Nenad Grahovac

Modelling of the hamstring muscle group by use of fractional derivatives

This paper deals with a viscoelastic model of the hamstring muscle group. The model includes fractional derivatives of stretching force and elongation, as well as restrictions on the coefficients that follow from the Clausius-Duhem inequality. On the basis of a ramp-and-hold type of experiment, four rheological parameters have been calculated by numerical treatment ab initio. Riemann-Liouville fractional derivatives were approximated numerically using the Grunwald-Letnikov definition. Obtained results were verified by use of the Laplace transform method. The stretching force in time domain involves Mittag-Leffler-type function.

ON IMPACT SCRIPTS WITH BOTH FRACTIONAL AND DRY FRICTION TYPE OF DISSIPATION

We study the dynamics of a mass, sliding on a dry surface and impacting against a rigid wall through a viscoelastic body, that we model as a straight rod of negligible mass. The problem comprises a constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and discontinuous inequality constraint conditions imposed by the Coulomb friction model. We show that the dynamics of the problem is governed by a single integro-differential inclusion. By use of the slack variable algorithm the problem was solved numerically. The predictions of the model concerning the duration of the impact, maximal values of the impacting force and deformation, as well as the restitution coefficient are determined for several values of system parameters. Depending on the dry friction coefficient three different impact scripts are identified: rebound after the impact, capture in the approaching phase, and capture in the reb...

Multivalued fractional differential equations as a model for an impact of two bodies

Impact models of two colliding bodies are usually generated by fundamental physical and geometrical principles. Initially, the number of unknown variables is not the same as the number of equations expressing the principles, so the models should be complemented with an appropriate set of constitutive equations which contain enough information on the physical properties of the system and therefore allow accurate predictions of its behavior. Along these lines we here discuss a problem when a block, moving along a line on a dry surface, impinges against another block being at rest, through a deformable straight rod of negligible mass. Among the variety of all possible choices that can be used, we suggest the constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and the Coulomb friction law given in the set-valued form. Owing to the presence of dry friction and the proposed fractional model, known as the fractional Zener model, the problem belongs to the class of set-valued fractional differential equations (or multivalued differential equations of arbitrary real order) leading to the equivalent Cauchy problem given in terms of two coupled integro-differential inclusions, for which the existence result ensuring the contractible solution set exists. By use of the combinatorial analysis of the problem, we identify 11 imaginable scripts and separate 10 feasible ones that were confirmed numerically by a procedure that combines nonlocal and nonsmooth modules.

Application of Fractional Calculus to Frontal Crash Modeling

Fractional derivative viscoelastic model is used in the analysis of a frontal impact of a vehicle against a rigid obstacle. The frontal part of the vehicle is first modeled as a viscoelastic fractional rod and then it is modeled as two different viscoelastic fractional rods with a different length. In the second model also the friction is taken into account. A motion is analyzed during several phases because of both different lengths of the rods and the presence of a dry friction force in the later model. Governing systems of differential equations together with the corresponding initial conditions are derived. Parameter identification is done on the basis of the existing experimental results using the solution of a posed impact problem. What makes the problem more complex, regarding the second model, is the fact that it belongs to the class of nonsmooth fractional order systems, which require special treatment when dealing with deformation history during different motion phases.

Earthquake response of adjacent structures with viscoelastic and friction dampers

We study the seismic response of two adjacent structures connected with a dry friction damper. Each of them consists of a viscoelastic rod and a rigid block, which can slide without friction along the moving base. A simplified earthquake model is used for modeling the horizontal ground motion. Energy dissipation is taken by the presence of the friction damper, which is modeled by the set-valued Coulomb friction law. Deformation of viscoelastic rods during the relative motion of the blocks represents another way of energy dissipation. The constitutive equation of a viscoelastic body is described by the fractional Zener model, which includes fractional derivatives of stress and strain. The problem merges fractional derivatives as non-local operators and theory of set-valued functions as the non-smooth ones. Dynamical behaviour of the problem is governed by a pair of coupled multi-valued differential equations. The posed Cauchy problem is solved by use of the Grünwald–Letnikov numerical scheme. The behaviour of the system is analyzed for different values of system parameters.