I have derived the equations of motion for a cylindrical shell using cylindrical coordinates. However, I would like to switch to a specific coordinate system that is commonly used in mechanical engineering literature. This system is defined on the middle surface of the cylindrical shell and is usually denoted by $ξ_1, ξ_2, ξ_3$ (as shown in the image below).
Is there a straightforward way to transform the equations from the cylindrical coordinate system to this $ξ_1,ξ_2,ξ_3$ system? Or do I need to derive all the internal and external forces from scratch to rewrite the equations in the new coordinate system?