Green’s Dictionary of Slang

strum v.

1. to have sexual intercourse; thus strumming n., sexual intercourse.

[UK]Grose Classical Dict. of the Vulgar Tongue.
[UK]Lex. Balatronicum.
[UK]Byron Beppo ii: Prudery flings aside her fetter [...] Giggling with all the gallants who beset her; And there are songs and quavers, roaring, humming, Guitars, and every sort of strumming.
[UK]Egan Grose’s Classical Dict. of the Vulgar Tongue.
[UK] ‘There’s Somebody Coming’ in Ri-tum Ti-tum Songster 37: O lor, deary me, how you are stuffing and strumming.
[UK] ‘The Chaffing Family’ in Nobby Songster 13: There’s Dick he chaffs Poll, about strumming all night.
[UK]‘Walter’ My Secret Life (1966) II 247: While the strumming was going on in the parlour, people bought cigars, and tobacco [...] little did they guess the fun going on behind that red curtain.

2. to masturbate.

[UK]‘The Amiable Family’ in Fal-Lal Songster in Spedding & Watt (eds) Bawdy Songbooks (2011) III 8: My daughter Selina Jane Anna, / Practizes upon the pianner, / She is such a rum ’un, / She is always a strumming, / Such a inweterat manner.
[UK]Cythera’s Hymnal There’s my daughter Paulina Jane Anna, / Plays very well on the piano, / But sahe’s such a rum ’un, / She’s always a strumming / In such an inveterate manner.

3. (US, also strum heads, strum someone’s head) to hit; to fight; thus head-strumming n., hitting someone.

[US]B. Jackson Thief’s Primer 72: When they was strumming my head, the sheriff there said, ‘Cut it out, you can’t do that here.’.
[US]C. Shafer ‘Catheads [...] and Cho-Cho Sticks’ in Abernethy Bounty of Texas (1990) 215: strum heads, v. – to fight.
[US]Jackson & Christian Death Row xvii: Head-strumming is hitting someone.

In derivatives

strummer (n.)

1. the penis.

[UK]‘Fleet Street Blowing’ in Out-and-Outer in Spedding & Watt (eds) Bawdy Songbooks (2011) IV 130: None now I need withold / However large the strummer / [...] / If fellows can but pay.

2. (Aus.) a lit. or fig. masturbator.

[Aus]‘Thommo’ Dict. Aus. Swearing & Sex Sayings 126: STRUMMER — A weak minded half wit, who tells inconsistant bullshit stories to impress other people. He’s dead from the neck up and needs an operation to separate his hand from his cock.

In phrases

strum oneself off (v.)

to masturbate.

[UK]K. Sampson Powder 467: Sorry, I just had to go and strum myself off.