Mechanical Design Handbook

Figure 1: The Chebyshev linkage converts rotary input into approximate straight-line output. Introduction to the Chebyshev Linkage The Chebyshev linkage is a four-bar mechanical linkage that converts rotational motion into approximate straight-line motion . It was invented by the 19th-century Russian mathematician Pafnuty Chebyshev , who was deeply involved in the theoretical problems of kinematic mechanisms. His goal was to improve upon existing designs, such as the Watt Straight-line Mechanism , which James Watt had used to revolutionize the steam engine. While Watt's design produces a lemniscate (figure-eight) curve with a straight section, the Chebyshev linkage is often preferred in specific machinery because the straight-line portion of the path is parallel to the line connecting the two fixed ground pivots. Search for Mechanism Design & Robotics Books Advertisement Design Ratios and Geometry The gen...

Figure 1: Watt's linkage example geometry and path generation. Introduction to Watt's Linkage The Watt's linkage (also known as the parallel motion linkage) is a cornerstone in the history of mechanical engineering. It is a type of four-bar linkage originally invented by James Watt in the late 18th century to solve a critical problem in steam engine design: constraining the piston rod to move in a straight line without using high-friction guideways. Before this invention, engines used chains to connect the piston to the beam, which meant they could only pull, not push. Watt's rigid linkage allowed for double-acting engines (pushing and pulling), doubling the power output. He was immensely proud of this kinematic solution, describing it in a 1784 letter to his partner Matthew Boulton: "I have got a glimpse of a method of causing a piston rod to move up and down perpendicularly by only fixing it to a piece of iron u...

Figure 1: A modern linear ball slide (like this THK model) is the contemporary solution for precise straight-line motion. Many modern engineering applications require components to move in a precise linear fashion, known as " straight-line motion ". Today, we take this for granted. We can simply purchase an off-the-shelf Linear Motion Guide that moves a device accurately along a rail with low friction. The Historical Challenge: Making a Straight Line However, in the late 17th and early 18th centuries—before the development of high-precision milling machines—it was extremely difficult to manufacture long, perfectly flat surfaces. Creating a sliding joint without significant backlash was nearly impossible. During that era, engineers had to rely on Linkages . Much thought was given to the problem of attaining a straight-line motion using only revolute (hinge) connections, which were much easier to manufacture. The most famous early result was...

In [ 3-Position Synthesis with Inversion Method - Part 2 ], we successfully determined the locations of the moving pivots (G and H) relative to our fixed ground pivots (O 2 and O 4 ). However, finding the points is only half the battle. Before we commit to manufacturing or detailed 3D modeling, we must verify that the mechanism actually moves smoothly between all three positions without locking up (toggle positions) or deviating from the path. Advertisement Constructing the Kinematic Chain Now that we have our four critical points (O 2 , O 4 , G, H), we need to "build" the mechanism links within the CAD Sketcher environment: Input Link (Link 2): Draw a solid line connecting the fixed ground O 2 to the moving pivot G. Output Link (Link 4): Draw a solid line connecting the fixed ground O 4 to the moving pivot H. Coupler Link (Link 3): This is the most important part. You must draw a rigid triangle connecting G, H, and the ...

In the previous introduction , we established the problem: We have fixed mounting points (O 2 and O 4 ) on our machine base, and we need to design a linkage to hit 3 specific positions. Standard synthesis moves the pivots to fit the motion. In Kinematic Inversion , we do the opposite: we virtually move the ground to fit the coupler. By "freezing" the coupler in Position 1 and moving the ground relative to it, we can geometrically find the required link lengths. Advertisement Step 1: Setup the Constraints Start by drawing your known constraints in the CAD Sketcher (NX, SolidWorks, etc.): 1. The Fixed Ground Pivots (O 2 and O 4 ). 2. The 3 Desired Coupler Positions (A 1 B 1 , A 2 B 2 , A 3 B 3 ). Figure 1: The setup showing fixed grounds (bottom circles) and the target motion path (red lines). Step 2: Inverting Ground Pivot O 2 Now we perform the "Inversion." We need to find where the ground pivot...

In our previous tutorials, such as [ 3-Position Motion Generation Synthesis with Alternate Moving Pivots ], we used a "standard" synthesis approach. We defined the moving coupler first, and the geometric construction dictated where the ground pivots (O 2 and O 4 ) had to be. But what if you don't have that freedom? Advertisement In real-world machine design, you often have a pre-existing frame or base. You cannot drill holes just anywhere; the ground pivots must be located at specific, available points. In this scenario, the standard method fails because it gives you valid kinematic solutions that might require mounting a pivot in thin air or inside a motor. The Solution: Kinematic Inversion To solve this, we use the Inversion Method . The Core Concept Instead of looking at the mechanism from the perspective of a stationary ground and a moving coupler, we invert our perspective. We pretend the Coupler is stationary...

In the previous post [ 3-Position Motion Generation Four-Bar Linkage Synthesis ], the locations of the fixed ground pivots (O 2 and O 4 ) were mathematically determined by the positions of points A and B. The Problem: Sometimes, these calculated fixed pivots land in impossible locations—inside another machine part, off the machine base, or too far away. The Solution: We use Alternate Moving Pivots . Instead of using the endpoints of the line AB, we create new points (C and D) that are rigidly attached to the moving body. By adjusting the location of C and D, we can steer the fixed pivots (O 2 and O 4 ) to desirable locations. Advertisement Step 1: Define the Desired Motion Draw the coupler link AB in its three design positions: A 1 B 1 , A 2 B 2 , and A 3 B 3 . Figure 1: Defining the three target positions. Sometimes standard pivot locations are invalid or obstructed. Step 2: Define Alternate Moving Pivots (C and D) ...

In real-world engineering, a mechanism often needs to guide a part through more than just a start and end point. It usually requires passing through 3 specified positions to clear obstacles or perform complex tasks. This technique is known as 3-Position Motion Generation . We can extend the logic from our previous post [ Four-bar linkage Synthesis using CAD Sketcher ] to solve this problem geometrically within a modern CAD environment like Siemens NX, SolidWorks, or CATIA. Advertisement The Design Challenge Assume we must design a mechanism to move Link AB through three specific positions (A 1 B 1 , A 2 B 2 , A 3 B 3 ) while avoiding an obstacle (represented by the rectangle below). Figure 1: Defining the three target positions (A1B1, A2B2, A3B3) relative to the obstacle. Step-by-Step Synthesis 1. Define the Positions: Draw Link AB in its three design positions: A 1 B 1 , A 2 B 2 , and A 3 B 3 . 2. Geometric Synthes...

In advanced Mechanism Design , engineers often face the challenge of moving a rigid body from one specific position to another. This process is known as Motion Generation Synthesis . While sophisticated solver software exists, you can perform this synthesis geometrically using the Constraint-Based Sketcher found in any modern CAD package like Siemens NX, SolidWorks, or CATIA. Advertisement The Goal: Moving a Line in a Plane Assume we need to design a 4-bar linkage that moves a coupler link from position AB (Start) to position A'B' (Target). Figure 1: Defining the Start Position (AB) and the Target Position (A'B'). Step-by-Step Geometric Synthesis The logic relies on finding the center of rotation for the moving points. 1. Locate the First Pivot (O 2 ): Draw a construction line connecting point A to A'. Then, create a Perpendicular Bisector of line AA'. Theory: Any point located on this...

This is the moment of truth. In the previous posts, we moved from abstract mathematical derivations in Excel to the concrete setup of a 3D Digital Twin . Advertisement The result of our timing diagram design—utilizing overlapping motion with Fifth-Degree (3-4-5) Polynomial and Linear cam functions—is now fully integrated into the 3D model. We are no longer just guessing; we are validating the Mechatronics Design Workflow . The Power of "Spreadsheet Run" The simulation below was executed using the Kinematics environment in the Unigraphics (UG) NX4 Motion Simulation Module (now known as Simcenter 3D). By utilizing the "Spreadsheet Run" command, we are not just animating the assembly; we are driving the geometry with pure, precise data. Every frame of movement corresponds to a specific calculation row in our Excel sheet. This creates a direct data bridge, confirming that the complex polynomial curves we designed will phy...

In [ Part 3 of this series ], we set up the kinematic joints for our machine. However, the drivers are currently set to "Constant Velocity," which does not reflect reality. Advertisement Now, we execute the most powerful part of the Digital Twin workflow: injecting our precise timing diagram data from Excel directly into the 3D simulation. Step 1: Exporting Joint Data to Excel 1. Select the "Graphing" command. This tool is typically used to view results, but we will use it to open the data channel. 2. Select "Spreadsheet" . This tells NX to bridge the data into Microsoft Excel. 3. Click OK. NX will automatically launch Excel. You will see columns for "drv J_Mill" and "drv J_Die" with linear values. These are the default placeholders we created earlier. We must replace these with our optimized curves. Step 2: Preparing the Data ...