Catalogue
Calculators
Focused engineering tools for fast, well-defined problems.
152 calculators
Dynamics & Control
19 calculatorsDomain overview →
Time and frequency response of dynamic systems, vibrations and control design.
System Dynamics
- First-Order System CalculatorPole, DC gain, time constant, 10–90% rise time and settling estimates of the standard lag G(s) = K/(τs + 1).
- Damping Ratio CalculatorDamping ratio ζ implied by a percent overshoot of the canonical second-order step response.
- Percent Overshoot CalculatorPercent overshoot of the unit-step response for a given damping ratio ζ, with the damping regime.
- Settling Time CalculatorApproximate 2% and 5% settling times from ζ and ωn, within the underdamped envelope approximation.
- Damped Natural Frequency CalculatorDamped natural frequency ωd from undamped natural frequency ωn and damping ratio ζ.
- Second-Order System CalculatorAll canonical second-order response characteristics for a given ωn and ζ: regime, poles, ωd, overshoot, peak time and settling estimates.
Transfer Functions & Stability
- Transfer Function CalculatorOrder, relative degree, properness, DC gain, poles, zeros and stability of a transfer function entered as numerator and denominator coefficients.
- Poles and Zeros CalculatorPoles and zeros with multiplicities, half-plane classification, right-half-plane zero count and an s-plane map.
- Bode Plot CalculatorExact magnitude and phase against frequency, with break frequencies, asymptotic slopes, resonance and bandwidth where they are defined.
- Routh–Hurwitz Stability CalculatorRouth array, first-column sign changes, right-half-plane root count and Hurwitz determinants for a real characteristic polynomial.
Control
- Gain and Phase Margin CalculatorEvery gain and phase crossover of a loop transfer function, with finite, infinite or undefined margin semantics and multiple-crossing warnings.
- Steady-State Error CalculatorSystem type, Kp, Kv, Ka and the steady-state error of a unity-feedback loop to step, ramp and parabolic references.
- Root Locus CalculatorRoot locus of 1 + K·G(s) = 0: branches, asymptotes, breakaway points, imaginary-axis crossings, stable gain range and the closed-loop poles at a chosen gain.
- PID Controller CalculatorClosed-loop poles, stability verdict, overshoot, settling time, steady-state step error and margins of a unity-feedback loop with C(s) = Kp + Ki/s + Kd·s.
Vibrations & Modal
- Mass–Spring–Damper CalculatorNatural frequency, critical damping, ζ, regime, damped frequency and period, logarithmic decrement, Q, poles and X/F for an SDOF system.
- Natural Frequency CalculatorUndamped natural frequency, natural period, critical damping and damping ratio of a mass–spring–damper system from m, c and k.
- Vibration Isolation CalculatorAbsolute displacement transmissibility, force transmissibility and relative displacement ratio of an SDOF mount at a given excitation frequency.
- 2DOF Natural Frequencies CalculatorUndamped natural frequencies and mass-normalized mode shapes of a two-degree-of-freedom mass–spring chain, from K φ = ω² M φ.
State Space
Engineering Mechanics
23 calculatorsDomain overview →
Forces, motion, mass properties, rigid-body mechanics and vibrations.
Statics & Equilibrium
- Resultant Force CalculatorVector sum of coplanar forces: resultant components, magnitude and direction, with the zero-resultant case reported honestly.
- Moment of a Force CalculatorMoment of a planar force about a chosen point, M = r × F, positive counter-clockwise, with the perpendicular lever arm.
- 2D Equilibrium CalculatorSupport reactions of a planar rigid body from ΣFx = 0, ΣFy = 0 and ΣM = 0, with an explicit determinate, indeterminate or inconsistent verdict.
- Static Friction CalculatorMaximum static friction μs·N against the required equilibrium force, distinguishing sticking, impending slip and no static equilibrium.
Mass Properties
- Centroid CalculatorArea and geometric centroid of a single shape or a composite section, with cutouts as negative signed areas.
- Center of Mass CalculatorTotal mass and centre of mass of discrete planar point masses, or of a composite body built from shapes with densities, with each contribution shown.
- Area Moment of Inertia CalculatorCentroidal Ix, Iy and Ixy in m⁴ for standard shapes, with parallel-axis translation and principal values.
- Mass Moment of Inertia CalculatorMass moment of inertia in kg·m² for standard rigid bodies, with the axis stated for every case.
- Huygens–Steiner (Parallel-Axis) Theorem CalculatorTransfer a centroidal inertia to a parallel axis with the Huygens–Steiner theorem, also known as the parallel-axis theorem, in the area form I_c + A·d² or the mass form I_G + m·d².
- Radius of Gyration CalculatorRadius of gyration k = √(I/A) for an area or k = √(I/m) for a mass, undefined when the divisor is not positive.
Particle Mechanics
- Constant Acceleration CalculatorExact constant-acceleration kinematics: the standard 1D relations with explicit solve modes, and 2D propagation of position, velocity and speed.
- Projectile Motion CalculatorIdeal projectile motion: launch components, apex, flight time to a chosen landing elevation, range, impact velocity and impact angle.
- Circular Motion CalculatorParticle circular motion: v = rω, normal and tangential acceleration, total acceleration, period, frequency and the vectors at a chosen angle.
- Newton's Second Law CalculatorΣF = m·a for a planar particle, with explicit modes for the resultant force, the mass or the acceleration from several known forces.
- Work–Energy CalculatorWork of a constant force, particle kinetic energy and the work–energy theorem, with an honest verdict when no physical final speed exists.
- Impulse–Momentum CalculatorPlanar impulse and linear momentum: p = m·v, J = F·Δt and J = Δp, kept as distinct quantities throughout.
- Collision CalculatorDirect 1D impact of two particles from momentum conservation and the coefficient of restitution, with the kinetic-energy loss.
- Curvilinear Motion CalculatorPlanar curvilinear kinematics in normal–tangential or polar coordinates, with the components, the total acceleration and the Coriolis-like polar term named explicitly.
Rigid-Body Mechanics
- Rotational Kinematics CalculatorConstant angular-acceleration relations between θ, ω, α and t, with one explicit unknown per mode.
- Rotational Dynamics CalculatorΣM_G = I_G·α for a planar rigid body, solved for the moment, the mass moment of inertia or the angular acceleration.
- Rigid-Body Work–Energy CalculatorT = ½m|v_G|² + ½I_Gω² with the translational and rotational parts kept apart, and T₂ − T₁ = W_net solved either way.
- Rigid-Body Impulse–Momentum CalculatorJ = m(v_G2 − v_G1) and K_G = I_G(ω₂ − ω₁) for one planar rigid body, including constant loads acting over a duration.
- Rolling Without Slipping CalculatorPure-rolling kinematics with the contact and top point velocities, and the incline case with the static friction the no-slip constraint demands.
Engineering Mathematics & Numerical Methods
19 calculatorsDomain overview →
Numerical methods engineers actually run: root finding, linear algebra, interpolation, differentiation and integration, and ordinary differential equations.
Root Finding
- Bisection Method CalculatorBracketed root of f(x) = 0 by repeated halving, with the sign-change requirement, bracket history and guaranteed error bound.
- Newton–Raphson CalculatorRoot of f(x) = 0 by the tangent update x − f(x)/f'(x), with an analytic or finite-difference derivative and explicit near-zero-derivative failure.
- Secant Method CalculatorDerivative-free root of f(x) = 0 from two starting values, with the secant update, step history and denominator failure reported.
- Fixed-Point Iteration CalculatorIteration x = g(x) with cobweb geometry, an optional residual f(x), and the local |g'(x*)| contraction diagnostic.
Linear Algebra & Matrix Computation
- Linear System SolverSolve Ax = b by Gaussian elimination with partial pivoting: unique, infinitely many or no solution classified by rank, with the residual and the full elimination trace.
- LU Decomposition CalculatorLU factorization with partial pivoting: P, L and U, the row swaps, pivot growth and the verified factorisation residual ‖PA − LU‖_F.
- Matrix Determinant CalculatorCalculate det(A) for a real square matrix from a numerically stable pivoted factorisation, with the sign, the logarithmic magnitude, row-swap parity and singularity diagnostics.
- Matrix Inverse CalculatorCalculate A⁻¹ for a real square matrix by solving A X = I against one pivoted factorisation, verified from both sides and reported with the conditioning of A.
- Matrix Condition Number CalculatorMatrix conditioning from singular values: κ₂ = σmax/σmin with κ₁ and κ∞, the reciprocal condition number, and why a small residual is not the same as a well-conditioned problem.
- Eigenvalues & Eigenvectors CalculatorEigenvalues and eigenvectors of a real square matrix, including complex conjugate pairs and repeated or defective eigenvalues, each validated by its own residual.
Interpolation & Approximation
- Linear Interpolation CalculatorInterpolate between two known points along the straight line joining them, with the fraction t, the slope and an explicit interpolation-or-extrapolation verdict.
- Lagrange Interpolation CalculatorThe unique polynomial of degree ≤ n − 1 through n points, evaluated in stable barycentric form, with degree, data span and extrapolation reported.
- Cubic Spline CalculatorNatural cubic spline through your data: piecewise cubic with C² continuity, S″ = 0 at both ends, segment coefficients and an explicit extrapolation verdict.
Numerical Differentiation & Integration
- Numerical Derivative CalculatorEstimate f'(x) by forward, backward or central difference with an automatic or manual step h, and see the stencil values and the step actually used.
- Trapezoidal Rule CalculatorDefinite integral by the composite trapezoidal rule, with the step size, node table, evaluation count and the O(h²) validity conditions.
- Simpson's Rule CalculatorDefinite integral by composite Simpson's 1/3 rule, with the even-n requirement, the 1-4-2-4-1 coefficients and the O(h⁴) validity conditions.
- Gaussian Quadrature CalculatorDefinite integral by Gauss-Legendre quadrature of order 2 to 8, with mapped nodes and weights, per-node contributions and the degree of exactness.
Ordinary Differential Equations
- Euler Method CalculatorSolve a scalar first-order initial-value problem with the explicit Euler method: signed fixed step, derived end time t₀ + N·h, full node table and an optional supplied reference.
- Runge-Kutta 4 (RK4) CalculatorSolve a scalar first-order initial-value problem with classical fourth-order Runge-Kutta: four stages per step, derived end time, node table and an optional supplied reference.
Structural Mechanics
20 calculatorsDomain overview →
Analyze how structural members carry loads, develop internal forces and stresses, deform, and lose stability.
Axial Loading
- Normal Stress CalculatorAverage normal stress σ = N/A on a cross-section from an internal axial force and an area, with the tension or compression sense preserved.
- Normal Strain CalculatorEngineering normal strain ε = ΔL/L from a change in length and an original length, in both plain and microstrain form.
- Hooke's Law CalculatorSolve σ = Eε for stress, strain or Young's modulus in the uniaxial linear-elastic range.
- Axial Deformation CalculatorElongation δ = NL/(AE) of a prismatic axially loaded member, signed so that tension lengthens and compression shortens.
Stress & Strain
- Factor of Safety CalculatorFactor of safety n = S/|σ| against a declared yield, ultimate or custom reference strength, with the matching utilisation.
- Plane Stress Transformation CalculatorTransform a plane-stress state σx, σy, τxy through a counter-clockwise rotation θ and read the rotated components.
- Principal Stresses CalculatorPrincipal stresses σ1 ≥ σ2, their orientations, and the maximum in-plane shear of a plane-stress state.
- Mohr's Circle CalculatorCentre, radius, principal stresses and maximum in-plane shear of a plane-stress state, drawn as a true Mohr circle with τ positive upward.
- Von Mises Stress CalculatorVon Mises equivalent stress for a plane-stress state, with an optional yield strength giving utilisation and a yield-based factor of safety.
Beams & Bending
- Beam Reactions CalculatorSupport reactions of a statically determinate beam — cantilever, simply supported or overhanging — from point loads, couples and uniform distributed loads, with equilibrium residuals and a determinacy verdict.
- Shear Force CalculatorInternal shear force at any station of a determinate beam, with both one-sided limits and the exact jump wherever a concentrated force or a support reaction acts.
- Bending Moment CalculatorInternal bending moment at any station of a determinate beam, sagging positive, with the exact −Mz jump at an applied couple and the analytically located extremes.
- Bending Stress CalculatorExtreme-fibre flexural stress σ = −My/I taken from the bending moment of the beam itself, with tension and compression named.
- Transverse Shear Stress CalculatorTransverse shear stress τ = VQ/(It) in a beam section, using the shear force the beam engine computes for the chosen station.
- Beam Deflection CalculatorSlope and deflection of a determinate beam by direct integration of EI v'' = M, with continuity at every load event and exact support conditions.
Torsion
- Torsional Shear Stress CalculatorShear stress in a solid or hollow circular shaft under torsion, τ = Tρ/J, at the outer surface and at any radius, with the polar second moment reported alongside.
- Angle of Twist CalculatorRelative rotation of one end of a circular shaft with respect to the other, φ = TL/(GJ), in radians and degrees, with the torsional stiffness GJ/L.
Columns & Buckling
- Slenderness Ratio CalculatorEffective length KL and slenderness ratio λ = KL/r of a column, from a radius of gyration entered directly or derived from I and A, with the theoretical end-condition factors.
- Euler Buckling Load CalculatorElastic critical load Pcr = π²EI/(KL)² of an ideal slender column, with critical stress and slenderness, reported explicitly as a bifurcation load and not a design capacity.
Trusses
Fluid Mechanics
22 calculatorsDomain overview →
Analyze fluids at rest and in motion through pressure, flow, energy, momentum, pipe losses and flow measurement.
Fluid Properties & Hydrostatics
- Density & Specific Gravity CalculatorDensity from mass and volume, specific gravity against a reference fluid, and specific weight γ = ρg.
- Hydrostatic Pressure CalculatorPressure at a depth in a static fluid, p = p_surface + ρgh, or between two elevations, with the pressure reference stated explicitly.
- Pressure Head CalculatorPressure expressed as a head of a chosen fluid, h = p/(ρg), and the pressure a given head represents.
- Manometer CalculatorPressure difference along a multi-fluid manometer path, built from ordered up and down moves through fluids of stated density.
- Hydrostatic Force CalculatorResultant pressure force on a plane submerged surface, F = ρg h_c A, for a rectangle or a circle at any inclination.
- Center of Pressure CalculatorDepth at which the hydrostatic resultant acts on a plane submerged surface, h_cp = h_c + I_G sin²θ/(A h_c).
- Buoyant Force CalculatorArchimedes buoyant force F_B = ρgV on a displaced volume, with an optional comparison against the body weight.
Conservation & Energy
- Continuity Equation CalculatorSteady incompressible continuity along one stream, A₁V₁ = A₂V₂, solved for either area, either speed or the common flow rate.
- Flow Rate & Velocity CalculatorQ = A V for one cross-section, from a diameter, a rectangle or a stated area, with the mass flow rate when a density is given.
- Bernoulli Equation CalculatorThe extended energy equation H₁ + h_p − h_t − h_L = H₂ between two stations, solved for a pressure, a speed, an elevation or a required head.
- Dynamic Pressure CalculatorDynamic pressure q = ½ρV² of a moving fluid, the speed it implies, and the velocity head it corresponds to.
- Pump Head & Power CalculatorHydraulic power P = ρgQh_p delivered to the fluid, the shaft power an efficiency implies, or the head or flow a stated power supports.
Internal Pipe Flow
- Reynolds Number CalculatorRe = ρVD_h/μ for internal flow, from a kinematic or a dynamic viscosity, with the laminar, transitional or turbulent regime it names.
- Hydraulic Diameter CalculatorD_h = 4A/P for a duct of any shape, with direct, circular and rectangular forms, for use in the Reynolds number and the Darcy–Weisbach loss.
- Darcy Friction Factor CalculatorThe Darcy friction factor f_D from f_D = 64/Re in laminar flow or the Colebrook correlation in turbulent flow, with the Fanning counterpart stated explicitly.
- Darcy–Weisbach Pressure Drop CalculatorMajor head loss h_f = f_D (L/D_h) V²/(2g) along a pipe run and the pressure drop Δp = ρgh_f it represents.
- Minor Loss CalculatorLocal head loss h_m = K V²/(2g) of a fitting, bend or valve, and the pressure drop that loss represents.
Momentum
Flow Measurement & Basic External Flow
- Venturi Meter CalculatorDischarge from a measured differential across a Venturi, or the differential a stated discharge produces, with the throat ratio, elevations and a prescribed discharge coefficient.
- Pitot Tube CalculatorSpeed magnitude from the stagnation-minus-static pressure difference, V = sqrt(2Δp/ρ), for steady incompressible flow along the stagnation streamline.
- Orifice & Torricelli CalculatorEfflux under a head: the Torricelli velocity sqrt(2gh) and the orifice discharge a prescribed discharge coefficient and area give it.
- Drag Force CalculatorF_D = ½ ρ V_rel² C_D A_ref with a prescribed drag coefficient, reported as a magnitude and, when a direction is stated, as a force opposing the relative motion.
Thermal Engineering
24 calculatorsDomain overview →
Ideal-gas properties and processes, energy balances, cycles and heat transfer.
Thermodynamic Properties & Processes
- Ideal Gas Law CalculatorpV = mRT solved for pressure, volume, mass, temperature or the gas constant, with absolute pressure and temperature only.
- Ideal-Gas Internal Energy & Enthalpy CalculatorΔu = cᵥΔT and Δh = cₚΔT with totals for a mass, or the final temperature a stated Δu or Δh implies.
- Specific Heat & Heat-Capacity Ratio CalculatorComplete R, cₚ, cᵥ and γ from any two, with redundant inputs cross-checked and conflicts refused.
- Polytropic Process Calculatorpvⁿ = const for an ideal gas: final state, boundary work, ΔU, ΔH and Q, with the isobaric, isothermal and isentropic cases named.
First Law & Energy Devices
- Closed-System First Law CalculatorΔU = Q − W for a fixed mass, solved for ΔU, Q or W, or all three checked for consistency, with Q > 0 into and W > 0 done by the system.
- Steady-Flow Energy Equation Calculatorq − wₛ = Δh + ΔV²/2 + gΔz per unit mass or as rates, solved for q, wₛ, h1, h2 or the outlet speed with every term shown.
- Turbine Power CalculatorTurbine shaft power and specific work from mass flow, enthalpy drop, speeds and elevations, adiabatic or with a stated heat rate.
- Compressor Power CalculatorCompressor power input and specific work input from the same steady-flow balance, with heat transfer stated rather than assumed.
- Nozzle & Diffuser CalculatorNozzle exit speed from an enthalpy drop or diffuser exit enthalpy from a speed drop, with no shaft work and impossible kinetic-energy states refused.
Thermodynamic Cycles
- Carnot Efficiency CalculatorThe reversible limit η = 1 − T_C/T_H between two reservoirs, with the work and rejected heat of a stated heat input.
- Otto Cycle CalculatorIdeal air-standard Otto efficiency from compression ratio and γ, with the four corner states and heat terms.
- Diesel Cycle CalculatorIdeal air-standard Diesel efficiency from compression ratio, cutoff ratio and γ; r_c → 1 recovers the Otto cycle.
- Brayton Cycle CalculatorIdeal air-standard Brayton efficiency from pressure ratio and γ, with the four corner states from inlet and turbine-inlet temperatures.
- Refrigerator & Heat-Pump COP CalculatorReversible refrigerator and heat-pump COPs between two reservoirs, with COP_HP = COP_R + 1 kept visible.
Conduction & Convection
- Plane-Wall Conduction CalculatorSteady plane-wall resistance, signed heat rate and heat flux from constant conductivity and boundary temperatures.
- Cylindrical-Wall Conduction CalculatorLogarithmic radial resistance and signed heat rate through a cylindrical shell.
- Thermal Resistance Network CalculatorSeries or parallel equivalent of conduction, convection and prescribed resistances, with topology-qualified temperatures.
- Convection Heat Transfer CalculatorNewton cooling with prescribed h: resistance, signed surface-to-fluid heat rate and heat flux.
- Fin Efficiency & Heat Transfer CalculatorStraight uniform adiabatic-tip fin parameter, efficiency and signed heat rate, including the zero-length limit.
Radiation & Heat Exchangers
- Stefan–Boltzmann Radiation CalculatorStefan–Boltzmann Radiation from frozen Thermal primitives with explicit physical validity.
- Radiation Exchange CalculatorRadiation Exchange from frozen Thermal primitives with explicit physical validity.
- Overall Heat-Transfer Coefficient CalculatorOverall Heat-Transfer Coefficient from frozen Thermal primitives with explicit physical validity.
- LMTD Heat Exchanger CalculatorLMTD Heat Exchanger from frozen Thermal primitives with explicit physical validity.
- Effectiveness–NTU CalculatorEffectiveness–NTU from frozen Thermal primitives with explicit physical validity.
Electrical Engineering
25 calculatorsDomain overview →
DC networks, capacitors and transients, AC and RLC circuits, power and three-phase, magnetics and transformers.
DC Networks & Kirchhoff
- Ohm's Law CalculatorSolve voltage, current or resistance from the other two, with signed values and absorbed power.
- Series & Parallel Resistor CalculatorEquivalent resistance and conductance of up to 8 resistors, with each resistor's share of voltage or current.
- Voltage Divider CalculatorUnloaded two-resistor divider: output voltage, current, drops and dissipated powers.
- Current Divider CalculatorConductance-based split of a total current across up to 8 parallel branches.
- Thévenin & Norton Equivalent CalculatorV_th, R_th and I_N between two terminals of a DC network, with explicit polarity.
Capacitors & Transients
- Capacitance Calculator (C = Q/V)Solve capacitance, charge or voltage from the other two, with stored energy and signed charge/voltage.
- Parallel-Plate Capacitor CalculatorIdeal parallel-plate capacitance from area, gap and relative permittivity, with optional charge, field and energy.
- Capacitors in Series & Parallel CalculatorEquivalent capacitance of up to 8 capacitors with per-capacitor charge, voltage and energy.
- Capacitor Energy CalculatorStored energy and the remaining capacitor quantities from any sufficient pair of C, V, Q and E.
- RC Charge & Discharge CalculatorFirst-order RC response from any V0 to Vf: τ, Vc(t), i(t), Q(t), stored energy and progress.
AC & RLC
- Inductive Reactance CalculatorXL = 2πfL with the impedance jXL, its magnitude and +90° phase; f = 0 gives a short circuit.
- Capacitive Reactance CalculatorXC = 1/(2πfC) with the impedance −jXC and −90° phase; f = 0 is reported as an open circuit, never infinity.
- Series RLC Circuit CalculatorImpedance, phase and inductive/capacitive character of a series RLC circuit, with f0, Q and optional RMS current and element voltages.
- RLC Resonant Frequency Calculatorf0 = 1/(2π√LC) and ω0, with the quality factor only when R is supplied.
- RLC Transient Response CalculatorSeries RLC step or natural response from explicit Vc(0) and i(0): α, ω0, ζ, damping class, ωd, Vc(t) and i(t).
Power & Three-Phase
- Single-Phase AC Power CalculatorApparent, real and reactive power from RMS voltage, current and power factor, with phase angle and lagging/leading nature.
- Power Factor & Power Triangle CalculatorSolve the power triangle from any two of P, Q, S and |PF|; the sign of Q is never invented and inconsistent sets are refused.
- Power Factor Correction CalculatorSingle-phase shunt-capacitor correction: initial and target Q, required Qc and capacitance C; real power unchanged.
- Three-Phase Power CalculatorBalanced star/delta power: line and phase voltages and currents, P, Q and S, with consistency checks on redundant inputs.
- Star–Delta Line & Phase Relations CalculatorBalanced line/phase voltage and current relations for star and delta connections, solved independently per side.
Magnetics & Transformers
- Magnetic Circuit CalculatorLinear magnetics and ideal transformer analysis with shared SI state.
- Solenoid Inductance CalculatorLinear magnetics and ideal transformer analysis with shared SI state.
- Faraday’s Law / Induced EMF CalculatorLinear magnetics and ideal transformer analysis with shared SI state.
- Ideal Transformer CalculatorLinear magnetics and ideal transformer analysis with shared SI state.
- Reflected Impedance CalculatorLinear magnetics and ideal transformer analysis with shared SI state.