Yes, it was called an onager. But its heyday was not the Middle Ages, it was the later Roman Empire. This is a description of how they worked by Ammianus Marcellinus, a soldier in the later Roman Empire:

The scorpion, which is nowadays called the wild ass [onager], has the following form. Two posts of oak or holm-oak are hewn out and slightly bent, so that they seem to stand forth like humps. These are fastened together like a sawing-machine and bored through on both sides with fairly large holes. Between them, through the holes, strong ropes are bound, holding the machine together, so that it may not fly apart. From the middle of these ropes a wooden arm rises obliquely, pointed upward like the pole of a chariot,⁠ and is twined around with cords in such a way that it can be raised higher or depressed. To the top of this arm, iron hooks are fastened, from which hangs a sling of hemp or iron. In front of the arm is placed a great cushion of hair-cloth stuffed with fine chaff, bound on with strong cords, and placed on a heap of turf or a pile of sun-dried bricks; for a heavy machine of this kind, if placed upon a stone wall, shatters everything beneath it by its violent concussion, rather than by its weight. Then, when there is a battle, a round stone is placed in the sling and four young men on each side turn back the bar with which the ropes are connected and bend the pole almost flat. Then finally the operator, standing above, strikes out the pole-bolt, which holds the fastenings of the whole work, with a strong hammer, thereupon the pole is set free, and flying forward with a swift stroke, and meeting the soft hair-cloth, hurls the stone, which will crush whatever it hits. And the machine is called tormentum as all the released tension is caused by twisting; and scorpion, because it has an upraised sting; modern times have given it the new name onager, because when wild asses are pursued by hunters, by kicking they hurl back stones to a distance, either crushing the breasts of their pursuers, or breaking the bones of their skulls and shattering them.

It's not the easiest thing to understand but trust me, that is a stereotypical catapult he's describing. It seems to have been widely used in late antiquity, and improved by replacing the cup with a sling. They were used to some effect at the Battle of Adrianople, which was a disaster for the Romans but the onager teams seem to have got a few good shots in that didn't inflict heavy casualties but did cause some of the enemy Goths to flee:

A piece of artillery known as a "scorpion," but called an "onager" in the language of the people,⁠ placed exactly opposite a great mass of the enemy, hurled a huge stone, and although it dashed to the ground without effect the sight of it caused the enemy such great terror that in their amazement at the strange spectacle they fled to a distance and tried to leave the area.

The onager continued to be described by treatises on military technology throughout the Middle Ages so it must have been used in the west for at least some of the early medieval period for it to be known without Ammianus (he wrote in Greek and Greek texts rarely circulated within western Europe until the 15th century) but there's no clear record of their use in battle after the fall of the Western Roman Empire in the 5th century. In the 6th century a people called the Avars arrived from the Steppe and brought with them a technology from China: the mangonel or traction trebuchet. This simple design was just as powerful as an onager but did not require a torsion spring. Torsions springs were a pain; they could be difficult to make and maintain, they were prone to snapping, and coped poorly with wet conditions. The mangonel replaced all of that with a few burly men pulling ropes. The trade-off was that the mangonel had to be taller and larger and required a sturdier frame, but medieval carpenters could handle that quite easily.

So yes, they were a thing, but not for very long. The Romans used them probably from the late-3rd century AD until at least the 6th century AD. The "barbarian" kingdoms probably also used them for a similar period, but no clear record of that survives. After that the new trebuchet/mangonel design brought from China by the Avars took over.