The essential condition of locating two masses in order to system becomes dynamically equivalent is:

(where I₁ & I₂ = Distance of two masses from the center of gravity of the body, K_G = Radius of gyration of the body)

Explanation:

Dynamically Equivalent System

• In mechanical systems, a dynamically equivalent system refers to a simplified system of masses that has the same dynamic behavior (i.e., same inertia properties) as the original complex body. This concept is especially useful in analyzing and designing mechanical systems because it allows engineers to replace a complicated body with a simpler, equivalent system without changing the dynamic response of the system.

Essential Condition:

• The essential condition of locating two masses such that the system becomes dynamically equivalent to the original body is given by the relationship between the distances of the two masses from the center of gravity of the body (I₁ and I₂) and the radius of gyration of the body (K_G). This condition ensures that the simplified system has the same moment of inertia as the original body.

The moment of inertia (I) of a body about an axis is a measure of the body's resistance to rotational motion about that axis. The radius of gyration (K_G) is a parameter that represents the distribution of the body's mass relative to the axis of rotation. It is defined such that:

I = M × K_G²

where M is the mass of the body.