Mathematical Modelling in Engineering Design.

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Mathematical Modeling in Engineering Design
MEC
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Contents

• Definition and Principles.
• Abstraction.
• Mathematical Tools.
• Physical Dimensions.
• Figures of Merit.
• Dimensional Analysis.
• Balance and Conservation.
• Analogies.
• Design Criteria.

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Model

• Miniature representation of something.
• A pattern of some thing to be made.
• An example for imitation or emulation.
• A description or analogy used to help visualize
  something (e.g., an atom) that cannot be directly
  observed.
• A system of postulates, data and inferences
  presented as a mathematical description of an
  entity or state of affairs.

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Mathematical Model

• Representation in mathematical terms of the
  behavior of real devices and objects.

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Mathematical Model

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Principles of Mathematical Modeling

• An activity with underlying principles and a host
  of methods and tools.
• Ability to predict the behavior of devices or
  systems that we are designing.
• Overarching principles are almost philosophical
  in nature.
• Individual questions recur often during the
  modeling process.

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Principles of Mathematical Modeling

• Why do we need a model?
• For what will we use the model?
• What do we want to find with this model?
• What data are we given?
• What can we assume?
• How should we develop this model, what are the
  appropriate physical principles we need to apply?

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Principles of Mathematical Modeling

• What will our model predict?
• Can we verify the models predictions (i.e., are
  our calculations correct?)
• Are the predictions valid (i.e., do our
  predictions conform to what we observe?)
• Can we improve the model?

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Mathematical Modeling
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Why Mathematical Models?

• Developing a mathematical model allows estimation
  of the quantitative behavior of the system.
• Quantitative results from mathematical models
  compared with observational data to identify a
  model's strengths and weaknesses.
• An important component of the final complete
  model of a system which is actually a collection
  of conceptual, physical, mathematical,
  visualization, and possibly statistical
  sub-models.

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Abstraction

• More general than specific.
• Thinking about finding the right level of
  abstraction or detail means identifying the right
  scale for our model.
• Means thinking about the magnitude or size of
  quantities measured with respect to a standard
  that has the same physical dimensions.

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Abstraction

• Choosing the right level of detail for the
  problem very important.
• Dictates the level of detail for the model.
• Requires a thoughtful approach to identifying the
  phenomena to be emphasized.
• To answer the fundamental question about why a
  model is being developed and how we intend to use
  it.

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Lumped Model Abstraction

• The actual physical properties of a real object
  or device are aggregated or lumped into less
  detailed, more abstract expressions.
• What we lump into lumped elements depends on the
  scale on which we choose to model, which depends
  in turn on our intentions for that model.
• Eg Aircraft (mass as point mass, effect of
  surrounding atmosphere as a drag force on the
  point mass) modeled in different ways depending
  on the goals.

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Mathematical Tools for Design Modeling

• Tools used to apply the big picture principles
  to develop, use, verify, and validate
  mathematical models.
• Dimensional analysis.
• Approximations of mathematical functions.
• Linearity.
• Conservation and balance laws

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Dimensional Homogeneity

• Rule of dimensional homogeneity not to be
  violated.
• Properly constructed equations representing
  general relationships between physical variables
  to be dimensionally homogeneous.
• Dimensions of terms that are added or subtracted
  to be the same.
• Dimensions on the right side of an equation to be
  the same as those on the left side.

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Physical Dimensions

• Every independent term in every equation to be
  dimensionally homogeneous or dimensionally
  consistent.
• Every term to have the same net physical
  dimensions.
• Ensure dimensional consistency (also called
  rational equations).
• Attach numerical measurements or values to
  physical quantities representing objects.

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Classes of Physical Quantities

• Fundamental or primary quantities - measured on a
  scale independent of those chosen for other
  fundamental quantities.
• Derived quantities - follow from definitions or
  physical laws, expressed in terms of the
  dimensions chosen as fundamental.
• Force a derived quantity derived from Newtons
  Laws of motion. If M, L, and T stand for mass,
  length and time respectively force F (M x L)/T2

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Units of a Physical Quantity

• An arbitrary multiple or fraction of a physical
  standard.
• Numerical aspects of dimensions of a quantity
  expressed in terms of a given physical standard.
• Use of most widely accepted international
  standard.
• Choice of units to facilitate calculation or
  communication.
• Magnitude or size of the attached number depends
  on the unit chosen.

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Physical Dimensions

• Physical dimensions of a quantity are constant.
• Must exist numerical relationships between the
  different systems of units used to measure the
  amounts of quantity.
• Conversion may be needed.
• Equality of units for a given dimension allows
  units to be changed or converted with a
  straightforward calculation.

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Physical Dimensions

• Each independent term in a rational equation has
  the same net dimensions.
• Can add quantities having the same dimensions,
  expressed in different units.
• Cannot add length to area in the same equation,
  or mass to time.
• Equations to be rational in terms of their
  dimensions.

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Figures of Merit

• Sets of units of interest for the scales of
  metrics to be used for assessing the achievement
  of objectives.
• To construct mathematical objective functions
  that represent figures of merit to optimize a
  design.
• All independent terms in an objective function to
  be rational, to have the same net dimensions.

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Number of Significant Figures

• Equal to the number of digits counted from the
  first nonzero digit on the left to either (a) the
  last nonzero digit on the right if there is no
  decimal point, or
• (b) the last digit (zero or nonzero) on the
  right when there is a decimal point.
• Eg 5415 four significant figures, 0.054
  two significant figures.
• NSF not determined by the placement of the
  decimal point.

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Dimensionless Quantities

• Often ratios of same kind.
• No dimensions.
• Intended to compare the value of a specific
  variable with a standard of obvious relevance.
• Eg Current Gain, Soil Porosity etc.

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Method of Dimensional Analysis

• List all of the variables and parameters of a
  problem and their dimensions.
• Anticipate how each variable qualitatively
  affects quantities of interest, ie does an
  increase in a variable cause an increase or a
  decrease?
• Identify one variable as depending on the
  remaining variables and parameters.
• Express that dependence in a functional equation.

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Method of Dimensional Analysis

• Choose and then eliminate one of the primary
  dimensions to obtain a revised functional
  equation.
• Repeat until a revised, dimensionless functional
  equation is found.
• Review the final dimensionless functional
  equation to see whether the apparent behavior
  accords with the behavior anticipated.

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Physical Idealization

• Idealize or approximate situations or objects so
  that we can model them and apply those models to
  find behaviors of interest.
• Two kinds of idealizations - physical and
  mathematical, order in which we make them is
  important.
• To have an initial physical idealization, then
  translate the physical idealization into a
  consistent mathematical model.

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Linear Models

• Nonlinear problems harder to solve.
• Linear models work extraordinarily well for many
  devices and behaviors of interest.
• Linear model approximations wherever possible.
• Eg Approximating sin a to a for small angles.

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Balance and Conservation

• Laws of conservation to be obeyed.
• Conservation laws are special cases of balance
  laws.
• Balance or conservation principles applied to
  assess the effect of maintaining levels of
  physical attributes.
• To count or measure both what goes in to and what
  comes out of the boundary of the domain under
  observation.

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Series and Parallel Connections

• Notion of division/deflection.
• Applying constitutive laws to series and parallel
  connections.
• To gain insights into design behavior when we
  link characterizations to appropriate balance or
  conservation laws.

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Series and Parallel Connections

Resistance
Spring
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Mechanical - Electrical Analogies

• To represent the function of a mechanical system
  as an equivalent electrical system by drawing
  analogies between mechanical and electrical
  parameters or vice versa.
• Wide use in electromechanical systems where there
  is a connection between mechanical and electrical
  parts.
• Analogizing - process of representing information
  about a particular subject (the analogue or
  source system) by another particular subject (the
  target system).

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Mechanical Electrical Analogies

• Analogical awareness a good habit of thought that
  experienced designers often exploit.
• Use of analogy between the elementary mechanical
  and electrical circuits.
• Eg springs are the mechanical elements, which
  store energy, while capacitors are the electrical
  elements, which store energy.

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Mechanical Electrical Analogy

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Design Criteria

• Depends on objectives and constraints.
• Relative importance of objectives to vary with
  client and application.
• Eg light weight - minimize the mass of
  material used, inexpensive - minimize the cost.
• Against what requirements do we assess the
  performance of our designs?
• Designing for strength to know the stresses at
  which a material fails.

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Design Criteria

• Designing for stiffness to determine values of
  the design variables as to lie within and not to
  exceed specified deflection limits.
• Design for Aesthetics and Ergonomics?
• Codes or standards specify the limits.

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Methods of Mathematical Modeling

• Theoretical modeling - system described using
  equations derived from physics.
• White-box models - system modeling entirely based
  on physical principles (equations).
• Experimental modeling - also called system
  identification is based on measurements.
• Black-box models - system modeling entirely based
  on experimental data (input/output measurements).

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Methods of Mathematical Modeling
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Methods of Mathematical Modeling

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Why Experimental Modeling

• If the system is complex, deriving the
  mathematical equations can be very hard.
• Most of the parameters used in the mathematical
  equations are not known, so the overall behavior
  of the modeled system is uncertain.
• Not all physical phenomena are captured or well
  known.

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Statistical Modeling

• A mathematical representation (mathematical
  model) of observed data.
• Applying statistical analysis to a dataset.
• Use of mathematical models and statistical
  assumptions to generate sample data and make
  predictions about the real world.
• Collection of probability distributions on a set
  of all possible outcomes of an experiment.

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Comments

• Preliminary design requires careful mathematical
  modeling.
• Need for focused research to obtain relevant
  data.
• Doing research to identify appropriate materials
  a skill in its own right.
• Knowledge on dimensions, scaling, simplifying
  assumptions, and how a model answers only the
  questions asked of it can be applied to almost
  all modeling and design efforts.

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Thank You