Jerk (Rate of Acceleration Change)
Compute the jerk, J = Δacceleration/Δtime, the rate of change of acceleration over time — the third derivative of position. High jerk causes jolts, vibration and wear; controlling it (jerk-limited or S-curve profiles) makes motion smooth, protecting mechanisms and improving the finish on CNC machines and elevators. Enter the acceleration change and the time interval.
Result
—
Jerk (rate of acceleration change)
After position, velocity and acceleration comes jerk (also called jolt or lurch): the rate at which acceleration changes over time, J = Δa/Δt — the third derivative of position. It is what you feel when a car judders through a gear change, or when an elevator starts and stops sharply: the culprit is not acceleration itself, but its sudden change. High jerk causes vibration, resonance, wear on gears and belts, and chatter marks on machined parts. That is why modern motion controllers rely on jerk-limited profiles (S-curve), which smooth the transitions — acceleration ramps up and down gradually instead of switching on and off. Comfort in elevators, trains and roller coasters is, to a large extent, jerk control. Enter the change in acceleration and the time interval.
Related Tools
Trapezoidal Profile Time
Compute the total time of a motion with a trapezoidal velocity profile, t = d/Vmax + Vmax/a, adding the cruise-velocity time to the acceleration and deceleration phases. It is the most common motion profile in motors and robots: accelerate to maximum speed, hold constant and decelerate. Enter the distance, the maximum velocity and the acceleration (assuming equal acceleration and deceleration).
Follower Max Jerk (SHM)
Calculate the maximum jerk of a simple-harmonic-motion cam follower, j_max = (π³·h·ω³) ÷ (2·β³), from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Jerk is the RATE OF CHANGE of acceleration (the third time-derivative of displacement). Though less known than velocity and acceleration, jerk is decisive for the SMOOTHNESS and vibration of a cam mechanism: abrupt acceleration changes (high jerk) generate SHOCKS that excite the system's natural frequencies, causing vibration, noise, fatigue and wear — even if peak acceleration is within limits. In SHM, although acceleration is continuous inside the rise, it is DISCONTINUOUS at the ends, meaning INFINITE jerk there (the formula gives the interior jerk peak, but the end discontinuities are the real problem). It is precisely to eliminate these acceleration discontinuities (infinite jerk) that CYCLOIDAL motion and polynomial profiles were developed — they ensure finite, continuous jerk, the choice for high-speed, precision cams. Jerk grows with the CUBE of the rotation ω, becoming critical at high speeds. Considering jerk is the mark of advanced cam design. Enter the lift, the angular velocity and the rise angle.
Max Acceleration (Parabolic Cam)
Calculate the (constant) maximum acceleration of a parabolic-motion (constant-acceleration) cam follower, a_max = (4·h·ω²) ÷ β², from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Parabolic, or constant-acceleration, motion is the law producing the LOWEST possible maximum acceleration for a given lift and time — so it minimizes peak inertia forces. It consists of two halves: in the first, the follower accelerates with CONSTANT acceleration (rising parabolic displacement); in the second, it decelerates with the same constant (negative) acceleration, stopping at the top. The name 'parabolic' comes from the displacement diagram, formed by two parabolas. The great advantage is the low maximum acceleration; the drawback is that acceleration JUMPS abruptly — from +a_max to −a_max at the middle, and from zero to ±a_max at the ends — generating infinite JERK there, causing shocks, noise and vibration. So in practice pure parabolic is little used at high speed (despite low peak acceleration), and cycloidal or modified profiles that smooth these transitions are preferred. Parabolic is didactic and useful when peak acceleration is the limiting factor and speeds are moderate. Enter the lift, the angular velocity and the rise angle.
Max Acceleration (Cycloidal Cam)
Calculate the maximum acceleration of a cycloidal-motion cam follower, a_max = (2π·h·ω²) ÷ β², from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Cycloidal motion is considered the BEST cam motion law for HIGH SPEEDS, and is standard in precision, high-rpm cams. Its decisive feature is that acceleration is a FULL SINE wave starting at zero, rising to a maximum, passing through zero, going to a minimum and returning to zero — i.e., acceleration is CONTINUOUS and starts and ends smoothly at ZERO at the ends, WITHOUT the discontinuities of SHM and parabolic. This means finite, continuous JERK, eliminating shocks and minimizing vibration excitation — the follower 'glides' smoothly without jolts. The price is a slightly HIGHER maximum acceleration than parabolic (2π ≈ 6.28 vs 4 in the factor) and SHM (π²/2 ≈ 4.93), but the dynamic SMOOTHNESS amply compensates at high speed. The name comes from the cycloid curve describing the displacement. Racing-engine valve cams, fast textile and packaging machines use cycloidal or derived (polynomial) profiles precisely to run at high rpm with low vibration. Enter the lift, the angular velocity and the rise angle.
Harmonic Drive Reduction
Compute the reduction ratio of a harmonic drive (strain wave gear), R = Nf/(Nc − Nf), where Nf is the flexspline tooth count and Nc the circular spline's (usually Nc = Nf + 2). This ingenious mechanism reaches huge reductions (50:1 to 300:1) in a single compact stage with zero backlash — which is why it is the heart of industrial and collaborative robot joints. Enter the flexspline and circular spline tooth counts.
Train Movement Resistance (Davis)
Calculate a train's specific movement resistance by the Davis equation, R = A + B·V + C·V², from coefficient A (rolling resistance and mechanical friction, speed-independent), B (resistance proportional to speed, from flange friction and oscillations), C (aerodynamic resistance, proportional to speed squared) and the speed V (km/h). The Davis equation, from the 1920s and still standard in railway engineering, describes the total resistance to motion the locomotive must overcome on straight, level track, per unit weight (N/t or kgf/t). At low speed the constant and linear terms (friction) dominate; at high speed the quadratic aerodynamic term dominates, decisive for high-speed trains (hence their careful streamlining). Davis resistance, plus grade (gravity) and curve resistances, sets the required tractive effort, energy consumption and locomotive traction capacity. It is the basis of traction calculation and train performance. Enter coefficients A, B and C and the speed.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.