Swirl rate

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In classical mechanics , the principle of twist is a physical law that states that to change the angular momentum of a body, a torque must be applied to it. Other terms for the law of swirl are the law of moments , the law of angular momentum , momentum moment or angular momentum balance .

An application example is the playground carousel in the picture. To get this going, you have to push it. From a technical point of view, a moment is created that gives the carousel angular momentum. The conservation of angular momentum then ensures that the carousel continues to rotate for a while. Frictional moments in the bearing and air resistance, however, generate a counter-torque that consumes the angular momentum and finally brings the rotation to a standstill again.

The mathematical formulation of the twist law is:

${\displaystyle {\vec {M}}_{c}={\dot {\vec {L}}}_{c}}$

It contains the moment acting from outside, the angular momentum of the body and its time derivative , each related to a fixed point , for which the origin in an inertial system is often used. Therefore, the index c is no longer explicitly stated below. In the special case where the external moments disappear, it can be seen that the angular momentum is retained. Correspondingly, angular momentum and for the angular momentum conservation record . Furthermore, the set of moments also stands for the set of moments from the statics . The d'Alembert inertial force opposite to the change in angular momentum makes itself felt as a gyroscopic effect . ${\displaystyle {\vec {M}}_{c}}$ ${\displaystyle {\vec {L}}_{c}}$ ${\displaystyle {\dot {\vec {L}}}_{c}}$ ${\displaystyle {\vec {c}}}$

The principle of the equality of the assigned shear stresses or the symmetry of the (Cauchy's) stress tensor follows from the twist law . The Boltzmann axiom has the same consequence , according to which internal forces in a continuum are moment-free. Thus the theorem of twist, the symmetry of the stress tensor and the Boltzmann axiom are related terms in continuum mechanics .

The rate of swirl plays a central role, especially in gyro theory . In continuum mechanics, it is used to clearly determine the skew-symmetrical part of the stress tensor.

In addition to Newton's Laws, the principle of twist is a fundamental and independent principle and was first introduced as such by Leonhard Euler in 1775.

history

In 1703, Jakob I Bernoulli used the principle of twist - without formulating it explicitly - to find the center of oscillation of a pendulum, which he did in a first, somewhat incorrect attempt in 1686. Clifford Truesdell therefore assumed that the theorem of twist, as an independent law of mechanics and as a kinetic generalization of the static equilibrium principle of torques, was first used by Jakob I Bernoulli in 1686. The law of swirl precedes Newton's laws of 1687.

In a work from 1744, Leonhard Euler was the first to use the principles of momentum and angular momentum to set up the equations of motion of a system. In 1750 he published in the treatise "Discovery of a New Principle of Mechanics" the gyroscopic equations , which are derived today from the theorem of swirl, but which Euler was able to deduce for the rigid body from Newton's second law . It was not until 1775, after studies of flat elastic continua, for which the balance of moments is indispensable, that Euler elevated the principle of twist to an independent principle for calculating the movements of bodies.

Augustin-Louis Cauchy introduced the stress tensor in 1822, the symmetry of which, in combination with the law of momentum, ensures the fulfillment of the law of swirl in the general case of deformable bodies. The interpretation of the twist rate was first recognized by MP Saint-Guilhem in 1851. ${\displaystyle {\vec {M}}={\dot {\vec {L}}}}$

In 1905, Ludwig Boltzmann pointed out that when a body is broken down into smaller (infinitesimal) volume elements, the internal reactions have to meet all static equilibrium conditions. Georg Hamel coined the name Boltzmann axiom for this statement .

Kinetics of rotation

The kinetics deals with states which are not in equilibrium. According to Newton's 2nd law , a resulting external force on a body leads to a change in speed ( acceleration ). Analogously, a resulting external torque means a change in the angular velocity resulting in an angular acceleration . The inertia behavior with regard to the rotation depends not only on the mass of a body, but also on its spatial distribution. ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {\alpha }}={\dot {\vec {\omega }}}}$

In the case of a rigid body , this is expressed by the moment of inertia Θ. For a rotation around a fixed axis, the following applies to the torque in the direction of this axis:

${\displaystyle M=\Theta \,\alpha }$

It should be noted that the moment of inertia does not only depend on the position of the axis of rotation (see Steiner's theorem ), but also on its direction. If the above equation is to be formulated more generally for any spatial direction, the inertia tensor Θ must be used instead :

${\displaystyle {\vec {M}}=\mathbf {\Theta } \cdot {\vec {\alpha }}+{\vec {\omega }}\times \mathbf {\Theta } \cdot {\vec {\omega }}}$

see below. The arithmetic symbol "×" forms the cross product .

In the two-dimensional special case, a torque merely accelerates or decelerates a rotational movement. In the general three-dimensional case, however, it can also change the direction of the axis of rotation (see e.g. precession ).

The many analogies in the kinetics of translation and rotation are given for rotation .

Formulations

Theorem of twist in point mechanics

Interplay of force F, moment τ, momentum p and twist L during the rotary motion of a mass point (sphere)

The relationship between the principle of momentum and the rate of twist becomes clear in point mechanics .

A body is given by n mass points m _k at the locations . In this system, which is demarcated from other masses, two types of forces can be distinguished: On the one hand, there are the internal forces that act between two mass points m _j and m _k belonging to the system and therefore always appear opposite in pairs, see Actio and Reactio . On the other hand, there are the external forces that act between the system masses and a mass outside the system and therefore only occur once on the system. Then the law of momentum reads for each individual mass point ${\displaystyle {\vec {r}}_{k}}$ ${\displaystyle {\vec {F}}_{jk}^{\rm {int}}}$ ${\displaystyle {\vec {F}}_{k}^{\rm {ext}}}$

${\displaystyle m_{j}{\ddot {\vec {r}}}_{j}={\vec {F}}_{j}^{\rm {ext}}+\sum _{k=1}^{n}{\vec {F}}_{jk}^{\rm {int}}\,,\quad j=1,\ldots ,n}$

( ) The swirl of the body around the origin is the sum of the angular momentum of the mass points according to ${\displaystyle {\vec {F}}_{jj}^{\rm {int}}:={\vec {0}}}$

${\displaystyle {\vec {L}}:=\sum _{j=1}^{n}m_{j}{\vec {r}}_{j}\times {\dot {\vec {r}}}_{j}}$

and the time derivative of this results in

${\displaystyle {\dot {\vec {L}}}=\sum _{j=1}^{n}m_{j}{\dot {\vec {r}}}_{j}\times {\dot {\vec {r}}}_{j}+\sum _{j=1}^{n}m_{j}{\vec {r}}_{j}\times {\ddot {\vec {r}}}_{j}=\sum _{j=1}^{n}{\vec {r}}_{j}\times (m_{j}{\ddot {\vec {r}}}_{j})}$

The accelerations can be expressed with the law of momentum in terms of the acting forces:

${\displaystyle {\begin{aligned}{\dot {\vec {L}}}=&\sum _{j=1}^{n}{\vec {r}}_{j}\times \left({\vec {F}}_{j}^{\rm {ext}}+\sum _{k=1}^{n}{\vec {F}}_{jk}^{\rm {int}}\right)\\=&\sum _{j=1}^{n}{\vec {r}}_{j}\times {\vec {F}}_{j}^{\rm {ext}}+{\underline {\sum _{j=1}^{n-1}\sum _{k=j+1}^{n}({\vec {r}}_{j}-{\vec {r}}_{k})\times {\vec {F}}_{jk}^{\rm {int}}}}\end{aligned}}}$

because the internal forces according to always appear opposite in pairs at two interacting mass points m _j and m _k . ${\displaystyle F_{kj}^{\rm {int}}=-F_{jk}^{\rm {int}}}$

Mass points only allow central forces and Siméon Denis Poisson proved in 1833 that a system of central forces kept in equilibrium in pairs does not exert any resulting torque, which means that the underlined sum is omitted. With this requirement, which is often not mentioned, the principle of twist arises in point mechanics

${\displaystyle {\dot {\vec {L}}}=\sum _{j=1}^{n}m_{j}{\vec {r}}_{j}\times {\ddot {\vec {r}}}_{j}=\sum _{j=1}^{n}{\vec {r}}_{j}\times {\vec {F}}_{j}^{\rm {ext}}=:{\vec {M}}_{R}^{\rm {ext}}}$

where is the resulting external moment acting on the system. The theorem of twist appears in point mechanics as a consequence of the theorem of momentum, which, however, is the result of the idealization of the masses as mass points that can only absorb central forces. ${\displaystyle {\vec {M}}_{R}^{\rm {ext}}}$

Georg Hamel called point mechanics “an intellectual impurity” and meant “what is meant by point mechanics is nothing other than the principle of emphasis.” Point mechanics is completely inadequate for deriving the principle of twist. When these considerations are transferred to a continuum , the assumption of central forces is equivalent to an axiom, the Boltzmann axiom below, which leads to the symmetry of Cauchy's stress tensor.

Nowhere in his Principia did Isaac Newton claim that the forces of interaction were central forces. If the masses are not idealized by mass points, then the law of twist helps: According to it, the internal forces cannot change the angular momentum and thus the underlined sum of the internal moments must disappear. Of course, the principle of twist also applies in point mechanics, but it is not a consequence of Newton's second law.

Swirl on the rigid body

→ Main article : Euler's equations (gyro theory)

In the case of a rigid body, the mass points follow Euler's velocity equation, which has important consequences and the vector equation

${\displaystyle {\vec {M}}=\mathbf {\Theta } \cdot {\dot {\vec {\omega }}}+{\vec {\omega }}\times \mathbf {\Theta } \cdot {\vec {\omega }}}$

leads. This equation is sometimes called Euler's (gyro) equation . Any unaccelerated fixed point or any moving center of mass of the body are suitable as a reference point for the moment and the inertia tensor Θ . The first term on the right takes into account the Euler forces and the second the fictitious centrifugal forces . If the rigid body were to revolve around the instantaneous axis of rotation with an angular velocity kept constant , then the centrifugal forces would have a resulting moment that just corresponds. However, since the axis of rotation constantly changes its position during the real movement of the rigid body, Louis Poinsot suggested the name fictitious centrifugal forces for these centrifugal forces . ${\displaystyle {\vec {\omega }}}$ ${\displaystyle -{\vec {\omega }}\times \mathbf {\Theta } \cdot {\vec {\omega }}}$

proof

Similar to the above, the body is viewed as the union of rigid masses m i with centers of mass that are specified relative to a reference point . The acceleration of the masses is then the same for a rigid body movement ${\displaystyle \textstyle \bigcup _{i}m_{i}{\vec {r}}_{i}}$ ${\displaystyle {\vec {r}}_{i}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}({\vec {r}}_{i},t)={\ddot {\vec {b}}}(t)+{\dot {\vec {\omega }}}(t)\times {\vec {r}}_{i}+{\vec {\omega }}(t)\times \left[{\vec {\omega }}(t)\times {\vec {r}}_{i}\right]}$ The rigid body rotates around the reference point with angular velocity . ${\displaystyle {\vec {\omega }}}$ ${\displaystyle {\vec {b}}}$ According to Newton's second law "force equals mass times acceleration" is ${\displaystyle \sum _{i}{\vec {F}}_{i}=\sum _{i}m_{i}{\vec {a}}_{i}=\sum _{i}m_{i}\{{\ddot {\vec {b}}}(t)+{\dot {\vec {\omega }}}(t)\times {\vec {r}}_{i}+{\vec {\omega }}(t)\times \left[{\vec {\omega }}(t)\times {\vec {r}}_{i}\right]\}}$ There are external forces in it, and the internal forces cancel each other out according to the principle of action and reaction . ${\displaystyle {\vec {F}}_{i}}$ According to the principle of swirl, the internal forces are moment-free, see the previous section. The moments of the external forces F i acting on the individual masses then add up to the resulting external moment ${\displaystyle {\begin{aligned}{\vec {M}}=&\sum _{i}{\vec {r}}_{i}\times {\vec {F}}_{i}=\sum _{i}{\vec {r}}_{i}\times m_{i}{\vec {a}}_{i}\\=&\sum _{i}m_{i}{\vec {r}}_{i}\times \left\{{\ddot {\vec {b}}}(t)+{\dot {\vec {\omega }}}(t)\times {\vec {r}}_{i}+{\vec {\omega }}(t)\times [{\vec {\omega }}(t)\times {\vec {r}}_{i}]\right\}\end{aligned}}}$ The first term disappears if the reference point is fixed in an inertial frame ( ) or the center of mass of the body is chosen as the reference point (then is ), and this is assumed here. ${\displaystyle \textstyle \sum _{i}m_{i}{\vec {r}}_{i}\times {\ddot {\vec {b}}}(t)}$ ${\displaystyle {\ddot {\vec {b}}}\equiv 0}$ ${\displaystyle \textstyle \sum _{i}m_{i}{\vec {r}}_{i}={\vec {0}}}$ The second term contains Euler forces whose moments are based on the BAC-CAB formula for gyroscopic action ${\displaystyle -m_{i}{\dot {\vec {\omega }}}(t)\times {\vec {r}}_{i}}$ ${\displaystyle {\begin{aligned}{\vec {K}}_{\text{Euler}}=&\sum _{i}-m_{i}{\vec {r}}_{i}\times ({\dot {\vec {\omega }}}\times {\vec {r}}_{i})=-\sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i}){\dot {\vec {\omega }}}-({\vec {r}}_{i}\cdot {\dot {\vec {\omega }}}){\vec {r}}_{i}]\\=&-{\underline {\sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i})\mathbf {1} -{\vec {r}}_{i}\otimes {\vec {r}}_{i}]}}\cdot {\dot {\vec {\omega }}}=-\mathbf {\Theta } \cdot {\dot {\vec {\omega }}}\end{aligned}}}$ sum up. The underlined term is the inertia tensor Θ , which is formed for the rigid body with the unit tensor 1 and the dyadic product “⊗” of vectors. With any three vectors , the dyadic product is defined by . ${\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}$ ${\displaystyle ({\vec {a}}\otimes {\vec {b}})\cdot {\vec {c}}:=({\vec {b}}\cdot {\vec {c}}){\vec {a}}}$ The third and last term in the above equation of moments is formed from the centrifugal forces that make up the gyroscopic effect ${\displaystyle -m_{i}{\vec {\omega }}(t)\times [{\vec {\omega }}(t)\times {\vec {r}}_{i}]}$ ${\displaystyle {\begin{aligned}{\vec {K}}_{\text{Zentrifugal}}=&\sum _{i}{\vec {r}}_{i}\times \{-m_{i}{\vec {\omega }}(t)\times [{\vec {\omega }}(t)\times {\vec {r}}_{i}]\}\\=&-\sum _{i}{\vec {r}}_{i}\times m_{i}[({\vec {\omega }}\cdot {\vec {r}}_{i}){\vec {\omega }}-({\vec {\omega }}\cdot {\vec {\omega }}){\vec {r}}_{i}]\\=&-\sum _{i}{\vec {\omega }}\times m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i}){\vec {\omega }}-({\vec {\omega }}\cdot {\vec {r}}_{i}){\vec {r}}_{i}]=-{\vec {\omega }}\times \mathbf {\Theta } \cdot {\vec {\omega }}\end{aligned}}}$ results. The angular momentum is calculated with the inertia tensor and this is how the time derivative arises : ${\displaystyle {\vec {L}}=\mathbf {\Theta } \cdot {\vec {\omega }}}$ ${\displaystyle {\vec {M}}={\dot {\vec {L}}}=(\mathbf {\Theta } \cdot {\vec {\omega }}{\dot {)\;}}=-{\vec {K}}=-({\vec {K}}_{\text{Euler}}+{\vec {K}}_{\text{Zentrifugal}})}$ ${\displaystyle (\mathbf {\Theta } \cdot {\vec {\omega }}{\dot {)\;}}=\mathbf {\Theta } \cdot {\dot {\vec {\omega }}}+{\vec {\omega }}\times \mathbf {\Theta } \cdot {\vec {\omega }}}$ The gyroscopic effects are d'Alembert's inertial forces and as such are a moment of equal size opposite to an attacking moment, which leads to the gyro equations: ${\displaystyle {\vec {K}}}$ ${\displaystyle {\vec {M}}}$ ${\displaystyle {\vec {M}}=-{\vec {K}}=\mathbf {\Theta } \cdot {\dot {\vec {\omega }}}+{\vec {\omega }}\times \mathbf {\Theta } \cdot {\vec {\omega }}}$

With regard to an orthonormal basis ê _1,2,3, the component _equations read:

${\displaystyle {\begin{aligned}M_{1}=&\Theta _{11}{\dot {\omega }}_{1}+\Theta _{12}{\dot {\omega }}_{2}+\Theta _{13}{\dot {\omega }}_{3}+(\Theta _{33}-\Theta _{22})\omega _{2}\omega _{3}+\Theta _{23}(\omega _{2}^{2}-\omega _{3}^{2})+\Theta _{13}\omega _{1}\omega _{2}-\Theta _{12}\omega _{3}\omega _{1}\\M_{2}=&\Theta _{12}{\dot {\omega }}_{1}+\Theta _{22}{\dot {\omega }}_{2}+\Theta _{23}{\dot {\omega }}_{3}+(\Theta _{11}-\Theta _{33})\omega _{3}\omega _{1}+\Theta _{13}(\omega _{3}^{2}-\omega _{1}^{2})+\Theta _{12}\omega _{2}\omega _{3}-\Theta _{23}\omega _{1}\omega _{2}\\M_{3}=&\Theta _{13}{\dot {\omega }}_{1}+\Theta _{23}{\dot {\omega }}_{2}+\Theta _{33}{\dot {\omega }}_{3}+(\Theta _{22}-\Theta _{11})\omega _{1}\omega _{2}+\Theta _{12}(\omega _{1}^{2}-\omega _{2}^{2})+\Theta _{23}\omega _{3}\omega _{1}-\Theta _{13}\omega _{2}\omega _{3}.\end{aligned}}}$

Here, Θ _{ik are} the components of the inertia tensor: Θ _ii are the moments of inertia around the i-axis and Θ _ik with k ≠ i are the moments of deviation . In a body-fixed coordinate system, these components are constant over time, otherwise mostly time-dependent.

Plane movements and the rate of angular momentum around the momentary pole

In the case of a plane movement, for example in the 1-2 plane, the component equation is reduced to

${\displaystyle {\begin{aligned}M_{3}=&\Theta _{33}{\ddot {\varphi }}\\M_{1}=&\Theta _{13}{\ddot {\varphi }}-\Theta _{23}{\dot {\varphi }}^{2}=\left({\frac {\Theta _{13}}{\Theta _{33}}}+{\frac {\Theta _{23}^{2}}{\Theta _{13}\Theta _{33}}}\right)M_{3}-{\frac {\Theta _{23}}{\Theta _{13}}}M_{2}\\M_{2}=&\Theta _{23}{\ddot {\varphi }}+\Theta _{13}{\dot {\varphi }}^{2}=\left({\frac {\Theta _{23}}{\Theta _{33}}}+{\frac {\Theta _{13}^{2}}{\Theta _{23}\Theta _{33}}}\right)M_{3}-{\frac {\Theta _{13}}{\Theta _{23}}}M_{1},\end{aligned}}}$

where φ is the angle of rotation around the 3-axis. As before, the moments of inertia Θ _ij (except Θ ₃₃ ) in a reference system that is not fixed to the body are generally dependent on the orientation and thus on the angle of rotation φ . The last two equations are mostly used to determine the reaction torques in 1 and 2 direction for forced operation in the 1-2 plane.

If the 3-direction is a main axis of inertia , then with the associated main moment of inertia Θ ₃ results without such reaction moments

${\displaystyle M_{3}=\Theta _{3}{\ddot {\varphi }}.}$

In the case of a plane rigid body movement with existing rotational movement, there is always a point in space called a momentary pole , in which firstly a particle of the rigid body located there is stationary and secondly the movement is represented as a pure rotational movement around this point. Thus the velocity field with the normal unit vector of the plane of motion ê ₃ reads : ${\displaystyle {\vec {m}}}$

${\displaystyle {\vec {v}}({\vec {x}},t)={\dot {\varphi }}(t){\hat {e}}_{3}\times {\big (}{\vec {x}}-{\vec {m}}(t){\big )}}$

With regard to the instantaneous center of gravity, if the 3-direction is a main axis, the angular momentum law has a similar form as with regard to the center of mass:

${\displaystyle M_{m3}=\Theta _{m3}{\ddot {\varphi }},}$

where now the moment and the mass moment of inertia are calculated with respect to the instantaneous pole.

proof

To prove this, the angular momentum of the rigid body is calculated as an integral: ${\displaystyle {\begin{aligned}{\vec {L}}_{c}:=&\int _{v}({\vec {x}}-{\vec {c}})\times {\vec {v}}\,\rho \mathrm {d} v\\=&\int _{a}({\vec {x}}-{\vec {m}}+{\vec {m}}-{\vec {c}})\times [{\dot {\varphi }}{\hat {e}}_{3}\times ({\vec {x}}-{\vec {m}})]\,\rho \mathrm {d} v\\=&\int _{a}({\vec {x}}-{\vec {m}})\times [{\dot {\varphi }}{\hat {e}}_{3}\times ({\vec {x}}-{\vec {m}})]\,\rho \mathrm {d} v+({\vec {m}}-{\vec {c}})\times [{\dot {\varphi }}{\hat {e}}_{3}\times \int _{v}({\vec {x}}-{\vec {m}})\,\rho \mathrm {d} v]\\=&\int _{v}[{\vec {r}}\times ({\dot {\varphi }}{\hat {e}}_{3}\times {\vec {r}})\,\rho \mathrm {d} v+({\vec {m}}-{\vec {c}})\times [{\dot {\varphi }}{\hat {e}}_{3}\times (m{\vec {s}}-m{\vec {m}})]\\=&\mathbf {\Theta } _{m}\cdot {\dot {\varphi }}{\hat {e}}_{3}+m({\vec {m}}-{\vec {c}})\times {\vec {v}}_{s}.\end{aligned}}}$ The vector is the distance vector to the instantaneous pole, the mass, the center of mass and is its speed. If the instantaneous pole is chosen as the reference point , the second term is omitted: ${\displaystyle {\vec {r}}={\vec {x}}-{\vec {m}}}$ ${\displaystyle \textstyle m:=\int _{v}\rho \mathrm {d} v}$ ${\displaystyle \textstyle {\vec {s}}:=\int _{v}{\vec {x}}\,\rho \mathrm {d} v}$ ${\displaystyle {\vec {v}}_{s}={\dot {\varphi }}{\hat {e}}_{3}\times ({\vec {s}}-{\vec {m}})}$ ${\displaystyle ({\vec {c}}={\vec {m}})}$ ${\displaystyle {\vec {L}}_{m}:=\mathbf {\Theta } _{m}\cdot {\dot {\varphi }}{\hat {e}}_{3}.}$ The inertia tensor with respect to the momentary pole ${\displaystyle \mathbf {\Theta } _{m}:=\int _{v}\{[({\vec {x}}-{\vec {m}})\cdot ({\vec {x}}-{\vec {m}})]\mathbf {1} -({\vec {x}}-{\vec {m}})\otimes ({\vec {x}}-{\vec {m}})\}\,\rho \mathrm {d} v}$ depends on the currently occupied space and is therefore mostly not constant. ${\displaystyle v}$ Substantial time derivative of the angular momentum yields with : ${\displaystyle {\vec {v}}({\vec {m}},t)={\vec {0}}}$ ${\displaystyle {\begin{aligned}{\dot {\vec {L}}}_{c}=&{\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {\Theta } _{m}\cdot {\dot {\varphi }}{\hat {e}}_{3})+m({\vec {m}}-{\vec {c}})\times {\dot {\vec {v}}}_{s}\\=&\mathbf {\Theta } _{m}\cdot {\ddot {\varphi }}{\hat {e}}_{3}+{\dot {\varphi }}{\hat {e}}_{3}\times \mathbf {\Theta } _{m}\cdot {\dot {\varphi }}{\hat {e}}_{3}+m({\vec {m}}-{\vec {c}})\times {\dot {\vec {v}}}_{s}={\vec {M}}_{c}.\end{aligned}}}$ If the instantaneous center of gravity is chosen as the reference point , then the last summand is omitted, and if a main axis of the body is parallel to the angular velocity, i.e. perpendicular to the plane of motion, then the second summand is omitted. If both are true, the vector equation has only one non-trivial component ${\displaystyle ({\vec {c}}={\vec {m}})}$ ${\displaystyle M_{m3}=\Theta _{m3}{\ddot {\varphi }}.}$

Theorem on the continuum

→ Main article : Second Cauchy-Euler law of motion

The physical laws formulated in mechanics for extended bodies are expressed in continuum mechanics as global integral equations from which, with suitable assumptions, local differential equations can be derived, which must be fulfilled at every point in the body. The external forces and the activities they are moments like in reality surface with voltage vectors (with the dimension of force per area started) on the surface. In addition, there are also volume-distributed forces (with the dimension force per mass or an acceleration) such as weight . Then the law of swirl in global formulation is: ${\displaystyle {\vec {t}}}$ ${\displaystyle {\vec {k}}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}({\vec {x}}-{\vec {c}}){\times }\rho {\vec {v}}\,\mathrm {d} v=\int _{v}({\vec {x}}-{\vec {c}})\times \rho {\vec {k}}\,\mathrm {d} v+\int _{a}({\vec {x}}-{\vec {c}})\times {\vec {t}}\,\mathrm {d} a}$

Here, ρ is the density and the velocity at the location in the volume of the body that owns the surface . The integral on the left stands for the angular momentum of the body in relation to any fixed reference point and forms the change over time. On the right side are the moments of the external forces. The first integral determines the moment of the volume-distributed forces and the second integral the moment of the surface-distributed forces . The arithmetic symbol stands for the cross product . ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle v}$ ${\displaystyle a}$ ${\displaystyle {\vec {c}}}$ ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}}$ ${\displaystyle \rho {\vec {k}}}$ ${\displaystyle {\vec {t}}}$ ${\displaystyle \times }$

The external forces induce a stress tensor field σ that fills the whole body via (the superscript “„ ”means transposition and is the outwardly directed normal unit vector on the surface) and the divergence law . The part of the integrals that affects the orbital angular momentum of the particles is omitted due to the momentum balance. What remains is an ineffective moment that is caused by shear stresses between the particles, and for this contribution to disappear, the Cauchy stress tensor must be symmetrical: ${\displaystyle {\vec {t}}={\boldsymbol {\sigma }}^{\top }\cdot {\hat {n}}}$ ${\displaystyle {\hat {n}}}$

${\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {\sigma }}^{\top }}$

In the Lagrangeian perspective, this concerns the second Piola-Kirchhoff stress tensor . In combination with the momentum balance, the symmetry of the stress tensor is equivalent to the principle of twist. ${\displaystyle {\tilde {\mathbf {T} }}}$

Boltzmann axiom

Stresses on a volume element (blue) with width d x and height d y (dimensions not shown)

Ludwig Boltzmann pointed out in 1905 that when a body is broken down into (infinitesimally) small volume elements, each must be in static equilibrium. The resulting internal forces and internal moments must therefore vanish at the interfaces of each volume element. The Cauchy fundamental theorem treated former condition of the disappearance of the internal forces. Georg Hamel coined the name Boltzmann axiom for the demand for the disappearance of the inner moments , since Boltzmann was the first to emphasize the independence of this consideration. The Boltzmann axiom applies to rigid bodies and many deformable bodies. However, there are also continua to which the Boltzmann axiom cannot be applied, see the following section.

This axiom is equivalent to the symmetry of Cauchy's stress tensor . Because so that the tension resultant on the volume element, blue in the picture, does not exert a moment, the line of action of the resulting force must go through the center of the volume element. The individual forces result from the stresses multiplied by the area on which they act. The line of action of the inertia forces and the forces of the normal stresses σ _xx and σ _yy lead through the center of the volume element. So that the line of action of the shear stress _resultant with components τ _yx · d x in the x-direction and τ _xy · d y in the y-direction must also pass through the center

${\displaystyle {\frac {\tau _{xy}\mathrm {d} y}{\tau _{yx}\mathrm {d} x}}={\frac {\mathrm {d} y}{\mathrm {d} x}}\quad \rightarrow \quad \tau _{yx}=\tau _{xy}}$

be valid. The latter is precisely the statement of the principle of the equality of the assigned shear stresses in the xy plane .

Cosserat continuum

In addition to the moment-free classical continuum with a symmetrical stress tensor, Cosserat continua (polar continua) that are not moment-free were also defined. One application of such a continuum is the shell theory . In the polar continua there are, in addition to the momentum flows and sources , see above, also angular momentum flows and sources. The Boltzmann axiom does not apply here and the stress tensor can be asymmetrical. If these angular momentum fluxes and sources are treated as usual in continuum mechanics, field equations arise in which the skew-symmetrical part of the stress tensor has no energetic meaning. The law of swirl becomes independent of the law of energy and is used to determine the skew symmetrical part of the stress tensor. Truesdell saw here the "true basic meaning of the twist principle". ${\displaystyle {\vec {t}}}$ ${\displaystyle \rho {\vec {k}}}$

Area set

The area swept by the driving beam changes into a triangle at a small d t

The law of area is a consequence of the law of swirl in the form: The resulting moment is equal to the product of double mass and the derivation of the surface velocity.

It relates to the driving beam to a mass point with mass m . This has the angular momentum with the speed and the momentum ${\displaystyle {\vec {r}}}$ ${\displaystyle {\dot {\vec {r}}}}$ ${\displaystyle {\vec {p}}=m{\dot {\vec {r}}}}$

${\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}=m{\vec {r}}\times {\dot {\vec {r}}}=m{\vec {r}}\times \mathrm {d} {\vec {r}}/\mathrm {d} t}$.

In the (infinitesimal) time d t, the driving beam sweeps over a triangle, the content of which is, see image and cross product “×” . So it turns out ${\displaystyle \mathrm {d} {\vec {A}}={\tfrac {1}{2}}{\vec {r}}\times \mathrm {d} {\vec {r}}}$

${\displaystyle {\vec {L}}=m{\vec {r}}\times \mathrm {d} {\vec {r}}/\mathrm {d} t=2m\,\mathrm {d} {\vec {A}}/\mathrm {d} t=2m{\dot {\vec {A}}}}$

With the rate of swirl it becomes:

${\displaystyle {\vec {M}}={\dot {\vec {L}}}=2m{\ddot {\vec {A}}}}$

The special case of the plane, torque-free motion in a central force is field-treated second law of Kepler, which is also under the name Flächensatz is known.

literature

1. ↑ Dankert, Dankert: Technical Mechanics. Springer, 7th edition, 2013, p. 571.
2. ^ ^{A b} D. Gross, W. Hauger, J. Schröder , WA Wall: Technische Mechanik 3 . Kinetics. Springer Vieweg Verlag, Heidelberg 2015, ISBN 978-3-642-53953-4 , p. 61 , doi : 10.1007 / 978-3-642-53954-1 .
3. ^ Conrad Eller: Holzmann / Meyer / Schumpich. Technical mechanics. Kinematics and kinetics. Springer, 12th edition, 2016, p. 127.
4. ↑ Stefan Hartmann: Technical Mechanics . John Wiley & Sons, 2014, ISBN 978-3-527-68162-4 , pp. 491 .
5. ↑ ^{a b c d e} István Szabó : History of mechanical principles . and their main uses. Springer, Basel 1977, ISBN 978-3-0348-5998-1 , pp.  22nd ff ., doi : 10.1007 / 978-3-0348-5998-1 ( limited preview in Google Book Search [accessed January 12, 2018]).
6. ↑ ^{a b c d e} H. Bremer: Dynamics and control of mechanical systems . BG Teubner, Stuttgart 1988, ISBN 978-3-519-02369-2 , doi : 10.1007 / 978-3-663-05674-4 ( limited preview in Google Book Search [accessed January 12, 2018]).
7. ↑ ^{a b c d e f g} Clifford Truesdell : The development of the twist law . In: Society for Applied Mathematics and Mechanics (ed.): Journal for Applied Mathematics and Mechanics (=  Issue 4/5 ). tape  44 , April 1964, p. 149 - 158 , doi : 10.1002 / zamm.19640440402 ( wiley.com ).
8. ↑ ^{a b} Felix Klein , Conr. Müller: Encyclopedia of the Mathematical Sciences with inclusion of its applications . Mechanics. Ed .: Academies of Sciences in Göttingen, Leipzig, Munich and Vienna. 4th volume, 1st part volume. Springer Fachmedien Verlag, Leipzig 1908, ISBN 978-3-663-16021-2 , p.  581 ff ., doi : 10.1007 / 978-3-663-16021-2 ( limited preview in the Google Book Search [accessed on February 12, 2020] see also wikisource ).
9. ^ MP Guilhem: New study on the theory of forces . In: Joseph Liouville (ed.): Journal de mathématiques pures et appliquées . tape XVI . Bachelier, Paris 1851, p. 347–374 (French, bnf.fr [accessed on February 11, 2020] Original title: Nouvelle étude sur la théorie des forces . Equation (4) on page 363 is the rate of twist in the co-rotating system.). , see also Klein and Müller (1908), p. 587.
10. ↑ H. Oertel (ed.): Prandtl guide through fluid mechanics . Fundamentals and phenomena. 13th edition. Springer Vieweg, 2012, ISBN 978-3-8348-1918-5 , p. 15 .
11. ↑ Friedrich Pfeiffer , Thorsten Schindler: Introduction to Dynamics . Springer-Verlag, 2014, ISBN 978-3-642-41046-8 , pp.  24 ( limited preview in Google Book Search [accessed January 13, 2018]).
12. ^ Rainer Tiemeyer: Axioms of Classical Mechanics . Hilbert's problem and Hamel's attempted solution from an epistemological perspective. Logos Verlag, Berlin 2016, ISBN 978-3-8325-4292-4 , p. 166 ( limited preview in Google Book Search [accessed January 12, 2018]).
13. ^ Clifford Truesdell , Walter Noll , Stuart Antman: The nonlinear field theories of mechanics . tape  3 . Springer Science & Business Media, Berlin, Heidelberg 2004, ISBN 978-3-540-02779-9 , pp. 389 ff . (English, limited preview in Google Book Search [accessed on January 14, 2018] Original title: The Non-Linear Field Theories of Mechanics .).
14. ^ RA Toupin: Theories of elasticity with couple-stress . In: Archive for Rational Mechanics and Analysis . Volume 17, Issue 2. Springer-Verlag, June 1964, ISSN  0003-9527 , p. 85–112 , doi : 10.1007 / BF00253050 (English, springer.com [accessed January 14, 2018]).
15. ↑ Karl-Heinrich Grote, Jörg Feldhusen (Ed.): Dubbel . Paperback for mechanical engineering. Springer Vieweg Verlag, Berlin, Heidelberg 2014, ISBN 978-3-642-38891-0 , p. B26 , doi : 10.1007 / 978-3-642-38891-0 ( limited preview in Google Book Search [accessed January 13, 2018]).