Set tooth counts, input speed, torque, and efficiency to calculate gear ratio and output parameters in real time. Visualize meshing animation and rotation direction for spur, compound, and planetary gears.
Parameters
Gear train type
Stages
Planetary gear setParameters
Sun gear teeth Z_S
Ring Gear Teeth Z_R
Input speed n1
rpm
Input torque T1
N·m
Transmission Efficiency η (per stage)
Module m [mm]
mm
Results
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Gear ratio i
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Output speed [rpm]
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Output torque [N·m]
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Transmitted power [W]
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Output pitch diameter d2 [mm]
Visualization
Drag gears on the canvas to change tooth count (drag up/down = enlarge/shrink)
⏱ 0.000 s
0/5 Saved
Theory & Key Formulas
Simple two-gear pair：$i = Z_2/Z_1 = n_1/n_2$
$T_2 = T_1 \cdot i \cdot \eta$ (η：transmission efficiency)
What exactly is a gear ratio, and why does it matter in this simulator?
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Basically, the gear ratio tells you the relationship between the speeds and torques of two connected gears. In this tool, for a simple two-gear system, it's the ratio of the driven gear's teeth to the driving gear's teeth. For instance, if a small gear with 10 teeth drives a large gear with 30 teeth, the ratio is 3:1. Try moving the "Driving gear teeth Z" and "Driven gear teeth Z" sliders above to see the ratio and the animation change instantly.
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Wait, really? So if the ratio is 3:1, does that mean the big gear is three times slower? What happens to the force or turning power?
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Exactly right! Speed and torque have a trade-off. If the output gear is three times slower ($n_2 = n_1 / 3$), its output torque is roughly three times higher ($T_2 \approx T_1 \times 3$). This is how a car's transmission lets the engine turn fast with low torque, but the wheels turn slower with high torque to move the car. In practice, some power is lost to friction. That's what the "Transmission Efficiency η" slider controls-try reducing it from 100% and watch the output torque drop.
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That makes sense for two gears. But the simulator also shows "Planetary" systems. What's going on there with the sun and ring gears? It looks complicated!
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Planetary gear sets are fascinating! They pack a huge speed reduction into a very compact space, which is why they're used in automatic transmissions and robot joints. In this simulator, you control the teeth of the central "sun gear" (Sun Gear) and the outer "Ring Gear Teeth". The ratio depends on which part is driving, which is held, and which is the output. A common case is holding the ring gear: then the speed ratio is $1 + Z_R/Z_S$. Try setting $Z_S=20$ and $Z_R=60$-you'll get a 4:1 reduction in a single, compact stage!
Physical Model & Key Equations
The fundamental relationship for a simple two-gear pair relates the number of teeth (Z) to the rotational speed (n). The gear ratio (i) defines the speed reduction or increase.
$$i = \frac{Z_2}{Z_1}= \frac{n_1}{n_2}$$
Where:
$i$ = Gear Ratio (Driven / Driving)
$Z_1, Z_2$ = Number of teeth on the driving and driven gears
$n_1, n_2$ = Rotational speed of the driving and driven gears (e.g., RPM)
This change in speed directly affects the torque (T), which is the rotational force. Accounting for real-world power loss through efficiency (η) is crucial for accurate design.
$$T_2 = T_1 \cdot i \cdot \eta$$
Where:
$T_1, T_2$ = Input and output torque (e.g., Nm)
$\eta$ = Transmission efficiency per stage (a value between 0 and 1) Physical Meaning: This equation embodies the conservation of energy (minus losses). Reducing speed ($i > 1$) multiplies the available torque, which is the core principle behind gear reducers in machinery.
Frequently Asked Questions
The unit for input rotation speed is rpm (revolutions per minute), and the unit for torque is Nm (Newton meters). Efficiency should be set between 0 and 1. For general spur gears, a guideline is 0.95 to 0.98, and for helical gears, 0.97 to 0.99.
When the gear ratio is increased, the output rotation speed decreases inversely. For example, if the gear ratio is 2 and the input rotation speed is 1000 rpm, the output will be 500 rpm. In return, the output torque increases approximately twofold, considering efficiency.
In the meshing of external gears, the rotation direction of the driving gear and the driven gear is always opposite. The animation correctly visualizes this physical phenomenon. If you want the rotation direction to be the same, add an idler gear or use an internal gear.
The overall gear ratio of a compound gear train is calculated by multiplying the gear ratios of each stage. For example, if the first stage has a ratio of 2 and the second stage has a ratio of 3, the overall gear ratio is 2 × 3 = 6. In this tool, it is automatically calculated by setting the parameters for each stage individually.
Real-World Applications
Vehicle Transmissions: Gearboxes use multiple gear ratios to allow an engine to operate at efficient speeds while providing the right amount of torque for acceleration (low gear) or high-speed cruising (high gear). The compound gear train calculations in this simulator are directly used in their initial design.
Industrial Reducers & Gearmotors: Conveyor belts, mixers, and heavy machinery use gear reducers to convert the high speed of an electric motor into the low-speed, high-torque motion needed for work. Engineers use these exact torque and efficiency calculations to select the right motor and gearbox.
Robotics & Precision Actuators: Robotic joints and CNC machine axes require precise control of speed and torque. Planetary gearheads, which you can model here, are favored for their high torque density, low backlash, and compactness, fitting directly into the arm of an industrial robot.
CAE Simulation Input: Before running complex multibody dynamics simulations in software like Adams or Simpack, engineers must define accurate gear ratios and inertia properties. This tool is used to verify those key input parameters, ensuring the digital model behaves like the real mechanical system.
Common Misunderstandings and Points to Note
When you start using this tool, there are a few common pitfalls to watch out for. First, understand that a larger gear ratio does not always mean better performance. While a large reduction ratio amplifies torque, it also significantly reduces rotational speed. For example, if you transmit a motor's maximum speed of 3000 rpm through a mechanism with a gear ratio of 10, the output will be only 300 rpm. This can sometimes be too slow for your intended application. The first step in design is to determine the gear ratio by considering the balance between speed and torque.
Next, be aware that the theoretical torque differs from the actual torque produced. If you set the "Transmission Efficiency η" to 1.0 (ideal) in the tool, the torque doubles neatly, but reality is not so forgiving. Especially in compound gear trains, the efficiency (e.g., 0.98) multiplies at each stage, causing the overall efficiency to drop progressively. With three stages, 0.98 cubed is approximately 0.94. So, with a 100 Nm input, you'd get about 752 Nm instead of the theoretical 800 Nm. Ignoring these losses can lead to major mistakes in motor selection, so be careful.
Also, did you know that in planetary gear parameter settings, the "number of planet gear teeth" is not directly involved in the gear ratio calculation? In the tool, you only input the number of sun gear and ring gear teeth. The number of planet gear teeth is determined by the meshing condition (ensuring all gears fit together smoothly). For instance, with a 20-tooth sun gear and a 60-tooth ring gear, the condition for the planet gear teeth is $(60-20)/2 = 20$ teeth. If you don't check this "assemblability," you might face significant rework at the drawing stage, so keep this in mind.