There has been a lot of fanfare with Dyson products each time they are released. They stand out because of their unusual design. However, from what I have experienced and researched, not is all as it seems.
Product support turns out to be very poor, and parts become unavailable after a short time - assuming they were available in the first place. It seems to be a case of wanting you to replace rather than repair.
The plastic used for the DC05 is a fire hazard as will be shown later. Furthermore, there was a recall of the DC05 in 2010 after the handle came apart in some units exposing live connections.
I find the instructions for the DC05 very poor, not really telling you anything about the vacuum cleaner itself.

Dyson would consider this particular DC05 to be well past its use-by date but I keep finding ways to keep it going.

The DC05 came in two versions, the base model, and the Motorhead model, which is the one I have. In this, the vacuum head contains a motorised brush which can be switched off for smooth floors, or on for carpets. A third switch setting runs the brush at reduced speed and switches off the suction. This is for the application of "Zorb" - a carpet dry cleaning powder unique to Dyson. This is poured over the carpet and brushed in using the Motorhead.

My Dyson DC05 appeared in 2010, when it was given to me by an HRSA member who found it in a council clean up campaign. Knowing how expensive they were, I was rather pleased to become its new owner. It had been well used with both HEPA filters completely blocked. By the look of it, the DC05 had been used to clean up after house renovations - it looked like gyprock powder in the filters. The fault was the motor. Initially, I replaced the brushes but joy was short lived - the commutator was in poor condition and I suspect damaged windings. The motors are available on ebay, and after learning there were two types, I ordered the Panasonic type which is what was in my machine. The two kinds are not interchangeable.
I also ordered new HEPA filters.
Performance was astounding! I had never seen a vacuum cleaner perform so well, and I could see the Dyson hype was justified. To me, no other vacuum cleaner would ever be acceptable again.

Motorhead Repair.
After a few years, problems developed with the Motorhead brush drive. By July 2017, the belt had lost teeth and the motor pulley was wearing down. The pulley is made of plastic which appeared rather soft and not designed to last in this application. Unfortunately, neither part was available from Dyson or Dyson parts suppliers.
In fact, it appears that these parts were never actually available! It is bizarre that something like the brush belt was not available. Instead, one is expected to purchase a complete new Motorhead.

New Belt.
The belt is a toothed type commonly used in laser printers and similar items. It would seem that it should be a generic part, and indeed, many ebay sellers exist for such things. By measuring the distance between the belt teeth, it was determined
that the pitch was 3mm. Width was 6mm. Counting the teeth showed there to be 174 of them. Visible amongst the remaining markings on the original belt was "174". There was also a "3" and a "6"  which confirmed the pitch and width.
I bought the new belt from ebay seller "boltonbearings" in the UK. Size is 174-3M-6

New Pulley.
The pulley was more difficult - all the teeth had worn to the point of not being able to count them. However, some idea could be ascertained by the diameter of the remnant pulley.
Initially it appeared to be a 14 tooth pulley, and one was ordered. These pulleys are available on ebay under the title of "Timing belt pulley". The numbering system tells what the size is; for example 3M14T - 3mm pitch, 14 teeth.
Bore size is also available in different sizes; I chose 6.35mm.

The original plastic pulley is moulded over a knurled brass boss which threads onto the motor shaft. The plastic needs to be removed so the boss can be used with the new pulley. I had to drill out the bore of the new pulley to 7mm to fit the boss, and the boss also needed a slight turning down. Also, the new pulley is much wider than the original and part of it needs to be cut off. I did this turning down using my Unimat miniature lathe. I also used the lathe for drilling out the bore.

It all looked very good and the new belt had the right pitch - it fitted the brush pulley perfectly. Alas, there was too much slack. The new pulley was too small.
So, another was ordered; this time I decided to get both 16 and 17 tooth versions. Unfortunately, they never arrived. This appears to be not uncommon with ebay suppliers from China. Luckily they are not expensive.
A different supplier was used to place another order, and these ones did arrive. It was found that the 17 tooth pulley was the right size, and it was bored and turned down as before. These pulleys are secured with two grub screws. As the part that had to be turned off is the collar where the grub screws are, I had to drill and tap the toothed section to take the grub screws. This was easily done. The grub screws were 4mm.
One this was done, it was all like new again, and a lot more durable with the aluminium pulley.

The Motorhead runs off 240V AC from wires embedded in the vacuum cleaner hose. Inside the unit is a 230V DC motor and a PCB which has a dual colour LED among other parts. The circuit was traced out. The motor has never been available as a spare part.

The basic circuit is of a W08 bridge rectifier feeding the motor. In series with the negative supply of the motor is a PTC thermistor. If the brush motor is jammed up because of an obstruction, it draws excessive current which causes the PTC thermistor to heat up and go high resistance. Thus, the motor is overload protected. The dual colour LED is normally green when power is applied. Under overload conditions it changes to red. It's quite an ingenious but simple circuit that does this. The LED is a two terminal device with the actual LED's inside connected inversely. Thus, the LED shows green for one polarity, and red for the other. Under normal conditions, the LED is fed via one of two diodes (depending on mains polarity) and the 33K resistor connected to the junction of the diodes. Polarity is such that the LED glows green. The junction of the PTC and 2.2uF capacitor is negative. The PTC is low resistance, and therefore the second 33K resistor is effectively connected across the LED. At only 2V across the LED, it has no effect.
Under conditions of overload, 240V DC appears across the PTC. Now, because the PTC is in the negative return of the motor circuit, the junction of the PTC and 2.2uF is now positive, relative to the negative terminal of the bridge rectifier. The LED polarity is now reversed and it glows red. The second 33K now limits the current.
RFI is reduced by the two chokes and 0.1uF capacitor. This is actually a 275V AC type. The 2.2uF provides some partial filtering for the motor's DC. If this capacitor was of higher value, the motor would actually be fed with 340V DC (peak of the 240V mains supply) which would be excessive.
For the purposes of testing, the motor itself runs happily on 12V DC at low speed.

New belt and aluminium pulley fitted.
 

Some of the live PCB tracks can be seen.

Opening the handle is surprisingly simple with only one Torx screw to be removed.

Insides covered in black sticky substance.

Cause of the burn up. The switch is poor quality.

It had been noted that for a long time the switch had not provided reliable connection; it being necessary to move the slider critically to get both suction and the Motorhead to operate.
There is also a PCB in the handle which controls the Motorhead.

Brush speed controller.

The switch has four positions; Off, suction only, suction with brush, and brush only. In the brush only position, the brush motor runs at reduced speed. As such it was assumed this PCB was used to obtain the reduced speed.
The circuit was traced out.
Three wires are embedded in the hose between the vacuum cleaner body and the hose handle. The hose is detachable at the vacuum cleaner body. The wires connect to the switch in the handle. One wire is the neutral, required for the Motorhead, another is the incoming live supply from the mains, and the third is the live supply back to the suction motor.
Two wires leave the handle to feed the Motorhead. They are embedded in the telescopic wand. The wand is detachable, and the Motorhead can be connected directly to the handle if required.

N is the common neutral connection, common to the mains neutral, suction motor neutral, and Motorhead neutral. The Motorhead is connected to N and B when Zorb is selected. A is the mains active (live).
The circuit is unusual and I do not fully understand it. It is clearly a half wave device, and tests confirm this. Normally, a simple half wave speed reduced comprises a single diode and nothing else. It works by chopping off every alternate half of the mains cycle, with the result being the load is fed with half power.
There is an inline fuse (1A M205) in the neutral, located in the handle just behind the switch. This protects the Motorhead and control circuit only.

Normal half wave control provides 170Vrms to the load. To the right is the waveform fed to the Motorhead when this controller is switched in.

Here, the resultant output is much less than half power. Judging from the brightness of an incandescent lamp connected instead of the Motorhead, it looked to be about 70V.  Obviously, the MOSFET is the active component. Given no heatsink, it was clearly operating as a switch, and not as a variable resistor. Looking at the waveform, it appears the MOSFET switches on when a certain voltage on the mains sine wave is reached, and then switches off when the voltage reduces back to that level.
At this point, it is unclear just how the circuit switches at the correct time, despite the simplicity of the circuit. It seems to be load independent because the same waveform was obtained with a 7W light bulb.
There is a BZXC79C15 zener diode across the MOSFET gate which prevents the gate seeing negative voltage, and also ensuring the gate cannot go positive by more than 15V. This is standard practice with many circuits. The gate is biassed on with the 220K resistor.

As I have never used the "Zorb" function, and never will, the low speed function for the Motorhead brush is not actually required. This means the controller PCB can be removed, and all that is required is a simple single pole switch in the handle for the full 240V AC to the Motorhead. One room in my house has particularly long carpet which jams up the Motorhead. For this reason, it's necessary to be able to turn off the brush motor.

New Switches.

Motor wires emerge here. The black wires are connected together. Purple is mains live to the handle switch, black is neutral, and grey is switched live to motor. Note unused switch mounting.

First thing to do was install the main switch on the motor housing. Generic switches are readily available on ebay and one was purchased. Connection is via 4.8mm quick connect lugs. The switch fitted straight into the moulded clips, and the actuator worked perfectly.
At this point, the vacuum cleaner was now operative as the base model.

Mains switch installed. Machine is now functional as the base model DC05.

Next was to deal with the handle switch for the brush motor.  I filed away the burnt parts of the handle that were protruding so as to give a flat surface again. Looking through my collection of switches, I found one that was actually long enough to bridge the elongated hole.
As the speed controller was no longer going to be used, the PCB was removed. Thus, the switch only needed to be a simple SPST type. It was secured to the handle housing with miniature nuts and bolts.
It was found the inline fuse had blown, so this also had to be replaced.

Switch mounted in handle over burnt out hole.

The original switch slider was replaced to fill the hole in the top of the handle. It relies on the plastic moulding to keep it in position; the switch is not needed.

Not very pretty, the new Motorhead switch is functional.

All worked perfectly, and the DC05 Motorhead is now functional again. In fact, I would say it will be more reliable without that original slide switch - a poor choice for a high power inductive load.

This DC-54 came my way, and what a disappointment! It had been 'got at' with a couple of the Torx screws missing underneath where the steering mechanism attaches. The machine didn't power up at all, and before getting involved, experience tells that it's best to have a quick look at the motor before getting involved too far.
What a daft design! Dyson would have you believe their cyclone bin is perfect and is the reason that their vacuum cleaners 'never lose suction'. The reality is that it isn't, and the DC-54 is a good demonstration of that. You see, not everything is caught in the bin. In the DC-05 and other models, there's a filter between the bin and the motor. In the DC-54, the air from the bin passes straight into the turbine and out through the back of the motor. Then there's a filter - after the dirt has been sucked through the motor. This filter is not a user replaceable or washable part. It is interesting to note a label visible when the bin is removed to say that the machine should not be used with plaster, ashes, or rubble. How limiting is that!?
In my example, there was human hair right through the turbine, jamming it up, and it had passed through the motor into the exhaust filter. The motor itself was inoperative with the brushes worn to the point where no contact was made. The brush material was dispersed throughout the filter. I noted the nut securing the turbine was missing and the washer was found in the filter. Perhaps the nut is stuck in a turbine blade?
Anyway, all that didn't give me much enthusiasm to repair it. Dirt being sucked through a motor, all over the bearings and commutator and past the windings. Not very clever.