Animation – Crank-Rocker Mechanism
The articulated four-bar mechanism is a common and widely used mechanical structure. Hydraulic lifting mechanisms, rocker mechanisms, and other forms in machine tools, construction machinery, and automobiles are different manifestations of the articulated four-bar mechanism.
Figure – Schematic Diagram of Articulated Four-Bar Mechanism
The following images show a crane and a hydraulic lifting mechanism, both of which rely on the internal articulated four-bar mechanism.
Figure – Crane
Figure – Hydraulic Lifting Mechanism
In the design of the articulated four-bar mechanism, the motion curve is extremely important, directly related to whether the mechanism operates according to the predetermined design. Currently, engineers often use simulation methods to calculate the motion state of the four-bar mechanism.
The Simdroid platform independently developed by Yundao Intelligent Manufacturing uses finite element methods to conveniently calculate and obtain the motion trajectory curve of the articulated four-bar mechanism.
The motion of the mechanism brings about large displacements of component geometries, thus geometric nonlinearity must be considered during the simulation calculations of the mechanism’s motion.
In linear finite element analysis, rotation is usually considered as a small rotation vector, i.e., an infinitesimal rotation vector, which has vector properties and can be decomposed relative to the global Cartesian coordinate axis to obtain three rotation components. The rotation vector between two incremental steps can be superimposed or the order of rotation can be exchanged.
In geometric nonlinear analysis, rotation is considered as finite rotation, which is similar to coordinate transformation and is represented by an orthogonal matrix (ΛT=Λ-1, |Λ|=1), which does not have superposition properties and the order of rotation cannot be changed. The rotation vector corresponding to finite rotation is called the rotation pseudo-vector θ, which is defined with θ/||θ|| as the rotation axis and ||θ|| as the rotation angle. After each solution, a mapping relationship is needed to map the infinitesimal rotation increment to finite rotation, which is called the exponential map:
Where:
Directly using the exponential map can lead to some issues, such as the angle corresponding to sin||θ||/2 not being unique after exceeding 180 degrees. The Simdroid platform uses quaternion vectors (quaternions vector) for updates, where its components can also be referred to as Euler parameters. After one incremental calculation, the quaternion increment is obtained as:
Where:
At the end of the last iteration, it is known that θo, where the subscript o indicates old, and n indicates new, allowing for the calculation of the updated quaternion vector after finite rotation:
The definition of the operator . is:
Then the new rotation matrix Λn is:
The articulated four-bar mechanism motion simulation APP developed based on the Simdroid platform allows the lengths of the four bars to be set as parameters. The interface is shown below:
Figure – Interface of the Articulated Four-Bar Mechanism Motion Simulation APP
This APP can calculate the motion trajectory curves of different specifications of four-bar mechanisms by changing parameters, such as double rocker mechanisms, crank-rocker mechanisms, and double crank mechanisms.
Animation – Double Rocker Mechanism
Animation – Double Crank Mechanism
Bi Tan, Yundao Intelligent Research – Yundao Intelligent Manufacturing Simulation APP Development Team member, graduated from Zhejiang University with a degree in Mechanical and Energy Engineering, and has 10 years of structural simulation experience. He has completed multiple industrial application simulation APPs based on the Simdroid platform, and as a core member, won the third prize in the 2018 China Industrial APP Innovation Application Competition for General Products.
Yundao Intelligent Research – Yundao Intelligent Manufacturing Simulation APP Development Team, with the mission of “Inclusive Simulation”, possesses professional simulation capabilities and mature simulation APP packaging development technology. Relying on the self-controlled simulation platform Simdroid, which covers four major physical fields: electromagnetic, structural, fluid, and thermal, Yundao Intelligent Research can provide enterprises with professional and fast simulation technology services, including engineering calculations, simulation training, tool development, and customized simulation cloud platform solutions, solving practical engineering problems and assisting enterprises in research and development innovation and transformation upgrades. We welcome cooperation and communication from enterprises, universities, research institutes, and industry experts.
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