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Areal moments of inertia revisited: on the distinction between the principal directions

Three commonly used methods to determine the principal moments of inertia of a plane area and their directions are based on: (i) the stationarity condition for the axial moment of inertia, (ii) the eigenvalue analysis, and (iii) Mohr’s circle. In this paper we provide two new derivations, which are based on: (a) the matrix diagonalization and the invariant tensor properties, and (b) the conjugacy property of the moment of inertia vectors. A new general expression is derived which speciﬁes the principal directions of inertia, as well as the directions of the maximum and minimum product of inertia. A comparative study of the ﬁve presented approaches is given, which is of interest from both conceptual and methodological points of view. The connection between the deviatoric part of the moment of inertia tensor and Land’s circle of inertia is also given. The presented analysis applies to any two-by-two symmetric second order tensor.


Introduction
The determination of the principal moments of inertia of a plane area and their principal directions is of great importance in the mechanics of rigid and deformable bodies, as evidenced by its detailed coverage in every undergraduate textbook on these subjects. The three commonly used methods to accomplish this are based on: (i) the condition of stationary axial moment of inertia about an arbitrary axis, (ii) the eigenvalue analysis of the moment of inertia tensor, and (iii) Mohr's circle. The objective of this paper is to present two additional derivations, which have not been reported in the literature before. The first is based on the matrix diagonalization of the moment of inertia tensor and its invariant tensor properties, and the second on the conjugacy property of the moment of inertia vectors. These methods also apply to an analogous determination of the principal stresses and their directions in two dimensional continuum mechanics problems (or any other two-by-two symmetric second order tensor), with the main difference that the moment of inertia tensor is a positive definite tensor, while the stress tensor is not necessarily so.
In Sect. 2 we diagonalize the moment of inertia tensor and use its invariant tensor properties to derive the expressions for the principal moments of inertia, and the expressions for the tangens of the angle that specifies each principal direction separately. The well-known derivation based on the eigenvalue analysis is included for comparison. In Sect. 3 we introduce the conjugacy property of the moment of inertia vectors, previously not explored in the literature in the context of moments of inertia, but well-known in the context of stress analysis (conjugacy of traction vectors), although not used there to determine the principal stresses and their directions either. We derive the general expression which yields the principal directions of inertia, as well as the directions of the maximum and minimum product of inertia. The geometric interpretation of all analytical results is given based on Mohr's circle of inertia. The moment of inertia tensor is decomposed into its isotropic and deviatoric part in Sect. 4. Since the isotropic part does not have preferred directions, the moment of inertia tensor and its deviatoric part share the same principal directions. As a result, a simpler construction of Mohr's circle for the deviatoric part can be used to determine the principal directions. The connection to Land's circle of inertia is then given. The results are summarized and discussed in the concluding Sect. 5.

Diagonalization of the moment of inertia tensor
Consider a fixed coordinate system (x, y) and a coordinate system (u, v) obtained from (x, y) by the counterclockwise rotation by an angle ϕ. The two sets of coordinates are related by the matrix transformation The matrix Q is the rotation matrix that carries the unit vectors e x and e y , along the coordinate directions (x, y), into the unit vectors e u = Qe x and e v = Qe y , along the coordinate directions (u, v). In (1), and in the sequel, the abbreviations c = cos ϕ and s = sin ϕ are used. The tensor of the second moment of inertia of an area A is defined (e.g., [1, p. 417]) as the symmetric tensor whose rectangular components are where x 1 = x and x 2 = y, I 0 = A (x 2 1 + x 2 2 ) d A is the polar moment of inertia, and δ i j is the Kronecker delta (δ i j = 1 if i = j, and zero otherwise). The matrix associated with (2) is with the elements It can be easily verified that, under the rotation of the coordinate system (1), the tensor components in (3) transform according to such that where I x x = I x x , I yy = I yy , and I xy = −I xy do not constitute the components of the second-order tensor, because, under the rotation of the coordinate system (1), they do not transform according to the required tensor transformation rule Q T IQ, but according to QIQ T . The component I xy is commonly referred to as the product of inertia (sometimes also mixed or deviatoric moment of inertia). The subsequent derivation is based entirely on the tensor definition of the moment of inertia, so that there is a complete analogy of the resulting formulas with those for the stress tensor. If I xy is used, rather than I xy , the analogy holds if the correspondence σ x x ↔ I x x , σ yy ↔ I yy , and σ xy ↔ −I xy is made (e.g., [2,3], although this is sometimes overlooked; [4, p. 436]).

Principal moments of inertia
Theorem The maximum and minimum moments of inertia are the diagonal entries of the diagonalized form of the moment of inertia tensor.
Proof The tensor II is observed as a diagonal matrix, with positive 1 diagonal entries I 1 and I 2 , in the coordinate system (1, 2) obtained from the (x, y) coordinate system by the rotation ϕ, such that Since the trace and the determinant of the matrix remain preserved under orthogonal transformations, we can write Eliminating I 2 from these two equations one obtains which means that the principal direction I 1 is either greater or smaller than both I x x and I yy . Thus, one of the diagonal entries on the right-hand side of (7) is the maximum moment of inertia and the other is the minimum. This completes the proof. We label I 1 to be the maximum, and I 2 to be the minimum. The two invariants, the trace and the determinant, appearing in (8) and (9), are known as the principal invariants of II. An invariant (dependent on the previous two) is also the Frobenious norm of II, defined as the sum of the squares of its elements, i.e., By combining (11) with (9), i.e., by subtracting 2I 1 I 2 from I 2 1 + I 2 2 , there follows Since we labeled the principal directions so that I 1 > I 2 , (12) gives The principal moments of inertia follow by adding and subtracting (8) and (13),

Principal directions
The angles that define the principal directions follow from the expanded form of (7), which is By using the transformations the first of (15) can be rewritten as a quadratic equation for t, This has the unique solution because the discriminant of the quadratic equation (17) is equal to zero; see (10). Likewise, the second of (15) can be rewritten as which has the unique solution because Finally, the third of (15), requiring that the product of inertia for the principal directions vanishes, gives which is satisfied by both ϕ 1 and ϕ 2 . The expressions (18) and (20) are more appealing than (22), because they unambiguously specify the two principal directions separately, while (22) does not make distinction between the two directions.

Octant theorem
Expression (22) is a common outcome of the analysis based on the stationarity condition dI uu /dϕ = 0. Since dI uu /dϕ = 2I uv (which means that, if the area has its maximum or minimum moment of inertia about an axis, the product of inertia for that axis and the axis orthogonal to it vanishes), there follows (22). The analysis of the sign of the corresponding second derivative d 2 I uu /dϕ 2 is more lengthy and tedious, but can be summarized as the following (octant) theorem. 2 Theorem The principal direction of the maximum moment of inertia passes through: To make the distinction between the principal directions, it actually suffices to remember a weaker (quadrant rather than octant) form of the theorem: The direction of the maximum moment of inertia passes through the 1 st and 3 rd quadrant if I xy > 0, and through the 2 nd and 4 th quadrant if I xy < 0. Another weaker form of the theorem is also helpful: If I x x > I yy , the direction of the maximum moment of inertia is closer to the x-axis than the y-axis (and vice versa) [6, p. 555].
Alternatively, to identify the angles ϕ 1 and ϕ 2 from (22), we can check which of the corresponding values of I uu is greater and which one is smaller [7, p. 429]. Or, we can use (22) and the trigonometric relations between the sine, cosine, and tangens, to obtain The angle ϕ 1 can then be determined as the angle which satisfies both expressions in (23) [8, p. 550]. To circumvent this double calculation, the use of the octant theorem, or the analytical determination of the angles ϕ 1 and ϕ 2 based on (18) and (20), is clearly more convenient.

Maximum product of inertia
Theorem Two orthogonal directions for which an area has equal axial moments of inertia are the directions for which that area has the maximum or minimum product of inertia.
Proof The moment of inertia tensor is observed in the coordinate systems (u, v) and (1, 2) as The determinant I uu I vv − I 2 uv remains unchanged under the rotation of the coordinate system. Thus, if I 2 uv is at its maximum in some coordinate system, the product I uu I vv must be at its maximum in the same coordinate system, too. Since the trace remains invariant under the rotation of the coordinate system, I uu + I vv = I 0 , we search for the maximum of the function g = I uu I vv = I uu (I 0 − I uu ). The extremum condition for this is which gives I uu = I 0 /2. Thus, in the coordinate system in which I uu I vv , and thus I 2 uv , is at its maximum, the two diagonal entries of the moment of inertia tensor are equal to each other (I uu = I vv = I 0 /2). This completes the proof. 3 Denoting the coordinate system for which the product of inertia is maximum by (ū,v), we have 3 An alternative simple proof is based on the identity so that the stationarity condition dI uv /dϕ = 0 gives I uu = (I xx + I yy )/2.
The corresponding, maximum value of I 2 uv follows from the invariance of the determinant of the moment of inertia tensor, which gives because I 1 I 2 = I 2 0 /4 − I 2 , by (14). To find the directions (ū,v) for which Iūv = I, we equate the spectral representations of II in the (ū,v) and (1, 2) coordinate systems. Upon the cancelation of the isotropic parts on the two sides of the equation, 4 it follows This implies that eū = (e 1 − e 2 )/2 and ev = (e 1 + e 2 )/2, i.e., the (ū,v) axes are obtained from the principal axes (1, 2) by the 45 • clockwise rotation. An alternative derivation based on the stationarity condition of I uv is well-known; e.g. [9, p. 536]. To find the coordinate axes (u, v) for which the product of inertia I uv has its maximum or minimum value, we require that which gives By comparing (22) and (30), tan 2ϕ tan 2φ = −1, so that ϕ 1 andφ 1 and 45 • apart.

Eigenvalue analysis
For the sake of comparison, we summarize in this section the well-known derivation of the principal moments of inertia and their directions, based on the eigenvalue analysis of the moment of inertia tensor II. The moment of inertia about an axis along the unit vector n is I n = n T IIn. The principal direction of inertia is the direction that maximizes or minimizes I n , over all directions n subject to the constraint n T n = 1. Thus, introducing the Lagrange multiplier λ and the function = I n − λ n T n, the condition ∂ /∂n = 0 for the unconstrained extremum of gives Evidently, if (31) is premultiplied by n T , it follows that λ = n T IIn is the maximum or minimum moment of inertia. The corresponding n, satisfying (31), is the principal direction. The scalar λ and the vector n are referred to as the eigenvalue and eigenvector of II. By taking n T = [c s], which automatically satisfies the constraint condition n T n = 1, the expanded form of the eigenvalue problem (31) is In order that this homogeneous system of linear algebraic equations for c and s has a nontrivial solution, the determinant of the system must vanish. Upon expansion, this gives a quadratic (secular) equation for the 4 The isotropic parts are eigenvalues, whose solutions are the principal moments of inertia, given by (14). The corresponding principal directions follow from (32). For λ = I 1 , (32) yields while for λ = I 2 , in agreement with the results from Sect. 2.2, and in analogy with the eigenvalue analysis of the plane state of stress [10,11].
To make the connection with the analysis based on the diagonalization of the moment of inertia tensor, we recall from linear algebra [12,13] that, when diagonalizing a matrix II, the column vectors of the modal (diagonalizing) matrix are eigenvectors of II, and the diagonal elements of the diagonalized form are corresponding eigenvalues of II. Thus, the modal matrix in (7) is where c 1 = cos ϕ 1 , s 1 = sin ϕ 1 , and c 2 = −s 1 , s 2 = c 1 specify the eigenvectors of II.

Moment of inertia vectors and their conjugacy property
The moment of inertia vector for the direction u is defined as Its representation in the (u, v) coordinate system is The conjugacy property of the inertia vectors p n and p m is for any unit direction vectors m and n, which holds because of the symmetry of II. This conjugacy property, not explicitly stated or explored in the literature in the context of the moments of inertia, is well-known in the stress analysis, where it applies to traction vectors over the surface elements with the unit normals m and n; [14, p. 79]. If (38) is applied to unit vectors e u and e y , there follows The unit vector e u , expressed in the (x, y) coordinate system, is e u = ce x + se y , while the unit vector e y , expressed in the (u, v) coordinate system, is e y = se u + ce v . Thus, by evaluating the left-hand side of (39) in the (u, v) coordinate system, and the right-hand side in the (x, y) coordinate system, carrying in mind that the representation of p y = IIe y in the (x, y) coordinate system is p T y = [I yx I yy ], there follows sI uu + cI uv = cI yx + sI yy .
This gives a remarkable expression, not reported in the literature before, Its validity can be verified directly by using the expressions for I uu and I uv from the transformation formulas (6), while its geometric interpretation by means of Mohr's circle will be given in Sect. 3.3.
Similarly, if (38) is applied to unit vectors e u and e x , there follows Since e x = ce u − se v , and since the representation of p x = IIe x in the (x, y) coordinate system is p T x = [I x x I xy ], by evaluating the left-hand side of (42) in the (u, v) coordinate system, and the right-hand side in the (x, y) coordinate system, there follows cI uu − sI uv = cI x x + sI xy .
This gives an alternative or dual expression to (41), which is

Principal directions
The expression (41) delivers the angle ϕ 1 by taking I uv = 0 and I uu = I 1 , and the angle ϕ 2 by taking I uv = 0 and I uu = I 2 . The results are in agreement with (18). The expression (44) delivers the angle ϕ 1 by taking I uv = 0 and I uu = I 1 , and the angle ϕ 2 by taking I uv = 0 and I uu = I 2 , such that in agreement with (20). Both (45) and (46) can also be obtained directly from the conjugacy property (38), by applying it to the direction pairs (1, x) and (2, x), respectively, i.e., from e T 1 IIe x = e T x IIe 1 and e T 2 IIe x = e T x IIe 2 (the left-hand sides being evaluated in the (1, 2) coordinate system, and the right-hand sides in the (x, y) coordinates system, or vice versa).

Principal moments of inertia
Once (45) or (46) are established, the corresponding principal moments of inertia follow from either I 1 = e T 1 IIe 1 or I 2 = e T 2 IIe 2 . For example, since e T 1 = [c 1 s 1 ], we have The substitution of in conjunction with t 1 = (I 1 − I x x )/I xy , yields This is a quadratic equation for I 1 , whose greater root is the maximum, and smaller root is the minimum moment of inertia, given by (14). The same results follow from the analysis of the expression I 2 = e T 2 IIe 2 and t 2 = (I 2 − I x x )/I xy .

Mohr's circle interpretation
The derived results can be given a geometrical interpretation by means of Mohr's circle By geometric properties of the circle, and the triangles within the circle shown in Fig. 1a, it readily follows that tan ϕ 1,2 = I xy confirming the results (45) and (46). In addition, and Figure 1b illustrates the use of the pole P to identify the actual orientation of the principal directions e 1 and e 2 (in the physical space), and the moments of inertia corresponding to orthogonal directions along the unit vectors e u and e v [15, p. 227]. The unit vector e u is at an angle ϕ with respect to the horizontal direction e x , and e v is 90 • counterclockwise from e u . The pole P is the point on Mohr's circle obtained from the point A(I x x , I xy ) as its mirror image across the vertical diameter of the circle, so that P(I yy , I xy ). The tangens of the angle ϕ, defining the point B(I uu , I uv ) and the direction e u , is confirming the result (41). The expression (54) delivers both ϕ 1 and ϕ 2 , by taking I uv = 0, and I uu equal to I 1 in the first case (for ϕ 1 ), and I uu equal to I 2 in the second case (for ϕ 2 ), so that tan ϕ 1,2 = I xy It also delivers the analytical expressions for the anglesφ 1 andφ 2 , corresponding to maximum and minimum values of I uv . These are obtained by substituting in (54) I uu = (I x x + I yy )/2 and I uv = ±I. The results are with the connectionφ 1,2 = ϕ 1 ∓ 45 • . The magnitude of the maximum product of inertia is I max uv = J 1 , and it is for the pair of orthogonal axes at 45 • relative to the principal axes of inertia (point D 1 in Fig. 2 The presented analysis of the deviatoric part of the moment of inertia can be related to Land's circle, which is another graphical method, albeit less frequently utilized than Mohr's circle [4,16]. This is shown in Fig. 3. The center of Land's circle is at the point C on the y-axis, and its diameter is I 0 = I x x + I yy . The point A has the coordinates (J x x , J xy ) relative to the center, as indicated. By extending the line AC (of length J) to intersect the outer (Land's) circle, the points A 1 and A 2 are obtained, such that A A 1 = I 1 = I 0 /2 + J and A A 2 = I 2 = I 0 /2 − J. The direction O A 1 specifies the principal direction 1, at the angle ϕ 1 relative to the horizontal x-axis, while O A 2 specifies the principal direction 2.

Conclusions
We have presented two new derivations of the expressions for the principal moments of inertia and their directions, which also applies to any two-by-two symmetric second order tensor. The first is based on the diagonalization of the moment of inertia tensor and its invariant tensor properties, and the second on the conjugacy property of the moment of inertia vectors. Both derivations deliver separate expressions for the tangens of the angle that identifies the direction of the maximum and minimum moment of inertia, thus circumventing the ambiguity associated with the use of tan 2ϕ expression, which is a common outcome of the analysis based on the stationarity condition for the axial moment of inertia. New appealing expressions (41) and (44) are derived, which are capable of specifying the principle directions of inertia, as well as the directions of the maximum and minimum product of inertia. All analytical results are given geometric interpretations based on Mohr's circle.
Of the five presented derivations, the simplest is probably the derivation based on Mohr's circle, which delivers all relevant information by simple geometric considerations. The classical derivation based on the stationarity condition for the axial moment of inertia is most direct and conceptually least involved, but the analysis of the sign of its second derivative, which governs the distinction between the directions of the maximum and minimum moment of inertia, is lengthy and tedious. The derivation based on the diagonalization of the moment of inertia tensor is mathematically elegant and effective, but it requires a familiarity with the matrix algebra and the invariant tensor properties. Similar comments apply to derivations based on the eigenvalue analysis, and the conjugacy property of the moment of inertia vectors. Only the last derivation yields the general expressions (41) and (44).
We believe that the presented analysis sheds some new light on this classical topic, from both conceptual and pedagogical points of view, which may be of interest to the applied mechanics community, in particular to mechanics instructors and their students, and to other interested readers.