## 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: 1. **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). 2. **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: 1. **Poorly chosen precision points:** These can lead to infeasible solutions. 2. **Assembly impossibility:** The resulting bar lengths do not allow the loop to be closed. 3. **Grashof problem:** The solution is a double-rocker when a crank is required (solved with a driver dyad). 4. **Branching problem:** The mechanism may "jump" to its other assembly configuration (the "mirror image") during motion, which is unacceptable. 5. **Poor transmission angle:** The mechanism may have points where the transmission angle is very small, leading to locking (jamming) or very high reaction forces. 6. **Singularities:** The mechanism may pass through singular configurations where it momentarily gains or loses degrees of freedom.