Electronic – Relay coil in series with resistor

I want to control a relay RSB1A120BDS that works on 24V DC using my Arduino. But, my power source that I use for other parts is 41V AC connected to a bridge rectifier. So my idea was to put a resistor in series with the relay coil to get the right voltage. This is a bit my setup:

^{simulate this circuit – Schematic created using CircuitLab}

The transistor is a TIP 120 STM NPN-Darlington TO-220 60V 5A 65W. (I know now MOSFETs are better for switching, but I didn't know that when I bought the transistors) I added a 220Ω resistor, because I read somewhere that this is necessary to limit current. I measure 3.27 V over R2 and 1.37 V between the base and the emitter. So we have 3.27 / 220 * 1000 = 15 mA base current.

The datasheet says the average resistance of the relay coil is 1440 ohm and the rated operational voltage is between 19.2 V and 26.4 V. So then by using a resistor of 1 kΩ, the voltage over the relay coil should be 41 * 1440 / (1440 + 1000) = 24 and that should be alright.

The flyback diode is a 1N4001 50V 1A. It removed it and measured it with my multimeter, and it seems to be working. (low resistance one way, very high resistance the other way)

However, it didn't work. When I turn the transistor on, sometimes I measure 23.7 V over the relay coil, but most of the time I measure 0 V, and all 41 V are over R1. So maybe the relay coil is short circuited. I also hear a sound sometimes while I'm working with the multi meter, I think the realy is turning on and off. I thought maybe the issue is that I use pulsed DC instead of normal DC, and the peak voltage of 24 * sqrt(2) = 34 is too high. But this answer says that shouldn't be a problem.

A thing is that 26.4 / 19.2 = 1.375 < 1.4 = sqrt(2). So I can never pick a resistor R1 to keep both the peak voltage and the RMS within the rated operational voltage limits, but almost though.

The relay makes a sound when I touch it with scissors. See this video.

When I try the simpler circuit below (same diode and relay, different but same type resistor), it does work as expected. No noise when I touch it with the scissors.

^{simulate this circuit}

What is the problem here?