Lesson 6: Introduction to Mechanism Synthesis

Lesson 6: Introduction to Mechanism Synthesis#

Analysis vs. Synthesis: What is Synthesis?#

Up to this point, our study has focused on the Analysis of Mechanisms. This process starts from an already-defined mechanism (with known topology and dimensions) and seeks to determine its kinematic properties (position, velocity, acceleration) and dynamic properties (forces, torques).

Now we invert the problem to address Mechanism Synthesis. This is the true heart of machine design. The synthesis process starts from a set of motion requirements and aims to create a mechanism that satisfies those requirements.

Synthesis can be divided into two broad categories:

Type Synthesis (or Structural Synthesis): Decides what type of mechanism is best suited (e.g. a four-bar, a six-bar, a cam mechanism, a gear train).
Dimensional Synthesis: Once the type has been chosen, it determines the dimensions (link lengths, pivot angles) required for the mechanism to carry out the desired task.

In this lesson, we will focus exclusively on Dimensional Synthesis.

The Three Fundamental Problems of Dimensional Synthesis#

Dimensional synthesis is classically classified into three problems, depending on what is to be controlled:

1. Function Generation#

The goal is to coordinate the motion of an input link with that of an output link, according to a prescribed mathematical function.

Definition: The position of the output link ($\psi$) is required to be a function of the position of the input link ($\phi$), i.e. $\psi = f(\phi)$.
Example: Design a four-bar mechanism where, for a 30º rotation of the input crank, the output rocker rotates 15º, for a 60º input the output is 25º, etc.

2. Path Generation#

The goal is for a specific point of a link (generally a point on the coupler) to trace a path (a curve) defined in the plane.

Definition: The path that point $P$ must follow is prescribed.
Important: In this problem, the orientation of the link containing point $P$ is not a requirement.

3. Motion Generation (or Rigid-Body Guidance)#

This is the most complete and restrictive problem. The goal is for an entire link (a rigid body) to move through a series of prescribed positions and orientations in the plane.

Definition: A series of positions is specified for the body, defined both by the location of a point (e.g. $P_1, P_2, P_3...$) and by the angular orientation of the link ($\alpha_1, \alpha_2, \alpha_3...$).

Precision Points and Structural Error#

It is very difficult (often impossible) for a simple mechanism, such as a four-bar linkage, to generate an arbitrary function or path perfectly throughout its entire range of motion. The mathematical relationship that governs them (e.g. $\psi = f(\phi)$) is transcendental, not algebraic.

To address this, we resort to the concept of Precision Points:

Definition: These are a finite number of points (or positions) at which we require the designed mechanism to coincide exactly with the desired function, path, or motion.
Structural Error: Between two precision points, there will be a difference (an error) between the desired motion and the motion the mechanism actually generates. This error, inherent to the design, is called structural error.

The number of precision points we can define is limited by the number of free variables of the mechanism (its dimensions).

Dimensional Synthesis Methods#

There are two broad families of methods for solving synthesis problems:

1. Graphical Synthesis#

Uses geometric constructions to find the solutions. It is a very visual method that helps to understand the geometric nature of the problem.

Example (Motion Generation): This is clearly illustrated in the graphical synthesis example for 3 positions. To guide a coupler ($A_1B_1$) to positions $A_2B_2$ and $A_3B_3$, the fixed pivots (ground) $O_2$ and $O_4$ must lie at the intersection of the perpendicular bisectors.
- $O_2$ is the circumcenter of triangle $A_1A_2A_3$.
- $O_4$ is the circumcenter of triangle $B_1B_2B_3$.
Limitations: Its precision is limited, it is laborious, and it becomes impractical for more than 3 precision points.

2. Analytical Synthesis#

Uses equations (algebraic or vector) to model the mechanism and solve for its dimensions. This is an exact, systematic method, ideal for computer implementation (e.g. in MATLAB).

For Function Generation: The classic method is Freudenstein’s Equation.
- It is based on the equation $K_1 \cos \psi + K_2 \cos \phi + K_3 = \cos(\phi - \psi)$, where $K_1, K_2, K_3$ are constants that depend on the link lengths.
- This allows a linear system to be set up for $K_1, K_2, K_3$ when 3 precision points (three pairs $(\phi, \psi)$) are specified.
For Motion Generation (Dyad Method): This is the most powerful approach, and the one we will use in our examples.
- Concept: The four-bar linkage is modeled as two “dyads” (two-link chains) that start at the ground pivots ($O_2, O_4$) and meet at the coupler ($A, B$).
- Formulation: Using complex numbers, the vector equation of a dyad is $\vec{W} + \vec{Z} = \vec{\delta}$. For a motion between position 1 and a position $j$, the equation becomes: $$\vec{W}(e^{j\beta_j}-1) + \vec{Z}(e^{j\alpha_j}-1) = \vec{\delta_j}$$
- Solving: For 3 positions ($j=2, 3$), we have 4 scalar equations. The number of unknowns is 5 (the components of $\vec{W}$ and $\vec{Z}$, and the angle $\beta_2$). This gives us one free choice (we can choose a value for $\beta_2$) to solve the linear system.
- Application: This is precisely the method applied further on, in an example where a four-bar mechanism is designed for 3 prescribed positions, solving the equations for the two dyads.

The Driver-Dyad Problem#

A very common design problem arises when applying these synthesis methods:

The Problem: The four-bar mechanism resulting from synthesis is very often a double-rocker (it does not satisfy Grashof’s law), or even if it does satisfy it, its range of motion is restricted.
The Consequence: This mechanism cannot be driven by a continuous-rotation (CR) motor, since none of its links pivoted to the ground can complete a full revolution.
The Solution: Add a driver dyad.
- The double-rocker four-bar linkage ($O_2ABO_4$) we designed is kept unchanged.
- A new dyad is added (a two-bar chain, e.g. $O_6CO_2$), designed to be a crank-rocker (it satisfies Grashof’s law).
- The motor (CR) is coupled to the new crank (at $O_6$), which “pushes” the original rocker (at $O_2$), forcing the whole mechanism to move.
The Result: This transforms the original four-bar mechanism into a six-bar mechanism (a Stephenson chain).
Application: This is the core of another example we will see, in which, starting from the double-rocker mechanism designed in the previous example, a new crank-rocker dyad is attached, allowing a continuous-rotation motor to drive the whole assembly.

Limitations and Final Considerations of Synthesis#

As summarized in the final slides, success in synthesis is not guaranteed. Obtaining a mathematical solution does not imply that it is a valid engineering solution. We must watch out for:

Poorly chosen precision points: These can lead to infeasible solutions.
Assembly impossibility: The resulting bar lengths do not allow the loop to be closed.
Grashof problem: The solution is a double-rocker when a crank is required (solved with a driver dyad).
Branching problem: The mechanism may “jump” to its other assembly configuration (the “mirror image”) during motion, which is unacceptable.
Poor transmission angle: The mechanism may have points where the transmission angle is very small, leading to locking (jamming) or very high reaction forces.
Singularities: The mechanism may pass through singular configurations where it momentarily gains or loses degrees of freedom.