Relation between Mathematical Equations and Block Diagrams

Consider the second-order differential equation of a spring-mass-damper system. The system is transformed into the Laplace domain under zero initial conditions.

The equation is then rearranged to isolate the output, which can be interpreted as signals entering blocks with specific transfer functions.

The output is obtained by integrating twice or by post-multiplying accordingly.

To simplify, the signals on the right-hand side are connected, leading to the final block diagram representation of the system.

Further simplification can be achieved by factoring the term from the internal feedback loop, resulting in an alternative block diagram.

The block diagram model can also incorporate internal variables representing acceleration and velocity.

As 1/s corresponds to integration in the Laplace domain, the acceleration is initially integrated to obtain the velocity, and subsequently, the velocity is integrated to yield the displacement signal.

The system's transfer function is found by moving the block at the input and feedback signals to the right-hand side of the comparator and simplifying the internal feedback loop. The resulting equation is the transfer function of the system.