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I am making a PWM motor speed controller.
I used the Astable mode thinking it would work; it did not.
Using an oscilloscope I noticed that I am only changing the frequency not the duty cycle by varying R2 
.
After more investigation I found this second schematics that used diodes and it works.
For a simple astable oscillator \$\mathrm{T_{hi}=\ln2\cdot (R1+R2)\ C}\$ and \$\mathrm{T_{low}=\ln2\cdot (R2)\ C}\$, and \$\mathrm{T = T_{hi}+T_{low}}\$.
why the first one did not work?
If only R2 varies, Thi varies but also the period, T does. And the amount of change may not be enough to rate it as a duty-cycle change. Or at least, it may not be that distinguishable.
what is the role of the diodes?
is there a formula for this?
For the modified circuit (the one with diodes), the diodes determine which portion of R1 will be taken into account during charging and discharging of the capacitor, C:
During charging, D2 will be forward biased and D1 will be reverse biased. So only the left-side portion of R1 (let's call it R1a) will be used: \$\mathrm{T_{hi}=\ln2\cdot (R2+R1a)\ C}\$.
During discharging, D1 will be forward biased and D2 will be reverse biased. So only the right-side portion of R1 (let's call it R1b) will be used: \$\mathrm{T_{low}=\ln2\cdot (R1b)\ C}\$.
The period is:
$$\mathrm{T = T_{hi} + T_{low}=\ln2\cdot (R2+R1a)\ C + \ln2\cdot (R1b)\ C \\ \Rightarrow T= \ln2 \cdot (R2+R1)\ C}\ $$
And the duty cycle is:
$$\mathrm{D = \frac{T_{hi}}{T} =1-\frac{R1b}{R_{total}}\ \ \ \ \ \ ;\ R_{total}=R1+R2 }$$
As can be seen from the last two equations above, the duty cycle varies with the position of R1's wiper because T remains constant.
what is this mode called?
I don't know if it has a specific name. But still, it's an astable oscillator.
Question
/ to continue, ...
Answer
Introduction
The OP is not happy of the standard NE555 astable, because it is not easy to independently adjust frequency and duty cycle.
There are two types of NE555 modules in the market. The expensive one uses different sets of resistor pair values for different range of frquencies. The cheap one uses minimal resistors and only a narrow range of frequency can be set.
Actually both of the above two version does not use the improved versions using two diodes.
And all version using NE555 are analog devices and therefore not precise compared to digital PWM controller/signal generator (I will talk about it later)
My main aim here is to explain why the OP's improved diode version is good. I read about a variety of improved diode versions. Below is one by Electronics Tutorials. This version has an explanation. So I am mentioning it here, to complement the already selected, very good answer, make answers of this question more complete.
References
