Slider-crank calculator
Calculate the piston displacement, velocity and acceleration of a slider-crank mechanism as well as the static and dynamic torque acting on the crank – over a full revolution and exactly via the connecting-rod angle β, not via the λ harmonic approximation. Enter crank radius, connecting-rod length, speed, piston force and reciprocating mass; the calculator delivers the curves as a diagram and the peak magnitudes live.
Slider-crank calculator (exact kinematics)
Model: planar slider-crank mechanism with a rigid, massless connecting rod and a reciprocating point mass at the piston, constant speed. Exact kinematics via the connecting-rod angle β (no λ approximation). Friction, gas-pressure curve and elasticities are not considered; the piston force F is assumed constant.
Results
Calculating …
Formulas and fundamentals
The slider-crank mechanism converts the rotation of the crank (radius r) through the connecting rod (length l) into the reciprocating motion of the piston. The governing quantities are the crank ratio λ = r/l and the connecting-rod angle β with sin β = λ·sin φ, where φ is the crank angle and ω = 2π·n/60 the angular frequency. The piston displacement follows exactly from the geometry as s = r·(1−cos φ) + l·(1−cos β); the piston travels the stroke 2r between top (φ = 0°) and bottom dead centre (φ = 180°).
Differentiating with respect to time gives the velocity v = ω·r·sin(φ+β)/cos β and the acceleration a = (ω·r)²/r · [cos(φ+β)/cos β + λ·cos²φ/cos³β]. This exact β form is superior to the common approximation a ≈ ω²·r·(cos φ + λ·cos 2φ): already at φ = 90° and λ = 1/3 the results differ noticeably (exact −13.96 m/s² versus −13.16 m/s² from the approximation). The maximum piston acceleration occurs at top dead centre and, because of the (1+λ) term, is larger than at bottom dead centre.
The piston force F produces, through the inclined connecting rod, a tangential force at the crank: F_T = F·sin(φ+β)/cos β and thus the static torque M_stat = F_T·r. The reciprocating mass m generates the inertia force m·a, which analogously contributes a dynamic torque M_dyn = m·a·sin(φ+β)/cos β · r; the total torque is M_ges = M_stat + M_dyn. For drive and flywheel sizing the reduced inertia J_red = m·(v/ω)², which describes the mass effect referred to the crankshaft, is additionally important.
Worked example
A crank mechanism with crank radius r = 50 mm and connecting-rod length l = 200 mm (λ = 0.25) runs at n = 1500 rpm, i.e. ω = 157.08 rad/s. The circumferential velocity of the crank pin is v_U = ω·r = 7.85 m/s.
At top dead centre (φ = 0°) the piston acceleration is at its maximum: a = ω²·r·(1+λ) = 157.08²·0.05·1.25 ≈ 1542 m/s². This high acceleration is why the free mass forces of high-speed crank mechanisms must be handled by design.
With a piston force of F = 5000 N and a reciprocating mass of m = 1.2 kg the calculator delivers the complete torque curve over the revolution. The static moment peaks at about φ = 75°, while the dynamic part dominates near the dead centres – the superposition yields the curve that governs motor and flywheel sizing.
Frequently asked questions
Why the exact β formula instead of the λ approximation?
The approximation a ≈ ω²·r·(cos φ + λ·cos 2φ) expands the connecting-rod angle only up to the first harmonic and becomes inaccurate as λ = r/l grows. The exact calculation via β = asin(λ·sin φ) holds for any λ < 1 without an error term. At φ = 90° and λ = 1/3 the exact formula gives −13.96 m/s², the approximation only −13.16 m/s² – and this difference grows further with λ.
Where does the maximum piston acceleration occur?
At top dead centre (φ = 0°), because there the fundamental and the second harmonic caused by the connecting rod add up: a_max ≈ ω²·r·(1+λ). At bottom dead centre (φ = 180°) they subtract to ω²·r·(1−λ), so the magnitude is smaller there. It is exactly this asymmetry that the exact calculation captures correctly, whereas the pure fundamental does not.
What does the reduced inertia J_red mean?
J_red = m·(v/ω)² is the mass moment of inertia of the reciprocating mass referred to the crankshaft. It varies over the revolution between zero (at the dead centres, v = 0) and a maximum at crank positions transverse to the cylinder axis. For flywheel and cyclic-irregularity calculations it is added to the constant share of the rotating masses.
What are the signs of displacement, velocity and acceleration?
The piston displacement s is counted positive from top dead centre towards bottom dead centre (0 … 2r). The velocity is positive during the first half-revolution (piston moving away) and negative during the second. The acceleration is largest at the dead centres and changes sign in between; it points towards the crankshaft centre whenever the piston is being decelerated.
Does the calculator account for the connecting-rod mass?
No, the model assumes a purely reciprocating point mass m at the piston (rigid, massless connecting rod). In practice the connecting rod is split into a reciprocating and a rotating mass share; the reciprocating share is added to the piston mass and entered here as m. The rotating share belongs to the revolving masses and is balanced separately.
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