Mikael Enelund - Academia.edu (original) (raw)

Papers by Mikael Enelund

16th International CDIO Conference hosted online by Chalmers University of Technology, Gothenburg, Sweden, 9-11 June 2020, Jun 10, 2020

Change and individualization are two aspects that are important in innovative higher education. I... more Change and individualization are two aspects that are important in innovative higher education. In this paper, we argue for how the concept of curriculum agility can be used as a framework for engineering education that is able to meet societal, environmental, and technological challenges. In order to both anticipate and meet the needs of the rapidly changing world, engineering education needs to have an organization that allows for innovation, change and adaptation, with the capacity to respond within a (much) shorter timeframe than traditionally seen in higher education. The structure and processes of such organizations should include the time needed to establish and decommission new educational programmes, and the flexibility within the programmes. The CDIO's Curriculum Agility Working Group has defined seven principles for curriculum agility and has analysed how these relate to the CDIO Standards. This paper describes how the principles can provide guidance on both a curricular and institutional level. The principles are mapped against the CDIO Standards, relating to what is required for an agile curriculum, in order to indicate how the Standards can be utilized to assess the flexibility and agility of educational programmes.

The paper reports on a study of student satisfaction in CDIO project courses. The aims are to inv... more The paper reports on a study of student satisfaction in CDIO project courses. The aims are to investigate if there are statistically significant differences in levels and variation of student satisfaction metrics between CDIO project courses and “traditional” courses, and to identify possible causes for these differences. The study was carried out at Chalmers University of Technology and focused on courses in its mechanical, automation and industrial design engineering programs. In these programs, about 20 CDIO project courses and 235 traditional courses are offered each year. In the study, student satisfaction and some other quantified metrics collected from Chalmers’ course evaluation system are compared for the two groups of courses. Further, the paper examines in more detail selected CDIO project courses, with high and low student satisfaction ratings. The results of the study provide support for the hypothesis that there are significant differences in ratings. A number of cause...

Degree projects (DP) are currently intensively focused in Sweden: The future national model for e... more Degree projects (DP) are currently intensively focused in Sweden: The future national model for evaluation of higher education will place a major emphasis on the quality of degree projects as an indicator of the quality of the entire education, and their quality will influence the funding of a university. Moreover, DP:s are actively used in program development as a vehicle to develop not only in-depth subject matter knowledge but also professional skills such as planning and communication. Simultaneously, Constructive alignment (CA) is being widely applied as a general approach for improving educational quality. Potentially, CA might also contribute to improving the quality of degree projects. In this paper, we examine how CA can be applied to degree projects. We conclude that CA is indeed applicable to degree projects in the sense that intended learning outcomes as well as teaching and assessment activities can be identified and aligned. But objectives, activities and assessment ar...

In order to design competitive products that meet today’s challenges, future engineers need knowl... more In order to design competitive products that meet today’s challenges, future engineers need knowledge and experience in applying and integrating optimization theory in the engineering design process. The Chalmers course Engineering Design and Optimization addresses this need and was developed and first offered in the fall semester of 2013. This paper provides a detailed account of the course development experiences, teaching methods and course evaluations using CDIO Standards. It also includes a discussion of the learning objectives, required resources, instructional processes and student assessments. The course is analyzed to see to what extent the course aim is satisfied and highlight areas of improvement. Three projects are carried out in groups of two students. The projects are a cantilever design-build-test exercise, a redesign project that involves material selection, and a multi- disciplinary design project of an engine component using multi-physics software. The assignments ...

The rapid development of computers and the internet has given new opportunities for engineering w... more 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 mathemati...

This paper describes the early stages of the developments of Tracks, an initiative to create, imp... more This paper describes the early stages of the developments of Tracks, an initiative to create, implement and evaluate a new educational model where the structure of the education is developed to give students the opportunity to create multi- and interdisciplinary competencies, meet their expectations and need for a more individualized study plan and shorten the lead times for changing the education to embrace new technologies. The new education model is based on the creation of tracks with different themes lying between existing programs not belonging to a specific department or school. The idea is to create individual and flexible study opportunities by introducing Track-courses within the themes. These courses address specific challenges that may be broad societal and profound research-driven. Tracks also include large investments in Chalmers learning environment. The paper focuses on Tracks as a large change initiative, strategies to manage the complexity of this change as well as...

IFAC Proceedings Volumes, 2006

Here an integro-differential formulation for time domain poroelasticity is derived and included i... more Here an integro-differential formulation for time domain poroelasticity is derived and included into a finite element framework for structural dynamics. Further, an algorithm for the time integration of the spatially discretized finite element equations is developed. The Biot poroelastic theory (formulated in the frequency domain) is taken as a starting point. In this theory, the dissipative force is proportional to the relative motion between the solid phase and the viscous fluid in the pores with a constant of proportionality, drag force coefficient, that is frequency dependent. The dissipative force is described in the time domain by a convolution term with a kernel that is the Fourier inverse of the drag force coefficient. As the exact analytical inversion of the drag force coefficient seems highly intractable, we have combined an asymptotic expansion for high frequencies with a low frequency correction term, yielding an analytically invertible drag force coefficient. The convolution with the fourier inverse of the asymptotic expansion, can be identified as a semi-derivative.

Vehicle System Dynamics, 2012

International Journal of Solids and Structures, 1999

... be justified by the fact that D N − ρ can be taken as the analytical continuation of ... In t... more ... be justified by the fact that D N − ρ can be taken as the analytical continuation of ... In the time domain the presence of integro-differential operators makes the solution of the structural ... how to choose the branch of the root in the Fourier transform of the fractional derivative operator. ...

International Journal of Solids and Structures, 1999

... and the corresponding finite element equations are outlined, including the treatment of bound... more ... and the corresponding finite element equations are outlined, including the treatment of boundaryconditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. ...

37th Structure, Structural Dynamics and Materials Conference, 1996

Numerical procedures for the time integration of the spatially discretized finite element equatio... more Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional derivative operators are presented. To avoid difficulties concerning fractional order initial conditions, a form of the fractional calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Leffler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A certain property of the memory kernel is used, in connection with Griinwald's 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's 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.

International Journal of Solids and Structures, 1999

... (15). It is then assumed that q k(β) have homogeneous initial condition. Hence, the stronger ... more ... (15). It is then assumed that q k(β) have homogeneous initial condition. Hence, the stronger CDI expressing the condition of nonnegative dissipative power is a sufficient condition for DI, while the inverse does not hold. 2.3. Fractional order rate laws. ...

Lecture Notes in Computer Science, 2006

We study a dynamic model for viscoelastic materials based on a constitutive equation of fractiona... more We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integrodifferential 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.

19th AIAA Applied Aerodynamics Conference, 2001

A time domain viscoelastic model for large three dimensional responses under isothermal condition... more A time domain viscoelastic model for large three dimensional responses under isothermal conditions is presented. Internal variables with fractional orders evolution equations are used to model the time dependent part of the response. By use fractional orders rate laws, the characteristics of the time dependency of many polymeric materials can be described using relatively few parameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does not hold (e.g, suspensions, vibration isolators and flexible joints). A numerical algorithm for the constitutive response is developed and implemented into a finite element code for structural dynamics. The algorithm calculates the fractional derivatives by means of the Griinwald-Lubich approach. Analytical and numerical comparisons of the constitutive response in the nonlinear regime are presented. The dynamic structural response of a viscoelastic bar is computed, including both geometrically and materially nonlinear effects. Moreover it is shown that by applying relatively small load magnitudes, the responses of the linear viscoelastic model are recovered.

36th Structures, Structural Dynamics and Materials Conference, 1995

Mechanics of Time-Dependent Materials, 2005

Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers... more Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers. Time domain responses as stress relaxation and creep as well as frequency domain responses are well represented. The drawback of fractional order models is that the fractional order operators are difficult to handle numerically. This is in particular true for fractional derivative operators. Here we propose a formulation based on internal variables of stress kind. The corresponding rate equations then involves a fractional integral which means that they can be identified as Volterra integral equations of the second kind. The kernel of a fractional integral is integrable and positive definite. By using this, we show that a unique solution exists to the rate equation. A motivation for using fractional operators in viscoelasticity is that a whole spectrum of damping mechanisms can be included in a single internal variable. This is further motivated here. By a suitable choice of material parameters for the classical viscoelastic model, we observe both numerically and analytically that the classical model with a large number of internal variables (each representing a specific damping mechanism) converges to the fractional order model with a single internal variable. Finally, we show that the fractional order viscoelastic model satisfies the Clausius-Duhem inequality (CDI).

An infinitely long, cylindrically bent harmonicdly vibrating flexible Kirchhoff plate backed by a... more An infinitely long, cylindrically bent harmonicdly vibrating flexible Kirchhoff plate backed by a thin viscous fluid layer is studied analvticallv. The eovernine equations for the dvnam_ ” -ically coupled plate-fluid layer system are solved by using a spatial Fourier transformation. The transverse plate velocity is chosen as the primary unknown. Introducing of certain approximations, which are justified for practical structures and is consistent with the solution technique used, leads to a polynomial dispersion relation from which the wave-numbers of the system are determined. The total solution is then shown to be built-up of three types of coupled waves. The frequency variation of the corresponding wavenumbers and the contributions from the individual waxes to the total solution are examined. The level of damping is quantified through a loss factor which can be used in approximately accounting for energy losses provided by the fluid without modelling it. Two different loss factors are proposed, the first one being defined from wavenumbers of the system and the second one defined from energies of the system under a certain harmonic load. Numerical examples show that the amount of damping is governed mainly by the fluid layer and the frequency of the applied load. N O M E N C L A T U R E z speed of sound in fluid [m/s] flexural plate stiffness [N/m] d fluid layer thickness [m] E Young’s modulus [N/m*] E(w) complex modulus [N/m’] ES maximum stored energy [Nm] EL dissipated energy [Nm] FO line-force magnitude [N/m] f1(a), fz(a) functions [Ns/m3] h plate thickness [m] WI imaginary part of 1 imaginary unit i* = -1 kb free bending wavenumber [l/m] kr bending wavenumber [l/m] h, kz wavenumbers for 0uid [l/m] ?J4% 2) pressure amplitude [N/m’] WI real part of h..(G 21, ula(z, z) fluid velocity amplitudes [m/s] u=(x) plate velocity amplitude [m/s] a wavenumber space coordinate [l/m] ai wavenumbers of coupled system [l/m] 6 acoustic boundary layer thickness [m] q(w) l o s s factor Xb free bending wavelength [m] P Poison’s ratio ” kinematic viscosity [ml/s] PO fluid density [kg/m31 w angular frequency [rad/s]

Alle rechten voorbehouden. Niets uit deze uitgave mag worden vemenigvuldigd en/of openbaar gemaak... more Alle rechten voorbehouden. Niets uit deze uitgave mag worden vemenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher.

A new computationally oriented mathematics education is presented. The education combines traditi... more A new computationally oriented mathematics education is presented. The education combines traditional symbolic mathematics with computational mathematics and programming in the Matlab environment. Engineering applications are explored in computational exercises that are taught jointly with the courses in mechanics and thermodynamics. COURSES We list the mathematics courses in the first year of the Mechanical Engineering program together with the accompanying engineering courses. The year is divided into four periods (quarters of eight weeks).

16th International CDIO Conference hosted online by Chalmers University of Technology, Gothenburg, Sweden, 9-11 June 2020, Jun 10, 2020

Change and individualization are two aspects that are important in innovative higher education. I... more Change and individualization are two aspects that are important in innovative higher education. In this paper, we argue for how the concept of curriculum agility can be used as a framework for engineering education that is able to meet societal, environmental, and technological challenges. In order to both anticipate and meet the needs of the rapidly changing world, engineering education needs to have an organization that allows for innovation, change and adaptation, with the capacity to respond within a (much) shorter timeframe than traditionally seen in higher education. The structure and processes of such organizations should include the time needed to establish and decommission new educational programmes, and the flexibility within the programmes. The CDIO's Curriculum Agility Working Group has defined seven principles for curriculum agility and has analysed how these relate to the CDIO Standards. This paper describes how the principles can provide guidance on both a curricular and institutional level. The principles are mapped against the CDIO Standards, relating to what is required for an agile curriculum, in order to indicate how the Standards can be utilized to assess the flexibility and agility of educational programmes.

The paper reports on a study of student satisfaction in CDIO project courses. The aims are to inv... more The paper reports on a study of student satisfaction in CDIO project courses. The aims are to investigate if there are statistically significant differences in levels and variation of student satisfaction metrics between CDIO project courses and “traditional” courses, and to identify possible causes for these differences. The study was carried out at Chalmers University of Technology and focused on courses in its mechanical, automation and industrial design engineering programs. In these programs, about 20 CDIO project courses and 235 traditional courses are offered each year. In the study, student satisfaction and some other quantified metrics collected from Chalmers’ course evaluation system are compared for the two groups of courses. Further, the paper examines in more detail selected CDIO project courses, with high and low student satisfaction ratings. The results of the study provide support for the hypothesis that there are significant differences in ratings. A number of cause...

Degree projects (DP) are currently intensively focused in Sweden: The future national model for e... more Degree projects (DP) are currently intensively focused in Sweden: The future national model for evaluation of higher education will place a major emphasis on the quality of degree projects as an indicator of the quality of the entire education, and their quality will influence the funding of a university. Moreover, DP:s are actively used in program development as a vehicle to develop not only in-depth subject matter knowledge but also professional skills such as planning and communication. Simultaneously, Constructive alignment (CA) is being widely applied as a general approach for improving educational quality. Potentially, CA might also contribute to improving the quality of degree projects. In this paper, we examine how CA can be applied to degree projects. We conclude that CA is indeed applicable to degree projects in the sense that intended learning outcomes as well as teaching and assessment activities can be identified and aligned. But objectives, activities and assessment ar...

In order to design competitive products that meet today’s challenges, future engineers need knowl... more In order to design competitive products that meet today’s challenges, future engineers need knowledge and experience in applying and integrating optimization theory in the engineering design process. The Chalmers course Engineering Design and Optimization addresses this need and was developed and first offered in the fall semester of 2013. This paper provides a detailed account of the course development experiences, teaching methods and course evaluations using CDIO Standards. It also includes a discussion of the learning objectives, required resources, instructional processes and student assessments. The course is analyzed to see to what extent the course aim is satisfied and highlight areas of improvement. Three projects are carried out in groups of two students. The projects are a cantilever design-build-test exercise, a redesign project that involves material selection, and a multi- disciplinary design project of an engine component using multi-physics software. The assignments ...

The rapid development of computers and the internet has given new opportunities for engineering w... more 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 mathemati...

This paper describes the early stages of the developments of Tracks, an initiative to create, imp... more This paper describes the early stages of the developments of Tracks, an initiative to create, implement and evaluate a new educational model where the structure of the education is developed to give students the opportunity to create multi- and interdisciplinary competencies, meet their expectations and need for a more individualized study plan and shorten the lead times for changing the education to embrace new technologies. The new education model is based on the creation of tracks with different themes lying between existing programs not belonging to a specific department or school. The idea is to create individual and flexible study opportunities by introducing Track-courses within the themes. These courses address specific challenges that may be broad societal and profound research-driven. Tracks also include large investments in Chalmers learning environment. The paper focuses on Tracks as a large change initiative, strategies to manage the complexity of this change as well as...

IFAC Proceedings Volumes, 2006

Here an integro-differential formulation for time domain poroelasticity is derived and included i... more Here an integro-differential formulation for time domain poroelasticity is derived and included into a finite element framework for structural dynamics. Further, an algorithm for the time integration of the spatially discretized finite element equations is developed. The Biot poroelastic theory (formulated in the frequency domain) is taken as a starting point. In this theory, the dissipative force is proportional to the relative motion between the solid phase and the viscous fluid in the pores with a constant of proportionality, drag force coefficient, that is frequency dependent. The dissipative force is described in the time domain by a convolution term with a kernel that is the Fourier inverse of the drag force coefficient. As the exact analytical inversion of the drag force coefficient seems highly intractable, we have combined an asymptotic expansion for high frequencies with a low frequency correction term, yielding an analytically invertible drag force coefficient. The convolution with the fourier inverse of the asymptotic expansion, can be identified as a semi-derivative.

Vehicle System Dynamics, 2012

International Journal of Solids and Structures, 1999

... be justified by the fact that D N − ρ can be taken as the analytical continuation of ... In t... more ... be justified by the fact that D N − ρ can be taken as the analytical continuation of ... In the time domain the presence of integro-differential operators makes the solution of the structural ... how to choose the branch of the root in the Fourier transform of the fractional derivative operator. ...

International Journal of Solids and Structures, 1999

... and the corresponding finite element equations are outlined, including the treatment of bound... more ... and the corresponding finite element equations are outlined, including the treatment of boundaryconditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. ...

37th Structure, Structural Dynamics and Materials Conference, 1996

Numerical procedures for the time integration of the spatially discretized finite element equatio... more Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional derivative operators are presented. To avoid difficulties concerning fractional order initial conditions, a form of the fractional calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Leffler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A certain property of the memory kernel is used, in connection with Griinwald's 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's 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.

International Journal of Solids and Structures, 1999

... (15). It is then assumed that q k(β) have homogeneous initial condition. Hence, the stronger ... more ... (15). It is then assumed that q k(β) have homogeneous initial condition. Hence, the stronger CDI expressing the condition of nonnegative dissipative power is a sufficient condition for DI, while the inverse does not hold. 2.3. Fractional order rate laws. ...

Lecture Notes in Computer Science, 2006

We study a dynamic model for viscoelastic materials based on a constitutive equation of fractiona... more We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integrodifferential 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.

19th AIAA Applied Aerodynamics Conference, 2001

A time domain viscoelastic model for large three dimensional responses under isothermal condition... more A time domain viscoelastic model for large three dimensional responses under isothermal conditions is presented. Internal variables with fractional orders evolution equations are used to model the time dependent part of the response. By use fractional orders rate laws, the characteristics of the time dependency of many polymeric materials can be described using relatively few parameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does not hold (e.g, suspensions, vibration isolators and flexible joints). A numerical algorithm for the constitutive response is developed and implemented into a finite element code for structural dynamics. The algorithm calculates the fractional derivatives by means of the Griinwald-Lubich approach. Analytical and numerical comparisons of the constitutive response in the nonlinear regime are presented. The dynamic structural response of a viscoelastic bar is computed, including both geometrically and materially nonlinear effects. Moreover it is shown that by applying relatively small load magnitudes, the responses of the linear viscoelastic model are recovered.

36th Structures, Structural Dynamics and Materials Conference, 1995

Mechanics of Time-Dependent Materials, 2005

Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers... more Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers. Time domain responses as stress relaxation and creep as well as frequency domain responses are well represented. The drawback of fractional order models is that the fractional order operators are difficult to handle numerically. This is in particular true for fractional derivative operators. Here we propose a formulation based on internal variables of stress kind. The corresponding rate equations then involves a fractional integral which means that they can be identified as Volterra integral equations of the second kind. The kernel of a fractional integral is integrable and positive definite. By using this, we show that a unique solution exists to the rate equation. A motivation for using fractional operators in viscoelasticity is that a whole spectrum of damping mechanisms can be included in a single internal variable. This is further motivated here. By a suitable choice of material parameters for the classical viscoelastic model, we observe both numerically and analytically that the classical model with a large number of internal variables (each representing a specific damping mechanism) converges to the fractional order model with a single internal variable. Finally, we show that the fractional order viscoelastic model satisfies the Clausius-Duhem inequality (CDI).

An infinitely long, cylindrically bent harmonicdly vibrating flexible Kirchhoff plate backed by a... more An infinitely long, cylindrically bent harmonicdly vibrating flexible Kirchhoff plate backed by a thin viscous fluid layer is studied analvticallv. The eovernine equations for the dvnam_ ” -ically coupled plate-fluid layer system are solved by using a spatial Fourier transformation. The transverse plate velocity is chosen as the primary unknown. Introducing of certain approximations, which are justified for practical structures and is consistent with the solution technique used, leads to a polynomial dispersion relation from which the wave-numbers of the system are determined. The total solution is then shown to be built-up of three types of coupled waves. The frequency variation of the corresponding wavenumbers and the contributions from the individual waxes to the total solution are examined. The level of damping is quantified through a loss factor which can be used in approximately accounting for energy losses provided by the fluid without modelling it. Two different loss factors are proposed, the first one being defined from wavenumbers of the system and the second one defined from energies of the system under a certain harmonic load. Numerical examples show that the amount of damping is governed mainly by the fluid layer and the frequency of the applied load. N O M E N C L A T U R E z speed of sound in fluid [m/s] flexural plate stiffness [N/m] d fluid layer thickness [m] E Young’s modulus [N/m*] E(w) complex modulus [N/m’] ES maximum stored energy [Nm] EL dissipated energy [Nm] FO line-force magnitude [N/m] f1(a), fz(a) functions [Ns/m3] h plate thickness [m] WI imaginary part of 1 imaginary unit i* = -1 kb free bending wavenumber [l/m] kr bending wavenumber [l/m] h, kz wavenumbers for 0uid [l/m] ?J4% 2) pressure amplitude [N/m’] WI real part of h..(G 21, ula(z, z) fluid velocity amplitudes [m/s] u=(x) plate velocity amplitude [m/s] a wavenumber space coordinate [l/m] ai wavenumbers of coupled system [l/m] 6 acoustic boundary layer thickness [m] q(w) l o s s factor Xb free bending wavelength [m] P Poison’s ratio ” kinematic viscosity [ml/s] PO fluid density [kg/m31 w angular frequency [rad/s]

Alle rechten voorbehouden. Niets uit deze uitgave mag worden vemenigvuldigd en/of openbaar gemaak... more Alle rechten voorbehouden. Niets uit deze uitgave mag worden vemenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher.

A new computationally oriented mathematics education is presented. The education combines traditi... more A new computationally oriented mathematics education is presented. The education combines traditional symbolic mathematics with computational mathematics and programming in the Matlab environment. Engineering applications are explored in computational exercises that are taught jointly with the courses in mechanics and thermodynamics. COURSES We list the mathematics courses in the first year of the Mechanical Engineering program together with the accompanying engineering courses. The year is divided into four periods (quarters of eight weeks).