Four-Bar Linkage
A four-bar linkage is the simplest closed-loop kinematic mechanism consisting of four rigid links connected by four revolute (pin) joints, with one link fixed as the frame (ground link). It converts rotary input motion from a crank into complex output motions of the follower link, enabling a vast range of mechanical paths and oscillations. Four-bar linkages are fundamental in machine design, robotics, prosthetics, automotive suspensions, and mechanical toys due to their simplicity and versatility.
Key Formula
l + s <= p + q (Grashof condition for continuous rotation)
LaTeX: l + s \leq p + q \quad \text{(Grashof's Condition)}
| Symbol | Meaning | Unit |
|---|---|---|
| l | Length of the longest link | m |
| s | Length of the shortest link | m |
| p | Length of one intermediate link | m |
| q | Length of the other intermediate link | m |
Worked Example
Problem
A four-bar linkage has link lengths: crank = 20 mm, coupler = 80 mm, follower = 60 mm, and ground link = 90 mm. Determine if this linkage satisfies the Grashof condition and classify the mechanism.
Solution
Step 1: Identify link lengths. Crank (s) = 20 mm (shortest), Ground = 90 mm, Follower = 60 mm, Coupler = 80 mm (longest = l) Step 2: Apply Grashof condition. l + s ≤ p + q 80 + 20 ≤ 60 + 90 100 ≤ 150 ✓ (Grashof condition satisfied) Step 3: Classify mechanism. Shortest link is the crank (adjacent to fixed link). Result: Crank-Rocker mechanism (crank makes full rotation, follower oscillates).
Answer
Grashof condition is satisfied; mechanism is a Crank-Rocker linkage.
Classification of Four-Bar Linkages Based on Grashof Condition and Fixed Link
| Mechanism Type | Fixed Link | Input Motion | Output Motion | Example Application |
|---|---|---|---|---|
| Crank-Rocker | Link adjacent to shortest | Full rotation | Oscillation | Windshield wiper, piston engine valve |
| Double-Crank (Drag Link) | Shortest link | Full rotation | Full rotation (non-uniform) | Quick-return mechanisms |
| Double-Rocker | Link opposite to shortest | Oscillation | Oscillation | Pantograph, landing gear |
| Crank-Crank (Grashof) | Coupler (non-shortest) | Full rotation | Full rotation | Parallel-crank linkages |
| Non-Grashof Linkage | Any link | Oscillation only | Oscillation | Rocking mechanisms, toys |
Interactive Tools
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Related Terms
Cam Mechanism
A cam mechanism is a higher kinematic pair consisting of a specially shaped rotating or translating element (the cam) that imparts a prescribed, non-uniform motion to a follower through direct contact. By designing the cam profile, engineers can generate virtually any desired follower displacement, velocity, and acceleration profile, making cam mechanisms indispensable in internal combustion engine valvetrains, automatic screw machines, textile machinery, and printing presses.
Mechanical Gear
A mechanical gear is a rotating machine element with teeth that mesh with another toothed component to transmit torque and rotational motion, simultaneously changing speed and mechanical advantage. Gears are fundamental power transmission devices used to increase or decrease rotational speed, multiply torque, or change the direction of motion. They are ubiquitous in automobiles, industrial machinery, watches, and robotics.
Piston
A piston is a cylindrical mechanical component that reciprocates within a cylinder to transfer force from an expanding gas or fluid to a crankshaft (in engines) or to compress/displace a fluid (in pumps and compressors). The piston forms a movable seal with the cylinder walls through piston rings, enabling controlled pressure differentials. Pistons are the heart of internal combustion engines, steam engines, hydraulic actuators, and pneumatic cylinders.
From Latin "quattuor" (four) and Old English "barre" (bar, rod). The systematic study of linkages was pioneered by Franz Reuleaux (Germany, 1875) in his seminal work "Theoretische Kinematik," which laid the foundations of modern machine theory and mechanism classification.