In this chapter, examples are presented to illustrate engineering applications of systems of linear differential equations.

Mathematical Modeling of Mechanical Vibrations

In many engineering applications, such as vibration of mechanical systems, the systems are usually complex and have to be modeled as multiple degrees-of-freedom systems, resulting in systems of linear ordinary differential equations. Modeling a complex engineering system as an appropriate, mathematically tractable problem and establishing the governing differential equations are often the first challenging step. In this section, a free vibration problem of a simple two degrees-of-freedom system is first considered to illustrated the basic procedure. The equations of motion of a more complex problem, i.e., the vibration of an automobile, which is modeled as a four degrees-of-freedom system, are then established.

Example 8.1 – Two Degrees-of-Freedom System

Two uniform rods AB and CD of mass density per unit length ρ are hinged at A and C. Rotational springs of stiffnesses κ1 and κ2 provide resistance to rotations of end A and end C, respectively, as shown. The lengths of AB and CD are L1 and L2, respectively. Rod AB carries a concentrated mass M at end B. The rods are connected by a spring of stiffness k. When the rods are hanging freely, they are vertical and there is no force in the spring.