Lesson 11

Machines: assemblies of members that do work

🟦  11.1  Translational and Rotational Work

Let's define a machine as an assembly of members that do work.

You were introduced to work in Physics. In engineering mechanics, we'll consider two subtypes: translational work (force that moves an object over a distance) and rotational work (a moment that rotates an object over an angle).

Translational work:

In this image, a small yellow bar lies on a table, and is pushed with a force P. The bar translates (or displaces or moves) distance u. We take the dot product of the vectors P and u to compute the work done on the bar relative to the direction defined by u.

Rotational work:

An identical orange bar lies on a table, and is rotated with a moment M. The bar rotates an angle θ, which we express in radians. We can express both M and θ as vectors (using double-arrow notation). We take the dot product of these two vectors to compute the work done on the bar relative to the axis defined by θ.

If you'd like to review the dot product, feel free to jump ahead to Lesson 13 before returning to this page. You won't need to apply it until then.

Translational and rotational work

Work and virtual work

In engineering mechanics, we apply the idea of work when we are investigating rigid body translation and rotation. There is a closely-related idea called virtual work that is also used in some engineering mechanics courses.

In a nutshell, when we talk about work, we're talking about kinetic energy (the energy of motion). When we talk about virtual work, we're talking about strain energy (the energy that structures store through changing shape, or deforming). If you'd like to learn more about strain, head over to the Mechanics of Materials part of this site. It's not taught in Statics.

Author's note: Virtual work is rarely taught in Statics courses. For that reason, it isn't part of the scope of this text, but in the future I do plan to add virtual work resources to this site as part of the Structural Analysis course, hopefully in Spring 2025.

🟦  11.2 Simple machines

A machine is an assembly of members that does work. The pump jack shown here is a useful example of a modern machine.

In Statics, we won't study the system in motion (that would be a Dynamics problem). For instance, we won't study how a force may vary as a function of time.

We can (and will) analyze a Statics "snapshot" of a system in motion at one instant in time. We investigate static equilibrium at that instant.

Most modern machines, despite their complexity, are comprised of different combinations of the six simple machines:

🟦  11.3 Pins, Pin Connections, and Two-Force Members

In order to successfully answer Statics problems about machines, we have to apply what we learned previously about connections and two-force members. Here is a quick review.

❏ Pins and pinned connections

A pin connection can transfer a force to members that connect to it. 

A pinned connection works like a pin (it transfers a force at an angle) but it is constrained from translation.

Pins and pinned connections permit rotation.

Pins in a pump jack

Two-force (or axial) members

A two-force member (2FM) is connected by pins at both ends. It doesn't matter whether the pins are pinned or not.

The inclination (θ) of the force transferred by the pin is known. It matches the inclination of the member itself.

For instance, in the image, at this instant in time, the horizontal member of the pump jack is being subjected to a vertical force from the top pin of the 2FM shown in the image. A vertical member can only exert a vertical force.

Alternatively, we could also consider another instant in time in which the 2FM is oriented at 60 degrees. In that analysis, the force would also be oriented at 60 degrees.

Always look for 2FM at the beginning of your problem-solving process. The x-component and y-component of the unknown force are coupled, so even when we express them in component form, there is only one unknown. 

❏ A 2FM in a pump jack

Complicated pin assemblies

Dutch artist (and engineer) Theo Jansen builds machines that draw inspiration from biomechanics and are 100% powered by wind.

These machines are full of pin connections and 2FM. While they are far too complex for a Statics course, I have referenced them here for two reasons:

Straandbeest "Animaris Rex" by Theo Jansen

The mechanics of a Strandbeest

Attribution: MichaelFrey, CC BY-SA 4.0, via Wikimedia Commons

Strandbeest leg

Attribution: MichaelFrey, CC BY-SA 4.0, via Wikimedia Commons

Types of panels

Placeholder: still image by SMR. This will probably be done after June 2024.

Attribution: I adapted the work of MichaelFrey in accordance with the license CC BY-SA 4.0.

🟦  11.4 The art of the FBD

For a given machine, like the pump jack, there are many different FBDs that you could construct as part of your problem-solving process.

The key is to be strategic about which FBD you draw.

Below, let's assume the self-weight of the main lever (W1, W2, and W3) and tension in the vertical cable are known. Someone has asked you to solve for the reactions at B or the force in member AC. If you recognize that AC is a 2FM, you can deduct that the x- and y- components of F4 are coupled (they are related through ratio). Therefore the main lever FBD below is solvable because there are three unknowns (F4, Bx, and By) and three equations of equilibrium (summation of forces x-direction, summation of forces y-direction, and summation of moments about any z-axis).

The key takeaway is that your first goal is to cut a FBD that is solvable:

🟦  11.5 Pliers: two levers connected with a pin

Let's work through a simple example.

in the animation, you will see that pliers are a machine that does work. We input a small force (with a hand) and output a larger compression force. We call this idea mechanical advantage.

The animation shows the pliers opening and closing (in motion). We wish to analyze a "snapshot" of the pliers when they are compressing an object. Work through the flipbook and draw out all of the FBDs. 

Animated pliers

Flipbook: pliers in static equilibrium

🟦  11.6 Compound pulleys

A compound pulley is a pulley system that contains multiple pulley.

A single pulley changes the direction of a force. A compound pulley creates a mechanical advantage. Inspect the pre-programmed views in the interactive model.

Then, work through the diagrams below, answering the following question: how much of a pull is required on the cable to lift the heavy box (weight of W)? We will neglect friction.

Note that pulley sheaves (1) and (3) are pinned to a plane of fixity. They remain stationary. Pulley sheave (2) moves upward as the box is lifted. We are not studying the dynamic behavior of the system; we study a Statics "snapshot" of one instant in time. You can also think about this type of problem as studying the force that would initiate the motion from the given position.

Author's note: I'm planning on making an animation to show how the pulley system moves. I'll post it when it's ready, but that is not likely to happen in June 2024.

Note again that you must cut FBDs strategically in order to have success solving these types of problems. Cutting the proper FBD is an art, and practice is required to learn how to do this successfully.

🟦  11.7 Hydraulic cylinders

Hydraulic cylinders are used in some machines. They may be modeled as 2FM, since they are pin-connected at both ends.

They change in length by pumping fluid from one side of the cylinder to the other, as shown in the short video linked here.



Below are two views of a machine with a hydraulic cylinder. The machine's purpose is to hoist (lift) the box.

The green bar and yellow hydraulic cylinder are both 2FM. 

Since the hydraulic cylinder can change length, we could consider many different "snapshots" of the machine in different positions. Each snapshot would have its own unique force flow and geometry.

Hoisting machine, lowered position
Hoisting machine, hoist position

🟦  11.8 Demonstration problem: the skid steer loader

A skid steer loader is depicted below. It's a machine used to lift or hoist material. It features two pairs of hydraulic cylinders that are changed in length by the operator. The animation shows a two-dimensional projection (or elevation) of a skid steer loader. Please note that while the design in the photo isn't identical to the design in the animation, they are quite similar.

Attribution: Norbert Schnitzler, CC BY-SA 3.0, via Wikimedia Commons
Attribution: Pasimi, CC BY-SA 4.0, via Wikimedia Commons

At one instant in time, we take a Statics "snapshot" of the skid steer loader. The first thing we need to do is bound the analysis problem. What do we need to include in the loading idagram? What can we ignore? The set of images below depict an approach for solving such a problem. Note again that you must be very strategic when cutting a FBD. Try to disassemble (unpin) as few members as possible. Every time you remove disassemble a pin, you introduce at least one unknown (one unknown for pins at the end of a 2FM and two unknowns for all other pins).

One person might set up the loading diagram like this. The fixed plane at the bottom can transfer vertical force, horizontal force, as well as a reaction moment. That's three unknowns.
Another person might set up the loading diagram like this. Nodes A, T, and R are assumed to be pinned. 
The trick is to create a FBD with no more than three unknowns. Don't disconnect anything you don't have to disconnect. Every time you disconnect a member, you have to add in one or more unknowns. This FBD is the best one for solving this particular problem.
First, use qualitative equilibrium to deduce the directions of the three unknown forces. Then, use the dimensional information given to you in the problem statement (as well as the given weights W1 and W2) to complete the quantitative solution with the E.o.E.

You may have noticed the use of the Principle of Transmissibility in the FBDs above. Even though the pushes and pulls do have a point of application, if you draw vectors that accurately, the FBD may become illegible. It's good to develop a habit of drawing the line of action with dashes, and then draw the vector on top with a heavy line.

🟦  11.9 Closing thoughts and advice

There's not really any new theory in this lesson. Our main tools in Statics are still:

Machines are often challenging for Statics students because the FBDs can be complex, making these types of problems look quite intimidating. Also, sometimes these types of problems are given without engineering connections (like the first problem, below), and students must make choices in the best way to model the problem. The geometry is much more complex than in trusses or cables. If and when you feel stuck, go back to the four bullets above. That is all that is required to solve a given problem.

Finally, if you leave this unit excited and inspired, a Mechanical Engineering major may be a good choice for you. If you find yourself wishing that you could limit your studies to structures that aren't in motion (like trusses and cables), then Civil Engineering (and its sister field Structural Engineering) may be a good option for you.

➜ Practice Problems

Do you like to listen to music while solving practice problems? An appropriate choice for this problem set is When Will the Writer by Danielle Ate the Sandwich.

Scrambled answers to problems one through five (without units) are: 0.67, 1.12, 4.00, 71.6, and 300.

Problem 1.

Hungry for some pie, you decide to crush this pecan with a nutcracker.

If the pecan crushes at 10 pounds of force, how much force P do you have to supply at the handle?

In this problem, and throughout this set of problems, neglect self-weight of the machine unless you are specifically directed to account for it.

Problem 2.

You are in the process of storing your vintage Volkswagen car in your garage when you suddenly see the neighbor's cat inside. You apply an upwards force, Pc, to hold the door stationary while calling the cat. The cat does not move. Here is an animation of the garage door in motion.

The garage door (1) weighs W = 600 N.

The door is supported by a horizontal roller (2) in a track (4) and and vertical roller (3) in a track (5). Draw a FBD of the door and solve for all unknown forces.

Attribution: I adapted the work of MichaelFrey in accordance with the license CC BY-SA 4.0. The car was originally created by Marseille77 under the same license.

Problem 3.

You use some tweezers to pluck an errant eyebrow hair.

You apply pressure with your fingers at A and B to apply compression to the hair at C.

If you are pushing with 0.01 pounds of force, how much compression does the hair experience? Use the FBD provided. 

The force transfer at D is a bit more complex than this model may imply, but the resultant force shown at D is sufficient for the purpose of answering this question correctly.

Problem 4.

You have a binder clip. Bored, you apply equal and opposite forces at C to open up the binder clip as shown. You estimate the magnitude of the forces exerted by your fingers and decide it's about 1N. You are applying both normal force and also a shear force (friction between your fingers and the metal).

Calculate the normal force at B. Do not take the thickness of the material into account.

Problem 5.

You're in charge of restoring an old hand pump from the early 1800s. This device pumps well water up to the service.

A force is applied, Pc, to generate a tension force in the piston rod (2) that in turns lifts the piston (5). The water is lifted up the cylinder (4) so that it can flow through the outlet (6).

If the applied force, Pc, is equal to 10 pounds, how much tension is generated in the piston rod?

This problem is a derivative work of Hand_pump-en.svg: by Manco Capac and a derivative work by MichaelFrey, in accordance with the CC BY-SA 3.0 license, via Wikimedia Commons

Problem 6.

Now you're doing some glassblowing. You pick up an oblong piece of hot glass with pliers. You push with force P = 10N.

Solve for the normal force at A.

Problem 7.

This system of pulleys is called a "block and tackle." The six pulley wheels provide significant mechanical advantage to the person hoisting the weight.

Solve for the hoist force in terms of the weight W.

This is the last problem of this problem set for Summer 2024.

This problem is a derivative work of This work is in the public domain. It was originally authored in 1912 by Henry Carhart and Horatio Chute and published in"First Principles of Physics."

Problem 8.

Not for Summer 2024.

Problem 9.

Not for Summer 2024.

Attribution: MichaelFrey, CC BY-SA 4.0, via Wikimedia Commons

Problem 10.

Not for summer 2024.

Holding pen

Not for Summer 2024.