Mikael Enelund

Damping described by fading memory—analysis and application to fractional derivative models

Abstract Some damping models where the actual stress does not depend on the actual strain but also on the entire strain history are studied. Basic requirements in the frequency and time domain significant for the choice of damping model are outlined. A one-dimensional linear constitutive viscoelastic equation is considered. Three different equivalent constitutive equations describing the viscoelastic model are presented. The constitutive relation on the convolution integral form is studied in particular. A closed form expression for the memory kernel corresponding to the fractional derivative model of viscoelasticity is given. The memory kernel is examined with respect to its regularity and asymptotic behavior. The memory kernels relation to the fractional derivative operator is discussed in particular and the fractional derivative of the convolution term is derived. The fractional derivative model is also given by two coupled equations using an internal variable. The inclusion of the fractional derivative constitutive equation in the equations of motion for a viscoelastic structure is discussed. We suggest a formulation of the structural equations that involves the convolution integral description of the fractional derivative model of viscoelasticity. This form is shown to possess several mathematical advantages compared to an often used formulation that involves a fractional derivative operator form of constitutive relation. An efficient time discretization algorithm, based on Newmarks method, for solving the structural equations is presented and some numerical examples are given. A simplification of the fractional derivative of the memory kernel, derived in the present study, is then employed, which avoids the actual evaluation of the memory kernel.

Time-Domain Finite Element Analysis of Viscoelastic Structures with Fractional Derivatives Constitutive Relations

Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional calculus operators are presented. To avoid difficulties concerning fractional-order initial conditions, a form of the fractlonal calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Lefler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A simplification of the fractional derivative of the memory kernel is used, in connection with Grunwalds definition of fractional differentiation and a backward Euler rule, for the time evolution of the convolution term. A desirable feature of this process is that no actual evaluation of the memory kernel is needed. This, together with the Newmark method for time integration, enables the direct calculation of the time evolution of the nodal degrees of freedom. To illustrate the ability of the numerical procedure a few numerical examples are presented. In one example the numerically obtained solution is compared with a time series expansion of the analytical solution.

Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws

Abstract A physically sound three-dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented. It is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of Mittag–Leffler function type. A time integration scheme, based on the Generalized Midpoint rule together with the Grunwald algorithm for numerical fractional differentiation, for integration of the constitutive response is developed. The predictive capability of the viscoelastic model for describing creep, relaxation and damped dynamic responses is investigated both analytically and numerically. The algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code Abaqus. The algorithm is then used together with an explicit difference scheme for integration of structural responses. In numerical examples, the quasi-static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated.

Fractional Derivative Viscoelasticity at Large Deformations

A time domain viscoelastic model for large three-dimensional responses underisothermal conditions is presented. Internal variables with fractional orderevolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the timedependency of many polymeric materials can be described using relatively fewparameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does nothold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code forstructural dynamics. The algorithm calculates the fractional derivatives by means of the Grünwald–Lubich approach.Analytical and numerical calculations of the constitutive response in the nonlinearregime are presented and compared. The dynamicstructural response of a viscoelastic bar as well as the quasi-static response of athick walled tube are computed, including both geometrically and materiallynonlinear effects. Moreover, it isshown that by applying relatively small load magnitudes, the responses ofthe linear viscoelastic model are recovered.

Fractional integral formulation of constitutive equations of viscoelasticity

A fractional integral viscoelasticity model is discussed. The model has the same constitutive advantages as the fractional derivative viscoelasticity model. Both models need few parameters to model the weak frequency dependence of many engineering materials. Also, both models represent a causal relation between excitation and response. The fractional integral model, however, gives a unique relation between excitation and response, whereas the fractional derivative model needs initial conditions to give a unique relation. The fractional integral model is incorporated into structural equations of motion. A time-discretization scheme for the solution of these equations is outlined, and some examples are given.

Space-time discretization of an integro-differential equation modeling quasi-static fractional-order viscoelasticity

We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind, with a weakly singular kernel in the time variable, and which also involves partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used at each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, which can be used as the basis for an adaptive strategy.

Pareto optimisation of railway bogie suspension damping to enhance safety and comfort

This paper presents the optimisation of damping characteristics in bogie suspensions using a multi-objective optimisation methodology. The damping is investigated and optimised in terms of the resulting performances of a railway vehicle with respect to safety, comfort and wear considerations. A complete multi-body system model describing the railway vehicle dynamics is implemented in commercial software Gensys and used in the optimisation. In complementary optimisation analyses, a reduced and linearised model describing the bogie system dynamics is also utilised. Pareto fronts with respect to safety, comfort and wear objectives are obtained, showing the trade-off behaviour between the objectives. Such trade-off curves are of importance, especially in the design of damping functional components. The results demonstrate that the developed methodology can successfully be used for multi-objective investigations of a railway vehicle within models of different levels of complexity. By introducing optimised passive damping elements in the bogie suspensions, both safety and comfort are improved. In particular, it is noted that the use of optimised passive damping elements can allow for higher train speeds. Finally, adaptive strategies for switching damping parameters with respect to different ride conditions are outlined and discussed.

Discretization of integro-differential equations modeling dynamic fractional order viscoelasticity

We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integro-differential equation with a weakly singular convolution kernel. We discretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equation of motion. These are then used to prove a priori error estimates. A numerical experiment is included.

Integration of Education for Sustainable Development in the Mechanical Engineering Curriculum

This paper presents and analyses the integration with progression of education for sustainable development in Chalmers University of Technology’s MScEng programme in Mechanical Engineering. The program has an aim and structure that emphasises employability, integration of general engineering skills, authentic engineering experiences with a focus on holistic view of the complete lifecycle of products and systems. The realisation of these aims stress the need of an integrated and adaptable sustainable development education for mechanical engineering. To reach this goal, we applied a combined top-down and bottom-up education development process that started with the formulation of program vision and program level learning outcomes. Faculty meetings and workshop were used to formulate the course learning outcomes and to map the program level outcomes to courses in which the outcomes are satisfi ed followed this. The strategy integrated specifi c sustainability topics in courses where it is appropriate and to have a separate course in sustainable development to ensure that general aspects of sustainable development are included and that a team of faculty takes full responsibility for this. Design-build-test project courses are shown to be suitable arenas for integrating teaching and learning of sustainable development. Results from a student survey on perceptions of the relevance and quality of sustainability education are discussed. Finally, continuing challenges in the area are identified

Integration of a computational mathematics education in the mechanical engineering curriculum

The rapid development of computers and the internet has given new opportunities for engineering work as wells as for teaching and learning. The use of advanced modern mathematics is becoming increasingly more popular in the engineering community and and most problem solutions and developments incorporate high precision digital models, numerical analyses and simulations. However, this kind of mathematics has not been fully implemented into current engineering education programs. Students spend too much time on solving oversimplified problems that can be expressed analytically and with solutions that are already known in advance. Instead, we should be using computers to solve more general, real-world problems. Here we present the integration of a computationally oriented mathematics education into the CDIO-based MSc program in mechanical engineering at Chalmers. We found that the CDIO-approach was beneficial when designing a reformed mathematics education and integrating the mathematics in the curriculum. In the reform of the mathematics education, traditional symbolic mathematics is integrated with numerical calculations and the computer is used as a tool. Furthermore, the computer exercises and homework assignments are taken from applications of mechanical engineering and solutions are analyzed and discussed by means of simulations. The experience is very positive. The students’ interest for computation and simulation has increased. The students consider the the computer to be an important tool for learning and understanding of mathematics. Students spend more time training mathematics and solve more problems.