🧮Calculators

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Unbalance Force

Calculate the centrifugal force generated by an unbalanced rotor, F = m·e·ω², from the unbalanced mass m, the eccentricity e (distance from the centre of mass to the rotation axis) and the angular velocity ω (rad/s). The result, in newtons, is the rotating force that excites vibration in the bearings and structure — proportional to the square of speed, which is why unbalance becomes critical at high speeds. Quantifying it guides the balancing of rotors, fans, turbines and wheels, reducing vibration, noise and fatigue. Enter the unbalanced mass, the eccentricity and the angular velocity.

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Natural Frequency from Static Deflection

Calculate a system's natural frequency from its static deflection, f_n = (1 ÷ 2π)·√(g ÷ δ), where δ is the static deflection caused by self-weight and g the gravitational acceleration (9.81 m/s²). The result, in Hz, is a practical and elegant way to estimate the natural frequency without separately knowing mass and stiffness — you just measure how much the system sags under its own weight. Larger deflections (more flexible systems) give lower natural frequencies, desirable in vibration isolators. It is widely used in spring and mount design. Enter the static deflection (in metres).

⚙️

Shaft Critical Speed

Calculate the critical speed of a rotating shaft, ω_c = √(k ÷ m), from the shaft stiffness k and the rotor mass m. The result, in rad/s, is the rotational speed that coincides with the shaft's bending natural frequency — at it, any small unbalance causes large-amplitude resonant vibration that can damage the equipment. Shafts should run with a safe margin below the first critical speed (rigid rotors) or pass through it quickly to a range above (flexible rotors). It is an essential calculation in designing high-speed turbines, pumps and motors. Enter the stiffness and the mass.

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Energy Dissipated per Cycle

Calculate the energy dissipated per cycle in a viscous damper, ΔE = π·c·ω·X², from the viscous damping coefficient c, the excitation frequency ω (rad/s) and the vibration amplitude X. The result, in joules, is the mechanical energy converted to heat each oscillation cycle by the damper — proportional to frequency and to the square of amplitude. Quantifying it is essential to size dampers and dissipators (in suspensions, buildings under earthquakes, isolators) and to estimate the heat generated by vibration. The greater the dissipation, the faster free vibration decays. Enter the damping coefficient, the frequency and the amplitude.

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Crystallinity Degree (Enthalpy)

Calculate a polymer's crystallinity degree by the melting enthalpy method, X_c = (ΔH_m ÷ ΔH_m°) × 100%, dividing the melting enthalpy measured by DSC (ΔH_m) by the enthalpy of the 100% crystalline polymer (ΔH_m°, a tabulated reference). The result, in %, is the fraction of polymer mass organized in crystalline regions, as opposed to amorphous ones. Crystallinity governs key properties: higher crystallinity raises stiffness, strength, density, opacity and chemical resistance, but lowers transparency and impact resistance. Enter the measured melting enthalpy and that of the 100% crystalline material.

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Polydispersity Index (PDI)

Calculate a polymer's polydispersity index (PDI), PDI = M_w ÷ M_n, dividing the weight-average molar mass (M_w) by the number-average molar mass (M_n). The dimensionless result (always ≥ 1) measures the breadth of the molar mass distribution: PDI = 1 means a monodisperse polymer (all chains the same size, rare, typical of living polymers); larger values mean a wide range of sizes. Commercial polymers have PDI of 2 to 20, depending on the polymerization process. PDI affects processability, strength and flow properties. Enter M_w and M_n.

⛓️

Degree of Polymerization

Calculate a polymer chain's degree of polymerization, DP = M_n ÷ M₀, dividing the number-average molar mass of the chain (M_n) by the molar mass of the monomer or repeat unit (M₀). The dimensionless result is the average number of monomer units in each chain. The higher the degree of polymerization, the longer the chains and the more pronounced the polymer properties: increased mechanical strength, melt viscosity, transition temperature and toughness. Below a critical value, the material lacks typical polymer properties. Enter the chain molar mass and the monomer molar mass.

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Glass Transition Temperature (Fox)

Calculate the glass transition temperature (Tg) of a blend or copolymer by the Fox equation, 1 ÷ Tg = w₁/Tg₁ + w₂/Tg₂, from the mass fraction w₁ of component 1 (with w₂ = 1 − w₁) and the Tg of each pure component (in kelvin). The result, in K, is the temperature at which the mixture goes from the glassy (rigid) to the rubbery (flexible) state. The Fox equation predicts the Tg of miscible blends, random copolymers and plasticized systems, the basis for tuning a polymer's flexibility by adding plasticizers or comonomers. Enter the mass fraction of component 1 and the two Tg values.

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Melt Flow Index (MFI)

Calculate the melt flow index (MFI), MFI = (mass × 600) ÷ time, from the mass of polymer extruded (g) and the extrusion time (s), normalizing to the mass that flows in 10 minutes. The result, in g/10min, measures how easily the molten polymer flows under standardized load and temperature (plastometer test). High MFI indicates a low-viscosity, low-molar-mass polymer, easy to inject; low MFI indicates high viscosity, high molar mass, better for extrusion and blow molding. It is the most used quality control parameter in the plastics industry. Enter the extruded mass and the time.

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Intrinsic Viscosity (Mark-Houwink)

Calculate a polymer's intrinsic viscosity by the Mark-Houwink-Sakurada equation, [η] = K·Mᵃ, from the constants K and a (specific to the polymer-solvent-temperature system) and the viscosity-average molar mass M. The result, in dL/g, relates the viscosity of a dilute polymer solution to its molar mass — the basis of molar mass determination by viscometry, a simple and cheap technique. The exponent a (between 0.5 and 0.8) reflects the chain conformation in the solvent: 0.5 for a theta solvent (coiled chain) and up to 1.0 for an extended chain in good solvent. Enter the constants K, a and the molar mass.

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Crystallinity by Density

Calculate a polymer's crystallinity degree by the density method, X_c = [ρ_c·(ρ − ρ_a)] ÷ [ρ·(ρ_c − ρ_a)] × 100%, from the sample's measured density (ρ) and the densities of the 100% amorphous (ρ_a) and 100% crystalline (ρ_c) phases. The result, in %, relies on crystalline regions being more compact and dense than amorphous ones — the higher the sample density, the higher its crystallinity. It is an alternative to DSC (enthalpy), simple and accurate, using a density gradient column or pycnometry. Enter the sample, amorphous and crystalline densities.

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Fiber Volume Fraction

Calculate the fiber volume fraction of a composite, V_f = (W_f/ρ_f) ÷ [(W_f/ρ_f) + (W_m/ρ_m)] × 100%, from the masses (or mass fractions) and densities of the fiber (W_f, ρ_f) and matrix (W_m, ρ_m). The result, in %, converts the mass composition (easily measured) into the volume composition, which determines the composite's mechanical properties by the rule of mixtures. Fiber volume fraction is a laminate's most important parameter: the higher it is (up to the packing limit, ~60-70%), the greater the stiffness and strength in the fiber direction. Enter the fiber and matrix masses and densities.

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Composite Modulus (Rule of Mixtures)

Calculate the longitudinal elastic modulus of a composite by the rule of mixtures, E_c = E_f·V_f + E_m·(1 − V_f), from the fiber modulus (E_f), the fiber volume fraction (V_f) and the matrix modulus (E_m). The result, in the modulus unit (GPa), is the modulus in the fiber direction (Voigt upper bound), assuming equal strain in fiber and matrix. It shows the composite stiffness is a volume-weighted average — stiff fibers (carbon, glass) at high fraction greatly raise the modulus. In the transverse direction, the inverse rule of mixtures (Reuss bound) applies, much lower. Enter the fiber modulus, the fiber fraction and the matrix modulus.

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Cooling Time (Injection Molding)

Estimate the cooling time of a flat part in injection molding, t = h² ÷ (π²·α), from the wall thickness h and the polymer's thermal diffusivity α. The result, in seconds, is the dominant time of the injection cycle — the part can only be ejected after cooling enough to be rigid. The most critical factor is thickness squared: doubling the thickness quadruples the cooling time (and the cost per part). That is why thin, uniform walls are a golden rule in injection part design. Plastics' low thermal diffusivity makes cooling the productivity bottleneck. Enter the wall thickness and the thermal diffusivity.

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PID Tuning Ziegler-Nichols (Kp)

Calculate the proportional gain of a PID controller by the Ziegler-Nichols closed-loop method, K_p = 0.6·K_u, from the ultimate gain K_u (the proportional gain at which the system enters sustained oscillation). The result is the recommended proportional gain for a PID controller; the other parameters derive from the ultimate period T_u (T_i = 0.5·T_u and T_d = 0.125·T_u). It is the classic experimental controller-tuning method, giving a quick starting point for the response of industrial processes, later adjusted to the desired performance. Enter the ultimate gain.

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Servo Torque for Arm

Calculate the static torque a servomotor needs to hold a horizontal arm, T = m·g·L, from the tip mass m, gravity g (9.81 m/s²) and the arm length L. The result, in N·m, is the minimum torque the servo must provide to keep the arm horizontal against the load weight — the most unfavorable position. It is essential in designing robotic arms, grippers and servo-driven mechanisms, sizing the motor with a safety margin over this value. For arms with their own mass, the center of mass is used. Enter the mass and the arm length.

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PWM Resolution (Levels)

Calculate the number of levels of a PWM signal, levels = 2^bits, from the bit resolution of the PWM generator (timer). The result is the number of discrete duty-cycle steps available: an 8-bit PWM offers 256 levels (0 to 255), allowing power adjustment in 256 steps; a 10-bit one, 1024 levels, finer control. Higher resolution gives smoother control of motor speed and LED brightness, but reduces the maximum possible PWM frequency for a given timer clock. It is a central parameter in microcontroller configuration. Enter the resolution in bits.

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Robot Arm Reach

Calculate the maximum reach of a planar two-link robotic arm, R = L₁ + L₂, by adding the lengths of the two links (arm and forearm). The result, in the length unit, is the radius of the work envelope — the farthest distance the tip (end-effector) can reach when the arm is fully extended. It defines the robot's workspace and is the first sizing parameter of manipulators. The minimum reach (dead zone at the center) is |L₁ − L₂|, and the useful area is the annulus between the two radii. Enter the two link lengths.

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Stepper Motor Speed

Calculate the rotation speed of a stepper motor, RPM = (pps × 60 × step angle) ÷ 360, from the pulse frequency pps (steps per second) and the motor's step angle (degrees per step). The result, in revolutions per minute, relates the command frequency sent to the driver with the actual shaft speed. Stepper motors lose torque at high speeds, so there is a practical maximum rotation. It is essential for programming axis speeds in 3D printers, CNC and automation. Enter the pulse frequency and the step angle.

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Torque for Angular Acceleration

Calculate the torque needed to angularly accelerate a rotating body, T = I·α, from the moment of inertia I and the desired angular acceleration α (rad/s²). The result, in N·m, is the rotational version of Newton's second law (F = m·a): the greater the assembly's inertia or the faster the intended acceleration, the more torque the motor must provide. It is fundamental in sizing drives that must accelerate and decelerate loads quickly — robots, positioners, spindles — where the acceleration torque adds to the friction and load torque. Enter the moment of inertia and the angular acceleration.

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Motor Torque from Current

Calculate the torque produced by a DC motor, T = K_t·I, from the torque constant K_t (N·m/A) and the armature current I. The result, in N·m, shows a DC motor's torque is directly proportional to current — which is why measuring current is the simplest way to estimate (and limit) torque and detect overloads. The torque constant K_t is a motor characteristic (numerically equal to the back-EMF constant K_e in SI units). It is the basis of torque control in servomotors and robotics. Enter the torque constant and the current.

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PWM Output Voltage

Calculate the average output voltage of a PWM signal, V_out = (duty ÷ 100) × V_sup, from the duty cycle (in %) and the supply voltage V_sup. The result, in volts, is the effective average voltage delivered to a load (motor, LED, heater) by rapidly switching the supply on and off. Varying the duty cycle from 0 to 100% varies the average voltage from 0 to V_sup, allowing power control without dissipating energy in resistors — the basis of motor speed and LED brightness control in microcontrollers. Enter the duty cycle and the supply voltage.

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DC Motor No-Load Speed

Calculate the no-load speed of a DC motor, ω = V ÷ K_e, from the applied voltage V and the back-EMF constant K_e (V·s/rad). The result, in rad/s, is the speed the motor reaches with no load, when the generated back-EMF nearly equals the applied voltage and the current drops to a minimum. It is the upper speed limit of the motor for a given voltage, the basis of the torque-speed curve (running from stall torque at zero speed to no-load speed at zero torque). It lets you estimate the operating range of servos and DC motors. Enter the voltage and the constant K_e.

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DC Motor Stall Current

Calculate the stall current of a DC motor, I = V ÷ R, from the applied voltage V and the armature resistance R. The result, in amperes, is the maximum current the motor draws when the shaft is locked (zero speed, no back-EMF) — far higher than normal operating current. It is the most dangerous current: it can burn the motor and driver if sustained, so systems include stall protection. It also corresponds to the maximum (stall) torque point on the torque-speed curve. It is essential for sizing fuses, drivers and power supplies. Enter the voltage and the armature resistance.

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Effective Field Capacity

Calculate the effective field capacity of a mechanized farming operation, FC = (v × L × Ef) ÷ 10, from the working speed v (km/h), the effective working width L (m) and the field efficiency Ef (decimal). The result, in hectares per hour, is the area the machine actually works per hour, already discounting time losses with turns, refills and overlaps (efficiency). The factor 10 adjusts the units. It is central to mechanization planning: it sets how many machines and hours are needed to complete an operation (planting, spraying, harvesting) in the available window. Enter the speed, the width and the field efficiency.

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Tractor Wheel Slip

Calculate the wheel slip of a tractor, slip = (1 − D_loaded ÷ D_unloaded) × 100%, comparing the distance traveled in a number of wheel revolutions under load (D_loaded) and unloaded (D_unloaded). The result, in %, measures how much the wheels spin without advancing, by slipping on the soil. Excessive slip wastes power and fuel and compacts the soil; zero slip indicates lack of traction. The ideal range for farm tractors is typically 8 to 15% on firm soil, adjusted with ballast and tire pressure. It is a key traction efficiency indicator. Enter the loaded and unloaded distances.

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Drawbar Pull

Calculate the available drawbar pull of a tractor, F = W × μ, multiplying the weight on the driving wheels W (kN) by the traction coefficient μ of the tire-soil pair. The result, in kN, is the pulling effort the tractor can exert on implements (plow, harrow, planter) — limited by soil grip, not engine power. The traction coefficient depends on soil and tire type (0.5 to 0.7 on firm soil; much less on loose or wet soil). Increasing the adhesive weight (ballast) raises the available force. Enter the adhesive weight and the traction coefficient.

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Fuel Consumption per Hectare

Calculate the fuel consumption per hectare of a mechanized operation, consumption = hourly consumption ÷ field capacity, dividing the tractor's hourly consumption (L/h) by the effective field capacity (ha/h). The result, in liters per hectare, is the practical indicator to budget the fuel cost of a farming operation and compare the energy efficiency of machines and settings. Heavy operations (subsoiling) consume far more L/ha than light ones (spraying). Combined with the diesel price and the total area, it gives the season's fuel cost. Enter the hourly consumption and the field capacity.

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Spray Volume

Calculate the spray volume applied per hectare, volume = (q × 600) ÷ (L × v), from the total nozzle flow q (L/min), the boom width L (m) and the travel speed v (km/h). The result, in liters per hectare, is the application rate — a critical spraying parameter that must match the pesticide recommendation and the target. The factor 600 converts units. Increasing speed or width lowers the applied volume; increasing nozzle flow raises it. Calibrating correctly ensures the right agrochemical dose, avoiding underdosing (ineffectiveness) or overdosing (waste and phytotoxicity). Enter the nozzle flow, the boom width and the speed.

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Gross Irrigation Depth

Calculate the gross irrigation depth, D_gross = D_net ÷ Ef, dividing the required net depth (the water that must reach the roots, in mm) by the irrigation system's application efficiency (decimal). The result, in mm, is the depth the system must actually apply so that, after losses (evaporation, drift, percolation, runoff), the net depth remains in the soil. More efficient systems (drip, ~90%) require less gross depth than less efficient ones (conventional sprinkler, ~75%; surface, ~50-60%). It is the basis of irrigation design and management. Enter the net depth and the application efficiency.

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Irrigation Time

Calculate the required irrigation time, t = depth ÷ application intensity, dividing the depth to apply (mm) by the system's application intensity (mm/h). The result, in hours, is how long the system (sprinkler, pivot, drip) must run to apply the desired depth. The application intensity must not exceed the soil's infiltration rate, or it causes runoff and erosion. Multiplied by the flow, it gives the water volume; combined with the number of sectors, it sets the total irrigation time of the area. It is a routine calculation in daily irrigation management. Enter the depth and the application intensity.

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Worked Area per Machine

Calculate the area worked by a farm machine, A = field capacity × time, multiplying the effective field capacity (ha/h) by the available operating time (h). The result, in hectares, is how much the machine can cover in a shift — direct input for operational planning: how many hours (or days) are needed to complete the planting, spraying or harvesting of an area, and whether the machine fleet meets the agronomic window (the period in which the operation must occur). Undersizing the fleet delays critical operations and reduces yield. Enter the field capacity and the operating time.

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Sprayer Nozzle Count

Calculate the number of nozzles on a spray boom, N = boom width ÷ nozzle spacing, dividing the total boom width (m) by the desired spacing between nozzles (m). The result is how many spray tips the boom must have to cover the swath uniformly. The standard spacing is typically 0.5 m, with nozzles whose spray angle and height ensure correct fan overlap for homogeneous spray distribution. Wrong spacing and nozzle count cause gaps or excess application along the swath. It is a sprayer design and calibration calculation. Enter the boom width and the nozzle spacing.

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Field Efficiency

Calculate the field efficiency of a mechanized operation, Ef = (effective capacity ÷ theoretical capacity) × 100%, dividing the effective field capacity (area actually worked per hour) by the theoretical capacity (the one obtained with no time losses). The result, in %, measures how much of the time the machine actually works, as opposed to headland turns, refills, adjustments, travel and overlaps. Simple operations in large fields have high efficiency (80-90%); complex operations in small, irregular fields, low (60-70%). Improving field efficiency (larger fields, fewer stops) reduces costs. Enter the effective and theoretical capacities.

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Buck Converter (Step-Down)

Calculate the output voltage of a buck (step-down) DC-DC converter in continuous conduction, V_out = D × V_in, from the duty cycle D (0 to 1) and the input voltage V_in. The result, in volts, is always less than or equal to the input — the buck converter lowers voltage efficiently (without dissipating the excess, unlike a linear regulator), by switching rapidly and filtering with an inductor and capacitor. Varying the duty cycle adjusts the output from 0 to V_in. It is the most common topology in switching power supplies and point-of-load regulators. Enter the duty cycle and the input voltage.

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Boost Converter (Step-Up)

Calculate the output voltage of a boost (step-up) DC-DC converter in continuous conduction, V_out = V_in ÷ (1 − D), from the input voltage V_in and the duty cycle D (0 to 1). The result, in volts, is always greater than the input — the boost converter raises voltage by storing energy in an inductor and releasing it in series with the source. As D approaches 1, the output tends to infinity (limited by real losses). It is used in supplies that must step up voltage (LEDs, batteries, power factor correction) and in photovoltaic systems. Enter the input voltage and the duty cycle.

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Buck-Boost Converter

Calculate the output voltage of a buck-boost DC-DC converter in continuous conduction, V_out = V_in × D ÷ (1 − D), from the input voltage V_in and the duty cycle D (0 to 1). The result, in volts, can be lower (D < 0.5) or higher (D > 0.5) than the input — the buck-boost converter steps voltage down or up depending on the duty cycle, with inverted output polarity in the classic topology. It is used when the input voltage can vary above and below the desired output (discharging batteries, universal supplies). Enter the input voltage and the duty cycle.

〰️

Output Voltage Ripple

Calculate the output voltage ripple of a switching converter, ΔV = I ÷ (f × C), from the output current I, the switching frequency f and the output capacitance C. The result, in volts, is the residual oscillation superimposed on the DC output voltage, caused by the filter capacitor charging and discharging each switching cycle. Higher frequency and capacitance reduce the ripple. Keeping the ripple within limits (typically <1% of the output) is essential to supply sensitive circuits. Enter the current, the switching frequency and the capacitance.

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Inductor Current Ripple

Calculate the inductor current ripple of a buck converter, ΔI_L = (V_in × D) ÷ (L × f), from the input voltage V_in, the duty cycle D, the inductance L and the switching frequency f. The result, in amperes, is the peak-to-peak variation of the inductor current each cycle. It is a central design parameter: a typical ripple of 20-40% of the average current is a good compromise. Higher inductance and frequency reduce the ripple (larger inductor, more costly; higher frequency, more switching losses). It also sets the boundary between continuous and discontinuous conduction. Enter the voltage, duty cycle, inductance and frequency.

⚡

Rectifier Average Voltage

Calculate the average (DC) voltage of a full-wave rectifier, V_dc = 2 × V_p ÷ π, from the peak voltage V_p of the sinusoidal input. The result, in volts, is the mean value of the pulsating voltage at the rectifier output before filtering — about 63.7% of the peak. A full-wave rectifier (bridge or center-tap) flips the negative half-cycles, doubling the ripple frequency and raising the average voltage compared with a half-wave rectifier (V_p/π). It is the basis of designing DC power supplies from the AC mains. Capacitor filtering then raises the voltage further (close to the peak). Enter the peak voltage.

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Square-Wave RMS Current

Calculate the RMS value of a pulsing square-wave current, I_rms = I_p × √D, from the peak current I_p and the duty cycle D (fraction of the period the current flows). The result, in amperes, is the RMS current that determines the actual heating (I²R losses) of a component that conducts in pulses — such as a transistor or winding in a switching converter. Unlike the average value, the RMS is what matters for sizing conductors, resistances and dissipation. The smaller the duty cycle, the lower the RMS for the same peak current. Enter the peak current and the duty cycle.

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MOSFET Conduction Loss

Calculate the conduction loss in a MOSFET, P = I² × R_ds(on), from the current through it I and the on-state channel resistance R_ds(on). The result, in watts, is the power dissipated as heat while the transistor is on, due to the channel resistance. It is one of the two main losses in power switches (the other being switching loss). MOSFETs with lower R_ds(on) reduce these losses, important at high currents and low frequencies. The conduction losses, plus the switching losses, set the heating and the heatsink requirement. Enter the current and the on-state resistance.

⚙️

Switching Loss

Calculate the switching (commutation) loss in a power transistor, P_sw = 0.5 × V × I × (t_on + t_off) × f, from the voltage V, the current I, the total commutation time (rise + fall) and the switching frequency f. The result, in watts, is the energy lost during the on-off transitions, when voltage and current coexist in the transistor. Unlike conduction losses, switching losses grow with frequency — which is why there is a practical limit to raising the frequency (which would shrink inductors and capacitors). Fast transistors (short times) and proper drivers minimize these losses. Enter the voltage, current, commutation time and frequency.

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Junction Temperature

Calculate the junction temperature of a power semiconductor, T_j = T_a + P × R_th, from the ambient temperature T_a, the dissipated power P and the total junction-to-ambient thermal resistance R_th (°C/W). The result, in °C, is the device's internal temperature (silicon junction), which must not exceed the manufacturer's limit (typically 150 °C) on pain of failure. The thermal resistance adds the junction-to-case, case-to-heatsink and heatsink-to-ambient stages. Lowering R_th (larger heatsink, ventilation, thermal paste) lowers the junction temperature. It is the central calculation of power electronics thermal design. Enter the ambient temperature, the dissipated power and the thermal resistance.

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Ocean Wavelength

Calculate the wavelength of an ocean wave in deep water, L = g·T² ÷ (2π), from the wave period T (s) and gravity g (9.81 m/s²). The result, in meters, is the distance between two successive crests — in deep water, it depends only on the period. Long-period waves (swell from distant storms) have much larger wavelengths than local wind waves. The wavelength sets the depth at which the wave 'feels' the bottom (about L/2), starts to refract and shoal until it breaks. It is a fundamental parameter of linear wave theory and coastal engineering. Enter the wave period.

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Deep Water Wave Celerity

Calculate the celerity (phase velocity) of an ocean wave in deep water, c = g·T ÷ (2π), from the period T (s) and gravity g. The result, in m/s, is the speed at which the wave crest propagates. In deep water, longer-period waves travel faster — a phenomenon called dispersion, which makes long-period swell reach the coast before the short waves generated by the same storm. The celerity is half the group velocity (at which energy travels) in deep water. It is a base concept of wave hydrodynamics. Enter the wave period.

🏖️

Shallow Water Wave Celerity

Calculate the celerity of a wave in shallow water, c = √(g·h), from the water depth h (m) and gravity g. The result, in m/s, is the propagation speed when the depth is much smaller than the wavelength — a situation in which the wave 'feels' the bottom and its speed depends only on depth, no longer on the period. This is why waves refract as they approach the coast (the deeper part travels faster) and why tsunamis travel at hundreds of km/h in the deep ocean and slow down (piling up energy) as they reach the coast. Enter the water depth.

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Wave Energy

Calculate the energy density of an ocean wave, E = (1 ÷ 8)·ρ·g·H², from the water density ρ (kg/m³, ~1025 for seawater), gravity g and the wave height H (m). The result, in J/m² (energy per surface area), is the sum of the wave's kinetic and potential energy — proportional to the square of the height, so large waves carry far more energy. It is the basis for calculating wave energy generation potential and the impact on coastal structures and beach erosion. Enter the water density and the wave height.

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Wave Power

Estimate the power flux of an ocean wave per meter of wave front, P ≈ 0.5 × H² × T, from the wave height H (m) and the period T (s), in deep water (seawater). The result, in kW/m, is the power available per meter of wave front width — a key indicator of wave energy potential for converters (WECs). Coasts exposed to ocean swells (European Atlantic, Pacific) reach 30-70 kW/m, a significant renewable resource. The power grows with the square of the height and linearly with the period. Enter the wave height and period.

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Tidal Range

Calculate the tidal range, range = high tide height − low tide height, subtracting the lowest from the highest level observed in a tidal cycle. The result, in meters, is the vertical difference between high and low tide — a fundamental parameter for navigation (available draft), port and coastal structure design, tidal energy and intertidal ecology. Spring tides (new/full moon) have maximum range; neap tides (quarter moon), minimum. Some regions have ranges of a few centimeters; others (Bay of Fundy), over 15 meters. Enter the high and low tide heights.

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Wave Steepness

Calculate the steepness of a wave, s = H ÷ L, dividing the wave height H by the wavelength L. The dimensionless result is the relative steepness of the wave — the taller it is for its length, the steeper. Steepness has a physical limit: deep-water waves break when s exceeds about 1/7 (0.143), as the crest becomes unstable (120° angle). Young storm waves are steep; swell that travels long distances is gentle (low steepness). Steepness governs wave stability, breaking and vessel comfort. Enter the wave height and wavelength.

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Iribarren Number

Calculate the Iribarren number (surf similarity parameter), ξ = tan(β) ÷ √(H ÷ L), from the beach slope tan(β), the wave height H and the wavelength L. The dimensionless result classifies the wave breaking type and runup: ξ < 0.5 indicates spilling breakers (flat beaches); 0.5 < ξ < 3.3, plunging (the wave 'barrels'); ξ > 3.3, surging/collapsing (steep beaches). It is widely used in coastal engineering to predict runup, structure overtopping and riprap stability. Enter the beach slope, wave height and wavelength.

〰️

Wave Group Velocity

Calculate the group velocity of an ocean wave in deep water, c_g = g·T ÷ (4π), from the period T (s). The result, in m/s, is the speed at which the wave energy (and the 'envelope' of a wave group) propagates — exactly half the celerity (phase velocity) in deep water. This difference explains a curious phenomenon: within a wave group, individual crests appear at the rear, advance through the group (faster than it) and disappear at the front. The group velocity is what matters for energy transport and predicting swell arrival at the coast. Enter the wave period.

⏱️

Wave Period

Calculate the period of a wave, T = 1 ÷ f, from the frequency f (Hz). The result, in seconds, is the time between two successive crests passing a fixed point — one of the most important properties of an ocean wave. The period determines the wavelength and celerity (in deep water), the depth at which the wave interacts with the bottom, and classifies the sea state: local wind waves have short periods (3-8 s), while swell from distant storms has long periods (10-20 s), travels faster and penetrates deeper. The period is measured by buoys and used in wave forecasting. Enter the wave frequency.

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Solidification Time (Chvorinov)

Calculate the solidification time of a casting by Chvorinov's rule, t = B × (V ÷ A)², from the mold constant B (min/cm², depending on the mold material and metal) and the ratio of the part's volume V to its surface area A. The result, in minutes, is the time for the metal to fully solidify. The rule shows that parts with a higher volume/area ratio (more 'massive') solidify more slowly — a fundamental casting design principle: risers (metal reservoirs) must have a larger modulus than the part to solidify last and feed the shrinkage. Enter the mold constant, the volume and the part area.

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Casting Cooling Modulus

Calculate the cooling modulus (or geometric modulus) of a casting, M = V ÷ A, dividing the volume V by the surface area A in contact with the mold. The result, in cm (length unit), is the parameter governing solidification speed: the larger the modulus, the slower the solidification (Chvorinov's rule says the time is proportional to the modulus squared). It is the basis of riser sizing in foundry — the modulus rule requires the riser modulus to be about 1.2 times that of the part, so it solidifies later and feeds the shrinkage, avoiding shrinkage cavities. Enter the part volume and area.

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Casting Metal Yield

Calculate the metal yield of a casting process, η = (part mass ÷ total poured mass) × 100%, dividing the finished part mass by the total poured metal (part + risers + runners + spills). The result, in %, measures the metal utilization efficiency: the rest (runners, risers, flash) is remelted, but consumes energy and adds cost. Typical yields range from 50 to 80%, depending on the part and gating complexity. Maximizing yield (well-sized risers, optimized gating) cuts energy and remelting costs. Enter the part mass and the total poured mass.

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Shrinkage Allowance

Calculate the pattern (mold) dimension accounting for solidification shrinkage, dimension = part dimension × (1 + shrinkage ÷ 100), from the desired final part dimension and the metal's linear shrinkage coefficient (%). The result is the larger dimension the pattern must have so that, upon solidifying and cooling, the part shrinks to the correct size. Each metal has its linear solidification shrinkage: steel ~2%, gray cast iron ~1%, aluminum ~1.3%, bronze ~1.5%. Patternmakers use shrink rules ('contraction rules') already scaled up. Ignoring shrinkage results in undersized parts. Enter the part dimension and the shrinkage coefficient.

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Metallostatic Pressure

Calculate the metallostatic pressure exerted by molten metal at the bottom of a mold, P = ρ × g × h, from the molten metal density ρ (kg/m³), gravity g and the metal column height h (m). The result, in pascals, is the pressure the molten metal exerts on the mold walls and bottom due to its own weight — analogous to hydrostatic pressure, but with the high density of metals. It is essential to size the mold strength (which can 'burst' or deform under pressure), predict core flotation and metal penetration into gaps. Dense metals (iron, ~7000 kg/m³) generate high pressures. Enter the metal density and the column height.

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Mold Fill Time

Calculate the fill time of a casting mold, t = V ÷ Q, dividing the cavity volume V by the metal flow rate Q of the gating system. The result, in seconds, is the time to completely fill the mold with molten metal. It is a critical parameter: filling too slowly lets the metal cool and solidify before filling everything (cold shut, misrun defects), while too fast causes turbulence, gas entrapment, mold erosion and inclusions. The optimal time depends on the part's weight and thickness and the metal. Sizing the gating system for the right time is central to casting design. Enter the cavity volume and the flow rate.