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| {{classical mechanics}}
| | Hi there, I am Andrew Berryhill. For a whilst I've been in Mississippi but now I'm contemplating other choices. I am really fond of to go to karaoke but I've been taking on new issues recently. Invoicing is my occupation.<br><br>my website; [http://kpupf.com/xe/talk/735373 love psychic readings] |
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| In [[classical mechanics]], the '''Newton–Euler''' equations describe the combined translational and [[rotational dynamics]] of a [[rigid body]].<ref name=Hahn>
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| {{cite book
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| |title=Rigid Body Dynamics of Mechanisms
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| |author=Hubert Hahn
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| |page=143
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| |url=http://books.google.com/books?id=MqrN3KY7o6MC&pg=PA143&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U00jfE08smw1IqJt69QdcMSKvDIeA
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| |isbn=3-540-42373-7
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| |publisher=Springer
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| |year=2002
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| }}</ref><ref name=Shabana>
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| {{cite book
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| |title=Computational Dynamics
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| |author=Ahmed A. Shabana
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| |page= 379
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| |url=http://books.google.com/books?id=dGfcbOsm2PwC&pg=PA379&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U01BZBb84es37aiHVpdE33IdGze-A
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| |isbn=978-0-471-37144-1
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| |year=2001
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| |publisher=Wiley-Interscience
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| }}
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| </ref>
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| <ref name=Slotline>
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| {{cite book
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| |title=Robot Analysis and Control
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| |author=Haruhiko Asada, Jean-Jacques E. Slotine
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| |url=http://books.google.com/books?id=KUG1VGkL3loC&pg=PA94&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U3LiZyQRj0zYXQ8ON2zwuiiwQO7dA
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| |isbn=0-471-83029-1
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| |publisher=Wiley/IEEE
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| |year=1986
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| |pages=§5.1.1, p. 94
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| }}</ref><ref name=Bishop>
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| {{cite book
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| |title=Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling
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| |author= Robert H. Bishop
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| |url=http://books.google.com/books?id=3UGQsi6VamwC&pg=PT104&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U1DtQ2BGV_Q34yAj-WhnQ4tStxPCw#PPT104,M1
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| |isbn=0-8493-9258-6
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| |publisher=CRC Press
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| |year=2007
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| |pages=§7.4.1, §7.4.2
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| }}</ref><ref name=Lin>
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| {{cite book
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| |title=High Fidelity Haptic Rendering
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| |author= Miguel A. Otaduy, [[Ming C. Lin]]
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| |page=24 |url=http://books.google.com/books?id=lk0StvDRoEMC&pg=PA24&dq=EUler+equations+%22rigid+body%22&lr=&as_brr=0&sig=ACfU3U0iOPnq-nMrS34O40ZMt0EbJEqu6g#PPA24,M1
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| |isbn=1-59829-114-9
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| |publisher=Morgan and Claypool Publishers
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| |year=2006
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| }}</ref>
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| Traditionally the Newton–Euler equations is the grouping together of [[Euler's laws of motion|Euler's two laws of motion]] for a rigid body into a single equation with 6 components, using [[column vector]]s and [[Matrix (mathematics)|matrices]]. These laws relate the motion of the [[center of gravity]] of a rigid body with the sum of [[force]]s and [[torques]] (or synonymously [[moment (physics)|moment]]s) acting on the rigid body.
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| ==Center of mass frame==
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| With respect to a coordinate frame whose origin coincides with the body's [[center of mass]], they can be expressed in matrix form as:
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| : <math>
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| \left(\begin{matrix} {\bold F} \\ {\boldsymbol \tau} \end{matrix}\right) =
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| \left(\begin{matrix} m {\boldsymbol 1} & 0 \\ 0 & {\bold I}_{\rm cm} \end{matrix}\right)
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| \left(\begin{matrix} \bold a_{\rm cm} \\ {\boldsymbol \alpha} \end{matrix}\right) +
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| \left(\begin{matrix} {\boldsymbol \omega} \times m {\boldsymbol v_{\rm cm}} \\ {\boldsymbol \omega} \times {\bold I}_{\rm cm} \, {\boldsymbol \omega} \end{matrix}\right),
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| </math>
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| where
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| :'''F''' = total [[force]] acting on the center of mass
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| :''m'' = mass of the body
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| :'''1''' = the 3×3 [[identity matrix]]
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| :'''a'''<sub>cm</sub> = acceleration of the [[center of mass]]
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| :'''v'''<sub>cm</sub> = velocity of the [[center of mass]]
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| :'''τ''' = total torque acting about the center of mass
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| :'''I'''<sub>cm</sub> = [[moment of inertia]] about the center of mass
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| :'''ω''' = [[angular velocity]] of the body
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| :'''α''' = [[angular acceleration]] of the body
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| ==Any reference frame==
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| With respect to a coordinate frame that is ''not'' coincident with the center of mass, the equations assume
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| the more complex form:
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| : <math>
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| \left(\begin{matrix} {\bold F} \\ {\boldsymbol \tau} \end{matrix}\right) =
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| \left(\begin{matrix} m {\boldsymbol 1} & - m [{\bold c}]\\
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| m [{\bold c}] & {\bold I}_{\rm cm} - m [{\bold c}][{\bold c}]\end{matrix}\right)
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| \left(\begin{matrix} \bold a_{\rm cm} \\ {\boldsymbol \alpha} \end{matrix}\right) +
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| \left(\begin{matrix} {m \boldsymbol \omega} \times \left({\boldsymbol \omega} \times {\bold c}\right) \\
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| {\boldsymbol \omega} \times ({\bold I}_{\rm cm} - m [{\bold c}][{\bold c}])\, {\boldsymbol \omega} \end{matrix}\right),
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| </math>
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| where '''c''' is the location of the center of mass,
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| and
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| :<math>
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| [\mathbf{c}] \equiv
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| \left(\begin{matrix} 0 & -c_z & c_y \\ c_z & 0 & -c_x \\ -c_y & c_x & 0 \end{matrix}\right)
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| </math>
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| denotes a [[Skew-symmetric matrix|skew-symmetric]] [[cross product matrix]]. The left hand side (forces and moments) of the equation above describes a spatial [[Wrench (screw theory)#Wrench|wrench]], see [[screw theory]].
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| The inertial terms are contained in the ''spatial inertia'' matrix
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| : <math>
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| \left(\begin{matrix} m {\boldsymbol 1} & - m [{\bold c}]\\
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| m [{\bold c}] & {\bold I}_{\rm cm} - m [{\bold c}][{\bold c}]\end{matrix}\right),
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| </math> | |
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| while the [[fictitious forces]] are contained in the term:<ref name=Featherstone>
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| {{cite book
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| |title=Rigid Body Dynamics Algorithms
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| |author= Roy Featherstone
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| |url=http://books.google.ca/books?id=UjWbvqWaf6gC&printsec=frontcover&dq=Rigid+Body+Dynamics+Algorithms
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| |isbn=978-0-387-74314-1
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| |publisher=Springer
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| |year=2008
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| }}</ref>
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| :<math>
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| \left(\begin{matrix} {m \boldsymbol \omega} \times \left({\boldsymbol \omega} \times {\bold c}\right) \\
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| {\boldsymbol \omega} \times ({\bold I}_{\rm cm} - m [{\bold c}][{\bold c}])\, {\boldsymbol \omega} \end{matrix}\right) .
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| </math>
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| 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.
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| ==Applications==
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| The Newton–Euler equations are used as the basis for more complicated "multi-body" formulations ([[Wrench (screw theory)|screw theory]]) that describe the dynamics of systems of rigid bodies connected by joints and other constraints. Multi-body problems can be
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| solved by a variety of numerical algorithms.<ref name=Shabana /><ref name=Featherstone /><ref name=Balafoutis>
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| {{cite book
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| |title=Dynamic Analysis of Robot Manipulators: A Cartesian Tensor Approach
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| |author= Constantinos A. Balafoutis, Rajnikant V. Patel
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| |page=Chapter 5
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| |url=http://books.google.com/books?id=7BcpyUjmLpUC&pg=PT195&dq=%22Kane%27s+dynamical+equations%22&lr=&as_brr=0&sig=ACfU3U1m290WlCUy1101Oj9Z9w3j5a4Lww#PPT151,M1
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| |isbn=0-7923-9145-4
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| |publisher=Springer
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| |year=1991
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| |nopp=true
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| }}</ref>
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| ==See also==
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| * [[Euler's laws of motion]] for a rigid body.
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| * [[Euler angles]]
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| * [[Inverse dynamics]]
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| * [[Centrifugal force]]
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| * [[Principal axes]]
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| * [[Spatial acceleration]]
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| * [[Wrench (screw theory)|Screw theory]] of rigid body motion.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Newton-Euler equations}}
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| [[Category:Rigid bodies]]
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| [[Category:Equations]]
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Hi there, I am Andrew Berryhill. For a whilst I've been in Mississippi but now I'm contemplating other choices. I am really fond of to go to karaoke but I've been taking on new issues recently. Invoicing is my occupation.
my website; love psychic readings