Parallel axis theorem: Difference between revisions

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{{Short description|Theorem in planar dynamics}}

{{Redirect-distinguish|Steiner's theorem|Steiner's theorem (geometry)}}

The '''parallel axis theorem''', also known as '''Huygens&ndash;Steiner theorem''', or just as '''Steiner's theorem''',<ref>{{cite bookcitation | title=Introduction to theoretical physics | author=Arthur Erich Haas | year=1928}}</ref> named after [[Christiaan Huygens]] and [[Jakob Steiner]], can be used to determine the [[moment of inertia]] or the [[second moment of area]] of a [[rigid body]] about any axis, given the body's moment of inertia about a [[Parallel (geometry)|parallel]] axis through the object's [[center of gravity]] and the [[perpendicular]] [[distance]] between the axes.

==Mass moment of inertia==

[[File:Steiner.png|thumb|right|The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the centrecenter of mass.]]

Suppose a body of mass {{math|''m''}} is rotated about an axis {{math|''z''}} passing through the body's [[centrecenter of gravitymass]]. The body has a moment of inertia {{math|''I''_cm}} with respect to this axis.

The parallel axis theorem states that if the body is made to rotate instead about a new axis {{math|''z&prime;''}}, which is parallel to the first axis and displaced from it by a distance {{math|''d''}}, then the moment of inertia {{math|''I''}} with respect to the new axis is related to {{math|''I''_cm}} by

:<math> I = I_\mathrm{cm} + md^2.</math>

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===Derivation===

We may assume, without loss of generality, that in a [[Cartesian coordinate system]] the perpendicular distance between the axes lies along the ''x''-axis and that the center of mass lies at the origin. The moment of inertia relative to the ''z''-axis is then

:<math>I_\mathrm{cm} = \int (x^2 + y^2) \, dm.</math>

The moment of inertia relative to the axis {{math|''z&prime;''}}, which is aat perpendiculara distance {{math|''dD''}} from the center of mass along the ''x''-axis from the centre of mass, is

:<math>I = \int \left[(x +- dD)^2 + y^2\right] \, dm.</math>

Expanding the brackets yields

:<math>I = \int (x^2 + y^2) \, dm + dD^2 \int dm +- 2d2D\int x\, dm.</math>

The first term is {{math|''I''_cm}} and the second term becomes {{math|''mdMD''²}}. The integral in the final term is a multiple of the x-coordinate of the [[center of mass]]{{snd}}which is zero since the center of mass lies at the origin. So, the equation becomes:

:<math> I = I_\mathrm{cm} + mdMD^2.</math>

=== Tensor generalization ===

The parallel axis theorem can be generalized to calculations involving the [[Moment of inertia#The inertiaInertia tensor|inertia tensor]].<ref Let {{math|''I<subname="Abdulghany">ij</sub>''}} denote the inertia tensor of a body as calculated at the centre of mass. Then the inertia tensor {{math|''J_ij''}} as calculated relative to a new point iscitation

| last = Abdulghany | first = A. R.

| date = October 2017

| doi = 10.1119/1.4994835

| issue = 10

| journal = American Journal of Physics

| pages = 791–795

| title = Generalization of parallel axis theorem for rotational inertia

| volume = 85| doi-access = free

| bibcode = 2017AmJPh..85..791A

}}</ref> Let {{math|''I_ij''}} denote the inertia tensor of a body as calculated at the center of mass. Then the inertia tensor {{math|''J_ij''}} as calculated relative to a new point is

:<math>J_{ij}=I_{ij} + m\left(|\mathbf{R}|^2 \delta_{ij}-R_i R_j\right),</math>

where <math>\mathbf{R}=R_1\mathbf{\hat{x}}+R_2\mathbf{\hat{y}}+R_3\mathbf{\hat{z}}\!</math> is the displacement vector from the centrecenter of mass to the new point, and {{math|δ_''ij''}} is the [[Kronecker delta]].

For diagonal elements (when {{math|''i'' {{=}} ''j''}}), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.

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where '''E'''₃ is the {{nobr|3 × 3}} [[identity matrix]] and <math>\otimes</math> is the [[outer product]].

Further generalization of the parallel axis theorem gives the inertia tensor about any set of orthogonal axes parallel to the reference set of axes x, y and z, associated with the reference inertia tensor, whether or not they pass through the center of mass.<ref name="Abdulghany"/>A. R.In Abdulghanythis generalization, Americanthe Journalinertia oftensor Physicscan 85,be 791moved (2017);from doi:being https:reckoned about any reference point <math>\mathbf{R}_{ref}</math> to some final reference point <math>\mathbf{R}_F</dx.doi.org/10.1119/1.4994835math> via the relational matrix .<math>M</refmath> as:

:<math> I_{F} = I_\mathrm{ref} + m(M[\mathbf{R},\mathbf{R}] - 2M[\mathbf{R},\mathbf{C}])</math>

where <math>\mathbf{C}</math> is the vector from the initial reference point to the object's center of mass and <math>\mathbf{R}</math> is the vector from the initial reference point to the final reference point (<math>\mathbf{R}_F = \mathbf{R}_{ref} + \mathbf{R}</math>). The relational matrix is given by

:<math> M[\mathbf{r},\mathbf{c}] = \left[\begin{array}{rrr}(r_y c_y + r_z c_z) & -1/2(r_x c_y + r_y c_x) & -1/2(r_x c_z + r_z c_x) \\

-1/2(r_x c_y + r_y c_x) & (r_x c_x + r_z c_z) & -1/2(r_y c_z + r_z c_y) \\

-1/2(r_x c_z + r_z c_x) & -1/2(r_y c_z + r_y c_z) & (r_x c_x + r_y c_y) \end{array}\right] </math>

==AreaSecond moment of inertiaarea==

The parallel axes rule also applies to the [[second moment of area]] (area moment of inertia) for a plane region ''D'':

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:<math> I_S = I_R + Md^2, \, </math>

which is known as the parallel axis theorem.<ref>{{Citation |first=Burton |last=Paul |year=1979 |title=Kinematics and Dynamics of Planar Machinery |publisher=[[Prentice Hall]] |isbn=978-0-13-516062-6 |doi= }}</ref>

==Moment of inertia matrix==

The inertia matrix of a rigid system of particles depends on the choice of the reference point.<ref name="Kane">{{citation|first1=T. R. |last1=Kane and |first2=D. A. |last2=Levinson, [https://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460 |title=Dynamics, Theory and Applications],|publisher= McGraw-Hill, NY,New York|year= 2005.}}</ref> There is a useful relationship between the inertia matrix relative to the center of mass '''R''' and the inertia matrix relative to another point '''S'''. This relationship is called the parallel axis theorem.

Consider the inertia matrix [I_S] obtained for a rigid system of particles measured relative to a reference point '''S''', given by

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y^2+z^2 & -xy & -xz \\ -y x & x^2+z^2 & -yz \\ -zx & -zy & x^2+y^2 \end{bmatrix}.</math>

This product can be computed using the matrix formed by the outer product ['''R''' '''R'''^T] using the identifyidentity

:<math> -[R]^2 = |\mathbf{R}|^2[E_3] -[\mathbf{R}\mathbf{R}^T]=

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[[Category:Mechanics]]

[[Category:Physics theorems]]

[[Category:ChristiaanMoment Huygens(physics)]]

[[fr:Moment d'inertie#Théorème de transport (ou théorème d'Huygens ou théorème de -Steiner)]]