Create this chain and sprocket using Brushes. Yes, brushes. This is a lot easier than it looks. The chain links and the sprocket teeth are created and made into a pattern brush. Create the shape, apply the brush and you have a sprocket and chain.
Start a link of the chain with a circle.
Duplicate the circle. Leave less than one circles space between the two.
Add a rectangle between the two circles. Make the height of the rectangle 2/3 of the height of the circles.
Select all three shapes and use the Pathfinder Unite mode to create on shape.
Use the Direct Selection Tool to select the inner 4 corners and pull the corner widgets out from the center.
This will create a smooth inner curves.
Remove the stroke and add a solid fill.
Select the link and choose Object > Path > Offset Path and set the offset to 10 pt to make a bigger shape.
Select the new shape and add a heavy stroke and no fill.
Add circles to the ends of the links to create a chain link.
Duplicate the outer shape.
Use the Direct Selection Tool to select and remove the ends of the shape, leaving just the lines.
Move the lines to the end of the link, so the lines end up with the curves of the link. Duplicate the link and move it to the left to create a link set.
Create a rectangle starting at the exact centers of each link.
Set the stroke to none and send the rectangle to the back of the links. This rectangle will provide the area that the brush will duplicate. Make sure the rectangle is behind the links.
Open the Brushes panel under Window > Brushes. Select the links and rectangle. Click on the Brushes dropdown menu and choose New Brush… Choose Pattern Brush. Name it Chain and click OK.
To create the sprocket brush. create two circles.
Use the Direct Selection Tool to select and remove 3./4 of each circle, leaving just the the curved lines shown.
Select both lines and choose Object > Path >Join to join the tips.
Select the shape, choose New Brush from the brush dropdown menu. Choose Pattern Brush, name it Sprocket and click OK.
To apply a brush, draw a circle and click on the Sprocket in the Brush Panel.
Create two circles for the chain to run over.
To edit the size of the brush (and not effect other places the brush is applied) click on the Brush dropdown menu and choose Options of Selected Object…
The options allow for control of the size of the brush on the select object. Slide the Scale slider to make the brush larger or smaller.
Duplicate the two Sprocket circles. Set the brush to Basic in the Brush panel to remove the Sprocket.
With both circles selected choose Object > Blend Options. Set the Spacing to Specified steps and the number to 125.
Choose Object >Blend > Make to create the Blend.
Choose Object >Blend > Expand.
Click in the Unite mode in the Pathfinder panel to create one shape.
With the shape, add the link brush to the outline.
Adjust the size of the chains on the path by using the Options of Selected Object… and scale the size of the chain.
Move the chain over the sprockets. Add inner circles inside the sprockets.
Select a sprocket and choose Object > Expand Appearance. Choose Object > Join to make all the segments on shape.
Select the sprocket shape and the inner shape. Fill them with gray. Choose Object > Compound Path > Make. This will make the gear one shape and knock out the middle section.
Repeat the last step with the smaller gear to create a knock out sprocket.
Create a circle on the big sprocket.
Select the inner circle and choose the Rotate tool. Move the Rotate tool to the center of the Sprocket, hold Option/Alt and click to open the Rotate dialog box.
Type 360 / how many circles in the sprocket. Click on the COPY button.
Use Command + D to repeat the rotate/copy of the circle around the sprocket.
Nice chain and sprocket. And some fun with brushes as well.