Chapter 1 Notes: Mathematical Modeling, Numerical Methods, and Problem Solving

Chapter Objectives

• Learning how mathematical models can be formulated on the basis of scientific principles to simulate the behavior of a simple physical system.

• Understanding how numerical methods afford a means to generalize solutions in a manner that can be implemented on a digital computer.

• Understanding the different types of conservation laws that lie beneath the models used in the various engineering disciplines and appreciating the difference between steady-state and dynamic solutions of these models.

You’ve Got a Problem: Modeling Free Fall

• Problem setup: Predict jumper velocity during free fall; used to design bungee cord length and strength.

• Framework: Use Newton’s Law: $a = \frac{F}{m}.$ (In general, with gravity and drag, this leads to a velocity-dependent differential equation as shown later.)

• Note on modeling progression: Start with basic Newtonian dynamics and extend to include drag forces to capture realistic motion.

A Simple Mathematical Model

• A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms.

• Models can be represented by a functional relationship between:

  • Dependent variables

  • Independent variables

  • Parameters

  • Forcing functions

Model Function

• Dependent variable: a characteristic that reflects the system’s behavior or state (e.g., temperature of a room).

• Independent variables: dimensions along which the system’s behavior is determined (e.g., time and space). Example: time t controlling temperature changes.

• Parameters: constants reflecting the system’s properties or composition.

• Forcing functions: external influences acting upon the system (e.g., environmental inputs).

An Example of a Mathematical Model

• Newton’s second law: $F = ma,$ where F is the net force, m is mass, a is acceleration.

• Recast by dividing by m: $a = \frac{F}{m}.$

• Simple algebraic form makes some models solvable exactly. More complex models may require differential equations and calculus.

• Terminal velocity example: use Newton’s second law to determine the motion of a falling body (e.g., a bungee jumper) and form a differential equation for acceleration:

  • The acceleration is the rate of change of velocity: $\frac{dv}{dt} = \frac{F}{m}.$

  • With gravity and drag, a differential equation emerges describing the motion.

Expressing the Net Force and Governing Equation

• Forces in play near Earth:

  • Downward gravity: $F_D = mg,$ with g (9.81 m/s²).

  • Upward air resistance: $F_{U}=-c_{d}v^2$ where $c_{d}$ is the drag coefficient and v is velocity.

• Net force (taking downward as positive for sign convention):

  $F=F_{D}+F_{U}$

• The equation of motion (Newton’s second law):

  $\frac{dv}{dt} = \frac{F}{m} = g - \frac{c_d}{m} v^2.$

• Key points:

  • Gravity provides a constant downward pull.

  • Air resistance increases with velocity (quadratic in v in this model).

  • The net force and acceleration depend on v, making the equation nonlinear.

  • Sign conventions are important; here, downward is taken as positive.

• Values used in examples:

  • Gravity: $g = 9.81\ \text{m/s}^2$

  • Drag coefficient: $c_d$ (given for examples, e.g., 0.25)

Analytical Solution to the Velocity Model

• Consider the differential equation: $\frac{dv}{dt} = g - \frac{c_d}{m} v^2$ with initial condition (v(0) = 0).

• Analytical (closed-form) solution:

  $v(t) = \sqrt{\frac{g m}{c<em>d}}\ \tanh\left(\sqrt{\frac{g c</em>d}{m}}\ t\right).$

• Characteristics:

  • This is an exact solution for the given ODE with the specified forcing term.

  • Terminal velocity (steady state) occurs when (dv/dt = 0):

    $0 = g - \frac{c<em>d}{m} v^2 \ \Rightarrow\ v</em>{\text{term}} = \sqrt{\frac{g m}{c_d}}.$

• Initial velocity is zero ((v(0)=0)).

Example 1.1: Analytical Solution to the Bungee Jumper Problem

• Problem data (as in the example):

  • Mass: ($m = 68.1\ \text{kg}$), drag coefficient: ($c_d = 0.25$) (units absorbed into the coefficient as appropriate).

  • Gravitational acceleration: ($g = 9.81\ \text{m/s}^2$).

• Analytical results:

  • Velocity after 10 s: approximately ($v(10) \approx 49.4214\ \text{m/s}$).

  • Terminal velocity (as ($t \to \infty$)): approximately ($v_{\text{term}} \approx 51.6983\ \text{m/s}$) (about 115.6 mph).

• Calculation highlights:

  • Terminal velocity formula: $v<em>{\text{term}} = \sqrt{\frac{g m}{c</em>d}}.$

  • With given parameters, plug in values to obtain the terminal velocity.

Model Results

• Numerical or analytical results can be graphed to visualize velocity vs. time for a given mass and drag coefficient.

• For the 68.1 kg jumper with c_d = 0.25, the velocity curve approaches the terminal velocity over time.

Analytical or Closed-form solution vs. Numerical solution

The equation (1.9) represents an analytical, or closed-form, solution which precisely satisfies the original differential equation. While such exact solutions are ideal, many complex mathematical models cannot be solved analytically. In these instances, numerical solutions are employed to approximate the true solution directly on a computer. Therefore, numerical methods serve as essential alternatives when a closed-form solution is not available, providing a means to estimate the behavior of the system.

Numerical Modeling

• Numerical methods reformulate the mathematical problem so it can be solved by logical and arithmetic operations (addition, subtraction, multiplication, and division).

• Finite differences are used to approximate derivatives.

• Definitions:

  • Differences in velocity: $\Delta v$

  • Differences in time: $\Delta t$

  • Velocity at initial time: $v(t_i)$

  • Velocity at a later time: $v(t_{i+1})$

• The derivative is approximated by:

  $\frac{dv}{dt} \approx \frac{\Delta v}{\Delta t} = \frac{v(t<em>{i+1}) - v(t</em>i)}{t<em>{i+1} - t</em>i}.$

• This approximation becomes exact in the limit as $\Delta t \to 0.$

Euler’s Method

• Substituting the finite difference approximation into the differential equation yields the update rule:

  $\frac{v(t<em>{i+1}) - v(t</em>i)}{t<em>{i+1} - t</em>i} \approx g - \frac{c<em>d}{m} v(t</em>i)^2.$

• Solve for the next value of velocity:

  $v(t<em>{i+1}) = v(t</em>i) + \Big(g - \frac{c<em>d}{m} v(t</em>i)^2\Big) (t<em>{i+1} - t</em>i).$

• In short: old value plus slope at the old point times the step size.

• This is the Euler method (explicit, first-order).

• Notation:

  • ($v<em>i$) denotes ($v(t</em>i)$).

  • Step size: $\Delta t = t<em>{i+1} - t</em>i.$

Numerical Example 1.2: Numerical Solution to the Bungee Jumper Problem

• Problem statement: Use Eq. (1.12) to compute velocity with Euler’s method, with step size $\Delta t = 2\ s.$

• Update formula used:

  $v<em>{i+1} = v</em>i + \Big(g - \frac{c<em>d}{m} v</em>i^2\Big) \Delta t.$

• Given initial conditions: start at rest, so ($v_0 = 0$).

• Calculations:

  • At ($t = 0$): ($v<em>2 = v</em>0 + [g - (c<em>d/m) v</em>0^2] \cdot 2 = 0 + (9.81) \cdot 2 = 19.62\ \text{m/s}.$)

  • At ($t = 4$): using ($v<em>2 = 19.62$) in the update, $\ v</em>4 = 19.62 + \Big(9.81 - \frac{0.25}{68.1} (19.62)^2\Big) \cdot 2 \approx 36.4137\ \text{m/s}.$)

• General takeaway: Euler’s method provides a straightforward, iterative way to approximate velocity with a chosen step size.

Numerical Results

• Using Euler’s method with 2 s steps yields a trajectory that approximates the analytical solution; results improve as step size decreases.

• Trade-off: smaller $\Delta t$ yields higher accuracy but more computational work.

• Visual comparison typically shows numerical solution approaching the analytical curve as step size decreases.

Generalizing Model Principles

Beyond Newton's second law, primary organizational principles in engineering and science are conservation laws, which fundamentally describe how quantities change over time. For time-variable or transient computations, this change is expressed as increments minus decrements ($\text{Change} = \text{increases} - \text{decreases}$). In steady-state conditions, where no net change occurs, this equation simplifies to zero (increases = decreases). Core conservation principles that form the basis for numerical models across various disciplines include the conservation of mass, momentum, charge, and energy.

Recap and Workflow

To recap, mathematical models translate physical phenomena into equations, requiring clear distinction between dependent and independent variables, parameters, and forcing functions. While analytical (closed-form) solutions provide exact results, their availability is limited, necessitating numerical methods to approximate solutions computationally. Euler's method serves as a fundamental numerical approach for first-order ordinary differential equations, where accuracy is improved by reducing the step size ($\Delta t$), albeit at the cost of increased computation. This modeling framework, unified by conservation laws, applies to both transient (time-variable) and steady-state problems. The universal workflow for numerical modeling systematically guides this process: first, select a governing conservation law; then, formulate the mathematical model; next, choose a numerical scheme, discretize by approximating derivatives with finite differences, and define step sizes ($\Delta t$ or $\Delta x$). The process continues iteratively on a computer (new = old + slope \times step) until a stopping criterion is met. Finally, validate the results against known solutions or benchmarks and refine the model by adjusting step sizes or employing higher-order methods for enhanced