©2003 Chris Zwar and CreativeCOW.net. All rights reserved.

Begin by opening a new Project, creating a new composition and importing the necessary files. There are a few image files we’ll be using in the project, including the “Worm Gear 1 second loop” from tutorial 3, and a range of cogs which I created using the techniques in tutorial 2. You will need to complete Tutorial 3 and render the Worm gear movie. Also download the project files from above.



I used the PAL defaults of 768 x 576, at 25 fps.

Drag the Worm Gear loop into the timeline and use the “loop” function to extend the duration to 1 minute – detailed in Part 3.



In this case, the X position of the Worm Gear is 50.

The first thing we’ll add is the big cog we made in tutorial 2. It’s been rendered out as a still image and named “Cog 3 – 36 teeth”. Drag it into the composition and position it so the teeth mesh nicely with the worm gear. I used the position values of 219, 127.



Because the worm gear moves upwards, I made the teeth of the cog rest on top of the thread. If the worm gear moved in the opposite direction, I would have made the teeth of the cog rest against the bottom side of a thread.



The idea is to make the Cog rotate as the Worm Gear turns. We could manually keyframe the rotation, but this is a good opportunity to demonstrate a simple expression. The rotation of the cog relates to the rotation of the worm gear. We can use an expression to calculate this for us, and this way we’ll never have to change the settings if we extend the duration of the composition.

Select the Cog layer, and press R to reveal the rotation settings.



Click on “rotation” to select the property, and go to “Add Expression”, in the Animation menu.



After Effects now turns the numbers red, and has added a line of text to the timeline. This line of text is our expression. At this stage, it’s saying that rotation = rotation. It might be zen, but it’s also a bit redundant.



What we’re going to do is set up a rule which determines how the cog relates to the worm gear. We should probably start thinking about what this rule is going to be.

The worm gear is an image of a worm gear turning – the movie itself doesn’t rotate or change position. So we will use the time of the movie as the basis for our expression.

We designed the worm gear so that it revolves once per second. Each time it revolves, the teeth move up until after one second, they look like they’ve moved up one whole tooth. All of this was done in Part 3. The point is that 1 second = 1 tooth.

Our cog has 36 teeth – again, we designed it this way in Part 2. So for our cog to do a complete revolution, the worm gear has to turn 36 times – 1 revolution of the worm gear for every tooth on the cog. We can use this information to create a relationship between the time of the worm gear movie, and the rotation of the cog.

We know that 36 revolutions of the worm gear equal 1 revolution of the cog.
We know that 36 revolutions of the worm gear will take 36 seconds.
We know that 1 revolution of the cog is 360 degrees.

Therefore, our cog has to turn 10 degrees every second – 360 degrees divided by 36 seconds.

That’s what our expression will do- it will tell After Effects that for every second of the Worm Gear animation, the cog has to turn 10 degrees.

Notice these three icons, because they’ll become your new best friends:



In simple terms, an expression is a formula. But “formulas” sound complicated and also imply lots of typing, and typing can be tedious.

These icons are our saviours – they are designed to let us build up expressions with the minimal amount of typing and hassle.

We’ll get to the middle icon later, but for now we’ll use the arrow.

The arrow is the shortcut to a menu of common commands. Instead of having to learn complex programming terminology, all we have to do is click on the one we want from a list. It’s really simple.

In this case, we’re going to use the “time” command – so click on the arrow to reveal all the options, and from the “Global” menu select “time”



After Effects has now inserted the word “time” where it used to say “rotation”.



You might be thinking that this is an anti-climax, because you could have typed the word “time” faster than it took to read that last section – but trust me - not having to type in commands is great. Also, if you don’t know the correct terminology for a command, you can usually guess by looking at the options. But let’s get back to it.

The “time” command works in seconds, and as we need our cog to turn 10 degrees every second, all we have to do is add a “*10”.

Click in the text box and type “*10”.



Click outside the box and you’re done. That’s an expression. The rotation of the cog equals the time (in seconds) multiplied by 10.



Try scrubbing in the timeline and you’ll notice that the cog now rotates in perfect synchronization with the worm gear. We don’t have to think about it any more.

Expressions really are simple, so lets set up another one.

Drag the other cog – the one with 12 teeth – into the composition and place it at 383,64. At frame 0, the teeth should mesh nicely with the larger cog.





It’s obvious that the large cog will turn the smaller cog, so once again we can easily set up an expression to do this for us.

Select the Cog layer with 12 teeth, and press “R” to reveal the rotation properties. Click on the word “rotation” to select it, and “add expression” – just as we did earlier.



This time, the rotation of our cog will be directly related to the rotation of the first cog, so adding the expression will be even easier.

Remember our new best friends:



The centre icon is the “pick whip” – and allows us to click on anything we want to include in our expression. It saves us lots and lots of typing.

All we have to do is click on the pick whip icon, and drag it to the “rotation” of our first cog. If you can’t see the rotation of the 36-tooth cog, select the layer and press “r” so the rotation expression which we just typed is visible.

You can drag the pick whip to either the word “rotation” or to the numbers, even though they’re red and say 0.

You’ll notice that when you drag the pick whip over the word “rotation”, After Effects draws a box around it, showing that you’re selecting it. When you release the pick whip, After Effects adds the text to our new expression for us:



It might look complicated, but remember – we didn’t have to type a letter. The text which After Effects has inserted is telling it where to find the rotation of the 36-tooth cog. From left to right, the text “this_comp” means that we’re looking as this composition (and not a pre-composition, for example). The “layer(“Cog 3 – 36 teeth.pct”) is telling After Effects which layer we’re dealing with, and the “rotation” at the end is telling After Effects that we’re copying the rotation value. It might look like a jumble, but there’s a simple underlying structure.

Basically, this line of text is now an expression that makes the rotation of our 12 tooth cog equal that of the layer “Cog3 – 36 teeth.pct”.

Because our first cog has 36 teeth, and this one only has 12, our 12 tooth cog should rotate 3 times as fast as the big cog (36 / 12 is 3) but in the opposite direction. To adjust our expression to do this automatically, all we have to do is click in the text box and add *-3.



And there we have it – our second expression.

You can scrub in the timeline and you’ll see how the worm gear is turning the big cog, and the big cog is turning the smaller cog.

All without keyframes.

Let’s do another one just for fun, to emphasise how simple it is.

Drag the “Cog 7” into the composition – it has 16 teeth. Position it at 236,316. At frame 0, it will mesh with the 36 tooth cog nicely.



Just as we did with our second cog, we’ll add a simple expression so the large cog with 36 teeth will turn it.

Because we want to link the two rotation properties, we need to be able to see both of them in the timeline. If you can’t see the “rotation” of the 36 tooth cog (as in the screenshot above), select the 36 tooth cog layer and press “r” to reveal the rotation, then do the same for our new cog, with 16 teeth. This means the properties we want to link are visible.

You should know the deal by now – click on the word “rotation” for our 16 tooth cog, select “add expression” from the menu, then use the pick whip to drag a link to the 36 tooth cog. Notice that you can drag the pick whip to either the word “rotation” or to the red numbers – they both work.



The big cog has 36 teeth, and the smaller one has 16, so the ratio of turns will be 36:16.

Do you know what 36 / 16 is? I wouldn’t know without a calculator, but After Effects will figure it out. So click in the expression text and add “*-(36/16). The minus sign makes the cog turn in the opposite direction.



Wow. Now we’ve added three expressions. Have a quick scrub in the timeline to see how everything is coming along. If we don’t slow down a bit we might turn into programmers.

The pick whip lets us make simple expressions very easily. If you want, you can import the other cogs which I’ve already made and have a ball creating your own drive trains, and use expressions to link them all together (I did, and I’ve saved it as a separate project file.)

We’ve added three really simple expressions that aren’t complicated, were easy to do, and work really well.

But you might be thinking that they weren’t really necessary. You could have added rotation keyframes just as easily, even though they’re not as flexible.

So what we’ll do now is use expressions to create a piece of animation which would be very, very difficult to animate manually and even if you did – would be really difficult to adjust afterwards.

We’re going to add a piston to our composition, and have it linked to our 16 tooth cog in the centre. The expression will automatically calculate the position and rotation of the piston rod for us.


Here’s a sneak preview of what we’re about to do…

To keep things simple, we’ll begin by using a white solid as the rod – we can change it later.

Add a new solid to the composition and call it “Drive Rod 1”. Make it 200 pixels wide and 6 pixels high. I made it white for visibility.



In the timeline, set the anchor point to 0,3. We will be animating the rod from the left side, not the centre which is the default.



The first thing we’ll do is use an expression to set the position of the rod. We want the left side of the solid to be connected to the solid part of the 16 tooth cog.

We begin as we did before – press “p” to reveal the position properties, and click on the word “position”. Then add an expression.



Our Drive Rod needs two values for its position – an X value, or horizontal position, and a Y value, or vertical position.

These positions will have a direct relationship to the number of degrees that the cog it’s attached to has turned - the relation of the position of the rod to the cog is determined by using trigonometry. This might sound scary, but it’s nothing more than we learned in high school and in actual fact, I just did a Google search for “trigonometry” and found the information which I needed.


The rotation of the cog will determine the x& y position of the rod. I learned this on the internet, showing that it’s useful for more things than rude pictures.

Remember our best friends?



We’re going to use the menu (the arrow on the right) to add the commands, so we don’t have to type much. You’ll be amazed at how easy it is to create an expression that looks complicated without even typing anything.

The “X” position of the rod is related to the Cosine of the cog’s rotation. I learned this on the Internet so it must be true.

Click on the arrow to show all our commands, and under the JavaScript Math menu, choose the first one – Math.cos(val).



Notice how After Effects adds the command to our timeline.



The word “val” is short for value, and this is where we put in the bit that relates to us. Highlight the word “val”, because we’re going to replace it.



Now that it’s highlighted, we can use the pick whip to select what we want to replace it with – we don’t have to type anything. In this case, it’s the rotation of the 16 tooth cog.

If you can’t see the rotation value of the 16 tooth cog, you’ll need to select the Cog layer and press “r”. This will give you an error message because we haven’t finished our expression yet. But you can ignore it, because we’re coming right back to it.


Drag the pick whip down to the rotation of the 16 tooth cog, and when you release it After Effects will insert the correct text into your expression for you.



There’s one little catch. The JavaScript based commands work in Radians, while After Effects works in Degrees. In order for the Cosine command to work properly, we need to convert the degrees to radians first. But once again, we don’t have to worry about the format of the command or even type anything – just use the menu.

If there isn’t, you need an opening bracket after the “s” in Cos – then position the cursor after the bracket, and select Degrees to Radians.



After Effects has inserted the correct command for us.



However, where it says “degrees” in brackets is where we want our rotation value, so move the text up as follows, and close the brackets at the end:

This expression is now giving us the Cosine of the Cog’s rotation.

Whoever invented Cosines, and it was probably the Ancient Greeks, worked them out with circles that had a radius of 1. However, our circle (the cog) has a larger radius than 1, so we need to multiply our Cosine by the size of the cog. There’s a hole in the cog specifically designed to connect to the Drive Rod, and it’s 30 pixels out from the centre. So click after the last brackets and add “*30”.

At this stage our Drive Rod is in the wrong position. The Drive rod is rotating around a centre of 0,0, so we need to add the position of the Cog to the Position of the Drive Rod.

Once again, we don’t need to type stuff, we just use the pick whip.

You will need to click on the 16 tooth Cog layer and press “p”, to reveal its position.

Because a position value needs two “dimensions” – an x and a y – but we’ve only written one so far, you will receive an error message from After Effects when you click outside the layer.



Just ignore the error message, we’ll fix it up in a minute.

Once you can see the position values for the 16 tooth cog, click back in our expression, type a “+” after the 30, and drag the pick whip to the X position of the Cog layer.



After Effects does the dirty work for us, and we’re halfway there.

What we’ve written so far is for the X dimension, and we only need a small modification for the Y dimension. Enclose the entire line in another set of brackets, then select the entire line of text, and press Apple-C to copy it.



Press return to start a new line, and press Apple-V to paste.



All we need to do is tweak the second line to make it apply to the Y value.

The first thing to do is to change the word “cos” into “sin”. Then change the [0] to [1], because the X value is dimension 0, and the Y value is dimension 1. If you’re really lazy, you don’t even have to change the [0] to a [1] yourself – just highlight the text to be changed and use the pick whip to select the Y position of the cog.



I took the lazy option – I’m about to use the pick whip.

Our expression is basically complete, we just need to format it so After Effects knows where to look.

A Position value has two “dimensions” – X & Y. In an expression, these values must be enclosed in square brackets, separated by a comma – like this [x,y].

We should enclose our formulas in square brackets, and separate the two lines by a comma:



Notice the square brackets at the start and end, and the comma at the end of line 1.

And that’s it.

It might look complicated, but we certainly didn’t have to type it out by hand, or learn any complex programming.



And now for the trick – no matter where you move the 16 tooth cog, the drive rod will move with it, and will always rotate with it. The expression we just typed doesn’t have to be altered no matter how fast the circle rotates, or where we position it.

Try doing that with keyframes.

But we’re only half way there. The next step is to angle the rod, or in other words, have the rotation of the drive rod relate to the rotation of the cog.

Once again we’ll use high-school trigonometry, which I actually learned on the internet and not at high school (I did learn it at high school, but I forgot it).


In this diagram, sin(x) = b/c. The Ancient Greeks were good at this sort of stuff.

Select the Drive Rod layer and press “r” to reveal the rotation properties. Select the word “rotation” and “add expression”. You should be getting good at this by now.



It’s possible to condense our formula into single line, but that would look very confusing. What we’ll do is break down the expression line by line, so you can see how simple the underlying structure is.

To make everything clear and easy to follow, we’ll use some variables. The standard in trigonometry is to label the sides of a triangle a,b & c. We’ll build our formula using this convention, so everyone can understand what we’re doing.

The value of “c” is the hypotenuse of the triangle, in this case it’s the width of our Drive Rod. This is easy to do – just type in “c=width”. I didn’t even type in the word “width”, I selected it from the menu. To separate our commands we have to use semi-colons, so add a semi-colon and press return.

The “b” side of a triangle is its height. In our case, the height is the difference between the y-position of the centre of the cog, and the y-position of the drive rod.

In the text box, type in “b=position[1]. The 1 in square brackets means we only want the y value – if we had used a [0] we would have the x value.

We can use the pick whip to select the position of the cog, so select the 16 tooth cog layer (ignoring any error message) and press “p”.


In this shot, I can see the position values of the 16 tooth cog, and am about the drag the pick whip down to the number “316”.

Click back in our text box and type a minus (-), then drag the pick whip down to the Y value of the 16 tooth cog – in this case it’s the number 316.

After Effects does our work for us:

Just like in high school, we’ve now set up the letter “c” to equal the hypotenuse, and the letter “b” to equal the height of our triangle.

Continuing the high school tradition, we’ll use the letter “x” as our unknown value, although we do know from trigonometry that sin(x) = b/c. Remember that After Effects wants degrees, but the expression command will give us an answer in radians.

Our expression should look like this:

But don’t think you have to type it all, remember the menus…



The last line in the expression gives us the angle we need – at the top corner of the triangle – which is opposite the angle which is calculated by the Sin command, which is at the bottom corner of the triangle. This is demonstrated in the earlier diagram.

Your composition should look like this.



We need to make another slight adjustment to rotate the drive rod in the direction we want it. This is because in trigonometry, a horizontal line is 0 degrees, while in After Effects, a horizontal line is 90 degrees.

Adding a simple line “x=x-90” fixes this difference.

I also added the letter x in square brackets at the end of the expression, this isn’t needed but is tidy. It shows that the final outcome of the expression is the value of x, and by placing it in square brackets After Effects knows that this is the value to use.

Our completed expression

Your composition should look like this:



Now scrub in the timeline. Notice how the drive rod moves with the Cog.

Once again, the absolute beauty of this expression is that the Drive Rod will ALWAYS be in the perfect position, regardless of the position of the cog or the speed it rotates. You can move the cog anywhere you want, you can even remove the cog’s rotation expression and insert your own keyframes to make it spin as fast or as slow as you want– the Drive Rod will still work perfectly.

It’s wonderful (pause to wipe away tears of pride).

If you’ve followed the tutorial this far then you’ll be all fired up for the next bit. We’ll add another drive rod to the first one, but this will only move horizontally.

Our second drive rod will attach to the end of the first one – and the position can be deduced by the famous Pythagoras theorem.



The second drive rod has an X position equal to the “a” of the triangle.

This isn’t as daunting as trigonometry, so we can breeze through this part:

Create a new solid and call it “Drive Rod 2” – make it 100 pixels wide and 6 high.



Change the anchor point to 0,3 – again we’re animating it from the left side – and add an expression for the position,



This is the expression we need to use, hopefully you can see the bits which you don’t need to type yourself:





Let’s go through it step by step.

Pythagoras says that c squared = a squared plus b squared. I think he was Greek too.

We know that “c” is the width of Drive Rod 1, and that “b” is the Height of Drive Rod 1 (the difference between its Y position and the Cogs Y position).

These two values are exactly the same as in our last expression.

So in this Expression, we can set up a simple equation to determine the value of “a”- which is the distance from the centre of the 16 tooth cog to Drive Rod 2.

The first line assigns “c” the width of Drive Rod 1.
The second line calculates the Height of the Drive Rod 1, equal to “b”.
Then we square “c”, and square “b”, and find the square root of the difference.
Then we just add the value of “a”, which is the distance between the Cog and the Drive Rod – to the X position of the Cog.

The final line is the actual position – notice how it is in square brackets. The format of position[x,y] is being followed, with x equal to a, and y = to the y position of the cog.

Now we have this:

Once again, try scrubbing in the timeline to see the results:



It’s worth noting again how powerful these expressions are.

I’ve already mentioned that you can move the Cog anywhere you want and the Drive Rods will move with it.

I’ve also mentioned that no matter how fast the Cog rotates – you can even randomly rotate the cog with the wiggler if you want – the Drive Rods will always be in the correct position.

But even more impressive – you can change the length of the Drive Rods and they’ll still be in the correct spot. Drive Rod 1 currently has a width of 200. If you change it to 100, or 300, it doesn’t matter – After Effects will re-calculate the expression so everything looks perfect.



In the upper pictures, the Drive Rod 1 has been changed to a length of 100 – but the expression still works. In the lower picture, the position of the Cog has been moved and the two Drive Rods move with it…all without keyframes.

This is our completed expression:

I decided to add a little round metal join to cover the space where the two Drive Rods join. All I needed to do was add a metal texture, and apply a circular mask.



Drag a texture into the timeline, and apply a mask:



The position of our round metal button is exactly the same as the position of Drive Rod 2 – so we just “add expression” the the position of the texture, and use the pick whip to select the position of “Drive Rod 2”



The round bit (so small you can miss it…)



Just make sure that the metal texture is above the Drive Rods so we can see it.


We’ve just added an elbow joint in less than a minute. Real mechanics can’t do that.

The aim of this tutorial has been to introduce you to expressions, and demonstrate how they can be used to automatically generate animations which would be difficult to do with keyframes.

I’ll leave you with one more – just to give you something to think about.

This time we’ll be using the ratchet which we made in Part 3. For the time being, I’ve added rotation keyframes to turn the ratchet, but later on I’ll connect it up to some more cogs and use expressions.



The latch is just a metal texture with a mask applied – you can look at the project file to see but it’s very straightforward.

The aim was to animate the rotation of the latch so it appeared to be pushed out by the ratchet teeth as it rotated, but would swing back when the teeth were clear.

This is the expression – applied to the latch, not the actual ratchet:



Without going into detail, this is what it does:

We only need to know how many degrees the ratchet has turned between each tooth.

So the variable “a” is the total number of degrees the ratchet has turned.
By dividing this number by 15 (360 degrees divided by 24 teeth), rounding the answer down, then multiplying the integer-answer by 15 again, we can work out the number of degrees we are between each tooth.

This way, no matter how many degrees the ratchet has turned, we’ll always be working with a number from 0 to 15. This value is returned by the letter “b”.

As the ratchet turns, the variable “b” will cycle from 0 to 15, then back to 0 to 15 again as each tooth passes.

At this stage, we can simply make this value the rotation of the latch, but the latch isn’t pushed out far enough to clear the teeth. So I multiplied “b” by 1.5.

Now the expression pushes the latch out as the ratchet turns, but we need the latch to fall back to a resting position once the tooth has passed.

The two “if” statements basically work out whether the latch is still touching a tooth… if not, then the value of “b” is decreased so the latch falls back to a vertical position.

The final line (b=-b) just reverses the direction of the latch, because it’s on the right side of the ratchet.

So that’s it.



The Worm Gear’s connected to the 36 tooth cog, which is connected to the 12 tooth cog and the 16 tooth cog… the 16 tooth cog is connected to two moving Drive Rods, and the Ratchet pushes out the latch as it turns…. ALL WITHOUT KEYFRAMES!

Now that we’ve looked at the basic principles of expressions, in both simple and more complicated forms, we can go completely nuts and add as many cogs as we can.



Hope you have fun.

Please discuss this technique in the After Effects forum at Creativecow.net