What is a velocity ratio in a linkage?

The velocity ratio is the ratio of the output link's angular velocity to the input crank's angular velocity. A ratio greater than 1 means the output moves faster than the input; less than 1 means it moves slower.

What is the Grashof condition?

A four-bar linkage satisfies the Grashof condition when the sum of the shortest and longest link lengths is less than or equal to the sum of the other two. Only Grashof linkages can have a link that rotates fully (360°).

Why does the calculator show "no solution" at some angles?

At certain crank angles the linkage reaches a toggle (dead-center) position where the coupler and follower are collinear. The velocity ratio becomes undefined (mathematically infinite or zero) at these positions.

What units does the velocity ratio use?

The velocity ratio is dimensionless — it is the ratio ω_out / ω_in. Any consistent angular velocity unit (rad/s, rpm, °/s) cancels out. For the slider-crank, slider velocity is given as a multiple of crank tip speed (v_slider / (r·ω)).

Can I use any length unit?

Yes. All four link lengths must use the same unit (mm, inches, etc.). Because the ratio is computed from lengths, the actual unit does not matter as long as it is consistent.

Linkage Velocity Calculator

Calculate input/output velocity ratios for four-bar and slider-crank mechanical linkage systems.

Linkage Type

All lengths in the same unit (mm, in, etc.)

Ground Link (d)

Crank Link (a) — input

Coupler Link (b)

Follower Link (c) — output

Crank Angle θ₂ (degrees)

Assembly Mode

Enter link lengths and click Calculate.

Summary

This tool analyzes planar four-bar linkage and slider-crank mechanisms using kinematic equations. Enter the link lengths and crank angle to compute the output angular velocity ratio and mechanical advantage. Results update instantly in the browser with no server calls.

How it works

Select the linkage type: four-bar or slider-crank.
Enter the link lengths (crank, coupler, follower, and ground for four-bar; crank and connecting rod for slider-crank).
Set the crank angle in degrees.
The calculator solves the kinematic loop equations and computes the velocity ratio (output / input angular velocity).
Adjust any input to see the ratio update instantly.

Use cases

Verify gear ratios replaced by linkage mechanisms in machinery design.
Determine if a follower link moves fast enough for a given crank speed.
Check mechanical advantage at various crank positions during a design iteration.
Teach or learn Grashof condition and kinematic inversion concepts.
Estimate slider velocity for piston or pump mechanisms.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu

Linkage Velocity Calculator

Use the Linkage Velocity Calculator

Summary

How it works

Use cases

Frequently Asked Questions