Angular acceleration

Angular acceleration
Angular acceleration
Units
SI	rad / s 2
GHS	rad / s 2
Notes
	pseudovector

Angular acceleration - a pseudovector physical quantity equal to the first derivative of the angular velocity pseudovector with respect to time

${\vec {\varepsilon }}={\frac {d{\vec {\omega }}}{dt}}$ ${\ displaystyle {\ vec {\ varepsilon}} = {\ frac {d {\ vec {\ omega}}} {dt}}}$ ${\ vec \ varepsilon} = {\ frac {d {\ vec \ omega}} {dt}}$

Angular acceleration characterizes the intensity of changes in the module and the direction of the angular velocity during the motion of a solid .

Content

How they come to the concept of angular acceleration: acceleration of a point of a rigid body in free motion

The concept of angular acceleration can be arrived at by considering the calculation of the acceleration of a point in a rigid body making free motion. Body point speed $B$ ${\ displaystyle B}$ with free movement, according to Euler's formula , is equal to

${\vec {v}}_{B}={\vec {v}}_{A}+{\vec {\omega }}\times {\vec {AB}}$ ${\ displaystyle {\ vec {v}} _ {B} = {\ vec {v}} _ {A} + {\ vec {\ omega}} \ times {\ vec {AB}}}$

Where ${\vec {v}}_{A}$ ${\ displaystyle {\ vec {v}} _ {A}}$ - body point velocity $A$ ${\ displaystyle A}$ adopted as a pole; ${\vec {\omega }}$ ${\ displaystyle {\ vec {\ omega}}}$ - pseudovector of the angular velocity of the body; ${\vec {AB}}$ ${\ displaystyle {\ vec {AB}}}$ is a vector released from the pole to the point whose speed is being calculated. Differentiating this expression in time and using the Rivals formula ^[1] , we have

${\vec {a_{B}}}={\vec {a_{A}}}+{\vec {\varepsilon }}\times {\vec {AB}}+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {AB}})$ ${\ displaystyle {\ vec {a_ {B}}} = {\ vec {a_ {A}}} + {\ vec {\ varepsilon}} \ times {\ vec {AB}} + {\ vec {\ omega} } \ times ({\ vec {\ omega}} \ times {\ vec {AB}})$

${\vec {a_{B}}}={\vec {a_{A}}}+{\vec {a_{BA}^{rot}}}+{\vec {a_{BA}^{axis}}}$ ${\ displaystyle {\ vec {a_ {B}}} = {\ vec {a_ {A}}} + {\ vec {a_ {BA} ^ {rot}}} + {\ vec {a_ {BA} ^ { axis}}}}$

Where ${\vec {a}}_{A}$ ${\ displaystyle {\ vec {a}} _ {A}}$ - pole acceleration $A$ ${\ displaystyle A}$ ; ${\vec {\varepsilon }}={\frac {d{\vec {\omega }}}{dt}}$ ${\ displaystyle {\ vec {\ varepsilon}} = {\ frac {d {\ vec {\ omega}}} {dt}}}$ - pseudovector of angular acceleration. Component acceleration point $B$ ${\ displaystyle B}$ calculated through angular acceleration is called rotational acceleration of a point $B$ ${\ displaystyle B}$ around the pole $A$ ${\ displaystyle A}$

${\vec {a}}_{BA}^{\,rot}={\vec {\varepsilon }}\times {\vec {AB}}$ ${\ displaystyle {\ vec {a}} _ {BA} ^ {\, rot} = {\ vec {\ varepsilon}} \ times {\ vec {AB}}}$

The last term in the resulting formula, which depends on the angular velocity, is called the persistent acceleration of the acceleration of the point $B$ ${\ displaystyle B}$ around the pole $A$ ${\ displaystyle A}$

${\vec {a}}_{BA}^{\,axis}={\vec {\omega }}\times \left({\vec {\omega }}\times {\vec {AB}}\right)$ ${\ displaystyle {\ vec {a}} _ {BA} ^ {\, axis} = {\ vec {\ omega}} \ times \ left ({\ vec {\ omega}} \ times {\ vec {AB} } \ right)}$

The geometric meaning of the angular acceleration pseudovector

Pseudovector ${\vec {\varepsilon }}$ ${\ displaystyle {\ vec {\ varepsilon}}}$ directed tangent to the hodograph of angular velocity. Indeed, we consider two values of the angular velocity vector at the time $t$ ${\ displaystyle t}$ and at time $t+\Delta t$ ${\ displaystyle t + \ Delta t}$ . Let us estimate the change in the angular velocity for the considered period of time $\Delta t$ ${\ displaystyle \ Delta t}$

$\Delta {\vec {\omega }}={\vec {\omega }}(t+\Delta t)-{\vec {\omega }}(t)$ ${\ displaystyle \ Delta {\ vec {\ omega}} = {\ vec {\ omega}} (t + \ Delta t) - {\ vec {\ omega}} (t)}$

We attribute this change to the period of time during which it occurred

${\frac {\Delta {\vec {\omega }}}{\Delta t}}={\vec {\varepsilon }}^{\,\,'}$ ${\ displaystyle {\ frac {\ Delta {\ vec {\ omega}}} {\ Delta t}} = {\ vec {\ varepsilon}} ^ {\, \, '}}$

The resulting vector is called the mean angular acceleration vector. It occupies the position of a secant, crossing the hodograph of the angular velocity vector at points $M_{0}$ ${\ displaystyle M_ {0}}$ and $M_{1}$ ${\ displaystyle M_ {1}}$ . We pass to the limit at $\Delta t\to 0$ ${\ displaystyle \ Delta t \ to 0}$

$\lim _{\Delta t\to 0}{\frac {\Delta {\vec {\omega }}}{\Delta t}}={\frac {d{\vec {\omega }}}{dt}}={\vec {\varepsilon }}$ ${\ displaystyle \ lim _ {\ Delta t \ to 0} {\ frac {\ Delta {\ vec {\ omega}}} {\ Delta t}} = {\ frac {d {\ vec {\ omega}}} {dt}} = {\ vec {\ varepsilon}}}$

The vector of average angular acceleration will go into the vector of instantaneous angular acceleration and take the tangent position at the point $M_{0}$ ${\ displaystyle M_ {0}}$ to the hodograph of angular velocity.

Expression of the angular acceleration vector in terms of the final rotation parameters

When considering the rotation of the body through the parameters of the final rotation, the angular acceleration vector can be described by the formula

${\vec {\varepsilon }}=\left(1-\cos \varphi \right)\left({\vec {u}}\times {\frac {d^{2}{\vec {u}}}{dt^{2}}}\right)+{\dot {\varphi }}\left(1+\cos \varphi \right){\frac {d{\vec {u}}}{dt}}+{\dot {\varphi }}\sin \varphi \left({\vec {u}}\times {\frac {d{\vec {u}}}{dt}}\right)+\sin \varphi \,{\frac {d^{2}{\vec {u}}}{dt^{2}}}+{\ddot {\varphi }}\,{\vec {u}}$ ${\ displaystyle {\ vec {\ varepsilon}} = \ left (1- \ cos \ varphi \ right) \ left ({\ vec {u}} \ times {\ frac {d ^ {2} {\ vec {u }}} {dt ^ {2}}} \ right) + {\ dot {\ varphi}} \ left (1+ \ cos \ varphi \ right) {\ frac {d {\ vec {u}}} {dt }} + {\ dot {\ varphi}} \ sin \ varphi \ left ({\ vec {u}} \ times {\ frac {d {\ vec {u}}} {dt}} \ right) + \ sin \ varphi \, {\ frac {d ^ {2} {\ vec {u}}} {dt ^ {2}}} + {\ ddot {\ varphi}} \, {\ vec {u}}}$

Where ${\vec {u}}$ ${\ displaystyle {\ vec {u}}}$ - an ort specifying the direction of the rotation axis; $\varphi$ ${\ displaystyle \ varphi}$ - the angle by which rotation is made around the axis ${\vec {u}}$ ${\ displaystyle {\ vec {u}}}$ .

Angular acceleration when the body rotates around a fixed axis

When the body rotates around a fixed axis passing through the fixed points of the body $O_{1}$ ${\ displaystyle O_ {1}}$ and $O_{2}$ ${\ displaystyle O_ {2}}$ derivatives of the unit vector of the rotation axis are zero

${\frac {d{\vec {u}}}{dt}}={\frac {d^{2}{\vec {u}}}{dt^{2}}}=0$ ${\ displaystyle {\ frac {d {\ vec {u}}} {dt}} = {\ frac {d ^ {2} {\ vec {u}}} {dt ^ {2}}} = 0}$

In this case, the angular acceleration vector is determined trivially through the second derivative of the rotation angle

${\vec {\varepsilon }}={\ddot {\varphi }}\,{\vec {u}}$ ${\ displaystyle {\ vec {\ varepsilon}} = {\ ddot {\ varphi}} \, {\ vec {u}}}$

or

${\vec {\varepsilon }}=\varepsilon \,{\vec {u}}$ ${\ displaystyle {\ vec {\ varepsilon}} = \ varepsilon \, {\ vec {u}}}$

Where $\varepsilon ={\ddot {\varphi }}$ ${\ displaystyle \ varepsilon = {\ ddot {\ varphi}}}$ - algebraic value of angular acceleration. In this case, the pseudovector of angular acceleration, like angular velocity, is directed along the axis of rotation of the body. If the first and second derivatives of the angle of rotation have the same sign

${\dot {\varphi }}\,{\ddot {\varphi }}>0$ ${\ displaystyle {\ dot {\ varphi}} \, {\ ddot {\ varphi}}> 0}$

then the angular acceleration vector and the angular velocity vector coincide in direction (the body rotates rapidly). Otherwise, when ${\dot {\varphi }}\,{\ddot {\varphi }}<0$ ${\ displaystyle {\ dot {\ varphi}} \, {\ ddot {\ varphi}} <0}$ , vectors of angular velocity and angular acceleration are directed in opposite directions (the body rotates slowly).

In the course of theoretical mechanics, a traditional approach is that in which the concept of angular velocity and angular acceleration is introduced when considering the rotation of a body about a fixed axis. Moreover, as a law of motion, the time dependence of the angle of rotation of the body

$\varphi =\varphi (t)$ ${\ displaystyle \ varphi = \ varphi (t)}$

In this case, the law of motion of a point on a body can be expressed in a natural way, as the length of an arc of a circle passed by a point when the body rotates from a certain initial position $\varphi _{0}=\varphi (t_{0})$ ${\ displaystyle \ varphi _ {0} = \ varphi (t_ {0})}$

$s(t)=R\,\left(\varphi (t)-\varphi _{0}\right)$ ${\ displaystyle s (t) = R \, \ left (\ varphi (t) - \ varphi _ {0} \ right)}$

Where $R$ ${\ displaystyle R}$ - the distance from the point to the axis of rotation (the radius of the circle along which the point moves). Differentiating the last time relation, we obtain the algebraic velocity of a point

${\frac {ds}{dt}}=v_{\tau }=R\,{\frac {d\varphi }{dt}}=\omega \,R$ ${\ displaystyle {\ frac {ds} {dt}} = v _ {\ tau} = R \, {\ frac {d \ varphi} {dt}} = \ omega \, R}$

Where $\omega ={\frac {d\varphi }{dt}}$ ${\ displaystyle \ omega = {\ frac {d \ varphi} {dt}}}$ - algebraic value of the angular velocity. The acceleration of a body point during rotation can be represented as the geometric sum of tangential and normal acceleration

${\vec {a}}_{M}={\vec {a}}_{M}^{\,\tau }+{\vec {a}}_{M}^{\,n}$ ${\ displaystyle {\ vec {a}} _ {M} = {\ vec {a}} _ {M} ^ {\, \ tau} + {\ vec {a}} _ {M} ^ {\, n }}$

moreover, tangential acceleration is obtained as a derivative of the algebraic velocity of a point

$a_{M}^{\,\tau }={\frac {dv_{\tau }}{dt}}={\frac {d}{dt}}\left(\omega \,R\right)=R\,{\frac {d\omega }{dt}}=\varepsilon \,R$ ${\ displaystyle a_ {M} ^ {\, \ tau} = {\ frac {dv _ {\ tau}} {dt}} = {\ frac {d} {dt}} \ left (\ omega \, R \ right ) = R \, {\ frac {d \ omega} {dt}} = \ varepsilon \, R}$

Where $\varepsilon ={\frac {d\omega }{dt}}={\frac {d^{2}\varphi }{dt^{2}}}$ ${\ displaystyle \ varepsilon = {\ frac {d \ omega} {dt}} = {\ frac {d ^ {2} \ varphi} {dt ^ {2}}}}$ - algebraic value of angular acceleration. Normal acceleration of a body point can be calculated using the formulas

$a_{M}^{\,n}={\frac {v_{\tau }^{2}}{R}}=\omega ^{2}\,R$ ${\ displaystyle a_ {M} ^ {\, n} = {\ frac {v _ {\ tau} ^ {2}} {R}} = \ omega ^ {2} \, R}$

The expression of the pseudovector of angular acceleration through the body rotation tensor

If the rotation of a rigid body is given by a rank tensor $\left(1,\,1\right)$ ${\ displaystyle \ left (1, \, 1 \ right)}$ ( linear operator ), expressed, for example, in terms of the final rotation parameters

$B_{\,m}^{\,p}=\left(1-\cos \varphi \right)\,u^{\,p}\,u_{\,m}+\cos \varphi \,\delta _{\,m}^{\,p}+\sin \varphi \,g^{\,pl}\,\epsilon _{\,lkm}\,u^{\,k}$ ${\ displaystyle B _ {\, m} ^ {\, p} = \ left (1- \ cos \ varphi \ right) \, u ^ {\, p} \, u _ {\, m} + \ cos \ varphi \, \ delta _ {\, m} ^ {\, p} + \ sin \ varphi \, g ^ {\, pl} \, \ epsilon _ {\, lkm} \, u ^ {\, k}}$

Where $\delta _{\,m}^{\,p}$ ${\ displaystyle \ delta _ {\, m} ^ {\, p}}$ - Kronecker symbol ; $\epsilon _{\,lkj}$ ${\ displaystyle \ epsilon _ {\, lkj}}$ - Levi-Civita tensor , then, the pseudovector of angular acceleration can be calculated by the formula

$\varepsilon ^{\,i}={\frac {1}{2}}\,\epsilon ^{ikl}\,g_{\,lp}\,\left(B_{\,m}^{'\,p}\,{\ddot {B}}_{\,k}^{\,m}+{\dot {B}}_{\,m}^{'\,p}\,{\dot {B}}_{\,k}^{\,m}\right)$ ${\ displaystyle \ varepsilon ^ {\, i} = {\ frac {1} {2}} \, \ epsilon ^ {ikl} \, g _ {\, lp} \, \ left (B _ {\, m} ^ {'\, p} \, {\ ddot {B}} _ {\, k} ^ {\, m} + {\ dot {B}} _ {\, m} ^ {' \, p} \, {\ dot {B}} _ {\, k} ^ {\, m} \ right)}$

Where $B_{\,m}^{'\,p}$ ${\ displaystyle B _ {\, m} ^ {'\, p}}$ is the inverse transformation tensor equal to

$B_{\,m}^{'\,p}=\left(1-\cos \varphi \right)\,u^{\,p}\,u_{\,m}+\cos \varphi \,\delta _{\,m}^{\,p}-\sin \varphi \,g^{\,pl}\,\epsilon _{\,lkm}\,u^{\,k}$ ${\ displaystyle B _ {\, m} ^ {'\, p} = \ left (1- \ cos \ varphi \ right) \, u ^ {\, p} \, u _ {\, m} + \ cos \ varphi \, \ delta _ {\, m} ^ {\, p} - \ sin \ varphi \, g ^ {\, pl} \, \ epsilon _ {\, lkm} \, u ^ {\, k} }$

Notes

↑ V.I. Drong, V.V. Dubinin, M.M. Ilyin et al .; under the editorship of K.S. Kolesnikova, V.V. Dubinin. The course of theoretical mechanics: a textbook for universities. - 2017 .-- S. 101, 111. - 580 p. - ISBN 978-5-7038-4568-4 .

Literature

Targ S. M. A Short Course in Theoretical Mechanics - 10th ed., Rev. and add. - M .: Higher. school., 1986 - 416 S.
Pogorelov D. Yu. Introduction to the modeling of the dynamics of body systems: Textbook. - Bryansk: BSTU, 1997 .-- 197 p.

[1] V.I. Drong, V.V. Dubinin, M.M. Ilyin et al .; under the editorship of K.S. Kolesnikova, V.V. Dubinin. The course of theoretical mechanics: a textbook for universities. - 2017 .-- S. 101, 111. - 580 p. - ISBN 978-5-7038-4568-4 .

Angular acceleration
${\boldsymbol {\varepsilon }}={\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}={\boldsymbol {\dot {\omega }}}$ ${\ displaystyle {\ boldsymbol {\ varepsilon}} = {\ frac {\ mathrm {d} {\ boldsymbol {\ omega}}} {\ mathrm {d} t}} = {\ boldsymbol {\ dot {\ omega} }}}$ ${\ boldsymbol \ varepsilon} = {\ frac {{\ mathrm d} {\ boldsymbol \ omega}} {{\ mathrm d} t}} = {\ boldsymbol {\ dot \ omega}}$
Units
SI	rad / s ²
GHS	rad / s ²
Notes
pseudovector