Inertia and Elastic Factors


The natural frequency depends on 'inertia factors' & 'elastic factors'. What are these factors? Is it that inertia factors are related to an object without any force acting on it but has the tendency to continue moving? And elastic factors are found in springs? In your last reply, you say 'a natural frequency depends on the charateristics of the vibration system. It increases with the stiffness of the elements and decreases with their mass'. Is that stiffness the elastic factor while the mass is the inertia factor? Are there any other example of these two types of factors?

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Vibrations, or oscillations as they are sometimes called result when a system has some sort of restoring force which tends to move the system to a certain state. One example is a spring with a weight hanging from it. If we find the weight at rest at a certain position, with the spring partially stretched, and pull the weight down a bit, the spring will tend to bring the weight back to where is was. If instead of pulling the weight down, we lift it up a bit, the spring force will no longer be sufficient to hold the weight in the new position and gravity will tend to move the weight back to its original position. The system has what I called a restoring force.

In a system as described above, the magnitude of the restoring force is a property of the spring and how much the weight was displaced. For a given spring, the magnitude of the restoring force is proportional to the displacement of the weight from the equilibrium position.

When we stretched the spring a bit by pulling the weight down, and released the weight, the spring got the weight moving back towards the equilibrium position and continued to exert extra force on it until that position was reached by the weight. At that point the inertia of the moving weight caused an overshoot of the equilibrium position, compressing the spring until the upward inertia was overcome. Of course at that point the weight is now too high, so the spring can not hold it there and the movement continues, with the weight bobbing up and down until energy losses (damping) kill the motion.

It is the mass of the weight, the "inertia factor" in this situation, that tends to slow things down. A heavy weight will bob up and down more slowly that a light one. The stiffness of the spring, the "elastic factor" in this situation, will tend to make things happen more quickly. With a stiff spring the weight will bob up and down faster than with a soft spring.

People talk about "elastic factors and "inertial factors", rather than mass and stiffness because these same ideas can be applied to systems other than masses and springs. Another example is in an electronic circuit where the role of mass is played by a property called inductance and the elastic factor is the circuit property called capacitance. It turns out that once you work out the math for the mass and spring thing you can apply the same equations to other situations where a system has a tendency to restore itself to some equilibrium position. It works not only in mechanics and electronics but also in chemistry, biology, ecology and many other applications.