Parallel axis theorem: Difference between revisions

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{{Short description|Theorem in planar dynamics}}

In [[physics]], the '''parallel axis theorem''' can be used to determine the [[moment of inertia]] of a [[rigid object]] about any axis, given the moment of inertia of the object about the [[Parallel (geometry)|parallel]] axis through the object's [[center of mass]] and the [[perpendicular]] [[distance]] between the axes.

{{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>{{citation | 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==

Let ''I''_''CM'' denote the moment of inertia of the object about the center of mass, ''M'' the object's mass and ''d'' the perpendicular distance between the two axes. Then the moment of inertia about the new axis ''z'' is given by:

[[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 center of mass.]]

Suppose a body of mass {{math|''m''}} is rotated about an axis {{math|''z''}} passing through the body's [[center of mass]]. 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_z I = I_\mathrm{cm} + Mdmd^2.\,</math>

Explicitly, {{math|''d''}} is the perpendicular distance between the axes {{math|''z''}} and {{math|''z&prime;''}}.

This rule can be applied with the [[stretch rule]] and [[perpendicular axes rule]] to find moments of inertia for a variety of shapes.

The parallel axis theorem can be applied with the [[stretch rule]] and [[perpendicular axis theorem]] to find moments of inertia for a variety of shapes.

[[Image:Parallelaxes.jpg|thumb|right|250px|Parallel axes rule for area moment of inertia.]]

[[Image:Parallelaxes-1.png|thumb|right|Parallel axes rule for area moment of inertia]]

The parallel axes rule also applies to the [[second moment of area]] (area moment of inertia);

:<math>I_z = I_x + Ad^2.\,</math>

where ''I_z'' is the area moment of inertia through the parallel axis, ''I_x'' is the area moment of inertia through the center of mass of the [[area]], ''A'' is the surface of the area, and ''d'' is the distance from the new axis ''z'' to the center of gravity of the area.

===Derivation===

The parallel axis theorem is one of few theorems referred to as '''Steiner's theorem''', after [[Jakob Steiner]].

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 at a distance {{math|''D''}} from the center of mass along the ''x''-axis, is

[[Category: Mechanics]]

[[Category: Introductory physics]]

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

[[cs:Steinerova věta]]

[[de:Steinerscher Satz]]

Expanding the brackets yields

[[it:Teorema di Huygens-Steiner]]

[[he:משפט שטיינר]]

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

[[nl:Stelling van Steiner]]

[[pl:Twierdzenie Steinera]]

The first term is {{math|''I''_cm}} and the second term becomes {{math|''MD''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:

[[pt:Teorema de Steiner]]

[[sl:Steinerjev izrek]]

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

[[sv:Steiners sats]]

=== Tensor generalization ===

The parallel axis theorem can be generalized to calculations involving the [[Moment of inertia#Inertia tensor|inertia tensor]].<ref name="Abdulghany">{{citation

| 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 center 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.

The generalized version of the parallel axis theorem can be expressed in the form of [[Component-free treatment of tensors|coordinate-free notation]] as

:<math> \mathbf{J} = \mathbf{I} + m \left[\left(\mathbf{R} \cdot \mathbf{R}\right) \mathbf{E}_{3} - \mathbf{R} \otimes \mathbf{R} \right],</math>

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"/> In this generalization, the inertia tensor can be moved from being reckoned about any reference point <math>\mathbf{R}_{ref}</math> to some final reference point <math>\mathbf{R}_F</math> via the relational matrix <math>M</math> 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>

==Second moment of area==

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

:<math>I_z = I_x + Ar^2,</math>

where {{math|''I_z''}} is the area moment of inertia of ''D'' relative to the parallel axis, {{math|''I_x''}} is the area moment of inertia of ''D'' relative to its [[centroid]], {{math|''A''}} is the area of the plane region ''D'', and {{math|''r''}} is the distance from the new axis {{math|''z''}} to the [[centroid]] of the plane region ''D''. The [[centroid]] of ''D'' coincides with the [[centre of gravity]] of a physical plate with the same shape that has uniform density.

==Polar moment of inertia for planar dynamics==

[[File:Steiners sats.PNG|thumb|right|Polar moment of inertia of a body around a point can be determined from its polar moment of inertia around the center of mass.]]

The mass properties of a rigid body that is constrained to move parallel to a plane are defined by its center of mass '''R'''&nbsp;=&nbsp;(''x'',&nbsp;''y'') in this plane, and its polar moment of inertia ''I''_''R'' around an axis through '''R''' that is perpendicular to the plane. The parallel axis theorem provides a convenient relationship between the moment of inertia I_S around an arbitrary point '''S''' and the moment of inertia I_R about the center of mass&nbsp;'''R'''.

Recall that the center of mass '''R''' has the property

:<math> \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}) \, dV=0, </math>

where '''r''' is integrated over the volume ''V'' of the body. The polar moment of inertia of a body undergoing planar movement can be computed relative to any reference point&nbsp;'''S''',

: <math> I_S = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{S})\cdot (\mathbf{r}-\mathbf{S}) \, dV,</math>

where '''S''' is constant and '''r''' is integrated over the volume&nbsp;''V''.

In order to obtain the moment of inertia ''I''_''S'' in terms of the moment of inertia ''I''_''R'', introduce the vector '''d''' from '''S''' to the center of mass '''R''',

: <math>

\begin{align}

I_S & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}+\mathbf{d})\cdot (\mathbf{r}-\mathbf{R}+\mathbf{d}) \, dV \\

& = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R})\cdot (\mathbf{r}-\mathbf{R})dV + 2\mathbf{d}\cdot\left(\int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}) \, dV\right) + \left(\int_V \rho(\mathbf{r}) \, dV\right)\mathbf{d}\cdot\mathbf{d}.

\end{align}

</math>

The first term is the moment of inertia ''I''_''R'', the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector&nbsp;'''d'''. Thus,

:<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 }}</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|first2=D. A.|last2=Levinson|title=Dynamics, Theory and Applications|publisher= McGraw-Hill, 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

:<math> [I_S] = -\sum_{i=1}^n m_i[r_i-S][r_i-S],</math>

where '''r'''_''i'' defines the position of particle ''P''_''i'', ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n''. Recall that [''r''_''i''&nbsp;&minus;&nbsp;''S''] is the skew-symmetric matrix that performs the cross product,

:<math> [r_i -S]\mathbf{y} = (\mathbf{r}_i - \mathbf{S})\times \mathbf{y},</math>

for an arbitrary vector&nbsp;'''y'''.

Let '''R''' be the center of mass of the rigid system, then

:<math> \mathbf{R} = (\mathbf{R}-\mathbf{S}) + \mathbf{S} = \mathbf{d} + \mathbf{S},</math>

where '''d''' is the vector from the reference point '''S''' to the center of mass '''R'''. Use this equation to compute the inertia matrix,

:<math> [I_S] = -\sum_{i=1}^n m_i[r_i- R + d][r_i - R+ d].</math>

Expand this equation to obtain

: <math> [I_S] = \left(-\sum_{i=1}^n m_i [r_i - R][r_i - R]\right) + \left(-\sum_{i=1}^n m_i[r_i - R]\right)[d] + [d]\left(-\sum_{i=1}^n m_i[r_i - R]\right) + \left(-\sum_{i=1}^n m_i\right)[d][d].</math>

The first term is the inertia matrix [''I''_''R''] relative to the center of mass. The second and third terms are zero by definition of the center of mass '''R''',

:<math> \sum_{i=1}^n m_i(\mathbf{r}_i -\mathbf{R}) = 0.</math>

And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix [''d''] constructed from&nbsp;'''d'''.

The result is the parallel axis theorem,

:<math> [I_S] = [I_R] - M[d]^2,</math>

where '''d''' is the vector from the reference point '''S''' to the center of mass '''R'''.<ref name="Kane"/>

===Identities for a skew-symmetric matrix===

In order to compare formulations of the parallel axis theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful.

Let [''R''] be the skew symmetric matrix associated with the position vector '''R'''&nbsp;=&nbsp;(''x'',&nbsp;''y'',&nbsp;''z''), then the product in the inertia matrix becomes

:<math> -[R][R]= -\begin{bmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{bmatrix}^2 = \begin{bmatrix}

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 identity

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

\begin{bmatrix} x^2+y^2+z^2 & 0 & 0 \\ 0& x^2+y^2+z^2 & 0 \\0& 0& x^2+y^2+z^2 \end{bmatrix}- \begin{bmatrix}x^2 & xy & xz \\ yx & y^2 & yz \\ zx & zy & z^2\end{bmatrix},</math>

where [''E''₃] is the 3&nbsp;×&nbsp;3 identity matrix.

Also notice, that

:<math> |\mathbf{R}|^2 = \mathbf{R}\cdot\mathbf{R} =\operatorname{tr}[\mathbf{R}\mathbf{R}^T],</math>

where tr denotes the sum of the diagonal elements of the outer product matrix, known as its trace.

==See also==

* [[Christiaan Huygens]]

* [[Jakob Steiner]]

* [[Moment of inertia]]

* [[Perpendicular axis theorem]]

* [[Rigid body dynamics]]

* [[Stretch rule]]

==References==

{{reflist}}

==External links==

{{commons category|Steiner's parallel axis theorem}}

*[http://scienceworld.wolfram.com/physics/ParallelAxisTheorem.html Parallel axis theorem]

*[http://mathworld.wolfram.com/MomentofInertia.html Moment of inertia tensor]

*[https://www.youtube.com/watch?v=mFVz7iCc45I Video about the inertia tensor]

[[Category:Mechanics]]

[[Category:Physics theorems]]

[[Category:Moment (physics)]]

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