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Mechanism Simulator

Crank-Slider Mechanism Simulator

Adjust crank radius, connecting rod length, and angular velocity to visualize piston kinematics in real time. Explore position, velocity, and acceleration waveforms with engine design applications.

Parameters
Crank Radius r 50 mm
Connecting Rod L 150 mm
Angular Velocity ω 3.0 rad/s
Initial Phase φ₀ 0°
Statistics
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Stroke (mm)
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R/L Ratio
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Max Velocity (mm/s)
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Max Accel. (mm/s²)

Theory

Piston position:

$$x(\theta) = r\cos\theta + \sqrt{L^2 - r^2\sin^2\theta}$$

Velocity (approx.):

$$v \approx -r\omega\!\left(\sin\theta + \frac{r}{2L}\sin 2\theta\right)$$

Acceleration (approx.):

$$a \approx -r\omega^2\!\left(\cos\theta + \frac{r}{L}\cos 2\theta\right)$$
CAE Application: Piston acceleration determines inertia forces, which directly affect bearing loads and crankshaft fatigue design in IC engines. The R/L ratio governs the degree of non-harmonic motion.

Position x(t)

Velocity v(t)

Acceleration a(t)

Phase θ vs Position

What is a Crank-Slider Mechanism?

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What exactly is this simulator showing? I see a circle, a rod, and a block moving back and forth.
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Basically, you're looking at the core of an engine. The circle is the crankshaft, the rod is the connecting rod, and the block is the piston. The simulator shows how spinning the crank (rotary motion) makes the piston slide back and forth (reciprocating motion). Try moving the "Crank Radius" slider above to see how a bigger crank makes the piston travel farther.
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Wait, really? So the piston's speed isn't constant? The graph shows it speeding up and slowing down.
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Exactly! That's the key point. The piston moves fastest near the middle of its stroke and slows to a stop at each end before reversing. This is because of the geometry of the linkage. For instance, in a car engine, this changing speed creates the inertia forces we need to manage. Change the "Rod Length" parameter and watch how the velocity curve changes shape.
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So why is acceleration so important? The graph shows it spikes even higher than velocity.
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Great observation. Acceleration is crucial because force is mass times acceleration ($F = m a$). Those high acceleration spikes mean huge inertia forces are acting on the piston and rod. In practice, this determines the loads on bearings and the stress on the crankshaft. Crank up the "Angular Velocity" slider and see how acceleration increases dramatically—this is why high-speed engines are so challenging to design!

Physical Model & Key Equations

The exact position of the piston (x) is found from the geometry of the triangle formed by the crank radius (r), connecting rod length (L), and the crank angle (θ).

$$x(\theta) = r\cos\theta + \sqrt{L^2 - r^2\sin^2\theta}$$

Where:
$x$ = Piston position from crank center
$r$ = Crank radius
$L$ = Connecting rod length
$\theta$ = Crank angle ($\theta = \omega t + \phi_0$)

Taking the derivative of position gives velocity, and a second derivative gives acceleration. A common simplified formula for velocity uses the ratio $r/L$ and is accurate for most practical designs.

$$v \approx -r\omega\!\left(\sin\theta + \frac{r}{2L}\sin 2\theta\right)$$

Where:
$v$ = Piston velocity
$\omega$ = Crankshaft angular velocity
The term $\frac{r}{2L}\sin 2\theta$ is the correction due to the finite connecting rod length. If the rod were infinitely long ($L \to \infty$), this term vanishes and the motion would be a simple sine wave.

Real-World Applications

Internal Combustion Engines: This is the most common application. Every piston in your car's engine is part of a crank-slider mechanism. The kinematics determine valve timing, inertial balance, and the forces that the crankshaft and bearings must withstand.

Steam Engines and Pumps: The classic reciprocating steam engine uses this mechanism to convert the linear push of steam pressure into useful rotary motion for factories, trains, and ships. Reciprocating pumps use it in reverse, taking rotary input to create linear pumping action.

Presses and Manufacturing Machinery: Mechanical presses use a crank-slider to deliver a powerful, controlled linear force at a specific point in the stroke, ideal for stamping metal, punching holes, or forging parts.

CAE and Engine Design: Computer-Aided Engineering software uses these exact kinematic equations to calculate inertial loads for finite element analysis (FEA) of crankshafts and connecting rods. The R/L ratio you adjust in the simulator is a key design variable that affects engine smoothness, size, and bearing wear.

Common Misconceptions and Points to Note

First, is the assumption that "the smaller the R/L ratio, the better." It's true that increasing the connecting rod length L (decreasing the R/L ratio) makes the piston velocity and acceleration waveforms closer to a sine wave, which is advantageous from a vibration perspective. However, it also increases the overall engine height and weight, as well as frictional losses. In practice, within the trade-off between overall engine packaging and performance, an R/L ratio around 1/3.5 to 1/4.5 is often chosen. For example, with a crank radius r=45mm, a typical connecting rod length L would be in the range of 160mm to 200mm.

Next, is simply equating the simulator's "angular velocity" with real-world engine RPM. While increasing the angular velocity ω in the tool causes acceleration to spike, in actual engine design the metric of "mean piston speed" is more critical. The formula is $V_m = 2 \times S \times N$ (S: Stroke, N: RPM), and this determines material and lubrication limits. For instance, mass-production gasoline engines are often designed so that $V_m$ does not exceed 20 m/s.

Finally, is considering only static balance. While the static balance of the crankshaft (using counterweights to balance rotation) is certainly important, the couple (shaking force) caused by the inertia forces of the reciprocating piston and connecting rod shakes the engine block. To reduce this vibration, countermeasures such as using a V-type engine with a bank angle or employing balance shafts are necessary. The acceleration calculated by the simulator is the first step in calculating this "shaking force."

Related Engineering Fields

The calculations in this simulator are fundamental to mechanism theory (kinematics), but they open the door to vast engineering fields. First is Noise, Vibration, and Harshness (NVH) engineering. Piston acceleration determines inertial forces, which transmit through the engine block to the vehicle body, becoming the source of "booming" noise and vibration. In CAE, this force is used as input for modal analysis and frequency response analysis to predict which components resonate at which engine speeds.

Next is material mechanics and fatigue strength analysis. The connecting rod and crank pin are subjected not only to combustion pressure but also to the massive inertial forces from the acceleration you saw in the simulator, applied repeatedly. This alternating stress causes fatigue failure in the metal. In CAE, this dynamic load is applied as a time history, and fatigue analysis software is used to evaluate the critical locations and lifespan of components in detail.

Furthermore, it is deeply connected to Fluid Dynamics (CFD). The piston's motion (position, velocity) influences the efficiency of gas exchange (intake/exhaust) within the cylinder. Particularly, when the piston speed slows near top dead center (more pronounced with a larger R/L ratio), it affects turbulence formation in the combustion chamber, changing combustion efficiency and knocking characteristics. In engine simulation, this piston motion is given as a boundary condition to calculate the internal flow.

For Further Learning

First, start by experiencing the physical meaning of "differentiation" alongside graphs. The simulator's continuous display from position → velocity → acceleration visually represents the relationship: "differentiating position with respect to time gives velocity, and differentiating velocity gives acceleration." Once you grasp this intuition, understanding higher-order physical quantities like vibration (the second derivative of displacement) and jerk (the derivative of acceleration) becomes easier.

Next, I recommend following the derivation of the approximation formulas. Take the exact formula for piston position $$x(\theta) = r\cos\theta + \sqrt{L^2 - r^2\sin^2\theta}$$ and expand the square root part into a power series of $r/L$ using the binomial theorem. For example, transforming it as $\sqrt{L^2 - r^2\sin^2\theta} = L\sqrt{1 - (\frac{r}{L}\sin\theta)^2} \approx L(1 - \frac{1}{2}(\frac{r}{L}\sin\theta)^2 + \cdots)$ and rearranging it as a function of $\theta$ allows you to derive the approximation formulas for velocity and acceleration. This technique of "linearizing or series-expanding nonlinear terms" is an approach frequently used in the CAE world.

Once you're comfortable with this tool, a good next step is to learn about primary and secondary inertial force balance in multi-cylinder engines and dynamic torque fluctuation of the crankshaft. Understanding single-cylinder motion and then considering how these motions combine and cancel out in multiple cylinders allows you to appreciate the interesting challenges of real engine design. All of these topics exist as extensions of this fundamental crank-slider model.