What is the difference between 振る and 揺る?

[揺]{ゆ}る appears in dictionaries, but I'd say it's an old Japanese. As a word "to shake", [揺]{ゆ}する is common in today's Japanese.

I kind of agree with DD_Owl's opinion "振る is 'higher frequency' shaking. 揺する is more like a swaying or rocking motion, 'lower frequency'."

振る has a nuance that you shake something relatively small with one hand (of course not always), while 揺する has a nuance that you sway something relatively big holding it with both hands.

Ex.

よく振ってからお飲みください。＝ Shake well before drinking.

犬がしっぽを振った。＝ The dog wagged its tale.

バットを振る ＝ swing a bat

[看護師]{かんごし}は子供たちを[腕]{うで}に[抱]{だ}いて揺する。＝ A nurse takes children in her arms and rocks them.

ぶらんこに[座]{すわ}る子供を揺すった。＝ I pushed the child in the swing.

ダンサーたちは，音楽のリズムに合わせて体を揺すっている。＝ Dancers sway to the rhythm of music.

I'll need a native speaker to back me up on this one, but from what I came up with during a brief search, 振る more commonly describes a motion that originates from oneself (and this could include a non-human animal). For example:

牛は尾を振った. The cow switched its tail.

私は首を横に振った. I shook my head.

彼女は銃を振った. She waved her gun.

On the other hand, 揺る and 揺する seem to have the nuance of one thing causing another thing to shake or sway. For example:

がたがた揺られながら乗っていく. To have a bumpy ride.

彼らの船は波に揺られていた. Their ship was at the mercy of the waves.

藤の花が風に揺られている.　The wisteria flowers are being swayed by the wind.

彼は息子の肩をつかんで揺すった. He shook his son by the shoulders.

Another important distinction is that, although both are transitive verbs, 揺る is used significantly more in its passive form,　揺られる。A brief search of sample sentences on Weblio containing 「揺られ」yielded over 400 hundred results, whereas a search for 「揺っ」and 「揺すっ」resulted in less than 30 combined.

Addendum

Sorry, but in the original post, I meant to discuss the frequency of motion you were asking about. And again, I would need a native speaker to back me up on this, but it doesn't seem like either verb is restricted to a particular frequency of motion, per se. Which is to say, someone could be waving quickly or slowly and it wouldn't alter the usage of 「振る」. Similarly, someone could shake or be shaken more or less violently, but it wouldn't alter the usage of 「揺る/揺する・揺られる」.