# How to calculate rotations from displacements?

I'm trying to develop a finite conical shell element with six dof at each node. I want to express the rotations $\phi$, $\theta$ and $\psi$ in terms of the displacements $u$, $v$ and $w$. The displacements are functions of $x$ and $y$. Using the geometric relations and small angle approximation, I have found $\phi\approx \tan(\phi)=\frac{dw}{dv}$, $\theta\approx \tan(\theta)=\frac{dw}{du}$ and $\psi\approx \tan(\psi)=\frac{dv}{du}$. But I'm not yet convinced that this is the way to go.

Do you know any other general approach to express the rotations in terms of the displacements?

Edit: I have attached a picture of the geometry of the cone and its coordinate system. $x$ is the meriodional coordinate along the generator of the cone which is presumed to be zero at the apex of the cone. $y$ is the circumferential coordinate. • "The displacements are functions of $x$ and $y$" ... and $z$, I take it.
– Wasabi
Oct 1, 2016 at 16:55
• The displacements functions are only functions of $x$ and $y$ because they are defined for a shell element. The deformation of the shell can be calculated from the deformation of the middle surface and therefore the displacements are biparametric. Oct 1, 2016 at 18:03
• A "conical" shell element presumably has an axis of symmetry, but you didn't tell us whether that is the x, y, or z axis. Also it is not clear if you are considering only axisymmetric deformations of the shell, or attempting to do something more general. From the geometric relations you wrote down you are really just considering an arbitrary curved shell, not a conical shell? Oct 1, 2016 at 19:48
• @alephzero I have edited my post and attached a picture of the geometry. Oct 1, 2016 at 21:23
• I think you will find that $\psi = \frac{1}{2} \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right)$ (maybe I have the signs flipped). For the other angles look at en.wikipedia.org/wiki/Infinitesimal_strain_theory Mar 22, 2017 at 19:11

There is no way to connect rotations and translations. That is the reason why we need the kinematic differential equation to relate velocity and angular velocity with each other.

If you only apply infinitesimal rotations, then this can be represented by infinitesimal translations. It even turns out that for infinitesimal rotations do commute, which is not true for general rotations.

The rotation matrix $\mathbf{R}$ for an angle $\beta$ around the direction $\mathbf{u}$ is given by the formula:

$$\mathbf{R}(\mathbf{u},\beta)=\cos \beta \mathbf{I}+\sin \beta \tilde{\mathbf{u}}+(1-\cos \beta)\mathbf{u}\mathbf{u}^T.$$

The skew-symmetric matrix is represented by $\tilde{\mathbf{u}}$, this is used to express the cross product as a matrix product.

Now, assume that we apply an infinitesimal rotation with the angle $\beta \rightarrow d \beta.$ Using small angle approximations we get:

$$\mathbf{R}(\mathbf{u},d \beta)=\mathbf{I}+d\beta \tilde{\mathbf{u}}.$$

From this expression, it is easy to see that two rotations indeed do commute (after neglecting higher order terms), hence infinitesimal rotations do behave like vectors, which are commutative, and not like matrices, which in general do not commute.

Thin shells can by modeled using Kirchhoff-Love or Mindlin approaches; the geometrical relations would be different in those cases. You will find exact geometrical relations in the following book, devoted to analysis of shells (including conical) based on the Kirchhoff-Love approach; please see the reference below. The reason why I recommend the following book is that it also contains a CD-ROM with complete source codes for the analysis of shells, including conical shells. In particular ,the example on the webpage stated below (also presented in the book)consists of several shells, including a conical component: http://members.ozemail.com.au/~comecau/quad_shell.htm