Relation between Mathematical Equations and Block Diagrams

In a spring-mass-damper system, the second-order differential equation describes the dynamic behavior of the system. When transformed into the Laplace domain under zero initial conditions, this equation can be effectively analyzed and manipulated. The transformation into the Laplace domain converts differential equations into algebraic equations, simplifying the process of isolating the output.

Applying the Laplace transform to the standard differential equation of the spring-mass-damper system gives the output as follows:

In constructing the block diagram, the signals on the right-hand side can be connected to simplify the representation. The block diagram can be further refined to incorporate internal variables like acceleration and velocity. Since 1/s corresponds to integration in the Laplace domain, the acceleration signal is integrated to obtain velocity, and the velocity is integrated to yield the displacement signal.

The block diagram simplification involves factoring terms from the internal feedback loop. This process leads to an alternative block diagram visually representing the relationships among acceleration, velocity, and displacement.

To derive the transfer function of the system, the block representing the input and feedback signals is moved to the right-hand side of the comparator, and the internal feedback loop is simplified. The resulting equation provides the transfer function:

This transfer function is essential for analyzing the system's behavior, predicting its response to various inputs, and designing control strategies to achieve desired performance characteristics.