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 classical ...

, the Newton–Euler equations describe the combined translational and rotational dynamics of a

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 force ...

. 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 vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...

s and

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

. These laws relate the motion of the

center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...

of a rigid body with the sum of

force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

s and

torques 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 the ...

(or synonymously moments) acting on the rigid body.

Center of mass frame

With respect to a coordinate frame whose origin coincides with the body's

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

for τ(

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 ...

) and an

inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...

for F(

), they can be expressed in matrix form as: :

\left(\begin  \\  \end\right) =
\left(\begin m  & 0 \\ 0 & _ \end\right)
\left(\begin \mathbf a_ \\  \end\right) +
\left(\begin 0 \\  \times _ \,  \end\right),

where :F = total

acting on the center of mass :''m'' = mass of the body :I₃ = the 3×3

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...

:a_cm = acceleration of the

:v_cm = velocity of the

:τ = total torque acting about the center of mass :I_cm =

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...

about the center of mass :ω =

angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

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  \\ _ \end\right) =
\left(\begin m  &  -m []^\\
m []^ & _ - m[]^[]^\end\right)
\left(\begin \mathbf a_ \\  \end\right) +
\left(\begin m[]^[]^  \\
^\times (_ - m []^\times[]^\times)\,  \end\right),

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

\equiv \left(\begin 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end\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 A wrench or spanner is a tool used to provide grip and mechanical advantage in applying torque to turn objects—usually rotary fasteners, such as nuts and bolts—or keep them from turning. In the UK, Ireland, Australia, and New Zealan ...

, see

screw theory Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...

. The inertial terms are contained in the ''spatial inertia'' matrix :

\left(\begin m  & - m []^\\
  m []^ & _ - m []^[]^\end\right),

while the fictitious forces are contained in the term: :

\left(\begin m^\times ^\times  \\
  ^\times (_ - m []^\times[]^\times)\,  \end\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 (

) 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.

References

{{DEFAULTSORT:Newton-Euler equations Rigid bodies Equations

Center of mass frame

Any reference frame

Applications

See also

References