Mechanical Design Handbook

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 Design Ratios and Geometry The genius of the Chebyshev linkage lies in its specific geometric proportions. The mechanism confines the coupler point P (the midpoint on...

Introduction to Watt's Linkage The Watt's linkage (often called the parallel linkage) is a crucial mechanism in the history of engineering. Originally invented by James Watt to constrain the piston movement of a steam engine, it remains a fundamental study in the kinematics of machinery today. Watt described this invention in a letter to Matthew Boulton in 1784 with great pride: 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 upon the beam, without chains or perpendicular guides [...] and one of the most ingenious simple pieces of mechanics I have invented. Search for Best Books on Kinematics & Linkage Design Kinematics and Geometry It is important to note that the Watt mechanism does not generate a mathematically perfect straight line. Instead, it generates a lemniscate curve (a figure-eight shape). However, for small ranges of motion, the path i...

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 (like the THK model shown to the right) that guides a device accurately along a rail. The manufacturing know-how of linear guide manufacturers has allowed us to expand the range of linear guidance into everything from CNC machines to 3D printers. These Linear Ball Slides are lightweight, compact, and operate with very low sliding resistance and low inertia. The Historical Challenge: Making a Straight Line However, in the late 17th and early 18th centuries—before the development of the milling machine or the planer—it was extremely difficult to machine long, perfectly straight, flat surfaces. For this reason, creating good prismatic pairs (sliding joints) without significant backlash was nearly impossible. During that era, eng...

In [ 3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - 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. 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 NX 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 original c...

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. Step 1: Setup the Constraints Start by drawing your known constraints in the NX Sketcher: 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 ). (The red lines in the image below represent the moving coupler in its three positions). Step 2: Inverting Ground Pivot O 2 Now we perform the "Inversion." We need to find where the ground pivot O 2 would be relative to Position 1 if the coupler stayed still. ...

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 geometry dictated where the ground pivots (O 2 and O 4 ) had to be. But what if you don't have that freedom? 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 and the Ground is moving . By fixing the coup...

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. 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 . Step 2: Define Alternate Moving Pivots (C and D) This is the critical step. We attach a "virtual" rigid shape to line AB to define new points C and D. Procedure: 1....

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 Unigraphics NX4 Sketch ] to solve this problem geometrically within the CAD environment. 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). 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 Synthesis for Pivot O 2 : To find the fixed pivot that allows point A to move through all three locations, we must find the center of the circle that passes through A 1...

In advanced Mechanism Design , we 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 Unigraphics NX (Siemens NX), SolidWorks, or CATIA. 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 to position A'B' . Step-by-Step Geometric Synthesis 1. Define the Positions: Draw the link AB (Start Position) and A'B' (End Position) in the NX Sketcher. 2. Locate the First Pivot (O 2 ): Draw a construction line connecting point A to A'. Create a Perpendicular Bisector of line AA'. Theory: Any point located on this bisector is equidistant from A and A', meaning it can serve as a fixed pivot point. ...

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 . 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. The Power of "Spreadsheet Run" The simulation below was executed using the Kinematics environment in the Unigraphics (UG) NX4 Motion Simulation Module . By utilizing the "Spreadsheet Run" command, we are not just animating the assembly; we are driving the geometry with pure data. Every frame of movement corresponds to a specific calculation row in our Excel sheet. This confirms that the complex polynomial curves we designed will physically clear the tooling without collision. Video Analysis: Virtual Commissioning Watch the simulation below closely. Unlike the "Constant Velocity" test in Part 3 (which r...

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. 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 Pro Tip: Do ...

In [ Part 2 of this series ], we finished setting the driver for the revolute joint of the indexing mill. Now, we will set up the Punch Die . Step 1: Setting up the Slider Joint The movement of the punch die is different from the indexing mill. It moves only in linear motion along the Z-axis. The joint for this kind of movement is called a "Slider" joint. Procedure: 1. Select Joint command. 2. Select "Slider" joint icon. 3. Select the "Die" link we created earlier. 4. Click "Orientation on the first link" → Select "Point". 5. Select the center point of the cylinder to define the joint origin. 6. Select "Vector" → Click the bottom face of the cylinder (defines the downward Z-axis). 7. Rename to "J_Die". 8. Click Ok. Step 2: Defining the Linear Driver 1. Right-click joint "J_Die" → Edit. ...

Let's continue from the previous post . Now it's time to visualize our previous calculation for the timing diagram of the indexing mill and punch die using 3D Motion Simulation in Unigraphics (UG) NX4 . While we successfully created a 2D motion simulation in Excel , modern engineering demands a full Digital Twin . The UG NX4 Assembly model is prepared as shown below. The mating conditions of the assembly model follow the sketch shown in [ Timing Diagram (Part 1 - No Overlap Movement) ]. Step 1: Entering the Simulation Environment New to UG NX4 Motion Simulation ? No problem. Follow this step-by-step guideline. In the motion simulation environment , all commands are initially disabled. You must right-click on the assembly file and select New Simulation . This command creates the necessary files and organizes them automatically. Step 2: Defining the Kinematic Environment ...