Crank-Slider Mechanism

Calculate slider displacement, velocity, and acceleration for a reciprocating crank-slider mechanism at any crank angle

Inputs

r

Crank Length

l

Rod Length

\omega

Angular Velocity

rad/s

Constant angular velocity of the crank (rad/s)

\theta

Crank Angle

Angle from top-dead-centre (TDC) position, 0°–360°

Formula Interpretation

Fig 1 — Crank rotates at ω; slider P reciprocates along the x-axis

Displacement

x = r\cos\theta + l - \dfrac{r^2}{2l}\sin^2\theta

The displacement $x$ of the slider joint P measured from the crank pivot O. The first term $r\cos\theta$ is the horizontal projection of the crank; the second is the rod length $l$ ; the third is the small second-order correction proportional to the crank ratio $\lambda = r/l$ .

Velocity

v = -r\omega\sin\theta - \dfrac{r^2}{l}\,\omega\sin\theta\cos\theta

The velocity $v$ of slider P. Note that $v$ is zero at $\theta=0°$ and $\theta=180°$ (the dead-centre positions) and reaches its maximum near $\theta=90°$ . The second term represents the higher-order correction due to the finite rod length.

Acceleration

a = -r\omega^2\!\left(\cos\theta + \dfrac{r}{l}\cos 2\theta\right)

The acceleration $a$ contains a primary component at frequency $\omega$ and a secondary component at $2\omega$ (the second harmonic). The ratio $\lambda$ controls the magnitude of the harmonic distortion.

Knowledge Points

Dead-Centre Positions

At θ=0° (TDC — top dead centre) the slider is at its maximum distance r+l from the pivot. At θ=180° (BDC — bottom dead centre) it is at its minimum distance l−r. Velocity is zero at both positions; acceleration is maximum in magnitude. These are the stroke reversal points.

Crank Ratio $\lambda = r/l$

The dimensionless crank ratio $\lambda = r/l$ determines the degree of kinematic non-linearity. When λ → 0 (very long rod) all three formulas reduce to simple harmonic motion: x ≈ l + r·cosθ, v ≈ −rω·sinθ, a ≈ −rω²·cosθ. Real mechanisms have λ typically between 0.2 and 0.5.

Applications

Crank-slider mechanisms are the basis of piston engines (converts slider motion to rotation), reciprocating pumps and compressors (rotation to linear), and metal-cutting machines such as shapers and power saws. Understanding the velocity and acceleration profiles is essential for balancing, force analysis, and fatigue life calculations.

Worked Example

A crank-slider mechanism has $r = 0.3\,\text{m}$ , $l = 1.0\,\text{m}$ , $\omega = 10\,\text{rad/s}$ . Find the slider displacement, velocity, and acceleration at $\theta=0°$ , $\theta=90°$ , $\theta=180°$ , and $\theta=270°$ .

\lambda = r/l = 0.3/1.0 = 0.3

θ = 0° (Top Dead Centre)

x = r + l = 0.3 + 1.0 = 1.300\,\text{m}

v = 0

a = -r\omega^2\!\left(1 + \tfrac{r}{l}\right) = -0.3 \times 100 \times 1.3 = -39.0\,\text{m/s}^2

θ = 90°

x = l - \dfrac{r^2}{2l} = 1.0 - \dfrac{0.09}{2} = 0.955\,\text{m}

v = -r\omega = -0.3 \times 10 = -3.000\,\text{m/s}

a = \dfrac{r^2\omega^2}{l} = \dfrac{0.09 \times 100}{1.0} = +9.0\,\text{m/s}^2

θ = 180° (Bottom Dead Centre)

x = -r + l = -0.3 + 1.0 = 0.700\,\text{m}

v = 0

a = r\omega^2\!\left(1 - \tfrac{r}{l}\right) = 0.3 \times 100 \times 0.7 = +21.0\,\text{m/s}^2

θ = 270°

x = l + \dfrac{r^2}{2l} = 1.0 + \dfrac{0.09}{2} = 1.045\,\text{m}

v = +r\omega = +3.000\,\text{m/s}

a = \dfrac{r^2\omega^2}{l} = +9.0\,\text{m/s}^2

The stroke length (distance between TDC and BDC) is (r+l) − (l−r) = 2r = 0.600 m. Maximum speed is rω = 3.000 m/s, occurring at ≈90° and ≈270°.

Extended Knowledge

•In a lathe or reciprocating machine tool the crank-slider converts rotary motor output into the linear cutting or feeding motion. By analysing the acceleration profile, engineers design counterweights to balance the inertia forces and reduce vibration.
•In a piston engine the gas-pressure force on the piston is transmitted through the connecting rod to the crank pin, producing torque on the crankshaft. The torque is not constant but varies with θ; the flywheel smooths out these fluctuations by storing kinetic energy during the power stroke.
•Offset crank-slider mechanisms, where the slider axis does not pass through the crank pivot O, are used when a different ratio of forward-to-return stroke times (quick-return ratio) is needed, such as in shaper machines.