Lesson 6: Force Analysis and Mechanism Synthesis

Five lessons told you how a mechanism moves. This one tells you what it costs in force, and how to design for it. A toggle clamp turns a light hand pull into a heavy clamping force; a scissor lift needs a hydraulic cylinder whose force runs away as the platform nears the floor; every pin and link must carry its load without breaking. Force analysis is statics done on the mechanism in a chosen position: free-body diagrams and force polygons give the joint reactions, the transmission angle warns where the force transmits badly, and stress sizing turns those forces into metal. The course then closes by running the whole process backward, synthesis: choosing a mechanism and its dimensions to meet a target. #ForceAnalysis #TransmissionAngle #MechanismSynthesis

Learning Objectives

By the end of this lesson, you will be able to:

1. Find joint reactions with free-body diagrams and force polygons
2. Relate mechanical advantage to the velocity ratio through virtual work
3. Judge force quality with the transmission angle and size links and pins for stress
4. Synthesise a mechanism, choosing type and dimensions, to meet a force and motion target

Real-World System Problem: From Motion to Metal

A toggle clamp on a machining fixture must hold a part down with several hundred newtons, applied by hand and held without effort while the tool cuts. The designer must answer: what hand force gives the required clamping force, what loads do the pins and links then carry, and are they strong enough? The same questions decide the hydraulic cylinder on a scissor lift and the motor on an engine. Motion analysis found the speeds and accelerations; now we find the forces they imply and the sizes they demand.

The Force Problem

Engineering Question: For a given input force or torque, what force appears at the output, what reactions load each joint, and are the links and pins strong enough?

Why Force Analysis Closes the Course

Fundamental Theory: Statics on a Mechanism

Free-Body Diagrams and the Force Polygon

The Force Polygon

A link in static equilibrium has forces that sum to zero, so drawn tip to tail they close into a polygon. Two special cases do most of the work:

• A two-force member (forces at only two joints, no other load) carries a force directed along the line joining the joints. The coupler of a four-bar and the main link of a toggle are two-force members.
• A three-force member has three forces that must be concurrent (meet at one point) and close into a force triangle. Knowing the directions of all three and the magnitude of one, the triangle gives the other two by measurement.

The force polygon is the statics counterpart of the velocity and acceleration polygons: the same draw-to-scale-and-measure method, now for forces.

Mechanical Advantage by Virtual Work

Force Ratio is the Reciprocal of the Velocity Ratio

An ideal (lossless) mechanism conserves power: input power equals output power. With ,

The mechanical advantage is the reciprocal of the velocity ratio found in the velocity analysis. Where the output slows (a limit or toggle position, velocity ratio toward zero), the mechanical advantage grows large. This is why the same toggle position that stopped the output in the velocity analysis amplifies force here. Real mechanisms lose a little to friction, so an efficiency multiplies the ideal value.

Transmission Angle and Stress

Transmission Angle and Allowable Stress

The transmission angle is the angle between the coupler and the follower at their joint. The component of the coupler force that drives the follower scales with , so near transmits force well and near or transmits almost none. The usual guide is throughout the motion.

Once a joint force is known, a pin of diameter in single shear carries , and a rectangular link of thickness and width under a bending moment carries with . Both must stay under the allowable stress for a safety factor .

Application 1: Toggle-Clamp Force Amplification and Sizing

This is the capstone worked example: from hand force to clamping force by force polygon, then the pin and link stresses.

Simulator and hands-on lab

Open the Toggle Clamp Simulator

Hands-on lab: Continue in the Toggle Clamp Experiments lab (siwit.co/TCM). Experiments 2 and 4 cover force amplification and pin/link stress sizing.

Step 1: Force Polygon for the Clamp Arm

The main link is a two-force member, so its force is along the link. The clamp arm is a three-force member (pad reaction, main-link force, pivot reaction), so its force triangle closes.

Click to reveal the force-polygon construction

1. Identify the members. The main link carries a force along its own line (two-force member). The clamp arm then has three forces: the main-link push at one joint, the pad reaction at the workpiece, and the pivot reaction at . ✅

2. Draw the triangle. Choose a force scale and mark it (for example 1 cm = 20 N). Lay the known main-link force tip to tail with the pad-reaction direction; the pivot reaction closes the triangle. Measuring the sides gives the pad force and the pivot force. ✅

3. The over-centre amplification. Geometrically the ideal force ratio of the toggle is , where is the angle of the links from the collinear (dead-centre) line. The closer to centre, the larger the amplification. ✅

Step 2: Clamping Force and Stresses

Click to reveal the numbers

1. Mechanical advantage at the lock margin:

   ✅

2. Clamping force:

   ✅

   The closer the rest position is set to top-dead-centre, the higher this rises, the over-centre design from Lessons 1 and 3 seen as force.

3. Stress check at a representative link/pin force of N (the internal forces exceed the pad force near the joints):

   ✅

   With allowable MPa, both are well within limits. ✅

Step 3: Verify in the Simulator

Click to reveal the simulator check

1. Open the simulator (siwit.co/TCM) and set the hand force, lock margin, efficiency, and the pin and link sizes. ✅

2. Confirm the clamping force rises sharply as the lock margin shrinks toward centre, and read the pin-shear and link-bending stresses with their pass/fail verdict against the allowable. They match the hand calculation. ✅

Application 2: Four-Bar Transmission Angle

The transmission angle tells you where in its cycle a four-bar transmits force well, and where it wastes it.

Simulator and hands-on lab

Open the Four-Bar Linkage Simulator

Hands-on lab: Continue in the Four-Bar Linkage Experiments lab (siwit.co/FBL). Experiment 2 maps the transmission angle and mechanism quality.

Step 1: Plot the Transmission Angle

Click to reveal the transmission-angle behaviour

1. Measure as the angle between the coupler and follower at joint , taken as the value between and . At it is , an excellent transmission. ✅

2. Sweep the crank. Across a full turn rises to about and falls to about . The dips below the guide are the poor-force zones, where a large coupler force produces only a small useful drive on the follower. ✅

3. Design response. If those zones fall inside the working stroke, change the link lengths (Application 4 synthesis) or re-time the load so the heavy work happens where is large. ✅

Application 3: Scissor-Lift Actuator Force

The scissor lift shows mechanical advantage working against the designer: the actuator force runs away as the platform nears the floor.

Simulator and hands-on lab

Open the Scissor Lift Simulator

Hands-on lab: Continue in the Scissor Lift Experiments lab (siwit.co/SLM). Experiment 1 plots the actuator force and Experiment 8 the link stress.

Step 1: Actuator Force by Virtual Work

Click to reveal the actuator-force relation

1. Apply virtual work. The actuator does work as the base spread changes, the load rises through the height change. Equating, the horizontal-base actuator force to hold load at scissor angle is:

   ✅

   (the exact constant depends on the actuator placement; the simulator gives the precise value for each type).

2. Read the runaway. At , N; at , N; at , N. As the platform nears the floor, the mechanical advantage works against the actuator and the force spikes. ✅

3. Design response. This is why scissor lifts work over a limited low-angle band, use a diagonal or pantograph actuator placement to improve the low-angle advantage, and never start fully flat. ✅

Application 4: Synthesise a Four-Bar for a Target

Analysis takes a mechanism and finds its behaviour. Synthesis takes a required behaviour and finds the mechanism. This is where the whole course is put to use.

Simulator and hands-on lab

Open the Four-Bar Linkage Simulator

Hands-on lab: Use the Four-Bar Linkage Experiments lab (siwit.co/FBL) to test each candidate set of link lengths against the target.

Step 1: Type and Dimensional Synthesis

Click to reveal the synthesis process

1. Type synthesis. Choose the mechanism family from the task. A continuous input turning an oscillating output points to a crank-rocker four-bar; a straight-line output points to a slider-crank; a clamp pointed to a toggle. The degrees-of-freedom check from the mobility analysis confirms one input drives it. ✅

2. Dimensional synthesis. Choose the link lengths to meet the motion. For a required rocker swing, fix the ground and rocker, then size the crank and coupler so the rocker reaches both extreme (limit) positions at the wanted angles. The Grashof test from the position analysis must pass so the crank fully rotates. ✅

3. Check force quality. Run the position analysis and plot the transmission angle (Application 2). If it dips below inside the working stroke, adjust the link lengths and repeat. Synthesis is this loop: propose lengths, analyse, refine. ✅

4. Verify the whole design. Confirm mobility, positions, velocities and mechanical advantage, accelerations and inertia loads, and finally the forces and stresses. A design is finished only when all six checks pass. ✅

Design Guidelines for Force Analysis and Synthesis

Summary and Course Conclusion

Key Concepts Mastered

1. Force polygons: static link forces close into a polygon; two-force members carry force along their line, three-force members close a triangle.
2. Mechanical advantage: the reciprocal of the velocity ratio by virtual work, large at the toggle and limit positions.
3. Transmission angle: the force-quality gauge, kept above about ; the four-bar here dips to twice per turn.
4. Stress sizing: joint forces become pin shear and link bending, checked against the yield stress divided by a safety factor.
5. Synthesis: choose the mechanism type and dimensions, then verify against every analysis in the course.

Force Results at a Glance

Mechanism	What you solve for	Key relation	Simulator
Toggle clamp	clamping force, stresses		siwit.co/TCM
Four-bar	transmission angle	between coupler and follower	siwit.co/FBL
Scissor lift	actuator force		siwit.co/SLM
Crank-slider	crank torque		siwit.co/CSM

The Course in One Thread

You met four mechanisms and analysed each through six lenses: whether it moves, where its links sit, how fast they move, how hard they accelerate and what inertia that creates, how to program a motion with a cam, and what forces flow through it and how to design for them. Every result was drawn to scale by hand, confirmed by calculation, and checked in an interactive simulator. That triad, the drawing for intuition, the mathematics for precision, and the simulator for exploration, is how planar mechanisms are understood and designed.

A Note on Tools

The force polygons, transmission-angle and actuator-force curves here were drawn from the statics and reproduced with a few lines of Python (NumPy). The simulators confirm the forces and report the stress verdicts. No specialised software is required; statics on a mechanism is the whole method.

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