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Mechanical Engineering Cheatsheet

A quick reference guide covering fundamental concepts, formulas, and principles in mechanical engineering. Ideal for students, engineers, and professionals.

Thermodynamics

Basic Concepts

Zeroth Law:

If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other.

First Law:

Energy can neither be created nor destroyed. Conservation of energy: ΔU = Q - W, where U is internal energy, Q is heat added, and W is work done.

Second Law:

The total entropy of an isolated system can only increase over time or remain constant in ideal cases. Entropy is a measure of disorder.

Third Law:

As temperature approaches absolute zero, the entropy of a system approaches a minimum or zero value.

Enthalpy (H):

A thermodynamic property defined as H = U + PV, where U is internal energy, P is pressure, and V is volume.

Entropy (S):

A measure of the disorder of a system. ΔS = Q/T for a reversible process at constant temperature.

Thermodynamic Processes

Isothermal:

Constant temperature process. PV = constant.

Adiabatic:

No heat transfer. PVγ = constant, where γ is the heat capacity ratio (Cp/Cv).

Isobaric:

Constant pressure process. V/T = constant.

Isochoric (Isometric):

Constant volume process. P/T = constant.

Polytropic:

Process described by PVn = constant, where n is the polytropic index.

Throttling:

Adiabatic process where enthalpy remains constant. Used in refrigeration.

Heat Engines and Refrigerators

Heat Engine Efficiency (η):

η = W/QH = 1 - (QC/QH), where W is work done, QH is heat input, and QC is heat rejected.

Carnot Efficiency (ηCarnot):

ηCarnot = 1 - (TC/TH), where TC is the cold reservoir temperature, and TH is the hot reservoir temperature (in Kelvin).

Coefficient of Performance (COP) - Refrigerator:

COPR = QC/W = QC/(QH - QC)

Coefficient of Performance (COP) - Heat Pump:

COPHP = QH/W = QH/(QH - QC)

Fluid Mechanics

Fluid Properties

Density (ρ):

Mass per unit volume: ρ = m/V

Specific Weight (γ):

Weight per unit volume: γ = ρg, where g is acceleration due to gravity.

Specific Gravity (SG):

Ratio of a fluid’s density to the density of water: SG = ρfluid/ρwater

Viscosity (μ):

Measure of a fluid’s resistance to flow. Dynamic viscosity.

Kinematic Viscosity (ν):

Ratio of dynamic viscosity to density: ν = μ/ρ

Surface Tension (σ):

Force per unit length acting at the interface between two fluids.

Fluid Statics

Pressure (P):

Force per unit area: P = F/A

Hydrostatic Pressure:

P = ρgh, where h is the depth from the surface.

Buoyancy (FB):

Upward force exerted by a fluid that opposes the weight of an immersed object: FB = ρfluidVdisplacedg

Manometry:

Use of liquid columns to measure pressure differences.

Fluid Dynamics

Continuity Equation:

A1V1 = A2V2 (for incompressible fluids)

Bernoulli’s Equation:

P + (1/2)ρV2 + ρgh = constant (along a streamline)

Reynolds Number (Re):

Re = (ρVD)/μ, where V is flow velocity, D is characteristic length, and μ is dynamic viscosity. Indicates laminar (Re < 2100) or turbulent flow (Re > 4000).

Darcy-Weisbach Equation:

ΔP = f (L/D) (ρV2/2), where f is the friction factor.

Materials Science

Material Properties

Young’s Modulus (E):

Measure of stiffness or resistance to elastic deformation: E = Stress/Strain

Poisson’s Ratio (ν):

Ratio of transverse strain to axial strain.

Yield Strength (σy):

Stress at which a material begins to deform plastically.

Tensile Strength (σu):

Maximum stress a material can withstand before breaking.

Hardness:

Resistance to localized plastic deformation (e.g., indentation).

Ductility:

Ability of a material to deform plastically before fracture.

Stress and Strain

Stress (σ):

Force per unit area: σ = F/A

Strain (ε):

Deformation per unit length: ε = ΔL/L

Shear Stress (τ):

Stress acting parallel to a surface.

Shear Strain (γ):

Angular deformation.

Hooke’s Law:

σ = Eε (in the elastic region)

Material Types

Metals: High strength, ductility, and thermal/electrical conductivity. Examples: Steel, Aluminum, Copper.

Ceramics: High hardness, brittleness, and resistance to high temperatures. Examples: Alumina, Silicon Carbide.

Polymers: Low density, flexibility, and can be easily molded. Examples: Polyethylene, Polypropylene.

Composites: Combination of two or more materials to achieve enhanced properties. Examples: Carbon fiber reinforced polymer (CFRP).

Mechanics of Materials

Stress Analysis

Axial Stress (σ):

Stress due to axial force: σ = P/A, where P is the axial force and A is the cross-sectional area.

Bending Stress (σb):

Stress due to bending moment: σb = My/I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia.

Shear Stress in Beams (τ):

τ = VQ/Ib, where V is the shear force, Q is the first moment of area, I is the moment of inertia, and b is the width of the beam.

Torsional Shear Stress (τ):

τ = Tr/J, where T is the torque, r is the radius, and J is the polar moment of inertia.

Principal Stresses:

Maximum and minimum normal stresses at a point.

Beam Deflection

Deflection Formulas:

Vary based on loading and support conditions. Common cases include cantilever beams and simply supported beams with various loads.

Cantilever Beam with Point Load at End:

Maximum deflection (δ) = (PL3)/(3EI), where P is the load, L is the length, E is Young’s modulus, and I is the moment of inertia.

Simply Supported Beam with Uniform Load:

Maximum deflection (δ) = (5wL4)/(384EI), where w is the uniform load per unit length.

Failure Theories

Maximum Shear Stress Theory (Tresca): Failure occurs when maximum shear stress exceeds the shear strength of the material.

Distortion Energy Theory (von Mises): Failure occurs when the distortion energy reaches the distortion energy at yield in a simple tension test.

Maximum Principal Stress Theory: Failure occurs when the maximum principal stress exceeds the tensile strength of the material.
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