Mathematics in Aircraft Maintenance

As an aircraft mechanic, you have to work on any component of the aircraft — airframe, flight controls, engines, landing gear, avionics, and more. There are a lot of different areas of physics involved. So, you need to be proficient with the maths that corresponds with these topics.

Basic maths — add, subtract, multiply, divide — is a must. When dealing with hardware, you need plane geometry and trigonometry. You might not have to solve trigonometry problems, but you need to understand angles and, perhaps, read diagrams. Plane geometry is useful because it teaches you to use logic.

When you deal with aircraft hardware, you also deal with tolerances — measured lengths and diameters of parts. You need to be proficient with specific measuring instruments and know-how to be sure what unit of measurement is called for. When it comes to assembly work, you have to know what torque value each fastener requires.

Mathematics is woven into many areas of everyday life. Performing mathematical calculations with success requires an understanding of the correct methods, procedures, practice, and review of these principles. Mathematics may be thought of as a set of tools. The aviation mechanic needs these tools to successfully complete the maintenance, repair, installation, or certification of aircraft equipment. Many examples of using mathematical principles by the aviation mechanic are available. Tolerances in turbine engine components are critical, making it necessary to measure within a ten-thousandth of an inch. Because of these close tolerances, it is important that the aviation mechanic can make accurate measurements and mathematical calculations.

Mathematics skills for Aircraft Technician

Aircraft Technician must understand the –

• Measurements with units when replacing and fabricating fuselage parts
• Proportions and ratios when assessing the electrical needs of circuitry
• Solving equations with one or more variables when repairing hydraulic systems
• Calculating volume of spheres, parallelepipeds and cylinders when assessing fuel needs
• Modeling with linear equations when computing the center of gravity (CG) of an aircraft

The Use of Mathematics in Aircraft Maintenance

#1 Repairing a Crack in the Fuselage

Skill Required – Mathematical Reasoning and Problem Solving

During the preflight inspection, AMEs are responsible for inspecting the aircraft for any suspicious signs of wear and tear. Faced with a crack in an aircraft’s fuselage, AMT must first determine the most effective method of repair. To do this, AME depends on knowledge of the physical properties of metals and composites – their strength, flexibility, and durability – when put under the stress of flight. AMEs need to understand the relationship between the materials used to construct the aircraft and the parts needed to make the repair and know the best methods for securing these parts to the aircraft.

Calculating shear strength also will help the AMT select the most appropriate materials for mending the crack – such as using metal rivets, a composite mixture of resin and hardener, or the fabrication of a new part. If a new part is required, the shape of the fabricated part is another important variable – especially if the crack has appeared on a part of the airplane with significant curvature. In that situation, AMTs have little choice but to create a new part to replace the damaged section.

#2 Fabricating New Parts for Repair

Skill Required: Geometry and Algebra

Depending on the size of the crack, the weight restrictions, and the bend allowance of the fabricated material, AMTs must perform a sophisticated series of algebraic and geometric calculations to determine the amount of material needed for the replacement part, the shape it should take, and the maximum weight allowance and other tolerances. For example, creating a patch for a particularly curved section of the airplane’s fuselage (or wing) requires calculating the bend allowance of sheet metal – a dimensional adjustment that must be factored in to ensure the safety of the replacement part. When AMTs bend metal to the desired form, the material of the outside angle is stretched, while the material on the inside angle is compressed.

Determining the proper bend radius is essential to constructing a strong and sound part. Too small a radius can cause further cracking, while too large a radius can result in costly overruns of material, excess weight, and ultimately a disruption in the airplane’s CG. To properly calculate the bend allowance, AMTs must measure the length of the brackets and the material thickness along with the angle and radius of the bend. They also must determine the “neutral axis,” or the K-factor – the percent of the material thickness where there is no stretching or compressing. Once the bend allowance is calculated, technicians can fabricate a new part; choose the appropriate rivet size, rivet pitch, and tooling; and successfully repair the aircraft.

Bend allowance = Angle in degrees x (π/180) x (Radius + K-factor x Thickness)

#3 Maintaining Center of Gravity

Skill Required: Measurement and Proportions

A team of AMTs working together also must consider the effect the repair will have on the airplane’s CG. Because an aircraft is designed to be balanced forward and aft, the CG is located in a precise range along the chord of the wing – the distance between the leading and trailing edge of a wing. If the actual balance point is too far forward or too far aft, the pilot may experience difficulty maintaining level flight. When making adjustments to an airplane’s fuselage, AMTs must find the CG by weighing the aircraft with platform scales or by using load cells positioned at designated points along the fuselage. AMTs must calculate these specific weights and the dimensions of those points to determine “moments” along an axis and determine the range within which the CG can be located. If, after making the repair, the actual CG falls beyond the approved envelope, the technicians need to calculate where and how much ballast they should add to reposition the CG within the approved range.

Since ballast is useless weight and only wastes fuel, fabricating durable but lightweight replacement parts is key to minimizing the need for additional costly adjustments to the airplane’s CG.

Moment = Weight x Distance

Center of gravity = (Σ Moments) / Total weight

What is Mathematics

Mathematics has no generally accepted definition. It includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis).

Mathematics plays an important role in the development of science and technology, so it is also considered as the queen of science.

List of branches of Mathematics

Pure Mathematics

• Number Theory
• Arithmetic
• Algebra
• Geometry
• Combinatorics
• Topology
• Mathematical Analysis

Applied Mathematics

• Calculus
• Statistics and Probability
• Set Theory
• Trigonometry

The main branches of mathematics are number theory, arithmetic, algebra, and geometry. Based on these branches, other branches have been discovered.

Learn Basics

• Mathematical Symbols
• Mathematical Abbreviations

Basic Mathematics for Aircraft Mechanics

As an aircraft maintenance technician, you only need to learn basic mathematics.

1. Arithmetic

Arithmetic is the branch of mathematics that deals with the study of numbers using various operations on them. Basic operations of math are addition, subtraction, multiplication, and division.

Whole Numbers

The whole numbers are the numbers without fractions and it is a collection of positive integers and zero.

• Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10……}
• Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,…}
• Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…}

Facts

• All the natural numbers are whole numbers
• Natural numbers, also called the counting numbers.
• All positive integers including zero are whole numbers
• All whole numbers are real numbers

Fractions

A fraction is a number written in the form N/D where N is called the numerator and D is called the denominator. The fraction bar between the numerator and denominator shows that division is taking place.

• Finding the Least Common Denominator
• Addition of Fractions
• Subtraction of Fractions
• Multiplication of Fractions
• Division of Fractions
• Reducing Fractions

Mixed Numbers

A mixed number is a combination of a whole number and a fraction.

• Addition of Mixed Numbers
• Subtraction of Mixed Numbers

Decimal Number System

The number system that we use every day is called the decimal system. The prefix in the word decimal, dec, is a Latin root for the word “ten.” The decimal system probably originated from the fact that we have ten fingers (or digits). The decimal system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The decimal system is a base 10 system.

• Addition of Decimal Numbers
• Subtraction of Decimal Numbers
• Multiplication of Decimal Numbers
• Division of Decimal Numbers
• Rounding Off Decimal Numbers
• Converting Decimal Numbers to Fractions
• Converting Fractions to Decimals

Ratio

A ratio is the comparison of two numbers or quantities. A ratio may be expressed in three ways:

1. as a fraction,
2. with a colon, or
3. with the word “to.”

For example, a gear ratio of 5:7 can be expressed as any of the following:

5/7 or 5:7 or 5 to 7

Aviation Applications of Ratio

Ratios have a widespread application in the field of aviation.

Examples:

1. Compression Ratio – The compression ratio on a reciprocating engine is the ratio of the volume of a cylinder with the piston at the bottom of its stroke to the volume of the cylinder with the piston at the top of its stroke. For example, a typical compression ratio might be 10:1 (or 10 to 1).
2. Aspect Ratio – The aspect ratio is the ratio of the length (or span) of an airfoil to its width (or chord). A typical aspect ratio for a commercial airliner might be 7:1 (or 7 to 1).
3. Air-fuel Ratio – The air-fuel ratio is the ratio of the weight of the air to the weight of fuel in the mixture being fed into the cylinders of a reciprocating engine. For example, a typical air-fuel ratio might be 14.3:1 (or 14.3 to 1).
4. Glide Ratio – The glide ratio is the ratio of the forward distance traveled to the vertical distance descended when an aircraft is operating without power. For example, if an aircraft descends 1,000 feet while it travels through the air for two linear miles (10,560 feet), it has a glide ratio of 10,560:1,000 which can be reduced to 10.56: 1 (or 10.56 to 1).
5. Gear Ratio – Gear ratio is the number of teeth each gear represents when two gears are used in an aircraft component. If the pinion gear has 8 teeth and a spur gear has 28 teeth. The gear ratio is 8:28. Using 7 as the LCD, 8:28 becomes 2:7.
6. Speed Ratio – Speed ratio is when two gears are used in an aircraft component; the rotational speed of each gear is represented as a speed ratio. As the number of teeth in a gear decreases, the rotational speed of that gear increases, and vice-versa. Therefore, the speed ratio of two gears is the inverse (or opposite) of the gear ratio. If two gears have a gear ratio of 2:9, then their speed ratio is 9:2. Example: A pinion gear with 10 teeth is driving a spur gear with 40 teeth. The spur gear is rotating at 160 rpm. Calculate the speed of the pinion gear. 

Proportion

A proportion is a statement of equality between two or more ratios. For example, 3/4 = 6/8 or 3:4 = 6:8. This proportion is read as, “3 is to 4 as 6 is to 8.”

Percentage

Percentage means “parts out of one hundred.” The percentage sign is “%.” Ninety percent is expressed as 90% (= 90 parts out of 100). The decimal 0.90 equals 90⁄100, or 90 out of 100, or 90%.

• Expressing a Decimal Number as a Percentage
• Expressing a Percentage as a Decimal Number
• Expressing a Fraction as a Percentage
• Finding a Percentage of a Given Number
• Finding What Percentage One Number is of Another
• Finding a Number When a Percentage of it is Known

Positive and Negative Numbers (Signed Numbers)

Positive numbers are numbers that are greater than zero. Negative numbers are numbers less than zero. Signed numbers are also called integers.

• Addition of Positive and Negative Numbers
• Subtraction of Positive and Negative Numbers
• Multiplication of Positive and Negative Numbers
• Division of Positive and Negative Numbers

Powers

The power (or exponent) of a number is a shorthand method of indicating how many times a number, called the base, is multiplied by itself. For example, 3⁴ is read as “3 to the power of 4.” That is, 3 multiplied by itself 4 times. The 3 is the base and the 4 is the power. Examples:

2³ = 2 × 2 × 2 = 8

Read “two to the third power equals 8.”

Special Powers

• Squared
• Cubed
• Power of Zero
• Negative Powers
• Law of Exponents
• Powers of Ten
• Roots
• Square Roots
• Cube Roots
• Fractional Powers

Scientific Notation

Scientific notation is used as a type of shorthand to express very large or very small numbers. It is a way to write numbers so that they do not take up as much space on the page. The format of a number written in scientific notation has two parts. The first part is a number greater than or equal to 1 and less than 10 (for example, 2.35). The second part is a power of 10 (for example, 106). The number 2,350,000 is expressed in scientific notation as 2.35 × 106. It is important that the decimal point is always placed to the right of the first digit. Notice that very large numbers always have a positive power of 10 and very small numbers always have a negative power of 10.

• Converting Numbers from Standard Notation to Scientific Notation
• Converting Numbers from Scientific Notation to Standard Notation
• Addition, Subtraction, Multiplication, and Division of Scientific Numbers

2. Algebra

Algebra is the branch of mathematics that uses letters or symbols to represent variables in formulas and equations.

Example :

Equation – d = v × t

Where distance = velocity × time, 

Variables are: d, v, and t.

Things to learn –

• Equations
• Algebraic Rules
• Solving for a Variable
• Use of Parentheses
• Order of Operation – PEMDAS

Order of Operation for Algebraic Equations

A commonly used acronym, PEMDAS, is used for remembering the order of operation in algebra. PEMDAS is an acronym for parentheses, exponents, multiplication, division, addition, and subtraction. To remember it, many use the sentence, “Please Excuse My Dear Aunt Sally.” Always remember, however, to multiply/divide or add/subtract in one sweep from left to right, not separately.

1. Parentheses
2. Exponents
3. Multiplication
4. Division
5. Addition
6. Subtraction

3. Geometry

Geometric Shapes

Understanding geometric shapes are the most important for aviation mechanics. There are many different shapes associated with geometry.

TRIANGLE

A triangle obviously has 3 sides and 3 (internal) angles. The sides are often represented by the 3 (small) letters a, b, and c; the angles by the (large) letters A, B, and C. The 3 angles add up to 180°.

The sum of the three angles in a triangle is always equal to 180°.

Types of triangles

Triangles Based on Sides

1. Equilateral
2. Isosceles
3. Scalene

Triangles Based on Angles

1. Acute
2. Right
3. Obtuse

RECTANGLE

A rectangle is a four-sided figure with opposite sides of equal length and parallel to each other.

SQUARE

A square is a four-sided figure with all sides of equal length and opposite sides are parallel to each other. All angles are right angles. A right angle is a 90° angle.

PARALLELOGRAM

A parallelogram is a four-sided figure with two pairs of parallel sides. Parallelograms do not necessarily have four right angles.

TRAPEZOID

A trapezoid is a four-sided figure with one pair of parallel sides.

CIRCLE

A circle is a closed, curved, plane figure. Every point on the circle is an equal distance from the center of the circle. The diameter is the distance across the circle (through the center). The radius is the distance from the center to the edge of the circle. The diameter is always twice the length of the radius. The circumference, or distance around, a circle is equal to the diameter times π.

ELLIPSE

An ellipse is a closed, curved, plane figure and is commonly called an oval.

WING AREA

To calculate wing area, it is necessary to know “span” and “chord.”

The Span is the length of the wing from wingtip to wingtip.

The chord is the average width of the wing from leading edge to trailing edge. If the wing is a tapered wing, the average width, known as the mean chord.

Area of a wing = span × mean chord

POLYGON

A polygon is a geometric closed figure bounded by straight lines. The term poly means multi. A triangle has the least number of sides. Other multi-sided figures have names indicating the number of sides. Hence:

• Pentagon – 5 sided,
• Hexagon – 6 sided,
• Octagon – 8 sided

Calculating Area of Two-Dimensional Solids

Formulas to calculate the area

Calculating Volume and Surface Area of Three Dimensional Solids

Formulas to calculate volume and surface area

Trigonometric Functions

Trigonometry is the study of the relationship between the angles and sides of a triangle. The word trigonometry comes from the Greek trigonon, which means three angles, and metro, which means measure.

Sine, Cosine, and Tangent

The three primary trigonometric functions and their abbreviations are sine (sin), cosine (cos), and tangent (tan). These three trigonometric functions are actually ratios comparing two of the sides of the triangle as follows:

• Sin of angle A = side a / side c
• Cos of angle A = side b / side c
• Tan of angle A = side a / side b

Pythagorean Theorem

Pythagorean theorem is used to find the third side of any right triangle when two sides are known.

The Pythagorean Theorem states that a² + b² = c². Where “c” = the hypotenuse of a right triangle, “a” is one side of the triangle, and “b” is the other side of the triangle.