Introduction to Modeling and Simulation

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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Spring 2008

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Introduction to Modeling and Simulation Lecture No. 2 Raul ´ Radovitzky Aeronautics and Astronautics Massachusetts Institute of Technology

Raul ´ Radovitzky (MIT)

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Today’s lecture topics

1

Basic Concepts in Science and Engineering Analysis

2

Formulation of discrete mathematical models Equilibrium problems Propagation problems Eigenvalue problems


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Analysis of physical system idealization formulation of mathematical model solution of mathematical model interpretation of results

References: Engineering Analysis, A Survey of Numerical Procedures, Chapters 1-3. S. H. Crandall. Krieger Publishing Co., 1983. Finite Element Procedures, Chapter 3, K. J. Bathe. Prentice Hall, 1996.


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Two categories of mathematical models lumped-parameter models systems often described with adequate precision by finite number of state variables formulation results in system of simultaneous (coupled) algebraic equations (static problems)

continuum models describe system details with infinite spatial and temporal resolution by continuum fields formulation results in systems of differential equations governing system response exact solution only possible for relatively simple systems, numerical discretization procedures needed.


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Numerical discretization procedures reduce the continuous model to a discrete idealization which can be solved in same manner as lumped-parameter model new sources of error (discretization) are introduced.


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Formulation of discrete mathematical models

Equilibrium problems

Equilibrium problems Examples

Elastic spring system U1, R1

k3

U2, R2 2

k2

1

E

U3, R3

k4

k1

Pipe system

3

k5

q1 Q

R = 10b

A

θ2

Surface coefficient 3k

6R

3

Surface coefficient 2k

4

I3

θ4

I1 2R Conductance 3k

Figure by MIT OpenCourseWare.


D

Resistor-battery circuit

θ

θ0

Conductance 2k

R = 3b



θ1

q2

q4 C

Q

R = 5b

q3

q2

Heat Transfer problem

B

R = 2b R = 5b

1

2

3

2R

4R

B

A

2E

E

I2


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Equilibrium problems Structure of the problem

Structure of the problem System is made up of simple elements with assumed effective behavior Equilibrium requirements for each element are known In addition to equilibrium, it is also necessary to satisfy certain element interconnection requirements.


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Analysis of spring system: System idealization

Choice of state variables: Ui , i = 1, 3 Assumed element response: kΔU = F Element equilibrium requirements (see ﬁgure) U1 F1(1)

k1

k1 U1 = F1(1) U1 k2

F1(2)

k2

U2

[ [[ [ [ [ 1

-1

U1

-1

1

U2

=

F1(2)

1

-1

U1

-1

1

U3

1

-1

U1

-1

1

U2

=

F1(3) F2(3)

=

F1(4)

F3(4)

U2 F2(3)

[ [[ [ [ [

F3(4)

[ [[ [ [ [

U2 k3

U3 k4

F1(4)

k4

F2(2)

U1 F1(3)

k3

U1 F2(2)

k5

U3 k5

F2(5)

F3(5)

[ [[ [ [ [ 1

-1

U2

-1

1

U3

=

F2(5) F3(5)

Figure by MIT OpenCourseWare. Raul ´ Radovitzky (MIT)

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Analysis of spring system Element interconnection requirements

Enforced by requiring equilibrium of three carts: (1)

(2)

(3)

(2) F2 (4) F3

(3) F2 (5) F3

(5) F2

(4)

F1 + F1 + F1 + F1 + +

+

= R1

(1)

= R2

(2)

= R3

(3)

Can be expressed as system of linear equations: KU = R, where: UT = [U1 U2 U3 ] RT = [R1 R2 R3 ]   (k1 + k2 + k3 + k4 ) −(k2 + k3 ) −k4 −(k2 + k3 ) (k2 + k3 + k5 ) −k5  K= (k4 + k5 ) −k4 −k5 Raul ´ Radovitzky (MIT)

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Analysis of Spring system Energy and Variational Formulation

Strain Energy of individual springs: 1 U = k∆U 2 2 Potential energy of external load V = −W = PU Potential energy of system: Π=

X i

Ui +

X

Vl

l

Equilibrium achieved by invoking Principle of Minimum Potential Energy: δΠ = 0 X ∂Π δUi = 0 (4) δΠ = ∂Ui i


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Analysis of Spring system Energy and Variational Formulation

Example: U=

i 1h k1 U12 + (k2 + k3 )(U2 − U1 )2 + k4 (U3 − U1 )2 + k5 (U3 − U2 )2 2 V = −(R1 U1 + R2 U2 + R3 U3 ), Π = U + V

Equilibrium:

∂Π ∂Ui

= 0, ∀i

∂Π = k1 U1 − (k2 + k3 )(U2 − U1 ) − k4 (U3 − U1 ) − R1 = 0 ∂U1 ∂Π = (k2 + k3 )(U2 − U1 ) − k5 (U3 − U2 ) − R2 = 0 ∂U2 ∂Π = k4 (U3 − U1 ) + k5 (U3 − U2 ) − R3 = 0 ∂U3 Raul ´ Radovitzky (MIT)

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Remarks about variational approach: Obtained same equations as with direct “vector” approach Advantage: automatically generates element interconnectivity requirements Disadvantage: less physical insight into problem formulation


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Propagation problems


Main characteristics: system response and, thus, state variables change with time objective is to calculate state variables for all times. true propagation problem is when element equilibrium relations are time dependent Examples Elastic spring-mass-damper system Cooling/Heating of an object


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Example: Spring-mass system same problem as before but now forces change with time AND carts have mass mi

¡2-¿ Need to add inertia forces to cart equilibrium equations by invoking D’alembert’s principle. (1)

(2)

(3)

(2) F2 (4) F3

(3) F2 (5) F3

(5) F2

(4)

F1 + F1 + F1 + F1 + +

+

¨1 = R1 (t ) − m1 U

¨2 = R2 (t ) − m2 U

(5) (6)

¨3 = R3 (t ) − m3 U

(7)

d 2 Ui dt 2

¨i = ˙ (0) = V0 . Can also where U , and initial conditions U(0) = U0 , U be written as matrix system: ¨ + KU = R(t) MU where M is the system mass matrix:   m1 0 0 M =  0 m2 0  0 0 m3 Raul ´ Radovitzky (MIT)

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Eigenvalue problems

Eigenvalue problems

Examples natural frequencies of a system buckling loads


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Introduction to Modeling and Simulation - Lecture ... - DSpace@MIT

Introduction to Modeling and Simulation - Lecture ... - DSpace@MIT

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