left hand side of the equation—which includes the sum of external forces, and the sum of external moments about P—describes a spatial wrench, see screw theory.

The inertial terms are contained in the spatial inertia matrix

\left(\begin{matrix} m {\bold I_3} & - m ^{\times}\\ m ^{\times} & {\bold I}_{\rm cm} - m ^{\times}^{\times}\end{matrix}\right),

while the fictitious forces are contained in the term:[6]

\left(\begin{matrix} m^\times ^\times {\bold c} \\ ^\times ({\bold I}_{\rm cm} - m ^\times^\times)\, {\boldsymbol \omega} \end{matrix}\right) .

When the center of mass is not coincident with the coordinate frame (that is, when c is nonzero), the translational and angular accelerations (a and α) are coupled, so that each is associated with force and torque components.

Applications

The Newton–Euler equations are used as the basis for more complicated "multi-body" formulations (screw theory) that describe the dynamics of systems of rigid bodies connected by joints and other constraints. Multi-body problems can be solved by a variety of numerical algorithms.[2][6][7]

See also

References

  1. ^
  2. ^ a b
  3. ^
  4. ^
  5. ^
  6. ^ a b
  7. ^
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.