A universal relationship between spring constant and torsion constant

A universal relationship exists between the torsion constant of a wire and the spring constant of a close-coiled helical spring wound from the same wire. The relationship, which is verified experimentally, involves only the coil radius of the wound spring and is independent of the wire diameter, length, and material.

https://www.sciencedirect.com/science/article/pii/S0022369700002055

Equilibria and instabilities of a Slinky: Discrete model

By considering it as a system of coils that act to resist axial, shearing, and rotational deformations, they develop a two-dimensional discretized model to predict the equilibrium configurations of a Slinky via the minimization of its potential energy. Careful consideration of the contact between coils enables this procedure to accurately describe the shape and stability of the Slinky under different modes of deformation. In addition, they provide simple geometric and material relations that describe a scaling of the general behavior of flexible, helical springs.

https://www.sciencedirect.com/science/article/pii/S0020746214001206

Experimental study of coupled oscillations on a Slinky Wilberforce pendulum

can be used for independent measurements to determine the spring constant
the theory is probably a good starting point for further development

(see also IYPT 2021 task 12 )

https://www.scielo.br/j/rbef/a/vZDtGZQcQGRNjrS7shqpyMg/

Lagrange's method applied to a Slinky

The Lagrange method is applied to two dynamic models of a Slinky, one based on point masses and linear springs and a second where the Slinky is represented as a sequence of half hoops connected by torsion springs. The use of Lagrange's method applied to a Slinky has produced a multi-body dynamic model that can potentially with minor modification reproduce all the interesting behaviour Slinkies are well known for.

https://journals.sagepub.com/doi/full/10.1177/03064190231191251

Oscillations of a suspended slinky

This paper discusses the vertical (!) oscillations of a spring (slinky) under its own weight.

https://iopscience.iop.org/article/10.1088/1361-6404/abcddf/meta

The kinematics and static equilibria of a Slinky

In this paper a model for a Slinky is presented. The Slinky is represented as a sequence of rigid half coils connected by torsional springs. A range of Slinky configurations in static equilibrium are calculated. Where possible the torsion spring model is compared with the much simpler point mass model of a Slinky, and a more sophisticated model that includes axial and shear deformation.

https://www.tandfonline.com/doi/full/10.1080/0020739X.2022.2129498