Reference · 30 terms

Glossary

Short, precise definitions of the words and symbols used in this module, each linked to the lesson where it is taught. Filter the list as you type, jump to a letter, or look a symbol up in Symbols at a glance.

Showing all 30 terms

A

Angular momentum \(\Hvec\): For a rigid body turning about a fixed point \(O\) or about its center of mass, \(\Hvec = \Imat\wvec\), in kg·m²/s. In plane rotation it reduces to \(H = I\omega\) along the axis; in 3D it is generally not parallel to \(\wvec\).; See: Lesson 6Related: Inertia tensor \(\Imat\), Angular velocity \(\wvec\), Dynamic balance
Angular velocity \(\wvec\): The rate and axis of rotation of a rigid body, in rad/s. Its direction is along the axis of rotation by the right-hand rule. Plane rotation has \(\wvec = \omega\khat\).; See: Lesson 6Related: Angular momentum \(\Hvec\), Kinetic energy \(T\)
Axis of symmetry: A line about which a body is symmetric, such as the axis of a shaft or a cone. Every axis of symmetry is a principal axis at each of its points, and the products of inertia that involve it vanish.; See: Lesson 8Related: Principal axes, Plane of symmetry, Body of revolution

B

Bearing moment: The couple that bearings must apply to keep a body spinning about a fixed axis when \(\Hvec\) is not along the axis. For constant \(\wvec\), \(\sum\Mvec = \wvec \times \Hvec\); it turns with the body and shakes the bearings.; See: Lesson 6Related: Dynamic balance, Angular momentum \(\Hvec\)
Body of revolution: A body generated by rotating a shape about an axis (a wheel, a shaft, a cone). Its axis is principal, and every perpendicular axis through a point on it is principal too, with two equal principal moments.; See: Lesson 8Related: Axis of symmetry, Principal axes

C

Center of mass \(G\): The mass-weighted average position, \(\bar\rvec = \int \rvec\,dm / m\); for composite bodies \(\bar x = \sum m_i \bar x_i / \sum m_i\). The parallel-axis theorem always involves an axis through \(G\).; See: Lesson 4Related: Parallel-axis theorem, Composite body
Composite body: A body treated as a set of simple parts. Its moment of inertia about an axis is the sum of the parts' moments of inertia about that same axis, each found with the parallel-axis theorem. A hole is a part with negative mass.; See: Lesson 4Related: Hole, Parallel-axis theorem, Center of mass \(G\)

D

Dynamic balance: A rotor is dynamically balanced when its shaft is a principal axis through its center of mass: \(G\) on the shaft and \(I_{xz} = I_{yz} = 0\). Then \(\Hvec\) stays along the shaft and the bearings carry no rotating moment.; See: Lesson 8Related: Static balance, Bearing moment, Principal axes

E

Eigenvalue problem: Principal axes satisfy \(\Imat\uvec = I\uvec\). The principal moments are the roots of \(\det(\Imat - I\,\mathbf{1}) = 0\), a cubic \(I^3 - J_1 I^2 + J_2 I - J_3 = 0\); each root gives an axis. Software such as numpy.linalg.eigh solves it directly.; See: Lesson 8Related: Principal axes, Principal moments of inertia, Invariants of the tensor

H

Hole (in a composite body): A removed region, treated as a part with negative mass. Its mass is found from the density of the body it is cut from, and its moment of inertia (with its own parallel-axis term) is subtracted.; See: Lesson 4Related: Composite body

I

Inertia ellipsoid: The surface whose radius along any axis through a point is proportional to \(1/\sqrt{I}\) about that axis. Its three symmetry axes are the principal axes; it is longest along the axis of minimum moment of inertia.; See: Lesson 8Related: Principal axes
Inertia tensor \(\Imat\): The symmetric \(3\times3\) matrix with the moments of inertia on the diagonal and the negatives of the products of inertia off it. It gives \(\Hvec = \Imat\wvec\), \(T = \tfrac12\wvec^\mathsf{T}\Imat\wvec\) and \(I_{Oa} = \uvec^\mathsf{T}\Imat\uvec\).; See: Lesson 6Related: Product of inertia \(I_{xy}\), Angular momentum \(\Hvec\), Principal axes
Intermediate-axis theorem: Free rotation of a rigid body about its axis of maximum or minimum moment of inertia is stable; about the intermediate axis small disturbances grow and the body flips over and over. Energy dissipation also makes minimum-axis spin drift toward the maximum axis.; See: Lesson 8Related: Principal axes, Principal moments of inertia
Invariants of the tensor: Combinations of the tensor that do not change when the axes rotate: the trace \(J_1 = I_{xx} + I_{yy} + I_{zz}\), the sum of the \(2\times2\) principal minors \(J_2\), and the determinant \(J_3\). In the plane, \(I_{x'x'} + I_{y'y'}\) is invariant.; See: Lesson 7Related: Eigenvalue problem, Rotated axes

K

Kinetic energy \(T\) (of rotation): For rotation about a fixed point or about \(G\), \(T = \tfrac12\wvec^\mathsf{T}\Imat\wvec = \tfrac12\wvec\cdot\Hvec\), in joules. For plane rotation about a fixed axis, \(T = \tfrac12 I\omega^2\).; See: Lesson 6Related: Inertia tensor \(\Imat\), Angular velocity \(\wvec\)

M

Mass moment of inertia \(I\): A body's resistance to angular acceleration about an axis: \(I = \int r^2\,dm\), where \(r\) is the perpendicular distance from the axis, in kg·m². It depends on the axis as well as the body. Not to be confused with the second moment of area.; See: Lesson 1, Lesson 2Related: Radius of gyration \(k\), Second moment of area, Parallel-axis theorem
Mohr's circle (for inertia): A graphical form of the plane transformation: the point \((I_{x'x'},\ I_{x'y'})\) moves around a circle of center \(\tfrac12(I_{xx} + I_{yy})\) and radius \(\sqrt{(\tfrac12(I_{xx} - I_{yy}))^2 + I_{xy}^2}\) through \(2\theta\) as the axes turn through \(\theta\). It meets the horizontal axis at the principal moments.; See: Lesson 8Related: Principal moments of inertia, Rotated axes

P

Parallel-axis theorem: \(I = \bar I + md^2\): the moment of inertia about any axis equals that about the parallel axis through \(G\) plus the mass times the square of the distance between them. For products, \(I_{xy} = \bar I_{x'y'} + m\bar x\bar y\); for the whole tensor, \(\Imat_O = \Imat_G + m(|\bar\rvec|^2\mathbf{1} - \bar\rvec\bar\rvec^\mathsf{T})\).; See: Lesson 3, Lesson 5, Lesson 6Related: Center of mass \(G\), Composite body
Particle (point mass): A mass small compared with its distance from the axis, so its own size can be ignored: it contributes \(mr^2\) to the moment of inertia.; See: Lesson 1Related: Mass moment of inertia \(I\)
Perpendicular-axis theorem: For a thin plate in the \(xy\)-plane only, \(I_{zz} = I_{xx} + I_{yy}\). It does not hold for bodies with thickness.; See: Lesson 2Related: Mass moment of inertia \(I\)
Plane of symmetry: A plane that divides a body into mirror-image halves. The two products of inertia that contain the coordinate normal to it vanish (symmetry about the \(xz\)-plane gives \(I_{xy} = I_{yz} = 0\)), and that normal is a principal axis.; See: Lesson 5Related: Product of inertia \(I_{xy}\), Axis of symmetry
Principal axes: Three perpendicular axes through a point about which all the products of inertia vanish, so the inertia tensor is diagonal. Spin about a principal axis gives \(\Hvec\) along \(\wvec\). Every body has them at every point.; See: Lesson 8Related: Principal moments of inertia, Eigenvalue problem, Axis of symmetry
Principal moments of inertia: The moments of inertia \(I_1 \ge I_2 \ge I_3\) about the principal axes. \(I_1\) and \(I_3\) are the largest and smallest moments of inertia about any axis through the point.; See: Lesson 8Related: Principal axes, Mohr's circle
Product of inertia \(I_{xy}\): \(I_{xy} = \int xy\,dm\), \(I_{yz} = \int yz\,dm\), \(I_{zx} = \int zx\,dm\), in kg·m². They measure how the mass is placed relative to pairs of planes, can be positive, negative or zero, and enter the inertia tensor with a minus sign.; See: Lesson 5Related: Inertia tensor \(\Imat\), Plane of symmetry, Parallel-axis theorem

R

Radius of gyration \(k\): The distance at which the whole mass, concentrated as a ring or particle, would have the same moment of inertia: \(k = \sqrt{I/m}\). For parallel axes, \(k^2 = \bar k^2 + d^2\).; See: Lesson 1, Lesson 3Related: Mass moment of inertia \(I\)
Rotated axes (transformation): The tensor in axes turned relative to \(x, y, z\): \(\Imat' = R\,\Imat R^\mathsf{T}\), with the new unit vectors as the rows of \(R\). For a turn \(\theta\) about \(z\), \(I_{x'x'} = \tfrac12(I_{xx} + I_{yy}) + \tfrac12(I_{xx} - I_{yy})\cos 2\theta - I_{xy}\sin 2\theta\).; See: Lesson 7Related: Mohr's circle, Invariants of the tensor

S

Second moment of area: \(\int y^2\,dA\), in m⁴: a property of a cross-section used for beam bending. It is not the mass moment of inertia; for a thin plate of density \(\rho\) and thickness \(t\), \(I_\text{mass} = \rho t\,I_\text{area}\).; See: Lesson 1Related: Mass moment of inertia \(I\)
Standard body: A simple uniform shape (rod, ring, disk, cylinder, tube, sphere, shell, block, plate, cone) whose moments of inertia are tabulated. See the table in Lesson 2 or on the Formula Sheet.; See: Lesson 2Related: Composite body
Static balance: A rotor is statically balanced when its center of mass lies on the shaft, so it stays put at any angle on a horizontal shaft. It is necessary but not sufficient for dynamic balance.; See: Lesson 8Related: Dynamic balance

U

Unit vector \(\uvec\): A vector of length 1 that marks a direction; its components are the direction cosines. The moment of inertia about an axis through \(O\) along \(\uvec\) is \(I_{Oa} = \uvec^\mathsf{T}\Imat_O\uvec\). Unit vectors are bold in this module; \(\ihat, \jhat, \khat\) lie along \(x, y, z\).; See: Lesson 7Related: Inertia tensor \(\Imat\), Rotated axes

Symbols at a glance

Met a symbol in a lesson or in another textbook and not sure what it stands for? Find it here, then follow the link to its entry.

Symbols used in this module
Symbol	Meaning	Entry
\(I,\ \bar I\)	Moment of inertia; about an axis through \(G\)	Mass moment of inertia \(I\)
\(I_{xx},\ I_{yy},\ I_{zz}\)	Moments of inertia about the coordinate axes	Mass moment of inertia \(I\)
\(I_{xy},\ I_{yz},\ I_{zx}\)	Products of inertia	Product of inertia \(I_{xy}\)
\(\Imat,\ \Imat_O,\ \Imat_G\)	Inertia tensor (about \(O\), about \(G\))	Inertia tensor \(\Imat\)
\(I_1,\ I_2,\ I_3\)	Principal moments, largest to smallest	Principal moments of inertia
\(k\)	Radius of gyration	Radius of gyration \(k\)
\(m,\ G,\ \bar x\)	Mass, center of mass and its coordinate	Center of mass \(G\)
\(d\)	Distance between parallel axes	Parallel-axis theorem
\(\wvec,\ \omega\)	Angular velocity and its magnitude	Angular velocity \(\wvec\)
\(\Hvec\)	Angular momentum	Angular momentum \(\Hvec\)
\(T\)	Kinetic energy	Kinetic energy \(T\)
\(\uvec\)	Unit vector along an axis	Unit vector \(\uvec\)
\(\theta_p\)	Angle of a principal axis in the plane	Principal axes
\(J_1,\ J_2,\ J_3\)	Invariants of the tensor	Invariants of the tensor
\(\ihat,\ \jhat,\ \khat\)	Unit vectors along \(x, y, z\)	Unit vector \(\uvec\)

Some books define the product of inertia with the opposite sign, so their inertia tensor has \(+I_{xy}\) off the diagonal. This module follows Hibbeler: \(I_{xy} = \int xy\,dm\), and \(-I_{xy}\) in the tensor.