Lecture notes Models of Mechanics

Anders Klarbring

Division of Mechanics, Linköping University, Sweden

First Lecture: Introduction

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Contents

1 The ideas goals and organization of the course

2 Models

3 Summary

4 Tutorial

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1 The ideas goals and organization of the course

2 Models

3 Summary

4 Tutorial

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The ideas goals and organization of the course

Engineering mechanics (or applied mechanics) consist of a number ofseemingly diverse fields:

Mechanics

Solid Mechanics

Fluid Mechanics

Machine Elements

Biomechanics

etc.

In the present course we will find that all of these subjects have acommon base .

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The ideas goals and organization of the course

Identifying such a common base gives several benefits:

It brings order in what one already knows

Brings economy in understanding new applications

Possibility of creating new models

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The ideas goals and organization of the course

What is this common base in short?

Universal lawsConservation of massLaw of linear momentum “F = ma”Law of angular momentum “M = r × ma”

Particular laws (particular to the subject field). “Force laws.”“Represents how a piece of material behaves when subject toforces.”Examples: Hooke’s law, viscous fluid, Newton’s law of gravitation

Universal laws + particular laws result in mathematical models .

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The ideas goals and organization of the course

During the course we (you) will

Reflect on the relation between “reality” and mathematical models.Derive basic models

Discrete models (particle mechanics)1D models (cable, beams, bars, pipe flow)3D (linear elasticity, Navier-Stokes’)

Write a paper where you derive and test a model of shallow waterflow (“tsunami”).

Hand in solutions of larger exercises.

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The ideas goals and organization of the course

Organization: three types of meetings in the course:

Lectures

Problem solving (classes)

Tutorials – “solutions” are handed in to the teacher the day beforethe tutorial and then discussed at the tutorial (possibly presentedby students).These solutions are sometimes parts of the larger exercises andthe paper assignment.

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The ideas goals and organization of the course

Final grade in the course is decided on

The paper assignment

The three larger exercises

Note that “wrong” solution during the process (say, at tutorials) will notbe reflected in the grade. The paper assignment is three times moreimportant compared to the exercises when the final grade in course isdecided.

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1 The ideas goals and organization of the course

2 Models

3 Summary

4 Tutorial

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Models

Model = a simplified representation of reality (could be many things;physical model, say, a toy car)

Mathematical model = simplified mathematical (abstract)representation of reality

Example: Mass point of particle mechanics

Figure, see board.

Mathematical model

f = ma (1)

a =d2xdt2 (2)

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Models

A mathematical model has some relevance when it has aninterpretation

Real world Conceptual world

Interpretation

Figure: The connection between the real or physical world and theconceptual world of concepts and theory.

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Models

The interpretation is built on giving model concepts referents in thereal world:

Example: The concept of a mass point.

mass point −→ tellus

mass point −→ atom

mass point −→ golf ball

mass point −→ etc.

Thus, many referents are possible, and a mathematical model mayhave many interpretations.

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Models

When we create a mathematical model we move from reality to themodel world. This is called idealization :

Real world Conceptual world

Interpretation

Idealization

This step can be systematized, which is what the course is about.

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Models

In order to use a mathematical model in calculations we need to definea data-answer structure:

datamodel

answer

Figure: A complete mathematical model as an operator producing an answerfrom appropriate data.

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Models

In the case of a mass point:

Data f = f (x ,

dxdt , t)

Answer x = x(t)

Alternative data-answer structure:

Data x = x(t)

Answer f = f (t)

This is a much simpler problem to solve (numerically).

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Models

A mathematical model with a given data-answer structure is called acomplete model.

The first exercise for the first tutorial consists in writing down acomplete model that you know from previous courses.

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Models

The following examples will be discussed on the board:

Example 2: 1D flow

Universal law: Conservation of mass

Particular law: Incompressible fluid

Example 3: Elastic bar

Universal law: Law of linear momentum (equilibrium version)

Particular law: Hooke’s law

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1 The ideas goals and organization of the course

2 Models

3 Summary

4 Tutorial

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Summary

A complete mathematical model works as an operator thatdelivers an answer from data.In mechanics, complete mathematical models are built up from

Universal laws: balance of mass and momentumParticular laws: describe how the forces depends on motion anddeformation

The relation between a model and “reality” comes from identifyingmodel concept with referents in the real world. This makes itpossible to do idealization (modeling) and interpretation(evaluating results).

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1 The ideas goals and organization of the course

2 Models

3 Summary

4 Tutorial

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Tutorial

For the first tutorial you should do the following:

Write down a complete model that you know from a previouscourse. What are the data and the answer?Write down at least 5 relations (equations, formulas) from previouscourses and indicate for each one what type of relation it is:

A universal lawA particular lawA combination of a universal law and a particular lawSomething else?

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