21.3:
Mechanical Systems
In mechanical systems, springs, and masses are akin to the roles of inductors and capacitors in electrical networks, with the energy-dissipating function of a viscous damper corresponding to that of electrical resistance.
The forces acting on the mass include an applied force moving in the same direction and the forces from the spring, viscous damper, and acceleration acting against it.
Like the RLC network, translational mechanical systems are defined by a unique differential equation formulated by applying Newton's law which mandates that the sum of all forces acting on the mass must equal zero.
Further, the Laplace transform is used on the equation under zero initial conditions. This expression, when simplified, yields the transfer function.
The components in rotational mechanical systems mirror those of translational systems but experience rotation.
These are managed similarly to translational ones, with torque substituting force, angular displacement replacing translational displacement, and inertia replacing the mass.
In the second-order differential equation for the rotational system, the Laplace transform is applied and then further simplified to yield the transfer function.
21.3:
Mechanical Systems
Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass's acceleration. This interplay of forces is mathematically described using Newton's second law, which states that the sum of all forces acting on the mass must be zero.
In translational mechanical systems, the behavior is captured by a unique differential equation derived from Newton's law. This equation accounts for all the forces acting on the mass. To solve the system analytically, the Laplace transform is applied to this differential equation under zero initial conditions. The Laplace transform, a powerful mathematical tool, converts the time-domain differential equation into an algebraic equation in the Laplace domain. Simplifying this equation yields the system's transfer function, a crucial concept that relates the output response to the input force in the frequency domain. The transfer function is essential for analyzing system stability and dynamics.
Rotational mechanical systems parallel translational systems but involve rotational motion. In these systems, torque replaces force, angular displacement substitutes for translational displacement, and rotational inertia takes the place of mass. The analogous differential equation for a rotational system, derived similarly using Newton's second law for rotation, describes the dynamics of rotational motion. By applying the Laplace transform to this second-order differential equation, and simplifying, the transfer function for the rotational system is obtained. This function provides insights into the rotational system's behavior, similar to how the transfer function in translational systems aids in understanding linear motion dynamics.