2014年4月20日 · Choose the origin at the apex of the cone, and calculate the elements of the inertia tensor. Then make a transformation such that the center of mass of the cone becomes the origin, and find the principal moments of inertia. Hint: Steiner's parallel-axis theorem might be useful for the second part of the problem. Homework Equations

2020年3月26日 · Transformation of the inertia tensor under a rotation of the coordinate frame and adjoint representation $\mathrm{Ad}$ of a Lie group on its Lie algebra. From the Lie algebra $\so(V,g)$, the inertia tensor can be extended to a left- or right-invariant tensor field on the entire Lie group $\SO(V,g)$, establishing a metric on the configuration space.

2015年10月5日 · The inertia tensor is the object which tells us how angular velocity is converted into kinetic energy or angular momentum and therefore it plays a similar role mass plays in rectilinear motion. To physically understand why this conversion factor is just a number in one case but it is a tensor in the other we just have to note that both ...

2022年6月19日 · $\begingroup$ @peek-a-boo The answer given in that post is way too difficult for me, I just understand that the Inertia tensor is a $(0,2)$ tensor, so a map which takes two vectors and gives a scalar, but can't understand what vectors it takes. $\endgroup$

2019年1月2日 · The inertia tensor of a body will change with rotation. The easiest way to rotate a rod is about its axis, and if I turn the rod on its side the same thing will be true along the new axis. Here is a derivation of the inertia tensor:

2021年5月19日 · Suppose we have computed the inertia tensor of an object about its COM. Suppose the object is then rotated, is there a simple transformation that connects the new inertia tensor to the old one thro...

2019年3月4日 · I am trying to provide colleagues with a spreadsheet method of transforming the inertia properties of a complex shaped body to a different coordinate system, involving only rotation. I've read that this can be achieved by multiplying the inertia tensor by the transform of the matrix of direction cosines and then multiplying the matrix of ...

The moment-of-inertia (MOI) tensor is real (no imaginary terms), symmetric, and positive-definite. Linear algebra tells us that for any (3x3) matrix that has those three properties, there's always a set of three perpendicular axes such that the MOI tensor can be expressed as a diagonal tensor in the basis of those axes.

2022年4月4日 · Looking back at the construction of this problem, I may have been a bit clearer. You can either construct the moment of inertia tensor at time ##t## as $$ I_{ij}(t) = \int \rho(\vec x, t) [\delta_{ij} \vec x^2 - x_i x_j] dV $$ where ##x_i## are the integration variables and ##\rho(\vec x, t)## is the density at position ##\vec x## at time ##t ...

Inertia Tensor - a $3\times3$ matrix, which describes the object "mass" of rotation in relation to a certain point, helping us calculating rotations around any axis in 3D-space. Moment of Inertia - a scalar which describes the object "mass" of rotation in relation to a certain axis.