Your FREE FE Mechanical Flashcards 2026 – 150+ Cards

FE Mechanical Flashcards

Order of an ODE vs. arbitrary constants

The number of arbitrary constants in the general solution of an ordinary differential equation equals its order; a first-order ODE has one, a second-order ODE has two.

Simpson's one-third rule

A numerical integration rule that fits a second-degree (parabolic) polynomial through three points spanning two subintervals; it requires an even number of subintervals.

Trapezoidal rule

A numerical integration method that approximates the area under a curve with straight-line segments between points; less accurate than Simpson's rule because it ignores curvature.

Newton's method

An iterative root-finding method that refines a guess using $x_{n+1} = x_n - f(x_n)/f'(x_n)$; it converges quadratically near a simple root.

Determinant of a 2×2 matrix

For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ the determinant is $ad - bc$. A nonzero determinant means the matrix is invertible.

Eigenvalue

A scalar λ for which $A\mathbf{v} = \lambda\mathbf{v}$ has a nonzero solution vector v; found by solving $\det(A - \lambda I) = 0$.

Partial derivative

The derivative of a multivariable function with respect to one variable while holding the others constant; e.g. for $f = x^2 y$, $\partial f/\partial x = 2xy$.

Euler's identity (complex numbers)

$e^{i\theta} = \cos\theta + i\sin\theta$, the link between polar and rectangular forms of a complex number; $r(\cos\theta + i\sin\theta)$ converts to $r\cos\theta + i\,r\sin\theta$.

Gradient (∇f)

A vector of the partial derivatives of a scalar field; it points in the direction of steepest increase, with magnitude equal to the maximum rate of change.

Laplace transform (use)

An integral transform that converts a linear differential equation in time into an algebraic equation in the s-domain, simplifying the solution of ODEs and control systems.

Taylor series

A representation of a function as an infinite sum of terms from its derivatives at a point: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$.

Dot product vs. cross product

The dot product $\mathbf{a}\cdot\mathbf{b}=|a||b|\cos\theta$ yields a scalar; the cross product $\mathbf{a}\times\mathbf{b}$ yields a vector perpendicular to both with magnitude $|a||b|\sin\theta$.

Empirical (68-95-99.7) rule

For a normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Arithmetic mean

The sum of all observations divided by the number of observations; the most common measure of central tendency.

Standard deviation

A measure of dispersion equal to $\sqrt{\sigma^2}$ (the root of the variance); it quantifies the typical spread of data about the mean.

z-score (standard normal value)

The number of standard deviations an observation lies from the mean: $z = (x - \mu)/\sigma$.

Confidence interval (interpretation)

A range computed by a method that captures the true parameter a stated percentage of the time over repeated sampling; it is a statement about the procedure, not one future value.

Effect of sample size on interval width

Increasing the sample size narrows a confidence interval, because the margin of error scales as $\sigma/\sqrt{n}$.

Coefficient of determination (R²)

The fraction of the variation in the response y explained by the regression model; an $R^2$ of 0.96 means 96% of the variance in y is explained by x.

Expected value

The probability-weighted average of all possible outcomes of a random variable: $E[X] = \sum x_i\,p_i$; used in decision making.

Binomial distribution

The discrete distribution of the number of successes in n independent trials each with success probability p; mean $np$, variance $np(1-p)$.

Least-squares regression

A curve-fitting method that minimizes the sum of squared residuals between data points and the fitted line, producing the best linear unbiased estimate of the trend.

Hold paramount public safety

The first canon of engineering ethics: engineers must hold paramount the safety, health, and welfare of the public, above duties to client or employer.

Areas of competence

An ethics rule requiring engineers to perform services only in fields where they are qualified by education or experience.

Faithful agent or trustee

The duty to serve each client or employer honestly, which requires avoiding conflicts of interest and disclosing any that arise.

Responsible charge (sealing)

An engineer may seal and sign only work prepared by or under their direct supervisory control; sealing unreviewed outside work is a prohibited misrepresentation.

Objective and truthful statements

The rules of conduct require engineers to issue public statements that are objective and truthful and to include all relevant information.

Overruled professional judgment

When an engineer's safety judgment is overruled in a way that endangers the public, the engineer must notify the client/employer and the appropriate authority and refuse to proceed unsafely.

Intellectual property

Legal protections for original work — patents, copyrights, trade secrets, and trademarks — that engineers must respect and not misappropriate.

NCEES Model Law

The model legislation NCEES publishes for states to adopt, defining licensure requirements and the rules of professional conduct for engineers.

Sustainability (societal consideration)

An ethical and professional obligation to weigh long-term economic, environmental, and life-cycle impacts of engineering decisions on society.

Time value of money

The principle that a sum of money has different worth at different points in time; cash flows must be made equivalent using an interest rate before comparison.

Present worth

The value today of a future cash flow, found by discounting it at the interest rate: $P = F/(1+i)^n$.

Rate of return (ROR)

The interest rate that makes the present worth of all cash flows equal to zero; also the rate at which net present worth is zero.

Benefit-cost ratio

The equivalent worth of benefits divided by the equivalent worth of costs; a public project is generally justified when the ratio is ≥ 1.0.

Equivalent uniform annual cost (EUAC)

All cash flows converted to an equivalent uniform yearly amount; among alternatives the one with the lowest EUAC is preferred.

Sunk cost

Money already spent that cannot be recovered; it is irrelevant to future decisions and must be excluded from the analysis.

Opportunity cost

The value of the best alternative forgone when a choice is made; unlike a sunk cost, it does influence decisions.

Effect of inflation

Inflation reduces the real purchasing power of a fixed future sum, which is why economics distinguishes real (constant) dollars from actual (then-current) dollars.

Annuity (uniform series)

A series of equal cash flows at regular intervals; converted to present or future worth with the (P/A) and (F/A) interest factors.

Fixed vs. variable cost

Fixed costs do not change with output (e.g., rent); variable costs change in proportion to production volume (e.g., materials).

Break-even analysis

Finding the production level or time at which total revenue equals total cost, so neither profit nor loss occurs.

Ohm's law

Voltage equals current times resistance: $V = IR$; the foundational relationship of DC circuit analysis.

Kirchhoff's current law (KCL)

The sum of currents entering a node equals the sum leaving it — a statement of charge conservation at a junction.

Kirchhoff's voltage law (KVL)

The algebraic sum of voltages around any closed loop in a circuit equals zero — a statement of energy conservation.

Electrical power

The rate of energy delivery in a circuit: $P = VI = I^2 R = V^2/R$.

Series vs. parallel resistance

Resistors in series add ($R = R_1 + R_2 + \dots$); resistors in parallel add as reciprocals ($1/R = 1/R_1 + 1/R_2 + \dots$).

Coulomb's law

The electrostatic force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.

Magnetic flux density (B)

The amount of magnetic flux passing through a unit area, measured in teslas; describes how concentrated a magnetic field is.

Faraday's law of induction

The induced EMF in a loop is proportional to the rate of change of magnetic flux through it; a steady flux induces no EMF.

Capacitor (DC steady state)

A device that stores charge on an electric field ($Q = CV$); in DC steady state it acts as an open circuit (no current flows).

Inductor (DC steady state)

A device that stores energy in a magnetic field; in DC steady state it acts as a short circuit (no voltage drop).

Equilibrium equations (2D)

A rigid body in planar static equilibrium has three independent equations: ΣFₓ = 0, ΣFᵧ = 0, and ΣM = 0 about any point.

Couple

Two equal, opposite, parallel forces that produce a pure moment with zero net force; its moment is the same about every point.

Two-force member

A member loaded at only two points; in equilibrium the forces must be equal, opposite, and collinear, so the force acts along the line joining the two points.

Truss member: tension vs. compression

A member is in tension when its internal force pulls the joints toward its center, and in compression when the force pushes them apart; ideal trusses carry only axial force.

Method of joints

A truss-analysis technique applying ΣFₓ = 0 and ΣFᵧ = 0 at each pin joint (no moment equation needed, since all forces pass through the joint).

Method of sections

A truss-analysis technique that cuts through the truss and applies equilibrium to one portion, directly finding the force in a specific interior member.

Centroid

The geometric center of an area or volume; the point about which the first moment of area is zero. Found by $\bar{x} = \frac{\int x\,dA}{\int dA}$.

Moment of inertia (area)

The second moment of area about an axis, a measure of a cross-section's resistance to bending or buckling: $I = \int y^2\,dA$.

Parallel-axis theorem

Relates the moment of inertia about a centroidal axis to a parallel axis: $I = I_c + A d^2$, where d is the distance between the axes.

Static friction (maximum)

The maximum friction force before sliding equals the coefficient of static friction times the normal force: $F_{max} = \mu_s N$.

Free-body diagram

A sketch isolating a body and showing all external forces and moments acting on it; the essential first step in any statics or dynamics problem.

Newton's second law

The net force on a particle equals its mass times its acceleration: $\sum F = ma$, with acceleration in the direction of the net force.

Linear momentum

The product of mass and velocity, $\mathbf{p} = m\mathbf{v}$; conserved in any collision absent external impulses.

Elastic vs. inelastic collision

Both conserve momentum, but only a perfectly elastic collision also conserves kinetic energy; an inelastic collision loses kinetic energy to deformation or heat.

Coefficient of restitution (e)

The ratio of relative velocity of separation to relative velocity of approach; e = 1 is perfectly elastic, e = 0 is perfectly plastic (bodies stick).

Work of a force

Work equals the force times the displacement in the direction of the force: $W = \int \mathbf{F}\cdot d\mathbf{r}$; a perpendicular force does no work.

Work-energy theorem

The net work done on a particle equals its change in kinetic energy: $W_{net} = \Delta KE = \tfrac{1}{2}mv_2^2 - \tfrac{1}{2}mv_1^2$.

Impulse-momentum principle

The impulse of the net force equals the change in momentum: $\int F\,dt = m\Delta v$.

Period of oscillation

The time to complete one full cycle of motion; it is the reciprocal of frequency, $T = 1/f$.

Natural frequency

The frequency at which a system oscillates freely after a disturbance; for a spring-mass system $\omega_n = \sqrt{k/m}$.

Resonance

The large-amplitude response that occurs when a forcing frequency approaches a system's natural frequency, adding energy in phase each cycle; limited only by damping.

Damping

An energy-dissipating effect that reduces vibration amplitude over time; underdamped systems oscillate with decay, critically damped return fastest without oscillating.

Normal vs. tangential acceleration

In curvilinear motion, tangential acceleration $a_t = dv/dt$ changes speed, while normal (centripetal) acceleration $a_n = v^2/\rho$ changes direction toward the center.

dV/dx and dM/dx relations

On a beam, the slope of the shear diagram equals the negative distributed load (dV/dx = −w), and the slope of the moment diagram equals the shear (dM/dx = V).

Maximum moment location

The bending moment reaches a local maximum or minimum where the shear diagram passes through zero.

Normal stress (axial)

The internal force per unit area on a cross-section perpendicular to the load: $\sigma = P/A$.

Shear stress (direct)

The internal force per unit area parallel to a cross-section: $\tau = V/A$.

Hooke's law

In the elastic region, stress is proportional to strain: $\sigma = E\varepsilon$, where E is the modulus of elasticity.

Bending stress (flexure formula)

The normal stress from bending a beam: $\sigma = Mc/I$, where M is the moment, c the distance to the outer fiber, and I the moment of inertia.

Torsional shear stress

The shear stress in a circular shaft under torque: $\tau = Tr/J$, where T is torque, r the radius, and J the polar moment of inertia.

Mohr's circle (purpose)

A graphical construction giving the principal stresses (where it crosses the normal-stress axis) and the maximum in-plane shear stress (its radius).

Principal planes

The orientations on which shear stress is zero and the normal stresses reach their extreme (principal) values; planes of maximum shear lie 45° from them.

Thermal stress

Stress induced when thermal expansion or contraction is restrained: $\sigma = E\alpha\,\Delta T$ for a fully constrained member.

Euler buckling load

The critical axial load for a slender column: $P_{cr} = \frac{\pi^2 EI}{(KL)^2}$. Doubling the length quarters the buckling load.

Effective length factor (K)

Accounts for end conditions in buckling: K = 1.0 for pinned-pinned, K = 0.5 for fixed-fixed, K = 0.7 for fixed-pinned, and K = 2.0 for fixed-free.

Poisson's ratio (ν)

The ratio of lateral strain to axial strain under uniaxial load; for most metals ν ≈ 0.3.

Ultimate tensile strength

The highest point on the engineering stress-strain curve — the maximum stress a material sustains before necking and fracture.

Yield strength (0.2% offset)

The stress marking the onset of permanent deformation, found for metals without a sharp yield point by the 0.2% (0.002 strain) offset method.

Modulus of elasticity (Young's modulus)

The slope of the initial linear elastic portion of the stress-strain curve; a measure of material stiffness.

Percent elongation

A ductility measure: the change in gauge length over the original gauge length, ×100. From 50 mm to 62 mm gives 24%.

Ductile vs. brittle material

A ductile material undergoes large plastic deformation before fracture (e.g., mild steel); a brittle material fractures with little plastic strain (e.g., cast iron, ceramics).

Pearlite

The lamellar microstructure of alternating ferrite and cementite formed when eutectoid-composition austenite is slowly cooled on the iron-carbon diagram.

Martensite

A hard, brittle non-equilibrium phase formed by rapidly quenching austenite, which traps carbon and suppresses diffusion; it does not appear on the equilibrium diagram.

Steel vs. cast iron boundary

On the iron-carbon diagram, alloys below about 2.1% carbon are steels; above that they are cast irons. 0.76% C marks the eutectoid composition.

Quenching (hardening)

A heat treatment that rapidly cools austenite (in water or oil) to form martensite, producing a hard but brittle structure.

Tempering

Reheating quenched (martensitic) steel to a moderate temperature to reduce brittleness and internal stress, trading some hardness for toughness.

Annealing

A heat treatment that slowly cools steel to soften it, relieve internal stresses, and refine grain structure, yielding equilibrium phases.

Fatigue failure

Failure under repeated cyclic loading at stresses below the static strength; it initiates at a stress concentration and propagates as a growing crack.

Creep

Slow, time-dependent plastic deformation under sustained load, especially at elevated temperatures (above roughly 0.4 of the absolute melting temperature).

Hardness

A material's resistance to localized plastic deformation (indentation), measured on scales such as Rockwell, Brinell, and Vickers; correlates with tensile strength.

Reynolds number

A dimensionless ratio of inertial to viscous forces, $Re = \rho V D/\mu$; pipe flow is laminar below ~2300 and turbulent above ~4000.

Laminar vs. turbulent flow

Laminar flow moves in smooth parallel layers at low Re; turbulent flow has chaotic mixing and eddies at high Re with greater frictional losses.

Laminar pipe velocity profile

Fully developed laminar pipe flow has a parabolic profile, zero at the wall (no-slip) and maximum at the centerline, where it is twice the average velocity.

Bernoulli's equation

For steady, incompressible, inviscid flow along a streamline, $P + \tfrac{1}{2}\rho V^2 + \rho g z$ is constant — conservation of mechanical energy.

Continuity equation

Conservation of mass for incompressible flow: $A_1 V_1 = A_2 V_2$, so a smaller area means higher velocity.

Hydrostatic pressure

The pressure at depth in a static fluid: $P = \rho g h$; it acts equally in all directions at a point (Pascal's principle).

Buoyant force (Archimedes)

The upward force on a submerged or floating body equals the weight of the fluid it displaces; a floating body's buoyant force equals its weight.

Viscosity

A fluid's resistance to shear deformation; dynamic viscosity μ relates shear stress to velocity gradient: $\tau = \mu\,du/dy$.

Major head loss (Darcy-Weisbach)

Friction loss in a pipe: $h_f = f \frac{L}{D}\frac{V^2}{2g}$, where f is the friction factor from the Moody chart.

Mach number

The ratio of flow speed to the local speed of sound, $Ma = V/c$; flow is subsonic below 1, supersonic above 1.

Normal shock wave

An abrupt, irreversible compression in supersonic flow: across it the Mach number drops to subsonic, while static pressure, temperature, and entropy increase.

Pump affinity (scaling) laws

For a centrifugal pump, flow scales with speed ($Q \propto N$), head with speed squared ($H \propto N^2$), and power with speed cubed ($P \propto N^3$).

First law of thermodynamics

Energy is conserved: for a closed system, $\Delta U = Q - W$ — the change in internal energy equals heat added minus work done by the system.

Second law (Clausius statement)

Heat flows spontaneously from a hotter body to a colder body; the reverse cannot occur without external work input.

Second law (Kelvin-Planck)

No cyclic heat engine can be 100% efficient; some heat must always be rejected to a low-temperature reservoir.

Entropy

A property measuring the unavailability of a system's energy or its degree of disorder; it increases in any real (irreversible) process.

Carnot efficiency

The maximum efficiency of any engine between two reservoirs, $\eta = 1 - T_C/T_H$ (absolute temperatures), achieved only by fully reversible processes.

Compressed (subcooled) liquid

A state in which the temperature is below the saturation temperature at the given pressure, so the substance remains pure liquid.

Superheated vapor

A state in which the temperature is above the saturation temperature at the given pressure, so the substance is entirely vapor.

Quality (x)

In a two-phase mixture, the mass fraction that is vapor: $x = m_{vapor}/m_{total}$, ranging from 0 (saturated liquid) to 1 (saturated vapor).

Rankine cycle

The ideal steam power cycle: isentropic pump compression, constant-pressure boiler heat addition, isentropic turbine expansion, and constant-pressure condenser heat rejection.

Brayton cycle

The ideal gas-turbine cycle: two isentropic processes (compressor, turbine) and two constant-pressure processes (combustor, cooling).

Otto cycle

The ideal spark-ignition engine cycle: two isentropic and two constant-volume processes; efficiency rises with compression ratio.

Ideal gas law

The equation of state $PV = mRT$ (or $Pv = RT$), relating pressure, volume, mass, and absolute temperature for an ideal gas.

Coefficient of performance (COP)

The efficiency measure for refrigerators and heat pumps: the desired heat transfer divided by the work input; can exceed 1.

Enthalpy (h)

A property combining internal energy and flow work, $h = u + Pv$; convenient for analyzing open (flow) systems and constant-pressure processes.

Three modes of heat transfer

Conduction (molecular contact within a medium), convection (energy carried by fluid motion), and radiation (electromagnetic waves, needing no medium).

Radiation (distinguishing feature)

The only mode that transfers energy through a vacuum, carried by electromagnetic waves; how solar energy reaches Earth.

Fourier's law (conduction)

Conduction heat rate is proportional to the temperature gradient and thermal conductivity: $q = -kA\,dT/dx$.

Newton's law of cooling (convection)

Convective heat rate is proportional to the surface area and temperature difference: $q = hA(T_s - T_\infty)$, with h the convection coefficient.

Stefan-Boltzmann law (radiation)

Radiant emission scales with the fourth power of absolute temperature: $q = \varepsilon\sigma A T^4$.

Thermal resistance

An analogy to electrical resistance for steady conduction; series resistances add, allowing the total heat rate to be found from the overall temperature difference.

Biot number

A dimensionless ratio of internal conduction resistance to surface convection resistance; if $Bi < 0.1$, the lumped-capacitance method applies (uniform internal temperature).

LMTD method

Heat-exchanger analysis using the log mean temperature difference; it requires both outlet temperatures, so sizing problems with unknown outlets need iteration.

Effectiveness-NTU method

A heat-exchanger method preferred when inlet temperatures are known but outlets are not; it gives a direct solution without iteration.

Heat-exchanger effectiveness (ε)

The actual heat transfer rate divided by the maximum possible rate; a dimensionless value between 0 and 1.

Fins (extended surfaces)

Surfaces added to increase the area available for convection; most effective with high base conductivity and a low convection coefficient (e.g., air-cooled).

PID controller

A feedback controller summing three actions: proportional (present error), integral (accumulated past error), and derivative (predicted rate of change of error).

Integral action (offset)

The PID term responsible for eliminating steady-state error, because it keeps accumulating error and adjusting output until the residual offset reaches zero.

Proportional gain (effect)

Raising Kp shrinks steady-state offset but makes the response more oscillatory and can reduce stability; proportional action alone cannot eliminate offset.

Derivative action

The PID term that responds to the rate of change of error, adding anticipatory damping that reduces overshoot but amplifies measurement noise.

Transfer function

The ratio of the Laplace transform of the output to that of the input, with zero initial conditions; it characterizes a linear time-invariant system in the s-domain.

Feedback (closed loop)

A control configuration that measures the output and feeds it back to compute the error driving the controller, improving accuracy and disturbance rejection.

Block diagram

A graphical representation of a control system as interconnected blocks (transfer functions) and signal arrows, used to derive the overall system response.

Accuracy vs. precision

Accuracy is closeness of a reading to the true value; precision is the repeatability of readings to one another. An instrument can be precise yet inaccurate (biased).

Systematic vs. random error

Systematic error is a consistent one-directional bias (not removed by averaging); random error is unpredictable scatter (reduced by averaging many readings).

Measurement uncertainty

The quantified doubt about a measurement result; propagated through calculations from the uncertainties of each input quantity.

Significant figures

The digits in a measurement that carry meaning about its precision; a calculated result should not be reported more precisely than its least-precise input.

Factor of safety

A material strength divided by the actual (working) stress in a part; a value greater than 1 provides a design margin against failure.

Distortion-energy (von Mises) theory

The most accurate static-failure theory for ductile metals: yielding occurs when the von Mises equivalent stress (combining all stress components) reaches the yield strength.

Maximum-shear-stress theory

A ductile-failure theory (Tresca) predicting yielding when the maximum shear stress reaches half the yield strength; slightly more conservative than von Mises.

Maximum-normal-stress theory

A failure theory for brittle materials: failure occurs when the maximum principal stress reaches the material's ultimate strength.

Goodman diagram

A fatigue-design plot with mean stress on the horizontal axis and alternating stress on the vertical axis; the line runs from the endurance limit to the ultimate strength.

Endurance limit

The cyclic stress amplitude below which a material (notably steel) can endure effectively infinite cycles without fatigue failure.

Stress concentration factor (Kt)

A multiplier on nominal stress accounting for the local rise in stress at a geometric discontinuity such as a hole, notch, or fillet.

Rotating shaft design

A rotating shaft is designed against combined torsional and bending fatigue, because rotation turns a steady transverse load into a fully reversed bending stress.

Spring rate (stiffness)

The force per unit deflection of a spring, $k = F/\delta$; for a helical compression spring it depends on wire diameter, coil diameter, and number of active coils.

Bearings (rolling vs. journal)

Rolling-element bearings use balls or rollers to reduce friction and carry load; journal (sliding) bearings rely on a lubricant film between shaft and bushing.

Power transmission (shaft power)

The power transmitted by a rotating shaft equals torque times angular speed: $P = T\omega$.

GD&T

Geometric Dimensioning and Tolerancing — a symbolic system (per ASME Y14.5) on engineering drawings that specifies allowable form, orientation, and location of features.

Limits and fits

The system of tolerances defining clearance, transition, or interference between mating parts (e.g., a press fit), ensuring assembly and function.

Pressure vessel hoop stress

The circumferential stress in a thin-walled cylinder under internal pressure: $\sigma_h = pr/t$, twice the longitudinal stress $\sigma_l = pr/2t$.