When you think of a metal spring, does this make you think of a slinky or perhaps a vehicle shock absorber? Truth is, the underlying mechanics of a vibrating spring are not only simple but permeate all of science from the celestial to the subatomic. Springs play a critical role in heat capacity, mechanical clocks and watches, musical instruments and basically anything that vibrates, oscillates or in any way jiggles and jostles. All of these things in a sense can have their properties approximated by the action of a simple metal spring.
To understand any spring, you first start off with the spring constant which is a measure of the stiffness of the spring. Springs made of thicker metal have larger spring constants than those which are made of thin metal. Likewise springs with small spring constants are made with thinner metals. The actual spring constant itself is simply the ratio of the force applied to stretching a spring divided by the distance the spring was stretched. If for example a spring is stretched one meter using a Newton of force, the spring constant would be one Newton per meter.
Hopefully you have heard of the law that for every force, there is an equal and opposite force. This is a correct scientific principle discovered by Isaac Newton. When it comes to a spring, the force applied to stretch or compress a spring produces an equal and opposite force in the spring known as the restoring force. The spring is continually trying to move back to its equilibrium position where it is neither under tension nor compression. In a watch or clock, this property is utilized to cause the swinging back and forth of a mechanical timer. The back and forth of the spring is calibrated to give some fraction of a second allowing the clock to sequentially tick tock forward in time, not unlike the rotation of the earth or its orbit around the sun. The angular frequency for any spring is given by the square root of the ratio of the spring constant and the mass at the end of the spring. Using this, the spring effectively becomes a timer once it is placed in motion.
Atoms in a solid act like little weights on springs with each spring attaching each atom to all its bonded neighbors. Each atomic and molecular bond is effectively a spring with an associated spring constant. As the material heats up, the atoms simply vibrate harder and faster with increasing heat content. When the heat content gets high enough, the energy can be sufficient to break the bonds. When the bonds start getting broken, the material is going through a phase change such as from a solid to a liquid or a liquid to a gas.
On a larger scale, when the string of a guitar is pulled to the side, all the atoms in that string are pulled just slightly apart causing a restoring force to pull the string back towards the equilibrium position. As the string moves through its equilibrium position and back and forth, it vibrates the air around it according to the characteristic frequencies of the string. The characteristic frequencies of a string represent the simple natural rate at which a spring vibrates back and forth. A similar effect occurs with electrons in electrical circuits having any amount of inductance and capacitance in parallel.
These natural frequencies can be very important in structural materials to insure that the vibrations are strongly dampened. Shock absorbers in your vehicle suspension are a good example of dampened springs where you don’t want the springs to continually oscillate back and forth over and over. The characteristic frequencies of a tall building or a large bridge also require damping to prevent them from long term large oscillation which could lead to catastrophic failure of the structure.
Any object then has at least some spring like qualities. When the object is at rest and only slightly twisted, stretched or compressed it will spring back into place. If the spring is not damped at all, it will then oscillate (like jello) until the energy is completely converted into random heat. Like regular springs, too much stretching can permanently damage the spring by splitting, ripping, bending or breaking the atomic bonds in the material.
Being able to predict mechanical properties of any object or system then can very often require a good understanding of the characteristic frequencies of the system in order to predict is operational behavior.