Crank-Slider Mechanism
Compute slider displacement, velocity and acceleration of a crank-slider mechanism at a given crank angle θ and angular speed ω (approximate formulas).
Inputs
Formula
s = r(1−cosθ) + l−√(l²−r²sin²θ); v = ωr(sinθ + r·sin2θ/(2l)); a = ω²r(cosθ + r·cos2θ/l)
Fundamentals
The crank-slider mechanism converts rotation to linear motion — the core of engines, compressors and presses.
Slider displacement $s=r(1-\cos\theta)+\sqrt{l^2-r^2\sin^2\theta}$ ($r$ crank, $l$ conrod, $\theta$ angle). For $l\gg r$ it approximates simple harmonic motion.
History
Watt used the crank-slider in his 1769 steam engine to turn reciprocation into rotation; it is the basic four-bar mechanism.
Engineering applications
The core motion converter in internal-combustion engines, compressors, presses and piston pumps.
Glossary
| Crank $r$ | Short rotating link providing the drive. |
| Conrod $l$ | Connects crank to slider, transmitting motion. |
| Stroke | Max slider travel, about $2r$. |
How to use
- Fill in Crank radius r, Conrod length l, Crank angle θ, Ang. speed ω in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Displacement s, Velocity v, Acceleration a in the results area.
Formula notesSlider displacement s = r(1−cosθ) + √(l²−r²sin²θ); velocity v = rω(sinθ + r·sin2θ/2l); accel a = rω²(cosθ + r·cos2θ/l).
Formula · Worked Example · Knowledge
Formula
Slider displacement $s=r(1-\cos\theta)+l-\sqrt{l^2-r^2\sin^2\theta}$, velocity $v=\omega r(\sin\theta+r\sin2\theta/(2l))$, accel $a=\omega^2 r(\cos\theta+r\cos2\theta/l)$.
Worked Example
r=50, l=200, θ=45°, ω=10 rad/s → s≈15.4 mm, v≈435 mm/s, a≈3300 mm/s².
Key Points
- θ=0° and 180° are dead centres (zero velocity).
- Larger rod ratio l/r → closer to SHM.
- Used in engines, presses, reciprocating pumps.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Crank radius r (mm) | r | mm | 50 |
| Conrod length l (mm) | l | mm | 200 |
| Crank angle θ (°) | theta | ° | 45 |
| Ang. speed ω (rad/s) | omega | rad/s | 10 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Displacement s | s | mm |
| Velocity v | v | mm/s |
| Acceleration a | a | mm/s² |
Applications
- Linkage and cam kinematics design
- Pulley mechanical-advantage study
- Engine valve-train and crank analysis
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: s = r(1−cosθ) + l−√(l²−r²sin²θ); v = ωr(sinθ + r·sin2θ/(2l)); a = ω²r(cosθ + r·cos2θ/l)
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Linkage and cam kinematics design