5.2 Graphs of Equations and Functions

Graphs and Their Application in Aviation

In aviation maintenance and operations, graphs are crucial visual tools used to depict relationships between physical and mechanical variables. Whether monitoring engine health, assessing structural load limits, or reading aerodynamic performance parameters, an aircraft technician must be proficient in plotting, reading, and interpreting both linear and non-linear graphs.

Fundamentals of Graphing

Graphs are plotted on a two-dimensional grid called the Cartesian coordinate system. This system consists of two perpendicular axes:

• The horizontal axis ($x$-axis), representing the independent variable.
• The vertical axis ($y$-axis), representing the dependent variable.

The intersection of these axes is the origin $(0,0)$. The grid is divided into four quadrants, though in aviation, most physical parameters reside in the first quadrant where both values are positive.

Linear Equations and Slopes

A straight-line graph is defined by the linear equation:

$y = mx + c$

Where:

• $m$ is the slope or gradient, representing the rate of change of $y$ relative to $x$.
• $c$ is the $y$-intercept, which is the value of $y$ when $x = 0$.

The slope $m$ is calculated from two points on the line, $(x_1, y_1)$ and $(x_2, y_2)$, using the formula:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

If $m$ is positive, the line rises from left to right; if negative, it falls. The roots of an equation represent the points where the graphed line intersects the axes. The $x$-intercept is the root found by setting $y = 0$.

Worked Example: An aircraft has an initial fuel weight of 480 kg. The engine burns fuel at a constant rate of 80 kg/hour.

1. Write the equation for the remaining fuel $F$ over time $t$ in hours: $F = -80t + 480$.
2. On the graph, the $y$-intercept is 480 kg, representing the initial fuel state at $t = 0$.
3. The slope $m$ is $-80$, indicating a decrease of 80 kg of fuel for every hour of flight.
4. To find the root (the $x$-intercept where $F = 0$): $0 = -80t + 480 \implies 80t = 480 \implies t = 6\text{ hours}$ This root tells the technician that the maximum flight endurance is exactly 6 hours.

Non-Linear Graphs and Functions

Many physical properties in aviation do not change linearly. They are represented by non-linear curves:

• Quadratic Functions: Represented by $y = ax^2 + bx + c$, resulting in a parabola. An example is the stress distribution along a cantilever wing spar under bending loads, or the path of an object dropped from an aircraft.
• Exponential Functions: Represented by $y = ab^x$, where the variable changes at a rate proportional to its current value. For example, atmospheric pressure ($P$) decreases exponentially with altitude ($h$): $P = P_0 e^{-h/H}$ Where $P_0$ is sea-level pressure and $H$ is a scale height (approximately 8,000 meters).

Interpreting Aircraft Performance Curves

Technicians must frequently read complex performance charts to verify system calibration or check aircraft limits.

1. Lift Coefficient vs. Angle of Attack

The lift generated by a wing is plotted as the Lift Coefficient ($C_L$) on the $y$-axis against the Angle of Attack ($\alpha$) on the $x$-axis:

• Linear Range: For small angles of attack, the curve is a straight line. The slope of this line is the lift-curve slope.
• Maximum Lift ($C_{L,max}$): As the angle of attack increases, the curve flattens and reaches a peak. This peak represents the maximum lift the wing can produce.
• Critical Stall Angle ($\alpha_{crit}$): The angle of attack at the peak of the curve. Beyond this angle, the smooth airflow over the top of the wing separates, causing the lift coefficient to drop sharply. This is the aerodynamic stall.

2. Drag Polar and Lift-to-Drag Ratio

Total aircraft drag is the sum of two primary components:

• Parasite Drag: Drag caused by skin friction, form, and interference. It increases quadratically with airspeed ($D_p \propto V^2$).
• Induced Drag: Drag generated as a byproduct of lift. It decreases quadratically with airspeed ($D_i \propto 1/V^2$).

When total drag is plotted against airspeed, the resulting curve is U-shaped. The minimum point of the total drag curve represents the minimum drag speed ($V_{md}$). At this speed, the ratio of lift to drag ($L/D$) is at its maximum ($L/D_{max}$), which represents the most aerodynamically efficient speed for the aircraft. This speed provides the maximum glide range if the engines fail.

3. Material Stress-Strain Curves

In structural testing of aviation materials, a specimen is pulled until it fractures, and stress (force per unit area) is plotted against strain (proportional elongation):

• Elastic Region: The initial straight-line portion where deformation is reversible. The slope of this line is Young's Modulus of Elasticity ($E$), which measures the stiffness of the material.
• Yield Point: The point where the curve bends, indicating the transition from elastic (reversible) to plastic (permanent) deformation.
• Ultimate Tensile Strength (UTS): The highest point on the curve, representing the maximum stress the material can withstand.
• Fracture Point: The end of the curve where the material physically breaks.

4. Weight and Balance Envelope

To ensure safe flight characteristics, every aircraft has a Weight and Balance Envelope graph:

• The $x$-axis shows the Center of Gravity (CG) location (usually in inches aft of a datum line).
• The $y$-axis shows the total weight of the aircraft.
• The envelope is a closed polygon defining the safe limits. The technician calculates the total weight and CG coordinates of the loaded aircraft. If the resulting point falls outside the envelope boundary, the aircraft is unsafe to fly because it may lack sufficient elevator control to recover from maneuvers or may exceed structural landing limits.