classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, 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 negligible, when a deforming pressure or deforming force is applied on it. The distance between any two given points on a rigid body rema ...

. 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 elements is an m \times 1 matrix consisting of a single column of 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 , c ...

s and

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

. 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 barycenter or balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. For ...

of a rigid body with the sum of

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

s and torques (or synonymously moments) acting on the rigid body.

Center of mass frame

With respect to a

coordinate frame In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...

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 barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

for τ(

torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

) and an

inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...

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. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

: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/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...

about the center of mass :ω =

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

of the body :α =

angular acceleration In physics, angular acceleration (symbol α, alpha) is the time rate of change of angular velocity. Following the two types of angular velocity, ''spin angular velocity'' and ''orbital angular velocity'', the respective types of angular accele ...

of the body

Any reference frame

With respect to a

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 vector from P to the center of mass of the body 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, see

screw theory Screw theory is the algebraic calculation of pairs of Vector (mathematics and physics), vectors, also known as ''dual vectors'' – such as Angular velocity, angular and linear velocity, or forces and Moment (physics), moments – that arise in th ...

. 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