The phrase "analytical formulation the" is not correct in written English.
It appears to be a fragment and lacks proper structure, making it unusable without additional context or rephrasing.
Example: "The analytical formulation of the problem was complex and required careful consideration."
Alternatives: "the analytical model" or "the analytical approach".
In the analytical formulation, the problem of a laterally graded half-plane in sliding frictional contact with a rigid punch of an arbitrary profile is considered.
In the analytical formulation, the adherents and the adhesives are all modeled as shear walls elements, using the mixed finite element method.
In the analytical formulation the contacts between the rolling elements and the races are considered as non-linear springs, whose stiffness are obtained by using Hertzian elastic contact deformation theory.
Being of analytical or semi-analytical formulation, the grids may be generated quickly and without the need for advanced software.
Section 3 presents the analytical formulation of the problem and the algorithm of the proposed approach.
The analytical formulation solves the problem based on the elementary theory of longitudinal vibrations and without taking the transversal motions into account.
The propagation parameters of the intra-cavity beam were determined using the ABCD-matrix method, allowing the analytical formulation of the size of two beam waists occurring inside the cavity.
The implementation of the analytical formulation and the numerical procedure are illustrated using three nonlinear problems.
The two-dimensional half-plane problem is formulated using the Fourier integral transform method and the analytical formulation of the contact problem is reduced to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure.
Finally, a wide parametric study have been carried out in order to validate the analytical formulation in the most common configurations showing that such formulation predicts the strength of the E-stub with good accuracy.
