+ Modeling with Differential Equations
+
+
+
+ Differential equations are more than just mathematical abstractions—they are the language we use to describe how things change in the real world. From population growth to cooling coffee, from falling objects to chemical reactions, differential equations provide a framework for understanding and predicting dynamic behavior.
+
+
+
+ In this section, we'll explore how to translate a real-world scenario into a differential equation. This process, called mathematical modeling, involves identifying variables, making assumptions, and using logic and physical principles to construct an equation that captures the essence of the problem.
+
+
+
+
+ The Problem: Newton's Law of Cooling
+
+
+ Imagine you've just poured yourself a hot cup of coffee at temperature 180^\circ F and placed it on a table in a room where the temperature is 70^\circ F. You know from experience that the coffee will cool down over time, but can we predict exactly how the temperature changes?
+
+
+
+ This scenario can be modeled using Newton's Law of Cooling, which states that the rate at which an object's temperature changes is proportional to the difference between its temperature and the ambient (room) temperature.
+
+
+
+
+ Assumptions
+
+
+ Before we can build our model, we need to make some simplifying assumptions:
+
+
+
+ - The room temperature remains constant at 70^\circ F.
+ - The coffee is well-mixed, so its temperature is uniform throughout.
+ - Heat loss occurs only through the exposed surface of the coffee.
+ - The rate of cooling is proportional to the temperature difference between the coffee and the room.
+ - External factors like wind or humidity are negligible.
+
+
+
+
+ Building the Model
+
+
+ Let's define our variables:
+
+
+
+ - T(t): the temperature of the coffee at time t (in minutes)
+ - T_{\text{room}}: the constant room temperature (70^\circ F)
+ - k: a positive constant representing the cooling rate
+
+
+
+ The rate of temperature change is the derivative \frac{dT}{dt}. According to Newton's Law of Cooling, this rate is proportional to the temperature difference:
+
+ \frac{dT}{dt} = -k(T - T_{\text{room}})
+
+
+
+
+ The negative sign indicates that temperature decreases when T > T_{\text{room}}. For our specific problem:
+
+ \frac{dT}{dt} = -k(T - 70)
+
+
+
+
+ Parameters:
+
+
+ - k > 0: The cooling constant, which depends on properties like the surface area of the coffee, the material of the cup, and air circulation. Larger k means faster cooling.
+ - T_{\text{room}} = 70: The ambient temperature in degrees Fahrenheit.
+
+
+
+
+ Data Collection Activity
+
+
+ Objective: Estimate the cooling constant k by collecting real temperature data.
+
+
+
+ Materials Needed:
+
+
+ - A cup of hot water (or coffee)
+ - A thermometer (digital or analog)
+ - A timer or stopwatch
+ - A room-temperature environment
+
+
+
+ Procedure:
+
+
+ - Heat water to approximately 180^\circ F and pour it into a cup.
+ - Measure and record the room temperature.
+ - Record the initial temperature of the water at t = 0.
+ - Measure and record the water temperature every 2 minutes for 20 minutes.
+ - Create a table with columns: Time (min), Temperature (°F), Temperature Difference (T - T_{\text{room}}).
+
+
+
+ Analysis Questions:
+
+
+ - Plot the temperature T versus time t. What pattern do you observe?
+ - Plot the temperature difference (T - T_{\text{room}}) versus time. Does this appear exponential?
+ - Based on your data, estimate the cooling constant k. (Hint: Use the fact that T(t) - T_{\text{room}} = (T_0 - T_{\text{room}})e^{-kt}, which we'll learn to derive in later chapters.)
+ - How would the cooling rate change if you used a metal cup instead of a ceramic one?
+ - What happens to the temperature as t \to \infty? Does this match your intuition?
+
+
+
+
+ Analytical Questions
+
+
+ Work through these questions to deepen your understanding:
+
+
+
+ - If the room temperature were 80^\circ F instead of 70^\circ F, how would this affect the differential equation?
+ - What if the coffee were initially at 60^\circ F (colder than the room)? Would it warm up? How would the differential equation change?
+ - Explain why the parameter k must be positive. What would happen if k were negative?
+ - Suppose you have two cups of coffee, one at 180^\circ F and one at 160^\circ F. Which one cools faster initially? Why?
+
+
+
+
+ Final Report
+
+
+ Prepare a brief report (1-2 pages) that includes:
+
+
+
+ - A summary of the modeling process, including your assumptions and the derived differential equation.
+ - Your experimental data presented in both table and graph form.
+ - An estimate of the cooling constant k with an explanation of your estimation method.
+ - A discussion of how well the model fits your data and any discrepancies you observed.
+ - Suggestions for improving the model or the experiment.
+
+
+
+
+
+ This modeling exercise demonstrates how differential equations arise naturally from real-world phenomena. By identifying variables, making reasonable assumptions, and applying physical principles, we translated a everyday experience into a mathematical model. In future chapters, we'll learn techniques to solve this and other differential equations analytically.
+
+
+
+
diff --git a/source/c1-classification/class-model.ptx b/source/c1-classification/class-model.ptx
new file mode 100644
index 00000000..25a1487d
--- /dev/null
+++ b/source/c1-classification/class-model.ptx
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+
+
+ Modeling with Classification
+
+
+
+ Understanding the order and linearity of a differential equation isn't just an academic exercise—it has practical implications for modeling real-world systems. The classification of a differential equation tells us about the complexity of the underlying phenomenon and guides us toward appropriate solution methods.
+
+
+
+ In this section, we'll explore how different types of differential equations arise from different physical contexts and learn to classify equations that model real systems.
+
+
+
+
+ The Problem: Spring-Mass Systems
+
+
+ Consider a mass attached to a spring, suspended vertically. When you pull the mass down and release it, it oscillates up and down. The question is: can we predict the position of the mass at any given time?
+
+
+
+ This seemingly simple system can be modeled with differential equations of different orders and linearity, depending on which physical effects we include.
+
+
+
+
+ Assumptions and Model Variations
+
+
+ Basic Model (Linear, Second-Order):
+
+
+
+ Let y(t) represent the displacement of the mass from its equilibrium position at time t. The simplest model assumes:
+
+
+ - The spring force is proportional to displacement (Hooke's Law): F_{\text{spring}} = -ky
+ - There is no friction or air resistance
+ - The mass m is constant
+
+
+
+ By Newton's second law (F = ma), where a = \frac{d^2y}{dt^2} is acceleration:
+
+ m\frac{d^2y}{dt^2} = -ky
+
+ or equivalently:
+
+ \frac{d^2y}{dt^2} + \frac{k}{m}y = 0
+
+
+
+
+ This is a linear, homogeneous, second-order differential equation.
+
+
+
+ Enhanced Model (Linear, Second-Order with Damping):
+
+
+
+ If we include air resistance or friction, which opposes motion proportionally to velocity:
+
+
+ - Damping force: F_{\text{damping}} = -c\frac{dy}{dt}
+
+
+
+ The equation becomes:
+
+ m\frac{d^2y}{dt^2} + c\frac{dy}{dt} + ky = 0
+
+
+
+
+ This is still linear and second-order, but now includes a first derivative term representing damping.
+
+
+
+ Nonlinear Model (Second-Order):
+
+
+
+ For large displacements, the spring may not obey Hooke's Law perfectly. A more realistic spring force might be:
+
+ F_{\text{spring}} = -ky - \alpha y^3
+
+
+
+
+ This leads to a nonlinear differential equation:
+
+ m\frac{d^2y}{dt^2} + c\frac{dy}{dt} + ky + \alpha y^3 = 0
+
+
+
+
+ Parameters:
+
+
+ - m: mass (kg)
+ - k: spring constant (N/m), representing spring stiffness
+ - c: damping coefficient (N·s/m)
+ - \alpha: nonlinear spring coefficient
+
+
+
+
+ Classification Activity
+
+
+ Objective: Practice classifying differential equations from various physical contexts.
+
+
+
+ Part 1: Classify These Equations
+
+
+
+ For each differential equation below, determine:
+
+
+ - The order
+ - Whether it is linear or nonlinear
+ - If linear, whether it is homogeneous or nonhomogeneous
+
+
+
+ Equations to Classify:
+
+
+ - Population growth: \frac{dP}{dt} = rP
+ - Radioactive decay: \frac{dN}{dt} = -\lambda N
+ - Logistic growth: \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)
+ - Forced harmonic oscillator: m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F_0\cos(\omega t)
+ - Pendulum (small angles): \frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0
+ - Pendulum (large angles): \frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0
+ - RC circuit: RC\frac{dV}{dt} + V = V_{\text{source}}(t)
+
+
+
+ Part 2: Modeling Exercise
+
+
+
+ Working in groups, construct a simple spring-mass system:
+
+
+
+ Materials:
+
+
+ - A spring
+ - Various masses (washers, weights, etc.)
+ - A ruler or measuring tape
+ - A timer
+
+
+
+ Procedure:
+
+
+ - Hang the spring vertically and measure its natural (unstretched) length.
+ - Attach a known mass and measure the new equilibrium length. The difference gives the stretch \Delta y.
+ - Use mg = k\Delta y to estimate the spring constant k.
+ - Pull the mass down slightly and release it. Time 10 complete oscillations and calculate the period T.
+ - Repeat with different masses.
+
+
+
+ Data Analysis:
+
+
+ - Create a table with mass, stretch, and period for each trial.
+ - Calculate the spring constant k from each mass-stretch measurement.
+ - Plot the period T versus mass m. What relationship do you observe?
+ - The theoretical period for a spring-mass system is T = 2\pi\sqrt{\frac{m}{k}}. Does your data support this?
+
+
+
+
+ Analytical Questions
+
+
+ - Why is the basic spring-mass equation second-order? What physical quantity is represented by the second derivative?
+ - How does adding damping change the classification? Does it change the order or linearity?
+ - What makes the large-angle pendulum equation nonlinear? How might this affect its behavior compared to the small-angle case?
+ - If you wanted to model a spring-mass system on the surface of the Moon (where gravity is weaker), which parameters would change?
+ - Can you think of other physical systems that might be modeled by second-order differential equations?
+
+
+
+
+ Final Report
+
+
+ Write a comprehensive report (2-3 pages) that includes:
+
+
+
+ - A comparison of the three spring-mass models (basic, damped, nonlinear), discussing how assumptions affect the classification.
+ - Your experimental data, including calculations of the spring constant and analysis of the period-mass relationship.
+ - Answers to the classification questions in Part 1 of the activity, with detailed reasoning.
+ - A discussion of when a linear model might be sufficient versus when a nonlinear model is necessary.
+ - Real-world examples of systems modeled by equations of different orders and linearity.
+
+
+
+
+
+ Classification is the first step in understanding a differential equation. By recognizing whether an equation is first-order or second-order, linear or nonlinear, we can anticipate its behavior and select appropriate solution techniques. The spring-mass system demonstrates how the same physical scenario can yield different equations depending on our modeling assumptions—and how those choices affect the mathematical classification.
+
+
+
+
diff --git a/source/c10-lt/lt-model.ptx b/source/c10-lt/lt-model.ptx
new file mode 100644
index 00000000..053ccaa6
--- /dev/null
+++ b/source/c10-lt/lt-model.ptx
@@ -0,0 +1,250 @@
+
+
+ Modeling with Laplace Transforms
+
+
+
+ The Laplace transform converts differential equations into algebraic equations, making them easier to solve, especially when dealing with piecewise or discontinuous forcing functions. This transform-based approach is fundamental in engineering, particularly in control systems and signal processing.
+
+
+
+ In this section, we'll use the Laplace transform to analyze a pharmaceutical delivery problem where drug concentration must be carefully controlled.
+
+
+
+
+ The Problem: Drug Dosing and Pharmacokinetics
+
+
+ When a drug is administered (orally or by injection), it enters the bloodstream, distributes through the body, and is gradually eliminated by metabolism and excretion. The concentration must remain within a therapeutic window—high enough to be effective but not so high as to be toxic.
+
+
+
+ How should we schedule doses to maintain safe and effective drug levels?
+
+
+
+
+ Assumptions
+
+
+ - The drug distributes instantaneously and uniformly (one-compartment model).
+ - Elimination follows first-order kinetics: rate proportional to concentration.
+ - Drug is administered as instantaneous doses (impulses) or as infusions.
+ - The body's volume of distribution is constant.
+ - No drug-drug interactions or changing physiology.
+
+
+
+
+ Building the Model
+
+
+ Let C(t) be the drug concentration in blood (mg/L) at time t (hours). The rate of change is:
+
+ \frac{dC}{dt} = -kC + I(t)
+
+
+
+
+ where:
+
+
+ - k > 0: elimination rate constant (1/hour), related to half-life by t_{1/2} = \ln(2)/k
+ - I(t): input rate (mg/(L·hour)), representing drug administration
+
+
+
+ Case 1: Single Bolus Dose
+
+
+
+ For an instantaneous dose D mg at t = 0, distributed in volume V L:
+
+ I(t) = \frac{D}{V}\delta(t)
+
+ where \delta(t) is the Dirac delta function.
+
+
+
+ Taking the Laplace transform with C(0) = 0:
+
+ sC(s) \amp = -kC(s) + \frac{D}{V}
+ C(s) \amp = \frac{D/V}{s + k}
+
+
+
+
+ Inverse transform gives:
+
+ C(t) = \frac{D}{V}e^{-kt}
+
+
+
+
+ Case 2: Repeated Doses
+
+
+
+ Doses D given every \tau hours. After n doses, the concentration accumulates:
+
+ C(t) = \frac{D}{V}e^{-kt}\left(1 + e^{-k\tau} + e^{-2k\tau} + \cdots + e^{-(n-1)k\tau}\right)
+
+
+
+
+ At steady state (many doses), using geometric series:
+
+ C_{\text{max}} = \frac{D/V}{1 - e^{-k\tau}}, \quad C_{\text{min}} = C_{\text{max}}e^{-k\tau}
+
+
+
+
+
+ Simulation Activity
+
+
+ Objective: Determine optimal dosing regimen to maintain concentration in therapeutic range.
+
+
+
+ Given Parameters:
+
+
+ - Drug: Amoxicillin (common antibiotic)
+ - Elimination half-life: t_{1/2} = 1 hour (so k = \ln(2) \approx 0.693/hour)
+ - Volume of distribution: V = 30 L
+ - Therapeutic range: 5-20 mg/L
+ - Available doses: 250 mg or 500 mg
+
+
+
+ Tasks:
+
+
+ - Simulate a single 500 mg dose. Plot C(t) for 8 hours. How long does concentration stay in the therapeutic range?
+ - Test dosing regimens:
+
+ - 250 mg every 4 hours
+ - 500 mg every 6 hours
+ - 500 mg every 8 hours
+
+
+ - For each regimen, plot C(t) showing at least 5 doses.
+ - Calculate C_{\text{max}} and C_{\text{min}} at steady state for each regimen.
+ - Which regimen keeps concentration in the therapeutic window?
+
+
+
+ Advanced: Loading Dose
+
+
+
+ To reach steady state immediately, use a loading dose D_L followed by maintenance doses D_M. Calculate:
+
+ D_L = D_M \cdot \frac{1}{1 - e^{-k\tau}}
+
+
+
+
+ Simulate this strategy and compare with regular dosing.
+
+
+
+
+ Laplace Transform Application
+
+
+ Problem: Constant infusion I(t) = I_0 starting at t = 0, with C(0) = 0.
+
+
+
+ Solution using Laplace transform:
+
+
+ sC(s) \amp = -kC(s) + \frac{I_0}{s}
+ C(s) \amp = \frac{I_0}{s(s+k)}
+ \amp = \frac{I_0}{k}\left(\frac{1}{s} - \frac{1}{s+k}\right) \quad \text{(partial fractions)}
+
+
+
+ Inverse transform:
+
+ C(t) = \frac{I_0}{k}(1 - e^{-kt})
+
+
+
+
+ Steady-state concentration: C_{\infty} = I_0/k.
+
+
+
+ Exercise: For therapeutic range 5-20 mg/L and k = 0.693/hour, what infusion rate I_0 maintains C = 12 mg/L at steady state?
+
+
+
+
+ Analytical Questions
+
+
+ - Verify the solution C(t) = \frac{D}{V}e^{-kt} by substituting into the differential equation with impulse input.
+ - Show that the time to reach 95% of steady-state concentration during infusion is approximately 3/k.
+ - Derive the formula for C_{\text{max}} at steady state for repeated doses.
+ - How does increasing the dosing interval \tau affect the fluctuation between C_{\text{max}} and C_{\text{min}}?
+ - For a drug with longer half-life (t_{1/2} = 6 hours), how would the dosing schedule change?
+
+
+
+
+ Extensions and Applications
+
+
+ - Two-compartment model: Central (blood) and peripheral (tissue) compartments with exchange between them.
+ - Nonlinear kinetics: Saturable elimination (Michaelis-Menten kinetics) for drugs like phenytoin.
+ - Population pharmacokinetics: Variability in parameters across patients.
+ - Therapeutic drug monitoring: Using measured concentrations to adjust dosing.
+
+
+
+ Research one of these topics and explain how it complicates the modeling.
+
+
+
+
+ Final Report
+
+
+ Prepare a detailed report (3-4 pages) including:
+
+
+
+ - Derivation of the pharmacokinetic model with clear explanation of assumptions.
+ - Complete Laplace transform solutions for both impulse and infusion inputs.
+ - Simulation results for multiple dosing regimens with graphs showing concentration versus time.
+ - Analysis of which dosing regimen maintains therapeutic levels most effectively.
+ - Calculation of loading and maintenance doses for immediate steady state.
+ - Comparison of intermittent dosing versus continuous infusion.
+ - Answers to all analytical questions.
+ - Discussion of practical considerations (patient compliance, convenience, cost).
+ - Brief exploration of one extension topic.
+
+
+
+
+
+ The Laplace transform is particularly powerful for systems with discontinuous inputs, making it ideal for pharmacokinetics where drugs are administered as discrete doses. This mathematical framework enables rational drug dosing design, ensuring safety and efficacy. Understanding these models is crucial in pharmacy, medicine, and pharmaceutical research. The same techniques apply to chemical engineering (batch processes), environmental science (pollutant transport), and any system involving pulsed or interrupted inputs.
+
+
+
+
diff --git a/source/c11-ltm/ltm-model.ptx b/source/c11-ltm/ltm-model.ptx
new file mode 100644
index 00000000..cc90d82b
--- /dev/null
+++ b/source/c11-ltm/ltm-model.ptx
@@ -0,0 +1,306 @@
+
+
+ Solving Real-World Problems with the Laplace Transform Method
+
+
+
+ The Laplace transform method is particularly effective for solving initial value problems with complex or piecewise forcing functions. By transforming the problem into the s-domain, we can handle discontinuities and impulses that would be difficult to manage in the time domain.
+
+
+
+ In this section, we'll model an impact absorption system—relevant to automotive safety, packaging design, and protective equipment.
+
+
+
+
+ The Problem: Crash Test and Impact Absorption
+
+
+ When a vehicle crashes into a barrier, the impact creates a sudden force. The vehicle's crumple zones and safety systems are designed to extend the duration of the impact, reducing the peak force and protecting occupants.
+
+
+
+ How can we model and optimize impact absorption systems using differential equations?
+
+
+
+
+ Assumptions
+
+
+ - The vehicle's front can be modeled as a spring-damper system.
+ - The crash force is approximated as a rectangular pulse or exponential decay.
+ - Motion is one-dimensional (head-on collision).
+ - The vehicle mass is concentrated at a point.
+ - The barrier is rigid (fixed wall).
+
+
+
+
+ Building the Model
+
+
+ Let x(t) be the displacement of the vehicle's center of mass. The equation of motion is:
+
+ m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)
+
+
+
+
+ where F(t) represents the impact force.
+
+
+
+ Scenario 1: Rectangular Pulse
+
+
+
+ The impact force lasts for duration T:
+
+ F(t) = \begin{cases}
+ F_0 \amp 0 \leq t \leq T \\
+ 0 \amp t > T
+ \end{cases}
+
+
+
+
+ Using the unit step function u(t):
+
+ F(t) = F_0[u(t) - u(t-T)]
+
+
+
+
+ Scenario 2: Impulse (Instantaneous Impact)
+
+
+
+ For a very short impact (e.g., hammer blow):
+
+ F(t) = I\delta(t)
+
+ where I is the impulse (force × time) in N·s.
+
+
+
+ Parameters:
+
+
+ - m = 1500 kg: vehicle mass
+ - k: effective stiffness of crumple zone (N/m)
+ - c: damping coefficient (N·s/m)
+ - F_0: peak impact force (N)
+ - T: impact duration (s)
+
+
+
+
+ Solution Using Laplace Transforms
+
+
+ Example: Impulse Response
+
+
+
+ For F(t) = I\delta(t) with x(0) = 0, x'(0) = 0:
+
+
+
+ Taking Laplace transform:
+
+ m[s^2X(s)] + c[sX(s)] + kX(s) = I
+
+
+
+
+ Solving for X(s):
+
+ X(s) = \frac{I}{ms^2 + cs + k}
+
+
+
+
+ For underdamped case (c^2 < 4mk), using partial fractions and inverse transform:
+
+ x(t) = \frac{I}{m\omega_d}e^{-\zeta\omega_0 t}\sin(\omega_d t)
+
+ where \omega_d = \omega_0\sqrt{1-\zeta^2}.
+
+
+
+ Maximum Displacement:
+
+
+
+ The peak displacement occurs at t_{\text{max}} = \pi/\omega_d:
+
+ x_{\text{max}} = \frac{I}{m\omega_d}e^{-\pi\zeta/\sqrt{1-\zeta^2}}
+
+
+
+
+
+ Computational Activity
+
+
+ Objective: Compare impact responses for different system designs.
+
+
+
+ Base Parameters:
+
+
+ - m = 1500 kg
+ - v_0 = 15 m/s (impact speed, ~54 km/h)
+ - I = mv_0 = 22500 N·s (impulse)
+
+
+
+ Test Three Designs:
+
+
+ - Stiff System: k = 500000 N/m, c = 10000 N·s/m
+ - Soft System: k = 100000 N/m, c = 8000 N·s/m
+ - Optimized System: k = 200000 N/m, c = 15000 N·s/m
+
+
+
+ For each design:
+
+
+ - Calculate \omega_0, \zeta, and \omega_d
+ - Use Laplace transform to find x(t) (or use the formula above)
+ - Plot displacement and velocity for 0.5 seconds after impact
+ - Calculate maximum displacement and maximum deceleration
+ - Determine which design minimizes peak acceleration while keeping displacement reasonable
+
+
+
+ Safety Criterion: Peak deceleration should not exceed 50 g (490 m/s²).
+
+
+
+
+ Piecewise Forcing Activity
+
+
+ Scenario: A two-stage impact where the vehicle first crushes a soft bumper, then contacts a stiffer frame.
+
+
+
+ Force Profile:
+
+ F(t) = \begin{cases}
+ F_1 \amp 0 \leq t < t_1 \\
+ F_2 \amp t_1 \leq t < t_2 \\
+ 0 \amp t \geq t_2
+ \end{cases}
+
+
+
+
+ Express using step functions:
+
+ F(t) = F_1[u(t) - u(t-t_1)] + F_2[u(t-t_1) - u(t-t_2)]
+
+
+
+
+ Take Laplace transform and solve for X(s), then find x(t).
+
+
+
+ Tasks:
+
+
+ - Implement this two-stage model with F_1 = 50000 N, F_2 = 100000 N, t_1 = 0.05 s, t_2 = 0.1 s.
+ - Compare with a constant force equal to the average: F_{\text{avg}} = (F_1 + F_2)/2.
+ - Which profile results in lower peak acceleration?
+
+
+
+
+ Analytical Questions
+
+
+ - Verify the solution for impulse response by direct substitution into the differential equation (using properties of the delta function).
+ - Show that for a step input F(t) = F_0u(t), the steady-state displacement is F_0/k.
+ - Derive the relationship between impact duration T and peak force F_0 for a given impulse I = F_0 T.
+ - How does increasing damping affect maximum displacement? Is there an optimal \zeta?
+ - For a rigid barrier (infinite k), what happens to the solution? Does it make physical sense?
+
+
+
+
+ Design Challenge
+
+
+ Design a crash absorption system for a 1200 kg vehicle traveling at 60 km/h (16.7 m/s) such that:
+
+
+ - Peak deceleration ≤ 40 g
+ - Maximum intrusion (displacement) ≤ 0.6 m
+ - The system returns to near-equilibrium within 0.5 s
+
+
+
+ Determine appropriate values for k and c, and verify your design with simulations.
+
+
+
+
+ Real-World Context
+
+
+ Research modern vehicle safety systems:
+
+
+ - Crumple zones and energy absorption
+ - Airbag deployment timing
+ - Seatbelt pretensioners
+ - Crash test standards (NCAP ratings)
+
+
+
+ Explain how engineers use mathematical models to optimize these systems.
+
+
+
+
+ Final Report
+
+
+ Prepare a comprehensive report (4-5 pages) including:
+
+
+
+ - Derivation of the impact absorption model.
+ - Complete Laplace transform solution for both impulse and rectangular pulse inputs.
+ - Comparison of three design configurations with graphs and data tables.
+ - Analysis of two-stage impact model.
+ - Complete solution to the design challenge with justification of parameter choices.
+ - Answers to all analytical questions.
+ - Discussion of real vehicle safety systems and how they relate to the model.
+ - Limitations of the model and suggestions for improvement (e.g., nonlinear springs, multi-body dynamics).
+
+
+
+
+
+ The Laplace transform method excels at handling piecewise and impulsive forcing functions that arise in impact problems. This mathematical framework has been instrumental in automotive safety design, helping engineers create systems that protect lives. The same principles apply to packaging design, helmet design, earthquake protection for buildings, and any application where sudden forces must be managed. Understanding how spring stiffness, damping, and force profiles interact enables rational design decisions that balance competing objectives like maximum deceleration, intrusion distance, and cost.
+
+
+
+
diff --git a/source/c12-ltp/ltp-model.ptx b/source/c12-ltp/ltp-model.ptx
new file mode 100644
index 00000000..c5de40e7
--- /dev/null
+++ b/source/c12-ltp/ltp-model.ptx
@@ -0,0 +1,306 @@
+
+
+ Modeling Systems with Piecewise Forcing
+
+
+
+ Many real-world systems experience forcing that changes abruptly—switches turn on and off, loads are applied and removed, and control systems activate at specific times. Piecewise forcing functions model these discontinuous inputs, and the Laplace transform with unit step functions provides an elegant solution method.
+
+
+
+ In this section, we'll model a temperature control system with intermittent heating—a scenario found in thermostats, incubators, and industrial processes.
+
+
+
+
+ The Problem: Thermostat Control System
+
+
+ A room is heated by a system that turns on when temperature drops below a setpoint and off when it exceeds another setpoint. This on-off control creates a piecewise forcing function.
+
+
+
+ How does the room temperature evolve under this intermittent heating? Can we predict cycling behavior?
+
+
+
+
+ Assumptions
+
+
+ - Room temperature is uniform (well-mixed air).
+ - Heat loss to the outside is proportional to temperature difference (Newton's law of cooling).
+ - The heater provides constant power when on.
+ - Outside temperature remains constant.
+ - No internal heat sources (people, appliances, sunlight).
+
+
+
+
+ Building the Model
+
+
+ Let T(t) be the room temperature (°C) at time t (minutes). The rate of temperature change is:
+
+ C\frac{dT}{dt} = -k(T - T_{\text{out}}) + H(t)
+
+
+
+
+ where:
+
+
+ - C: thermal capacity (J/°C)
+ - k: heat loss coefficient (J/(min·°C))
+ - T_{\text{out}}: outside temperature (°C)
+ - H(t): heating rate (J/min), piecewise function
+
+
+
+ Simplifying by dividing by C:
+
+ \frac{dT}{dt} = -\alpha(T - T_{\text{out}}) + h(t)
+
+ where \alpha = k/C and h(t) = H(t)/C.
+
+
+
+ Piecewise Heating Schedule:
+
+
+
+ Scenario 1: Single On-Off Cycle
+
+
+
+ Heater on for interval [t_1, t_2]:
+
+ h(t) = h_0[u(t-t_1) - u(t-t_2)]
+
+
+
+
+ Scenario 2: Periodic Cycling
+
+
+
+ Heater on for time T_{\text{on}}, off for time T_{\text{off}}, repeating:
+
+ h(t) = h_0 \sum_{n=0}^{\infty} [u(t - nT) - u(t - nT - T_{\text{on}})]
+
+ where T = T_{\text{on}} + T_{\text{off}} is the period.
+
+
+
+ Parameters (Example):
+
+
+ - \alpha = 0.05 /min (time constant 20 min)
+ - T_{\text{out}} = 5°C
+ - h_0 = 2°C/min (heating rate)
+ - T(0) = 15°C
+
+
+
+
+ Solution Using Laplace Transforms
+
+
+ Example: Heater On from t_1 = 0 to t_2 = 10 min
+
+
+
+ The differential equation is:
+
+ \frac{dT}{dt} + \alpha T = \alpha T_{\text{out}} + h_0[u(t) - u(t-10)]
+
+
+
+
+ Taking Laplace transform with T(0) = 15:
+
+ s\mathcal{L}\{T\} - 15 + \alpha\mathcal{L}\{T\} \amp = \frac{\alpha T_{\text{out}}}{s} + h_0\left(\frac{1}{s} - \frac{e^{-10s}}{s}\right)
+ \mathcal{L}\{T\} \amp = \frac{15}{s+\alpha} + \frac{\alpha T_{\text{out}}}{s(s+\alpha)} + \frac{h_0(1-e^{-10s})}{s(s+\alpha)}
+
+
+
+
+ Using partial fractions and inverse transform yields:
+
+ T(t) = \begin{cases}
+ T_{\text{out}} + (15-T_{\text{out}})e^{-\alpha t} + \frac{h_0}{\alpha}(1-e^{-\alpha t}) \amp 0 \leq t < 10 \\
+ \text{(different expression)} \amp t \geq 10
+ \end{cases}
+
+
+
+
+
+ Simulation Activity
+
+
+ Objective: Explore temperature dynamics under various control strategies.
+
+
+
+ Part 1: Single Heating Cycle
+
+
+
+ - Simulate with parameters given above, heater on from t = 0 to t = 30 min.
+ - Plot temperature versus time for 0-120 minutes.
+ - What is the maximum temperature reached?
+ - After the heater turns off, how long until temperature drops back to initial value?
+
+
+
+ Part 2: Bang-Bang Control
+
+
+
+ Implement a simple thermostat:
+
+
+ - Turn heater ON when T < 18°C
+ - Turn heater OFF when T > 22°C
+ - Otherwise, maintain previous state
+
+
+
+ - Simulate for 300 minutes starting at T(0) = 15°C.
+ - Plot temperature and heater state (on/off) versus time.
+ - Does the system reach a steady oscillation? If so, what are the period and amplitude?
+ - What percentage of time is the heater on at steady state?
+
+
+
+ Part 3: Design Optimization
+
+
+
+ Desired: maintain temperature between 19°C and 21°C.
+
+
+
+ - Test different heating rates: h_0 = 1, 2, 3°C/min.
+ - For each, adjust the thermostat setpoints to minimize cycling frequency while staying in range.
+ - Which design provides the most stable temperature with minimal cycling?
+
+
+
+
+ Experimental Activity (Optional)
+
+
+ Simple Home Experiment:
+
+
+
+ Materials:
+
+
+ - Cup of hot water
+ - Thermometer
+ - Microwave or hot plate (for reheating)
+ - Timer
+
+
+
+ Procedure:
+
+
+ - Heat water to ~60°C and place in room temperature environment.
+ - Record temperature every 2 minutes as it cools.
+ - After 20 minutes, reheat briefly in microwave (10-20 seconds).
+ - Continue temperature measurements.
+ - Reheat again at 40 minutes.
+
+
+
+ Analysis:
+
+
+ - Plot temperature versus time.
+ - Identify the cooling time constant 1/\alpha from the exponential decay segments.
+ - Compare with a continuous heating model.
+
+
+
+
+ Analytical Questions
+
+
+ - For a step heating input starting at t=0, find the steady-state temperature (as t \to \infty).
+ - Show that after heating stops, the temperature decays exponentially toward T_{\text{out}}.
+ - For periodic heating with period T, derive the steady-state maximum and minimum temperatures.
+ - What is the optimal heating power h_0 to maintain average temperature at 20°C with 10% duty cycle (heater on 10% of time)?
+ - How does improving insulation (decreasing \alpha) affect cycling frequency?
+
+
+
+
+ Advanced Topic: Proportional Control
+
+
+ Instead of on-off control, use proportional heating:
+
+ h(t) = K_p(T_{\text{set}} - T(t))
+
+
+
+
+ This gives:
+
+ \frac{dT}{dt} = -(\alpha + K_p)T + \alpha T_{\text{out}} + K_p T_{\text{set}}
+
+
+
+
+ Tasks:
+
+
+ - Solve this equation for T(t).
+ - Find the steady-state temperature.
+ - Compare proportional control with bang-bang control: which is smoother? Which is more energy efficient?
+
+
+
+
+ Final Report
+
+
+ Submit a comprehensive report (4-5 pages) including:
+
+
+
+ - Derivation of the thermal model with piecewise heating.
+ - Solution using Laplace transforms for a single heating pulse.
+ - Simulation results for single heating cycle with analysis of temperature evolution.
+ - Bang-bang control simulation showing steady-state cycling behavior.
+ - Design optimization results comparing different heating rates and setpoints.
+ - If conducted, experimental data with parameter estimation.
+ - Answers to all analytical questions.
+ - Comparison of bang-bang versus proportional control strategies.
+ - Discussion of practical applications (HVAC systems, ovens, incubators, greenhouses).
+ - Recommendations for energy-efficient temperature control.
+
+
+
+
+
+ Piecewise forcing functions model the reality of intermittent control—switches, valves, and relays that turn on and off rather than operating continuously. The unit step function and Laplace transform provide powerful tools for analyzing these systems mathematically. Understanding how systems respond to discontinuous inputs is essential in control engineering, where trade-offs between stability, energy efficiency, and component wear must be balanced. The simple thermostat model illustrates principles that scale up to industrial process control, power grid management, and automated manufacturing.
+
+
+
+
diff --git a/source/c13-linsys/linsys-model.ptx b/source/c13-linsys/linsys-model.ptx
new file mode 100644
index 00000000..9b545d72
--- /dev/null
+++ b/source/c13-linsys/linsys-model.ptx
@@ -0,0 +1,257 @@
+
+
+ Modeling with Linear Systems
+
+
+
+ Many real-world systems involve multiple interacting components, each with its own state variable. Linear systems of differential equations model these coupled dynamics, where the rate of change of each variable depends on all the variables in the system.
+
+
+
+ In this section, we'll model predator-prey interactions—a classic ecological system where two populations influence each other's growth rates.
+
+
+
+
+ The Problem: Predator-Prey Dynamics
+
+
+ Consider a simplified ecosystem with rabbits (prey) and foxes (predators). Rabbits reproduce and provide food for foxes. Foxes need rabbits to survive and reproduce. How do these populations evolve over time?
+
+
+
+ This interaction can lead to oscillating population cycles—when rabbits are abundant, fox populations grow; as foxes increase, rabbit populations decline; with fewer rabbits, fox populations then decline; and the cycle repeats.
+
+
+
+
+ Assumptions
+
+
+ - Without predators, prey population grows exponentially.
+ - Predators encounter prey at a rate proportional to both populations.
+ - Without prey, predator population declines exponentially.
+ - The environment has unlimited resources (for a linear model).
+ - No age structure or spatial distribution.
+
+
+
+
+ Building the Model
+
+
+ Let R(t) = rabbit population and F(t) = fox population at time t (in months).
+
+
+
+ Linearized Lotka-Volterra Equations:
+
+
+
+ Near an equilibrium point, we can linearize the system:
+
+ \frac{dR}{dt} \amp = aR - bRF
+ \frac{dF}{dt} \amp = -cF + dRF
+
+
+
+
+ For small perturbations around equilibrium (R^*, F^*), let r = R - R^* and f = F - F^*:
+
+ \frac{dr}{dt} \amp = -bF^* r + aR^* f
+ \frac{df}{dt} \amp = dF^* r - cR^* f
+
+
+
+
+ Or in matrix form:
+
+ \frac{d}{dt}\begin{pmatrix} r \\ f \end{pmatrix} = \begin{pmatrix} -bF^* \amp -aR^* \\ dF^* \amp -cR^* \end{pmatrix}\begin{pmatrix} r \\ f \end{pmatrix}
+
+
+
+
+ Parameters (Example):
+
+
+ - a = 0.5: rabbit growth rate (/month)
+ - b = 0.02: predation rate (/(rabbit·month))
+ - c = 0.3: fox death rate (/month)
+ - d = 0.01: fox reproduction efficiency (/(fox·month))
+
+
+
+ Equilibrium: R^* = c/d = 30 rabbits, F^* = a/b = 25 foxes.
+
+
+
+
+ Solution via Eigenanalysis
+
+
+ The linearized system has coefficient matrix:
+
+ A = \begin{pmatrix} -0.5 \amp -15 \\ 0.25 \amp -15 \end{pmatrix}
+
+
+
+
+ Characteristic equation:
+
+ \det(A - \lambda I) = (\lambda + 0.5)(\lambda + 15) + 3.75 = 0
+
+
+
+
+ Solving gives eigenvalues (which may be complex), determining the nature of the equilibrium:
+
+
+ - Real, both negative: stable node (populations return to equilibrium)
+ - Real, opposite signs: saddle point (unstable)
+ - Complex with negative real part: stable spiral (damped oscillations)
+ - Pure imaginary: center (perpetual oscillations)
+
+
+
+
+ Computational Activity
+
+
+ Objective: Simulate predator-prey dynamics and analyze phase portraits.
+
+
+
+ Part 1: Time Series
+
+
+
+ - Solve the system numerically using the parameters above.
+ - Start with initial conditions: R(0) = 40, F(0) = 20.
+ - Plot R(t) and F(t) separately versus time for 0-50 months.
+ - Do the populations oscillate? If so, estimate the period.
+ - Do the oscillations grow, decay, or remain constant?
+
+
+
+ Part 2: Phase Portrait
+
+
+
+ - Plot F versus R (phase space trajectory).
+ - Mark the equilibrium point (R^*, F^*).
+ - Try several different initial conditions and plot their trajectories.
+ - Classify the equilibrium: node, spiral, center, or saddle?
+
+
+
+ Part 3: Parameter Exploration
+
+
+
+ - Increase predation efficiency: b = 0.03. How does this affect dynamics?
+ - Increase fox death rate: c = 0.4. How does the equilibrium shift?
+ - Can you find parameters that lead to population extinction (trajectory reaching zero)?
+
+
+
+
+ Data Collection Activity
+
+
+ Objective: Analyze real predator-prey data and fit a model.
+
+
+
+ Historical Data Set: Hudson Bay Company fur trading records (1900-1920) for lynx (predator) and hare (prey).
+
+
+
+ Tasks:
+
+
+ - Obtain or use provided data set of lynx and hare populations over time.
+ - Plot both populations versus time. Do you observe cyclical behavior?
+ - Estimate the period of oscillation.
+ - Plot hare vs. lynx in phase space. Does it form a closed loop?
+ - Estimate parameters a, b, c, d that roughly match the observed behavior.
+ - Simulate your model and compare with actual data.
+
+
+
+
+ Analytical Questions
+
+
+ - Find the equilibrium point (R^*, F^*) for the nonlinear Lotka-Volterra equations by setting both derivatives to zero.
+ - Calculate the eigenvalues of the linearized system matrix at equilibrium. What do they tell you about stability?
+ - Show that for the nonlinear system, the quantity E = dR - c\ln R + bF - a\ln F is conserved (constant along trajectories). What does this imply about the phase portrait?
+ - If the prey growth rate a doubles, how does the equilibrium fox population change?
+ - In the linearized model, under what parameter conditions would you expect:
+
+ - Both populations to return to equilibrium without oscillating?
+ - Damped oscillations?
+ - Sustained oscillations?
+
+
+
+
+
+
+ Extensions: More Complex Models
+
+
+ Modify the basic model to include:
+
+
+
+ - Prey carrying capacity: Replace aR with aR(1-R/K) (logistic growth)
+ - Alternative food source: Modify fox equation to allow survival without prey
+ - Multiple predators or prey: Extend to three or more species
+ - Harvesting: Add constant removal terms for hunting/fishing
+
+
+
+ Implement one extension and analyze how it changes the dynamics.
+
+
+
+
+ Final Report
+
+
+ Prepare a comprehensive report (4-5 pages) including:
+
+
+
+ - Derivation of the predator-prey system from biological principles.
+ - Linearization around the equilibrium point with full matrix formulation.
+ - Eigenvalue calculation and stability analysis.
+ - Time series plots showing population oscillations.
+ - Phase portraits for multiple initial conditions.
+ - Parameter exploration results with discussion of biological interpretation.
+ - Analysis of real lynx-hare data with model fitting.
+ - Answers to all analytical questions.
+ - Implementation and analysis of at least one model extension.
+ - Discussion of model limitations and suggestions for improvement.
+ - Brief review of other systems modeled by coupled linear equations (chemical reactions, competing species, disease spread, economics).
+
+
+
+
+
+ Linear systems of differential equations reveal the rich dynamics that emerge from coupled interactions. The predator-prey model demonstrates how simple rules—prey reproduce, predators consume prey—can generate complex oscillatory behavior. This mathematical framework extends far beyond ecology: coupled chemical reactions, multi-compartment pharmacokinetics, interacting markets in economics, and coupled oscillators in physics all follow similar structures. Understanding eigenvalues, phase portraits, and stability provides insight into whether systems converge to equilibrium, oscillate indefinitely, or spiral out of control. These tools are fundamental in biology, engineering, economics, and any field where multiple processes interact dynamically.
+
+
+
+
diff --git a/source/c14-nlinsys/nlinsys-model.ptx b/source/c14-nlinsys/nlinsys-model.ptx
new file mode 100644
index 00000000..1bcd9b0a
--- /dev/null
+++ b/source/c14-nlinsys/nlinsys-model.ptx
@@ -0,0 +1,290 @@
+
+
+ Modeling with Nonlinear Systems
+
+
+
+ Nonlinear systems exhibit behaviors impossible in linear systems: multiple equilibria, limit cycles, chaos, and extreme sensitivity to initial conditions. While harder to solve analytically, nonlinear models often capture reality more accurately than their linear approximations.
+
+
+
+ In this section, we'll model an epidemic outbreak using the SIR model—a nonlinear system that reveals how diseases spread through populations and how interventions can control outbreaks.
+
+
+
+
+ The Problem: Disease Outbreak and Control
+
+
+ When an infectious disease emerges in a population, public health officials must predict its spread and evaluate intervention strategies. Will it cause an epidemic? How many will be affected? What vaccination level prevents an outbreak?
+
+
+
+ The SIR (Susceptible-Infected-Recovered) model divides the population into three groups and tracks how individuals move between them.
+
+
+
+
+ Assumptions
+
+
+ - The population is well-mixed (everyone has equal contact probability).
+ - Total population is constant (no births, deaths, or migration).
+ - Infection occurs through contact between susceptible and infected individuals.
+ - Recovery confers permanent immunity.
+ - The disease has a constant transmission rate and recovery rate.
+
+
+
+
+ Building the Model
+
+
+ Let:
+
+
+ - S(t) = number of susceptible individuals
+ - I(t) = number of infected individuals
+ - R(t) = number of recovered individuals
+ - N = S + I + R = total population (constant)
+
+
+
+ SIR Differential Equations:
+
+
+ \frac{dS}{dt} \amp = -\frac{\beta SI}{N}
+ \frac{dI}{dt} \amp = \frac{\beta SI}{N} - \gamma I
+ \frac{dR}{dt} \amp = \gamma I
+
+
+
+ Parameters:
+
+
+ - \beta: transmission rate (contacts per day × probability of transmission per contact)
+ - \gamma: recovery rate (1/day), so 1/\gamma is the average infectious period
+ - R_0 = \beta/\gamma: basic reproduction number (average number of new infections from one infected person)
+
+
+
+ Key Insight: An epidemic occurs (infections increase) only if R_0 > 1 and S(0) is large enough.
+
+
+
+ Threshold Condition: Infections increase when:
+
+ \frac{dI}{dt} > 0 \quad \Rightarrow \quad \frac{\beta S}{N} > \gamma \quad \Rightarrow \quad S > \frac{N}{R_0}
+
+
+
+
+
+ Phase Plane Analysis
+
+
+ Since S + I + R = N is constant, we can eliminate R and analyze the (S,I) phase plane.
+
+
+
+ Nullclines:
+
+
+ - \frac{dS}{dt} = 0 when I = 0 or S = 0
+ - \frac{dI}{dt} = 0 when I = 0 or S = N/R_0
+
+
+
+ The vertical line S = N/R_0 separates regions where I increases (left side, epidemic) from regions where I decreases (right side, no epidemic).
+
+
+
+
+ Simulation Activity
+
+
+ Objective: Explore epidemic dynamics and intervention strategies.
+
+
+
+ Scenario: Flu Outbreak in a Town
+
+
+
+ Base Parameters:
+
+
+ - N = 10000 people
+ - \beta = 0.5 /day (so R_0 = 2.5 if \gamma = 0.2)
+ - \gamma = 0.2 /day (infectious period = 5 days)
+ - Initial: S(0) = 9990, I(0) = 10, R(0) = 0
+
+
+
+ Part 1: Natural Progression
+
+
+
+ - Simulate the system for 100 days.
+ - Plot S(t), I(t), and R(t) on the same graph.
+ - When does the infection peak? What is the maximum number of infected individuals?
+ - What fraction of the population is eventually infected (final R value)?
+ - Create a phase plane plot (I vs. S).
+
+
+
+ Part 2: Vaccination Campaign
+
+
+
+ Vaccinating a fraction p of the population reduces initial susceptibles: S(0) = N(1-p), R(0) = Np.
+
+
+
+ - Test vaccination levels: p = 0, 0.2, 0.4, 0.6, 0.8.
+ - For each, simulate and record the peak infection level and final attack rate (total infected).
+ - Calculate the herd immunity threshold: p_c = 1 - 1/R_0. Does this match your simulations?
+ - Plot peak infections versus vaccination coverage.
+
+
+
+ Part 3: Social Distancing
+
+
+
+ Reducing contact rate decreases \beta. Model intervention starting at day 30:
+
+
+
+ - Reduce \beta from 0.5 to 0.3 at t = 30.
+ - Compare with no intervention.
+ - How much does this "flatten the curve"?
+ - What if intervention starts at day 10 instead? Day 50?
+
+
+
+
+ Real-World Data Analysis
+
+
+ Objective: Fit SIR model to actual outbreak data.
+
+
+
+ Data Sources:
+
+
+ - COVID-19 data from a specific region (early outbreak phase)
+ - Historical flu pandemic data
+ - Measles outbreak in an unvaccinated community
+
+
+
+ Tasks:
+
+
+ - Obtain cumulative case data (or infected count) over time.
+ - Estimate population size N.
+ - Estimate \gamma from known infectious period for the disease.
+ - Fit \beta by comparing model simulations with data (trial and error or optimization).
+ - Calculate R_0 from fitted parameters.
+ - Plot data and model fit together.
+ - Discuss goodness of fit and model limitations.
+
+
+
+
+ Analytical Questions
+
+
+ - Show that \frac{d}{dt}(S+I+R) = 0, confirming constant total population.
+ - Prove that S(t) is always decreasing (never increases). What is the physical interpretation?
+ - Derive the final size equation: R_\infty = N - S_\infty where S_\infty satisfies S_\infty = S_0 e^{-R_0(R_\infty/N)}.
+ - For R_0 = 2.5, calculate the herd immunity threshold 1 - 1/R_0. Why is this the minimum vaccination coverage needed?
+ - How does doubling the infectious period (1/\gamma) affect R_0 and outbreak dynamics?
+ - The SIR model has no equilibrium with I > 0. Why does this mean epidemics are transient, not sustained?
+
+
+
+
+ Extensions: More Complex Models
+
+
+ Explore variations of the SIR model:
+
+
+
+ - SEIR model: Add exposed (E) compartment for latent period before infectiousness
+ - SIRS model: Allow temporary immunity (R individuals become S again)
+ - Age-structured model: Different transmission rates for different age groups
+ - Spatial model: Track disease spread across connected regions
+ - Stochastic effects: Include randomness in small populations
+
+
+
+ Implement one extension and analyze how it changes predictions.
+
+
+
+
+ Policy Analysis Activity
+
+
+ You are a public health advisor. A new disease with R_0 = 3 and 7-day infectious period emerges.
+
+
+
+ - What vaccination coverage is needed for herd immunity?
+ - If only 50% coverage is achievable, what contact reduction (decrease in \beta) is needed to compensate?
+ - Compare two strategies:
+
+ - Strategy A: 60% vaccination, no other measures
+ - Strategy B: 40% vaccination + 30% reduction in contacts
+
+ Which prevents more infections?
+
+ - What is the optimal timing for interventions if they can only be sustained for 60 days?
+
+
+
+
+ Final Report
+
+
+ Submit a comprehensive report (5-6 pages) including:
+
+
+
+ - Derivation of the SIR model from first principles.
+ - Analysis of equilibria and phase plane structure.
+ - Derivation of the epidemic threshold and R_0 interpretation.
+ - Simulation results for natural progression with detailed graphs.
+ - Analysis of vaccination campaign effectiveness with herd immunity calculations.
+ - Social distancing intervention results showing timing effects.
+ - Real-world data fitting with parameter estimates and discussion of fit quality.
+ - Answers to all analytical questions with complete derivations.
+ - Implementation and analysis of at least one model extension.
+ - Policy recommendations from the policy analysis activity with justifications.
+ - Critical discussion of model assumptions and limitations.
+ - Reflection on the role of mathematical modeling in public health decision-making.
+
+
+
+
+
+ Nonlinear systems capture phenomena that linear models cannot: thresholds, tipping points, and irreversible transitions. The SIR model demonstrates how mathematical analysis—even when exact solutions are unavailable—can guide critical decisions affecting millions of lives. Understanding R_0, herd immunity thresholds, and intervention timing has been crucial in managing diseases from smallpox to COVID-19. This framework extends beyond epidemiology to any system with thresholds: chemical reactions, opinion dynamics, technology adoption, financial contagion, and neural network activation. Nonlinear dynamics reveal that small changes in parameters or initial conditions can lead to dramatically different outcomes—a lesson essential for navigating complex systems in an uncertain world.
+
+
+
+
diff --git a/source/c2-solns/solns-model.ptx b/source/c2-solns/solns-model.ptx
new file mode 100644
index 00000000..f6d3fd74
--- /dev/null
+++ b/source/c2-solns/solns-model.ptx
@@ -0,0 +1,187 @@
+
+
+ Modeling and Verifying Solutions
+
+
+
+ Understanding what it means for a function to be a solution to a differential equation is crucial for modeling. In real-world applications, we often have data or observations and need to verify whether a proposed mathematical model actually describes the phenomenon we're studying.
+
+
+
+ In this section, we'll model a classic physics problem and learn how to verify that our solution makes sense both mathematically and physically.
+
+
+
+
+ The Problem: Free Fall with Air Resistance
+
+
+ When an object falls through air, two forces act on it:
+
+
+ - Gravity: F_g = mg (downward)
+ - Air resistance: F_r = -bv (upward, proportional to velocity)
+
+
+
+ At low speeds, air resistance is approximately proportional to velocity. We want to model how the velocity changes as the object falls.
+
+
+
+
+ Assumptions
+
+
+ - The object starts from rest at t = 0.
+ - Air resistance is proportional to velocity: F_r = bv where b > 0.
+ - The acceleration due to gravity is constant: g = 9.8 \text{ m/s}^2.
+ - The object's mass m remains constant.
+ - We ignore other factors like wind, buoyancy, or changes in air density.
+
+
+
+
+ Building the Model
+
+
+ Let v(t) be the velocity of the object at time t (positive downward). By Newton's second law:
+
+ m\frac{dv}{dt} = mg - bv
+
+
+
+
+ Dividing by m:
+
+ \frac{dv}{dt} = g - \frac{b}{m}v
+
+
+
+
+ Parameters:
+
+
+ - g = 9.8 \text{ m/s}^2: gravitational acceleration
+ - m: mass of the object (kg)
+ - b: drag coefficient (kg/s), depends on object shape and air density
+ - v_{\text{terminal}} = \frac{mg}{b}: the terminal velocity (maximum velocity reached)
+
+
+
+ As time goes on, the velocity approaches v_{\text{terminal}}, where the forces balance and \frac{dv}{dt} = 0.
+
+
+
+
+ Verification Activity: Coffee Filter Drop
+
+
+ Objective: Collect data on falling objects and verify whether our model accurately predicts their motion.
+
+
+
+ Materials:
+
+
+ - Coffee filters (identical, flat-bottom type)
+ - Stopwatch or video recording device
+ - Measuring tape or meter stick
+ - Optional: motion sensor or app for tracking position
+
+
+
+ Procedure:
+
+
+ - Drop a single coffee filter from a height of 2-3 meters and time how long it takes to hit the ground.
+ - Repeat 5 times and calculate the average fall time.
+ - Repeat the experiment with 2, 3, 4, and 5 stacked coffee filters (increasing mass while keeping shape constant).
+ - For each stack size, calculate the average velocity: v_{\text{avg}} = \frac{\text{height}}{\text{time}}.
+ - Record all data in a table: Number of filters, Mass, Height, Time, Average velocity.
+
+
+
+ Analysis Questions:
+
+
+ - Does a single coffee filter appear to reach terminal velocity during its fall? How can you tell?
+ - Plot average velocity versus mass. What pattern emerges?
+ - The terminal velocity formula is v_{\text{terminal}} = \frac{mg}{b}. If b is approximately constant (same shape for all stacks), how should terminal velocity depend on mass?
+ - Estimate the drag coefficient b from your data by assuming the coffee filters reached terminal velocity.
+ - How would the motion differ if you dropped the filters on the Moon (no air)?
+
+
+
+
+ Solution Verification
+
+
+ The general solution to our differential equation \frac{dv}{dt} = g - \frac{b}{m}v is:
+
+ v(t) = \frac{mg}{b}\left(1 - e^{-\frac{b}{m}t}\right)
+
+
+
+
+ Let's verify this is indeed a solution:
+
+
+
+ Verification Steps:
+
+
+ - Compute the derivative: \frac{dv}{dt} = \frac{mg}{b} \cdot \frac{b}{m}e^{-\frac{b}{m}t} = ge^{-\frac{b}{m}t}
+ - Substitute into the right side: g - \frac{b}{m}v = g - \frac{b}{m} \cdot \frac{mg}{b}(1 - e^{-\frac{b}{m}t}) = g - g(1 - e^{-\frac{b}{m}t}) = ge^{-\frac{b}{m}t}
+ - Since both sides equal ge^{-\frac{b}{m}t}, the solution is verified.
+ - Check initial condition: v(0) = \frac{mg}{b}(1 - e^0) = 0 ✓
+ - Check long-term behavior: As t \to \infty, e^{-\frac{b}{m}t} \to 0, so v(t) \to \frac{mg}{b} = v_{\text{terminal}} ✓
+
+
+
+
+ Analytical Questions
+
+
+ - What happens to the solution if b = 0 (no air resistance)? Does this match free fall?
+ - For what value of t does the velocity reach 95% of terminal velocity?
+ - How would doubling the mass affect the terminal velocity? The time to reach terminal velocity?
+ - If you drop two objects of the same mass but different shapes, which will have a higher terminal velocity? Why?
+ - Compare coffee filter data with the theoretical solution. What factors might explain any discrepancies?
+
+
+
+
+ Final Report
+
+
+ Prepare a report (2-3 pages) including:
+
+
+
+ - A derivation of the differential equation from Newton's laws.
+ - Your experimental data presented clearly with graphs.
+ - An estimate of the drag coefficient b based on your measurements.
+ - A verification that the proposed solution satisfies the differential equation and initial conditions.
+ - A comparison between your experimental results and the theoretical predictions.
+ - A discussion of the model's limitations and how it could be improved.
+
+
+
+
+
+ Verifying solutions is a critical skill in modeling. It ensures that our mathematical expressions actually represent the physical reality we're trying to describe. This example of free fall with air resistance shows how differential equations can capture the balance between competing forces and how solutions can be validated both analytically and experimentally.
+
+
+
+
diff --git a/source/c3-di/di-model.ptx b/source/c3-di/di-model.ptx
new file mode 100644
index 00000000..e6291376
--- /dev/null
+++ b/source/c3-di/di-model.ptx
@@ -0,0 +1,198 @@
+
+
+ Modeling with Direct Integration
+
+
+
+ Direct integration is our first systematic method for solving differential equations. It applies when the rate of change depends only on the independent variable, not on the function itself. While this might seem restrictive, many important real-world phenomena can be modeled this way.
+
+
+
+ In this section, we'll model projectile motion, a classic problem where direct integration naturally applies.
+
+
+
+
+ The Problem: Projectile Motion
+
+
+ When you throw a ball into the air, its motion is governed by gravity. If we ignore air resistance, the acceleration is constant and equals g = 9.8 \text{ m/s}^2 downward.
+
+
+
+ Can we predict the ball's height and velocity at any time, given its initial conditions?
+
+
+
+
+ Assumptions
+
+
+ - Air resistance is negligible.
+ - Gravitational acceleration is constant: g = 9.8 \text{ m/s}^2.
+ - The motion is vertical (one-dimensional).
+ - Initial height is h_0 and initial velocity is v_0 (positive upward).
+
+
+
+
+ Building the Model
+
+
+ Let h(t) represent the height at time t, and v(t) = \frac{dh}{dt} the velocity. The acceleration is:
+
+ a(t) = \frac{dv}{dt} = -g
+
+
+
+
+ This is a first-order differential equation for velocity. Since the right side depends only on time (in fact, it's constant), we can solve by direct integration:
+
+ \int \frac{dv}{dt}\ dt \amp = \int -g\ dt
+ v(t) \amp = -gt + C_1
+
+
+
+
+ Using the initial condition v(0) = v_0:
+
+ v(t) = v_0 - gt
+
+
+
+
+ Now v(t) = \frac{dh}{dt}, so we have another first-order equation:
+
+ \frac{dh}{dt} = v_0 - gt
+
+
+
+
+ Integrating again:
+
+ \int \frac{dh}{dt}\ dt \amp = \int (v_0 - gt)\ dt
+ h(t) \amp = v_0t - \frac{1}{2}gt^2 + C_2
+
+
+
+
+ Using h(0) = h_0:
+
+ h(t) = h_0 + v_0t - \frac{1}{2}gt^2
+
+
+
+
+ Parameters:
+
+
+ - g = 9.8 \text{ m/s}^2: gravitational acceleration
+ - h_0: initial height (m)
+ - v_0: initial velocity (m/s, positive upward)
+
+
+
+
+ Data Collection Activity: Ball Toss
+
+
+ Objective: Measure projectile motion and compare with the theoretical model.
+
+
+
+ Materials:
+
+
+ - A ball (tennis ball, baseball, etc.)
+ - Video recording device (smartphone camera works well)
+ - Measuring tape or meter stick
+ - Video analysis software or app (e.g., Tracker, PhyPhox, or manual frame-by-frame analysis)
+
+
+
+ Procedure:
+
+
+ - Set up a camera on a tripod to record vertical motion against a measured background (e.g., a wall with marked heights).
+ - Toss the ball straight up and record the motion.
+ - Using video analysis, track the ball's position at equal time intervals (e.g., every 0.1 seconds).
+ - Create a data table: Time (s), Height (m), Velocity (m/s - estimated from consecutive positions).
+ - Repeat 2-3 times for reliability.
+
+
+
+ Analysis Questions:
+
+
+ - Plot height versus time. Does it follow a parabolic path?
+ - Plot velocity versus time. Is it linear?
+ - From your velocity plot, estimate the acceleration (slope of the line). How does it compare to g = 9.8 \text{ m/s}^2?
+ - At what time does the ball reach its maximum height? What is the velocity at that instant?
+ - Use your position data to estimate the initial height h_0 and initial velocity v_0.
+ - Using these values, plot the theoretical curve h(t) = h_0 + v_0t - \frac{1}{2}gt^2 on the same graph as your data. How well do they match?
+
+
+
+
+ Analytical Questions
+
+
+ - If the ball is thrown with twice the initial velocity, how does this affect the maximum height reached? The time to reach maximum height?
+ - Derive a formula for the maximum height in terms of v_0 and g.
+ - If the ball is thrown downward (negative v_0), how does the model change?
+ - What happens to the solution if we include air resistance? Would direct integration still work?
+ - Explain why this problem requires integrating twice. What does each integration represent physically?
+
+
+
+
+ Extensions: Horizontal Projectile Motion
+
+
+ For a ball thrown at an angle, we can decompose the motion into horizontal and vertical components:
+
+
+ - Horizontal: \frac{d^2x}{dt^2} = 0 (constant velocity)
+ - Vertical: \frac{d^2y}{dt^2} = -g
+
+
+
+ Both equations can be solved by direct integration!
+
+
+
+
+ Final Report
+
+
+ Write a report (2-3 pages) that includes:
+
+
+
+ - Derivation of the position and velocity equations using direct integration.
+ - Your experimental setup description with diagrams or photos.
+ - Data tables and graphs comparing experimental data with theoretical predictions.
+ - Analysis of how well the model fits reality, including discussion of any discrepancies.
+ - Answers to the analytical questions.
+ - A brief exploration of what happens when assumptions are violated (e.g., air resistance).
+
+
+
+
+
+ Direct integration is powerful for problems where acceleration (or any higher derivative) is a known function of time. Projectile motion is the quintessential example, and the model we developed has been used for centuries—from predicting cannon trajectories to landing spacecraft. The success of this simple model, despite ignoring many factors, demonstrates the value of well-chosen assumptions in mathematical modeling.
+
+
+
+
diff --git a/source/c4-sov/sov-model.ptx b/source/c4-sov/sov-model.ptx
new file mode 100644
index 00000000..14a7770e
--- /dev/null
+++ b/source/c4-sov/sov-model.ptx
@@ -0,0 +1,209 @@
+
+
+ Modeling with Separation of Variables
+
+
+
+ Separation of variables is one of the most versatile techniques for solving first-order differential equations. It applies when the rate of change can be expressed as a product of separate functions of the independent and dependent variables. This structure appears naturally in many growth and decay processes.
+
+
+
+ In this section, we'll model population growth using the logistic equation, a classic example where separation of variables provides insight into how populations evolve over time.
+
+
+
+
+ The Problem: Bacterial Growth with Limited Resources
+
+
+ Consider a bacterial colony growing in a petri dish with limited nutrients. Initially, when the population is small, growth is approximately exponential. However, as resources become scarce and the dish becomes crowded, the growth rate slows down.
+
+
+
+ How can we model this transition from exponential growth to a steady state?
+
+
+
+
+ Assumptions
+
+
+ - The growth rate is proportional to both the current population and the remaining capacity.
+ - There is a maximum sustainable population (carrying capacity) K.
+ - The environment provides constant resources per unit time.
+ - There is no immigration or emigration.
+ - All individuals have equal reproductive potential.
+
+
+
+
+ Building the Model
+
+
+ Let P(t) represent the population at time t. The rate of change \frac{dP}{dt} should:
+
+
+ - Increase with P (more individuals produce more offspring)
+ - Decrease as P approaches K (limited resources slow growth)
+
+
+
+ The logistic equation captures this behavior:
+
+ \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)
+
+
+
+
+ This can be rewritten as:
+
+ \frac{dP}{dt} = \frac{r}{K}P(K - P)
+
+
+
+
+ Parameters:
+
+
+ - r > 0: intrinsic growth rate (growth rate when population is small)
+ - K > 0: carrying capacity (maximum sustainable population)
+ - P_0: initial population at t = 0
+
+
+
+ Solving by Separation of Variables:
+
+
+
+ Separate the variables:
+
+ \frac{dP}{P(K-P)} = \frac{r}{K}\ dt
+
+
+
+
+ Using partial fractions on the left side and integrating both sides yields:
+
+ P(t) = \frac{K}{1 + Ae^{-rt}}
+
+ where A = \frac{K - P_0}{P_0}.
+
+
+
+
+ Lab Activity: Yeast Population Growth
+
+
+ Objective: Observe logistic growth in a yeast culture and estimate model parameters.
+
+
+
+ Materials:
+
+
+ - Active dry yeast
+ - Sugar solution (nutrient source)
+ - Test tubes or small bottles
+ - Hemocytometer or turbidity meter (or use visual density scale)
+ - Warm water bath (optimal: 30-35°C)
+ - Timer
+
+
+
+ Procedure:
+
+
+ - Prepare a sugar solution: dissolve 5g sugar in 100mL warm water.
+ - Add 1g active dry yeast and mix well.
+ - Divide into multiple test tubes (for repeated measurements).
+ - Every 30 minutes for 4-6 hours, measure the yeast density (cell count or turbidity).
+ - Record: Time (hours), Population measure (cells/mL or turbidity units).
+
+
+
+ Alternative Simplified Version:
+
+
+ Use provided data from a controlled experiment (data set available from instructor).
+
+
+
+ Data Analysis:
+
+
+ - Plot population versus time. Describe the curve's shape.
+ - Does the population appear to level off? Estimate the carrying capacity K from your graph.
+ - In the early phase (first 2-3 measurements), does growth appear exponential? Estimate the initial growth rate r.
+ - Compute \ln\left(\frac{K-P_0}{P_0}\right) - rt for each data point. If the model is correct, this should equal \ln\left(\frac{K-P}{P}\right).
+ - Plot \ln\left(\frac{K-P}{P}\right) versus t. Is it approximately linear? The slope should be -r.
+
+
+
+
+ Analytical Questions
+
+
+ - What happens to P(t) as t \to \infty? Verify this using both the differential equation and the solution.
+ - Find the time at which the population reaches half the carrying capacity (P = K/2).
+ - At what population level is the growth rate \frac{dP}{dt} maximized? (Hint: find \frac{d}{dP}\left[\frac{dP}{dt}\right] and set it to zero.)
+ - If the initial population P_0 > K (starting above carrying capacity), what happens? Sketch the solution.
+ - How does doubling r affect the time to reach carrying capacity? How does doubling K affect it?
+ - Compare the logistic model to exponential growth (\frac{dP}{dt} = rP). When are they similar? When do they differ significantly?
+
+
+
+
+ Extensions: Real-World Applications
+
+
+ The logistic model appears in many contexts:
+
+
+ - Epidemiology: spread of disease in a population
+ - Ecology: fish populations in a lake, predator-prey dynamics
+ - Technology adoption: how innovations spread through a market
+ - Chemical reactions: autocatalytic processes
+
+
+
+ Research one of these applications and explain how the logistic equation applies.
+
+
+
+
+ Final Report
+
+
+ Submit a comprehensive report (3-4 pages) including:
+
+
+
+ - Derivation of the logistic equation from the stated assumptions.
+ - Complete solution using separation of variables (show all steps).
+ - Your experimental data with clear graphs and analysis.
+ - Parameter estimation: values of r, K, and P_0 with explanation of methods used.
+ - Comparison between your data and the theoretical logistic curve (include both on the same graph).
+ - Answers to analytical questions with detailed explanations.
+ - Discussion of model limitations and factors not accounted for (e.g., death rate, mutations, environmental changes).
+ - Brief exploration of one real-world application of the logistic model.
+
+
+
+
+
+ The logistic equation demonstrates the power of separation of variables to solve differential equations that model real-world constraints. Unlike exponential growth, which continues indefinitely, logistic growth captures the realistic scenario where growth eventually stabilizes. This model has been fundamental in understanding everything from bacterial colonies to human populations, and illustrates how a simple differential equation can capture complex behavior.
+
+
+
+
diff --git a/source/c5-if/if-model.ptx b/source/c5-if/if-model.ptx
new file mode 100644
index 00000000..c780f8d5
--- /dev/null
+++ b/source/c5-if/if-model.ptx
@@ -0,0 +1,209 @@
+
+
+ Modeling with Integrating Factors
+
+
+
+ The integrating factor method extends our ability to solve first-order linear differential equations beyond what separation of variables can handle. This technique is particularly useful when modeling systems where the rate of change depends linearly on the quantity itself plus an external forcing function.
+
+
+
+ In this section, we'll model a classic electrical circuit problem that naturally leads to a first-order linear differential equation.
+
+
+
+
+ The Problem: RC Circuit Charging
+
+
+ Consider a simple electrical circuit consisting of a resistor (R) and a capacitor (C) connected to a voltage source V_s(t). When the circuit is connected, current flows and the capacitor charges.
+
+
+
+ How does the voltage across the capacitor change over time?
+
+
+
+
+ Assumptions
+
+
+ - The resistor has constant resistance R (ohms, \Omega).
+ - The capacitor has constant capacitance C (farads, F).
+ - The voltage source V_s(t) can vary with time.
+ - The circuit operates at low frequencies where ideal component behavior holds.
+ - There are no other circuit elements (e.g., inductors).
+
+
+
+
+ Building the Model
+
+
+ Let V_C(t) be the voltage across the capacitor at time t. By Kirchhoff's voltage law, the sum of voltage drops around the circuit equals the source voltage:
+
+ V_s(t) = V_R + V_C
+
+
+
+
+ The voltage across the resistor is V_R = IR where I is the current. For a capacitor, I = C\frac{dV_C}{dt}. Substituting:
+
+ V_s(t) = RC\frac{dV_C}{dt} + V_C
+
+
+
+
+ Rearranging to standard form:
+
+ \frac{dV_C}{dt} + \frac{1}{RC}V_C = \frac{1}{RC}V_s(t)
+
+
+
+
+ This is a first-order linear differential equation with p(t) = \frac{1}{RC} and q(t) = \frac{1}{RC}V_s(t).
+
+
+
+ Parameters:
+
+
+ - R: resistance (typically 1-1000 k\Omega)
+ - C: capacitance (typically 1-1000 \mu F)
+ - \tau = RC: time constant (seconds), characterizes how quickly the capacitor charges
+ - V_s(t): source voltage (volts)
+
+
+
+ Solution for Constant Voltage Source:
+
+
+
+ For V_s(t) = V_0 (constant), using integrating factor \mu(t) = e^{t/(RC)}:
+
+ V_C(t) = V_0\left(1 - e^{-t/(RC)}\right) + V_C(0)e^{-t/(RC)}
+
+
+
+
+ If V_C(0) = 0 (capacitor initially uncharged):
+
+ V_C(t) = V_0\left(1 - e^{-t/\tau}\right)
+
+
+
+
+
+ Lab Activity: RC Circuit Experiment
+
+
+ Objective: Build an RC circuit, measure the charging behavior, and estimate the time constant.
+
+
+
+ Materials:
+
+
+ - Breadboard and wires
+ - Resistors (e.g., 1 k\Omega, 10 k\Omega)
+ - Capacitors (e.g., 100 \mu F, 470 \mu F)
+ - DC power supply or battery (5-9V)
+ - Multimeter or oscilloscope
+ - Switch
+
+
+
+ Procedure:
+
+
+ - Build the circuit: Connect resistor and capacitor in series with the voltage source and switch.
+ - Set the voltage source to a known value (e.g., V_0 = 5V).
+ - Discharge the capacitor completely by shorting it briefly (carefully!).
+ - Close the switch and immediately start timing.
+ - Measure and record V_C every 5-10 seconds for 1-2 minutes (or until V_C approaches V_0).
+ - Repeat with different combinations of R and C.
+
+
+
+ Safety Note: Use low voltages (5-12V). Discharge capacitors before handling.
+
+
+
+ Data Analysis:
+
+
+ - Plot V_C versus time for each R-C combination.
+ - For each trial, find the time when V_C = 0.632 V_0 (this equals \tau). Compare to \tau = RC.
+ - Transform the data: plot \ln(V_0 - V_C) versus t. This should be linear with slope -1/\tau.
+ - Use the slope to estimate \tau and compare with the theoretical value.
+ - How does changing R affect the charging time? How about changing C?
+
+
+
+
+ Analytical Questions
+
+
+ - Verify that the solution V_C(t) = V_0(1 - e^{-t/\tau}) satisfies the differential equation and initial condition V_C(0) = 0.
+ - What percentage of the final voltage is reached after one time constant (t = \tau)? After 2\tau? After 5\tau?
+ - If the voltage source suddenly switches off at time t_1, what differential equation governs the discharge? Solve it.
+ - For a sinusoidal voltage source V_s(t) = V_0\sin(\omega t), the differential equation becomes nonhomogeneous. What would you expect the steady-state solution to look like?
+ - In what real-world applications would you want a large time constant? A small one?
+
+
+
+
+ Extensions: Applications
+
+
+ RC circuits appear in many applications:
+
+
+ - Timing circuits: Used in electronics to create delays (e.g., camera flashes, LED blinkers).
+ - Filters: Block or pass certain frequency components of signals.
+ - Smoothing: Convert pulsed DC to steady DC (power supplies).
+ - Neural modeling: The charging/discharging of neurons is often modeled with RC circuit equations.
+
+
+
+ Choose one application and research how the RC circuit model is used in practice.
+
+
+
+
+ Final Report
+
+
+ Prepare a detailed report (3-4 pages) including:
+
+
+
+ - Derivation of the RC circuit differential equation from Kirchhoff's laws.
+ - Complete solution using the integrating factor method (show all steps).
+ - Circuit diagrams and photos of your experimental setup.
+ - Data tables and graphs for each R-C combination tested.
+ - Analysis comparing experimental time constants with theoretical predictions.
+ - Discussion of sources of error and model limitations.
+ - Answers to all analytical questions.
+ - A brief exploration of one real-world application of RC circuits with explanation of how the time constant affects performance.
+
+
+
+
+
+ The integrating factor method is essential for solving first-order linear differential equations with variable coefficients or forcing functions. The RC circuit model demonstrates how this mathematical technique applies to real electrical systems. Understanding this model is crucial in electronics, signal processing, and even biological systems where similar charge-discharge dynamics occur. The time constant \tau = RC provides immediate insight into system behavior without fully solving the equation—a valuable engineering concept.
+
+
+
+
diff --git a/source/c6-qm/qm-model.ptx b/source/c6-qm/qm-model.ptx
new file mode 100644
index 00000000..f618e44e
--- /dev/null
+++ b/source/c6-qm/qm-model.ptx
@@ -0,0 +1,221 @@
+
+
+ Qualitative Modeling and Phase Line Analysis
+
+
+
+ Sometimes we don't need an exact formula for a solution—understanding the qualitative behavior is enough. Qualitative methods allow us to analyze differential equations that may be difficult or impossible to solve analytically, yet still extract valuable information about long-term behavior, equilibria, and stability.
+
+
+
+ In this section, we'll model a predator-prey system and use phase line analysis to understand its dynamics without finding explicit solutions.
+
+
+
+
+ The Problem: Harvesting a Fish Population
+
+
+ A lake contains a fish population that grows logistically when left alone. However, commercial fishing removes fish at a constant rate. How does harvesting affect the long-term sustainability of the population?
+
+
+
+ This question can be answered using qualitative analysis without solving the differential equation explicitly.
+
+
+
+
+ Assumptions
+
+
+ - Without harvesting, the population follows the logistic model.
+ - Fish are harvested at a constant rate h (fish per year) regardless of population size.
+ - The carrying capacity K and growth rate r remain constant.
+ - Environmental conditions are stable.
+
+
+
+
+ Building the Model
+
+
+ Let P(t) be the fish population at time t. The logistic growth with constant harvesting is:
+
+ \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) - h
+
+
+
+
+ This can be rewritten as:
+
+ \frac{dP}{dt} = -\frac{r}{K}P^2 + rP - h
+
+
+
+
+ Parameters:
+
+
+ - r > 0: intrinsic growth rate
+ - K > 0: carrying capacity
+ - h \geq 0: harvesting rate
+
+
+
+ Equilibrium Solutions: Set \frac{dP}{dt} = 0 and solve:
+
+ rP\left(1 - \frac{P}{K}\right) = h
+
+
+
+
+ This gives a quadratic equation in P. The number and nature of equilibria depend on h:
+
+
+ - If h = 0: equilibria at P = 0 (unstable) and P = K (stable)
+ - If 0 < h < \frac{rK}{4}: two equilibria, one stable and one unstable
+ - If h = \frac{rK}{4}: one equilibrium (semi-stable)
+ - If h > \frac{rK}{4}: no equilibria (population crashes to zero)
+
+
+
+
+ Qualitative Analysis Activity
+
+
+ Objective: Use phase line diagrams to understand how harvesting affects population dynamics.
+
+
+
+ Part 1: Phase Line Construction
+
+
+
+ For each harvesting scenario below, construct a phase line diagram:
+
+
+ - No harvesting: h = 0, r = 0.5, K = 1000
+ - Light harvesting: h = 50, r = 0.5, K = 1000
+ - Moderate harvesting: h = 125, r = 0.5, K = 1000
+ - Heavy harvesting: h = 150, r = 0.5, K = 1000
+
+
+
+ For each:
+
+
+ - Find equilibrium points (where \frac{dP}{dt} = 0)
+ - Determine stability (check sign of \frac{dP}{dt} on either side)
+ - Draw the phase line with arrows indicating direction of change
+ - Sketch several solution curves P(t) for different initial conditions
+
+
+
+ Part 2: Critical Harvesting Rate
+
+
+
+ Calculate h_{\text{max}} = \frac{rK}{4} for your parameters. This is the maximum sustainable yield—the highest harvesting rate that maintains a stable population.
+
+
+
+ Questions:
+
+
+ - What happens if h > h_{\text{max}}? Why is this outcome independent of initial population?
+ - For h < h_{\text{max}}, which equilibrium is stable? What does this mean for management?
+ - Why is harvesting at exactly h_{\text{max}} risky even though an equilibrium exists?
+
+
+
+
+ Simulation Activity
+
+
+ Use a computational tool (spreadsheet, Desmos, Python, MATLAB) to simulate the differential equation:
+
+
+
+ Setup:
+
+
+ - Choose parameters: r = 0.5, K = 1000
+ - Test different harvesting rates: h = 0, 50, 100, 125, 130
+ - For each h, simulate with initial populations: P_0 = 100, 500, 900
+ - Plot P(t) for each case on the same axes (different colors)
+
+
+
+ Analysis:
+
+
+ - Compare your simulations with your phase line predictions. Do they match?
+ - For which h values does the population collapse regardless of P_0?
+ - Identify the threshold where behavior changes qualitatively.
+ - How long does it take the population to reach equilibrium (approximately) for different h values?
+
+
+
+
+ Analytical Questions
+
+
+ - Derive the formula for h_{\text{max}} = \frac{rK}{4} by finding the maximum of rP(1 - P/K).
+ - At what population level does the maximum growth rate occur (without harvesting)? How does this relate to h_{\text{max}}?
+ - If fishery managers want a safety margin, what harvesting rate should they choose (in terms of h_{\text{max}})? Justify your answer.
+ - How would climate change affecting r or K impact sustainable harvesting?
+ - Modify the model for proportional harvesting: \frac{dP}{dt} = rP(1-P/K) - eP where e is the effort (fraction of population harvested). Analyze equilibria and stability.
+
+
+
+
+ Real-World Context
+
+
+ Research a historical example of overfishing (e.g., Atlantic cod collapse, California sardine fishery). Analyze:
+
+
+ - What were the estimated parameters (K, r, harvesting rates)?
+ - How does the historical data compare with model predictions?
+ - What management strategies could have prevented collapse?
+ - What are the limitations of this simple model?
+
+
+
+
+ Final Report
+
+
+ Submit a comprehensive report (3-4 pages) that includes:
+
+
+
+ - Derivation of the harvesting model with clear explanation of assumptions.
+ - Complete phase line analysis for all four harvesting scenarios with diagrams.
+ - Simulation results with graphs comparing different harvesting rates and initial conditions.
+ - Calculation and interpretation of maximum sustainable yield.
+ - Answers to all analytical questions with detailed reasoning.
+ - Case study analysis of a real fishery with application of the model.
+ - Discussion of model limitations and potential improvements (e.g., age structure, multiple species, environmental variability).
+ - Policy recommendations for sustainable harvesting based on your analysis.
+
+
+
+
+
+ Qualitative methods reveal essential features of differential equations without requiring explicit solutions. The harvested logistic model demonstrates how small changes in parameters can lead to dramatically different outcomes—a lesson crucial for resource management. Phase line analysis provides intuitive understanding of stability and long-term behavior, making it an invaluable tool for applied mathematics, ecology, economics, and engineering.
+
+
+
+
diff --git a/source/c7-em/em-model.ptx b/source/c7-em/em-model.ptx
new file mode 100644
index 00000000..e2094555
--- /dev/null
+++ b/source/c7-em/em-model.ptx
@@ -0,0 +1,236 @@
+
+
+ Numerical Modeling with Euler's Method
+
+
+
+ Many differential equations cannot be solved analytically, yet we still need solutions for practical applications. Euler's method provides a straightforward numerical approach to approximate solutions by following the tangent line direction step by step.
+
+
+
+ In this section, we'll model a nonlinear pendulum—a system whose differential equation cannot be solved in closed form—and use Euler's method to understand its motion.
+
+
+
+
+ The Problem: The Nonlinear Pendulum
+
+
+ A pendulum consists of a mass attached to a rigid rod or string, free to swing. For large angles, the motion is governed by a nonlinear differential equation. Unlike the simple harmonic motion approximation (\sin\theta \approx \theta), the true pendulum exhibits complex behavior.
+
+
+
+ How does the period of oscillation depend on amplitude for large swings?
+
+
+
+
+ Assumptions
+
+
+ - The pendulum rod is rigid and massless.
+ - All mass is concentrated at the bob.
+ - There is no friction or air resistance.
+ - The pivot point is fixed.
+ - Motion is constrained to a vertical plane.
+
+
+
+
+ Building the Model
+
+
+ Let \theta(t) be the angle from vertical at time t. The torque equation gives:
+
+ \frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0
+
+
+
+
+ where L is the length of the pendulum and g = 9.8 \text{ m/s}^2.
+
+
+
+ This is a second-order nonlinear differential equation. To apply Euler's method, we convert it to a system of first-order equations:
+
+ \frac{d\theta}{dt} \amp = \omega
+ \frac{d\omega}{dt} \amp = -\frac{g}{L}\sin\theta
+
+
+
+
+ where \omega is the angular velocity.
+
+
+
+ Parameters:
+
+
+ - g = 9.8 \text{ m/s}^2: gravitational acceleration
+ - L: length of pendulum (m)
+ - \theta_0: initial angle (radians)
+ - \omega_0: initial angular velocity (radians/s)
+
+
+
+ Euler's Method for Systems:
+
+
+
+ Starting from initial conditions (\theta_0, \omega_0), update using:
+
+ \theta_{n+1} \amp = \theta_n + h\omega_n
+ \omega_{n+1} \amp = \omega_n - h\frac{g}{L}\sin\theta_n
+
+ where h is the time step size.
+
+
+
+
+ Numerical Simulation Activity
+
+
+ Objective: Implement Euler's method to simulate pendulum motion and investigate how period depends on amplitude.
+
+
+
+ Implementation:
+
+
+
+ Use a computational tool (spreadsheet, Python, MATLAB, or provided code template):
+
+
+
+ - Set parameters: g = 9.8, L = 1.0 (gives period \approx 2 seconds for small angles)
+ - Set initial conditions: \omega_0 = 0, various \theta_0 values
+ - Choose time step: h = 0.01 (or smaller for better accuracy)
+ - Implement the update formulas
+ - Run simulation for sufficient time to observe several complete oscillations
+
+
+
+ Part 1: Small Angle Approximation
+
+
+
+ - Simulate with \theta_0 = 0.1 radians (\approx 5.7°)
+ - Plot \theta(t) and \omega(t)
+ - Measure the period T (time for one complete oscillation)
+ - Compare with the small-angle formula: T_0 = 2\pi\sqrt{\frac{L}{g}}
+
+
+
+ Part 2: Large Amplitude Effects
+
+
+
+ - Repeat simulations for \theta_0 = 0.2, 0.5, 1.0, 1.5, 2.0 radians
+ - For each, measure the period T
+ - Create a table: \theta_0, T, T/T_0
+ - Plot period versus initial amplitude
+
+
+
+ Part 3: Phase Space Portrait
+
+
+
+ - For several initial amplitudes, plot \omega versus \theta (phase portrait)
+ - Observe the shape of the trajectories—they should be closed loops for oscillatory motion
+ - What happens as \theta_0 approaches \pi (upside-down position)?
+
+
+
+
+ Physical Pendulum Activity
+
+
+ Optional Experimental Validation:
+
+
+
+ Materials:
+
+
+ - Pendulum (string and mass, or rigid rod)
+ - Protractor
+ - Stopwatch
+ - Meter stick
+
+
+
+ Procedure:
+
+
+ - Measure the length L of your pendulum.
+ - For various starting angles (5°, 15°, 30°, 45°, 60°), release from rest and time 10 complete oscillations.
+ - Calculate the period: T = \text{total time}/10
+ - Compare experimental periods with your Euler's method simulations.
+
+
+
+
+ Analytical Questions
+
+
+ - How does the step size h affect the accuracy of Euler's method? Test with h = 0.1, 0.01, 0.001.
+ - For the small-angle approximation (\sin\theta \approx \theta), the equation becomes \frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0. Solve this analytically and compare with your numerical solution.
+ - Energy should be conserved: E = \frac{1}{2}mL^2\omega^2 + mgL(1-\cos\theta). Calculate E at each time step in your simulation. How well is energy conserved? Why might it drift?
+ - What happens if you start the pendulum with \theta_0 = \pi (exactly upside down)? Is this a stable or unstable equilibrium?
+ - For what initial angle does the period equal 1.1 T_0 (10% longer than the small-angle period)?
+
+
+
+
+ Extensions
+
+
+ Modify your code to explore:
+
+
+ - Damping: Add a term -b\omega to model air resistance. How does the amplitude decay?
+ - Driving force: Add F_0\cos(\omega_d t) to model a periodically driven pendulum. Can you observe resonance?
+ - Chaotic pendulum: For strong driving and damping, the pendulum can exhibit chaotic motion. Explore parameter ranges where chaos occurs.
+
+
+
+
+ Final Report
+
+
+ Submit a comprehensive report (4-5 pages) including:
+
+
+
+ - Derivation of the pendulum differential equation from Newton's laws or energy principles.
+ - Explanation of how the second-order equation is converted to a system of first-order equations.
+ - Description of your Euler's method implementation (include code snippets or pseudocode).
+ - Graphs of \theta(t) and \omega(t) for multiple amplitudes.
+ - Period versus amplitude data and graph, with comparison to small-angle theory.
+ - Phase space portraits showing closed orbits.
+ - Energy conservation analysis throughout your simulations.
+ - Discussion of numerical accuracy and how step size affects results.
+ - If conducted, comparison with experimental pendulum data.
+ - Exploration of at least one extension (damping, driving, or chaos).
+
+
+
+
+
+ Euler's method, though simple, is a powerful tool for understanding differential equations that resist analytical solution. The nonlinear pendulum demonstrates why numerical methods are essential in applied mathematics—the real world often gives us equations we cannot solve exactly, yet we can still extract meaningful predictions through careful numerical approximation. This approach extends to weather prediction, rocket trajectories, circuit simulation, and countless other applications where exact solutions are unattainable.
+
+
+
+
diff --git a/source/c8-lhcc/lhcc-model.ptx b/source/c8-lhcc/lhcc-model.ptx
new file mode 100644
index 00000000..f2f0f1f2
--- /dev/null
+++ b/source/c8-lhcc/lhcc-model.ptx
@@ -0,0 +1,245 @@
+
+
+ Modeling with Homogeneous Linear Equations
+
+
+
+ Linear homogeneous constant coefficient (LHCC) equations model systems where forces or rates depend linearly on the state and its derivatives, with no external driving. These equations appear in mechanical vibrations, electrical circuits, and many other applications where natural oscillations occur.
+
+
+
+ In this section, we'll model a damped mass-spring system and explore the three distinct types of motion that can result: overdamping, critical damping, and underdamping.
+
+
+
+
+ The Problem: Suspension System Design
+
+
+ Automobile suspension systems use springs and shock absorbers (dampers) to provide a smooth ride. When a car hits a bump, the suspension compresses and then returns to equilibrium. The question is: how quickly should it return, and should it oscillate?
+
+
+
+ Different damping levels produce qualitatively different behaviors, each appropriate for different applications.
+
+
+
+
+ Assumptions
+
+
+ - The spring obeys Hooke's law: F_s = -kx
+ - Damping is proportional to velocity: F_d = -c\frac{dx}{dt}
+ - The mass is constant
+ - No external forces act on the system
+ - Motion is vertical (one-dimensional)
+
+
+
+
+ Building the Model
+
+
+ Let x(t) represent the displacement from equilibrium. Newton's second law gives:
+
+ m\frac{d^2x}{dt^2} = -kx - c\frac{dx}{dt}
+
+
+
+
+ Rearranging to standard form:
+
+ \frac{d^2x}{dt^2} + \frac{c}{m}\frac{dx}{dt} + \frac{k}{m}x = 0
+
+
+
+
+ This is a second-order linear homogeneous equation with constant coefficients.
+
+
+
+ Parameters:
+
+
+ - m: mass (kg)
+ - k: spring constant (N/m)
+ - c: damping coefficient (N·s/m or kg/s)
+ - \omega_0 = \sqrt{k/m}: natural frequency (rad/s)
+ - \zeta = \frac{c}{2\sqrt{km}}: damping ratio (dimensionless)
+
+
+
+ The characteristic equation is:
+
+ r^2 + \frac{c}{m}r + \frac{k}{m} = 0
+
+
+
+
+ The discriminant determines the type of motion:
+
+
+ - Overdamped (\zeta > 1): Two distinct real roots, no oscillation
+ - Critically damped (\zeta = 1): Repeated real root, fastest return without overshoot
+ - Underdamped (0 < \zeta < 1): Complex roots, damped oscillation
+
+
+
+
+ Physical Activity: Spring-Mass-Damper System
+
+
+ Objective: Build physical systems demonstrating different damping regimes.
+
+
+
+ Materials:
+
+
+ - Springs (slinky or coil springs)
+ - Masses (washers, weights)
+ - Damping materials (cardboard, foam, or water)
+ - Stopwatch
+ - Ruler or motion sensor
+ - Optional: video recording device
+
+
+
+ Setup Variations:
+
+
+
+ - Underdamped: Mass-spring in air (minimal damping)
+ - Increased Damping: Attach cardboard fins or foam to increase air resistance
+ - Heavy Damping: Mass-spring system moving through water or oil
+
+
+
+ Procedure:
+
+
+ - For each setup, pull the mass down a fixed distance and release.
+ - Record the motion (video or position measurements over time).
+ - Count oscillations (if any) and measure the period.
+ - Measure the decay time (time to reduce amplitude by half).
+ - Note whether the system overshoots equilibrium.
+
+
+
+ Analysis:
+
+
+ - Plot displacement versus time for each damping level.
+ - Classify each system as underdamped, critically damped, or overdamped.
+ - For underdamped cases, measure the damped frequency and compare with the natural frequency.
+ - Estimate the damping ratio \zeta from the logarithmic decrement method (if applicable).
+
+
+
+
+ Computational Activity
+
+
+ Use the exact solutions to generate plots:
+
+
+
+ Case 1: Underdamped (\zeta = 0.2)
+
+
+ x(t) = e^{-\zeta\omega_0 t}(A\cos(\omega_d t) + B\sin(\omega_d t))
+
+ where \omega_d = \omega_0\sqrt{1-\zeta^2}
+
+
+ Case 2: Critically Damped (\zeta = 1)
+
+
+ x(t) = e^{-\omega_0 t}(A + Bt)
+
+
+
+ Case 3: Overdamped (\zeta = 2)
+
+
+ x(t) = Ae^{r_1 t} + Be^{r_2 t}
+
+ where r_1, r_2 are negative real roots.
+
+
+ For each case with \omega_0 = 1, x(0) = 1, x'(0) = 0:
+
+
+ - Solve for constants A and B
+ - Plot x(t) for 0 \leq t \leq 20
+ - Compare the three cases on one graph
+ - Which returns to equilibrium fastest? Which overshoots?
+
+
+
+
+ Analytical Questions
+
+
+ - Verify that x(t) = e^{-\zeta\omega_0 t}\cos(\omega_d t) satisfies the differential equation for underdamped motion.
+ - Show that critical damping occurs when c = 2\sqrt{km}.
+ - For underdamped motion, derive the relationship between consecutive peak amplitudes (logarithmic decrement).
+ - Why is critical damping preferred for car suspensions and door closers?
+ - In what applications would you want underdamping? Overdamping?
+ - How does doubling the mass affect the natural frequency? The damping ratio?
+
+
+
+
+ Design Challenge
+
+
+ You are designing a suspension system for a 1000 kg vehicle. The spring constant is k = 40000 N/m.
+
+
+
+ - Calculate the natural frequency \omega_0.
+ - For a comfortable ride, you want \zeta = 0.7 (underdamped but not too oscillatory). What damping coefficient c is needed?
+ - If the car hits a bump causing an initial displacement of 0.1 m, how long until the displacement is less than 0.01 m?
+ - Compare this design with a critically damped system. Which returns to equilibrium faster?
+
+
+
+
+ Final Report
+
+
+ Prepare a detailed report (4-5 pages) including:
+
+
+
+ - Derivation of the damped harmonic oscillator equation from Newton's laws.
+ - Solution of the characteristic equation for all three damping cases.
+ - Physical experiment description with photos/diagrams and data.
+ - Classification of your experimental systems by damping type.
+ - Graphs comparing the three damping regimes (both experimental and theoretical).
+ - Parameter estimation for your physical systems.
+ - Complete solution to the design challenge with justification.
+ - Discussion of real-world applications where each damping type is preferred.
+ - Analysis of how parameter changes affect system behavior.
+
+
+
+
+
+ Linear homogeneous constant coefficient equations provide exact, predictable solutions for a wide class of physical systems. The damped harmonic oscillator is fundamental in engineering design, from vehicle suspensions to building earthquake resistance, from electrical filters to quantum mechanics. Understanding how the characteristic equation's roots determine system behavior—whether oscillatory, exponential, or critical—is essential for controlling and optimizing these systems. The mathematical classification (overdamped, critically damped, underdamped) directly translates to observable physical phenomena, demonstrating the power of differential equations in engineering and science.
+
+
+
+
diff --git a/source/c9-uc/uc-model.ptx b/source/c9-uc/uc-model.ptx
new file mode 100644
index 00000000..2289d746
--- /dev/null
+++ b/source/c9-uc/uc-model.ptx
@@ -0,0 +1,266 @@
+
+
+ Modeling Forced Systems with Undetermined Coefficients
+
+
+
+ When a system is subjected to external forcing, the differential equation becomes nonhomogeneous. The method of undetermined coefficients allows us to find particular solutions when the forcing function has a special form. These forced systems exhibit both transient and steady-state behavior.
+
+
+
+ In this section, we'll model a driven mechanical oscillator and explore the phenomenon of resonance—one of the most important concepts in engineering and physics.
+
+
+
+
+ The Problem: Resonance in Bridges and Buildings
+
+
+ When soldiers march across a bridge in lockstep, their synchronized footsteps create a periodic driving force. If the frequency of their steps matches the bridge's natural frequency, the amplitude of oscillation can grow dangerously large—a phenomenon called resonance.
+
+
+
+ How can we predict when resonance will occur and how severe it will be?
+
+
+
+
+ Assumptions
+
+
+ - The system behaves as a linear spring-mass-damper.
+ - The external force is periodic: F(t) = F_0\cos(\omega t).
+ - Damping is present but may be small.
+ - Parameters remain constant.
+
+
+
+
+ Building the Model
+
+
+ The equation of motion with external forcing is:
+
+ m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F_0\cos(\omega t)
+
+
+
+
+ In standard form:
+
+ \frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = \frac{F_0}{m}\cos(\omega t)
+
+
+
+
+ where:
+
+
+ - \omega_0 = \sqrt{k/m}: natural frequency
+ - \zeta = \frac{c}{2\sqrt{km}}: damping ratio
+ - \omega: driving frequency
+
+
+
+ General Solution Structure:
+
+
+ x(t) = x_h(t) + x_p(t)
+
+
+ - x_h(t): homogeneous solution (transient, dies out due to damping)
+ - x_p(t): particular solution (steady-state response)
+
+
+
+ Steady-State Solution:
+
+
+
+ Using undetermined coefficients, the particular solution has the form:
+
+ x_p(t) = A\cos(\omega t) + B\sin(\omega t)
+
+ or equivalently:
+
+ x_p(t) = R\cos(\omega t - \phi)
+
+
+
+
+ The amplitude of steady-state oscillation is:
+
+ R = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2}}
+
+
+
+
+ Resonance occurs when \omega \approx \omega_0, giving maximum amplitude:
+
+ R_{\text{max}} \approx \frac{F_0}{2\zeta m\omega_0^2} = \frac{F_0}{c\omega_0}
+
+
+
+
+
+ Physical Activity: Driven Oscillator
+
+
+ Objective: Observe resonance in a physical system and measure frequency response.
+
+
+
+ Setup 1: Mass-Spring System with Oscillating Support
+
+
+
+ Materials:
+
+
+ - Spring and mass
+ - Mechanical oscillator or manual shaking apparatus
+ - Frequency control (metronome or signal generator)
+ - Ruler for amplitude measurement
+ - Video camera
+
+
+
+ Procedure:
+
+
+ - Suspend the mass-spring system.
+ - Determine the natural frequency by releasing from rest and timing oscillations.
+ - Drive the system by shaking the support at various frequencies (start far from \omega_0).
+ - For each driving frequency, wait for steady state and measure the amplitude.
+ - Record: Driving frequency, Steady-state amplitude.
+ - Test frequencies both below and above \omega_0, with extra measurements near \omega_0.
+
+
+
+ Setup 2: Resonance Demonstration (Simpler)
+
+
+
+ Use coupled pendulums of different lengths driven by one oscillating pendulum. Observe which pendulum responds most strongly based on length matching.
+
+
+
+ Analysis:
+
+
+ - Plot steady-state amplitude versus driving frequency (frequency response curve).
+ - Identify the resonant frequency (where amplitude is maximum).
+ - Compare the resonant frequency with the natural frequency \omega_0.
+ - Estimate the damping ratio from the sharpness of the resonance peak.
+
+
+
+
+ Computational Activity
+
+
+ Simulate the forced oscillator equation numerically:
+
+
+
+ - Set parameters: m = 1, k = 4 (so \omega_0 = 2), F_0 = 1
+ - Test three damping levels: \zeta = 0.1, 0.3, 0.7
+ - For each damping level, simulate with driving frequencies: \omega = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0
+ - Plot x(t) for several periods to see both transient and steady-state behavior
+ - Measure the steady-state amplitude for each case
+ - Create a frequency response plot (amplitude vs. \omega) for each damping level
+
+
+
+ Observations:
+
+
+ - How does damping affect the resonance peak?
+ - How long does it take to reach steady state?
+ - What happens when \omega = \omega_0 exactly?
+
+
+
+
+ Analytical Questions
+
+
+ - Verify that x_p(t) = R\cos(\omega t - \phi) satisfies the nonhomogeneous differential equation by direct substitution.
+ - Derive the condition for maximum amplitude by taking \frac{dR}{d\omega} = 0.
+ - Show that in the limit of zero damping (\zeta \to 0), the resonant amplitude becomes infinite.
+ - For very low frequencies (\omega \ll \omega_0), what is the approximate steady-state amplitude? How does it relate to the static displacement F_0/k?
+ - For very high frequencies (\omega \gg \omega_0), what happens to the amplitude?
+ - Calculate the quality factor Q = 1/(2\zeta). How does Q relate to the sharpness of the resonance?
+
+
+
+
+ Engineering Case Studies
+
+
+ Research one of these famous resonance-related incidents:
+
+
+ - Tacoma Narrows Bridge (1940): Collapsed due to wind-induced resonance
+ - Millennium Bridge, London (2000): Closed shortly after opening due to pedestrian-induced wobbling
+ - Tuned mass dampers: Used in skyscrapers (e.g., Taipei 101) to counteract wind and earthquake oscillations
+
+
+
+ Explain how the resonance phenomenon contributed to the problem and how engineers addressed it.
+
+
+
+
+ Design Problem
+
+
+ A machine with mass 100 kg sits on an elastic foundation with stiffness k = 10^5 N/m. The machine vibrates at 10 Hz during operation.
+
+
+
+ - Calculate the natural frequency of the system.
+ - Is the system near resonance? If so, what problems might occur?
+ - Design a damping system (c value) to limit the amplitude to less than 2 mm, given F_0 = 500 N.
+ - Alternatively, how much mass would you need to add to shift the natural frequency away from the operating frequency?
+
+
+
+
+ Final Report
+
+
+ Submit a comprehensive report (4-5 pages) including:
+
+
+
+ - Derivation of the forced oscillator equation and solution using undetermined coefficients.
+ - Analytical derivation of the resonance condition and maximum amplitude formula.
+ - Experimental setup description with frequency response data and graphs.
+ - Comparison of experimental resonance frequency with theoretical predictions.
+ - Computational simulation results showing frequency response for different damping levels.
+ - Answers to all analytical questions with detailed work.
+ - Case study analysis of a real resonance-related engineering problem.
+ - Complete solution to the design problem with recommendations.
+ - Discussion of how resonance is both a challenge (structural failure) and a tool (MRI machines, musical instruments).
+
+
+
+
+
+ Resonance is one of the most important phenomena in science and engineering. While it can be destructive when unwanted (bridge collapses, machine failures), it's also intentionally exploited in countless applications (radio tuning, MRI imaging, musical instruments, atomic clocks). The method of undetermined coefficients provides the tools to analyze forced systems and predict resonance behavior. Understanding how forcing frequency, natural frequency, and damping interact is essential for designing safe structures and efficient devices. This mathematical framework has saved lives and enabled technologies, demonstrating the real-world impact of differential equations.
+
+
+
+
From 2165fcbbc551e619e9cf4f4be33cd7ef68ea4a3b Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Wed, 4 Feb 2026 03:36:24 +0000
Subject: [PATCH 3/5] Update main.ptx to include all modeling sections
Co-authored-by: geoff-cox <63681414+geoff-cox@users.noreply.github.com>
---
source/main.ptx | 14 ++++++++++++++
1 file changed, 14 insertions(+)
diff --git a/source/main.ptx b/source/main.ptx
index 32725a3c..c3689db8 100644
--- a/source/main.ptx
+++ b/source/main.ptx
@@ -42,6 +42,7 @@