Mathematical and Computer
What is Engineering Modeling?
Model: A representation of a real object or
system of objects for purposes of visualizing its
appearance or analyzing its behavior.
Simulation: Transition from a mathematical or
computer model to a kinematics description
(motion) of the system behavior based on sets of
Engineering Modeling and Simulation
Experimentation might not be feasible due to:
• inaccessible inputs and outputs.
• experiment may be too dangerous.
• cost of experimentation might be too high.
• time constants of the system may not be
compatible with human dimensions.
• experimental behavior might be obscured by
Types of Models
• Visual models to simulate form and appearance: graphical
sketches, computer solid models.
• Physical models to simulate function: prototypes, mock-ups,
• Mathematical/Computational models to simulate function:
algebraic and/or differential equations used for computer
• Empirical models to simulate function: relationships between
variables established by direct measurement (equations, charts, or
Physical Principles in Mathematical Modeling
• Newton's Laws:
–Equation of static equilibrium
–Equation of dynamic equilibrium (motion)
–Conservation of energy
–Conservation of momentum
• Constitutive relations:
–friction constants and relations (force vs. normal)
–material constants and relations (stress vs. strain)
Projectile Toy Gun
When fired, the spring releases all of its
energy to the gun top and projectile. The
projectile then transfers rotational energy to
translational energy in two ways:
1) through an impulse on the projectile
when the gun stops moving; and
2) during flight, through aerodynamic lift
Forces Acting on Projectile
Equations of Motion:
Sum of the forces in the vertical direction:
Sum of moments about the center of mass of
Fn includes drag and lift on the blade.
Drag: resistance to air (function of velocity,
density, area, and shape); opposes the motion.
Lift: a function of the pressure difference
between top and bottom of the blade (e.g.,
airplanes wings) provided by high speeds, and
the surface area of the blade. It should exceed
the weight to rise (e.g., helicopters).
3-D Geometric Computer Models
– Design Visualization
– Design Analysis
– Design Simulation
– 3-D Section Views
– .STL Files for Prototyping
– Generate 2-D Drawings
Example Project Computer Model
Example Project Analysis
-------------- SOLIDS -----------
Bounding box: X: 1.7000 - 4.3000
Y: 1.7000 - 4.3000
Z: -1.2000 - 0.8349
Centroid: X: 3.0000
Moments of inertia: X: 23.9782
Products of inertia: XY: 22.6474
Radii of gyration: X: 3.0869
Computer Solid Modeling
• Build a Computer Solid Model of Each Individual
• Make an Assembly Model of all Your parts.
• Make a Shaded Color Hardcopy of Each Part and
• Make a Mass Properties Analysis Report of Each
• Make an .STL File of Each Part (for later use)
Save Your Files!