
brummen
to hum
To produce a deep, monotonous, vibrating sound, typical of machines, large insects, or bears.
Der alte Kühlschrank brummt ziemlich laut.
The old refrigerator is humming pretty loudly.
to hum / grumble / boom
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to hum
To produce a deep, monotonous, vibrating sound, typical of machines, large insects, or bears.
Der alte Kühlschrank brummt ziemlich laut.
The old refrigerator is humming pretty loudly.

to grumble
To speak indistinctly to oneself in a deep, often dissatisfied voice, or to mutter.
Er brummte nur eine unverständliche Antwort und widmete sich wieder seiner Zeitung.
He only grumbled an incomprehensible answer and turned back to his newspaper.

to boom
Colloquial for a very good economic situation; when a business or the economy is doing very well.
Dank des Weihnachtsgeschäfts brummt der Laden.
Thanks to the Christmas business, the store is booming.

one's head is pounding
To have a dull, throbbing headache.
Nach der langen Nacht im Club brummte ihm der Schädel.
After the long night at the club, his head was pounding.

to serve time
Colloquial for serving a prison sentence.
Wegen des Überfalls musste er drei Jahre brummen.
Because of the robbery, he had to serve three years.
'Brummen' (to hum) is an onomatopoeic word. The sound of the word imitates the deep, vibrating noise it describes.
Hörst du auch, wie der Transformator brummt?
Do you hear how the transformer is humming too?
You can almost always recognize the meaning of 'brummen' from the subject of the sentence. Machines, engines, and large insects 'brummen' (hum/buzz) as a sound. People 'brummen' (grumble). Businesses and the economy 'brummen' (are booming/doing well).
Die Klimaanlage brummt leise im Hintergrund.
The air conditioning hums quietly in the background.
When a person 'brummt' (grumbles), this can introduce a dissatisfied statement. Often a subordinate clause with 'dass' (that) follows, or what is said is put in quotation marks.
Er brummte, dass er keine Lust habe, aufzuräumen.
He grumbled that he didn't feel like tidying up.
Saying that a business or the economy 'brummt' (is booming) is colloquial. It means that everything is very successful and a lot of revenue is being generated.
Seit der Eröffnung der neuen U-Bahn-Station brummt unser Café.
Since the opening of the new subway station, our café has been booming.
The expression 'der Kopf brummt' (one's head is pounding) is a fixed phrase for a dull headache. The person who has the pain is often in the dative: 'Mir brummt der Kopf.' (My head is pounding.)
Ich habe zu wenig geschlafen, jetzt brummt mir der Kopf.
I didn't sleep enough; now my head is pounding.
The meaning 'eine Gefängnisstrafe verbüßen' (to serve a prison sentence) is colloquial and belongs to jargon. You can add the length of the sentence.
Er musste für den Diebstahl zwei Jahre brummen.
He had to serve two years for the theft.
to sit
Sitzen and brummen can both colloquially mean that someone is serving a prison sentence. In this context, they have the same core meaning; sitzen must not be understood as referring to an ordinary body position.

Wegen des Überfalls muss der Täter drei Jahre im Gefängnis brummen.
The perpetrator has to serve three years for the robbery.
Wegen des Überfalls muss der Täter drei Jahre im Gefängnis sitzen.
The perpetrator has to serve three years in prison for the robbery.