Planar Mechanics

Planar mechanics is the study of how machines move and transmit force in a two-dimensional plane. It is the analytical foundation behind robotic arms, engines, lift platforms, clamps, and most of the linkages inside everyday machines.

This course teaches that analysis through four mechanisms you return to in every lesson: the four-bar linkage, the slider-crank, the scissor lift, and the toggle clamp. Rather than meeting a new example each week, you study these four in depth, viewing each through a different question: first whether it moves, then where each link is positioned, how fast the links move, how hard they accelerate, and what forces that motion demands. Every result you work out by hand can be checked in an interactive simulator.

What Each Lesson Answers

Each lesson applies one analytical lens to the same mechanisms. The questions build on each other.

Lesson	The one question it answers	The skill you gain
1. Kinematic Joints and Constraint Analysis	Will it move, and how many motors does it need?	Count degrees of freedom, classify joints, recognise over-constraint
2. Position Analysis of Planar Linkages	Where is every link?	Solve vector loop equations for any configuration
3. Velocity Analysis and Instantaneous Centers	How fast is each part moving?	Find velocities by differentiation and by instantaneous centers
4. Acceleration Analysis and Dynamic Forces	How fast is the motion changing, and what force does that take?	Find accelerations and the inertia forces they create
5. Cam-Follower Systems and Motion Programming	How do I design a shape that produces a chosen motion?	Program follower motion and generate cam profiles
6. Force Analysis and Mechanism Synthesis	What forces flow through the mechanism, and how do I design for them?	Size actuators and links, then synthesise new mechanisms

How the Lessons Build on Each Other

Lessons 2 to 4 share a single thread. You write a mechanism’s vector loop equation once in Lesson 2, differentiate it once for velocity in Lesson 3, and differentiate it again for acceleration in Lesson 4. Lesson 6 then turns to the forces that motion requires, and to the reverse problem of designing a mechanism to meet a specification. Lesson 5 covers the one case where you design the motion directly: the cam profile.

Because the mechanisms stay the same, concepts are introduced once and reused. Lesson 1 defines the mechanisms and their joints; later lessons refer back to them instead of starting over.

The Mechanisms You Will Master

You analyse these four mechanisms from a new angle in each lesson. Each has an interactive simulator for checking your work.

Lesson 5 adds the cam-follower, the higher-pair mechanism used in valve trains, indexing tables, and automated machinery, where the goal is to design a surface that produces an exact, programmed motion.

Simulators and Hands-On Labs

Every calculation in this course can be verified in the browser. Each mechanism has an interactive simulator, paired with a set of guided Python experiments in the Mechanism Design and Simulation course.

How Each Lesson Is Structured

1. Real-world system problem A working machine that depends on the lesson’s analysis, with the specific question it raises.

2. Fundamental theory The mathematics and kinematics you need, derived from first principles.

3. Worked applications, drawn first Each problem is solved graphically before it is solved with equations: you draw the space diagram to scale, then the velocity diagram, then the acceleration diagram, and measure the answers off the page. The closed-form calculation and the simulator then confirm what you drew.

4. Design guidelines Practical rules for using the analysis in your own designs.

Prerequisites

Vector mechanics, basic calculus, and elementary physics. The graphical work needs only a drawing set; the analysis scripts use Python with NumPy and Matplotlib, so a working knowledge of Python is helpful but not required to follow the worked solutions.

Key Resources

• Interactive simulators and labs: the four simulators above and the Mechanism Design and Simulation experiments
• Programming: Python with NumPy and Matplotlib for the analysis scripts
• Textbooks: Theory of Machines and Mechanisms (Uicker, Pennock and Shigley), Kinematics and Dynamics of Machinery (Wilson and Sadler)

Start with Lesson 1: Kinematic Joints and Constraint Analysis to learn whether a mechanism moves before you analyse how it moves.