Harmonic excitation of One-Degree-of-Freedom Systems

Forced vibrations of a one-degree-of-freedom system occur when work is being done on the system while the vibrations occur. For example forced excitation includes ground motion during an earthquake or the motion caused by an unbalanced reciprocating component.

A mechanical or structural system is said to undergo forced vibration whenever external energy is supplied to the system during the vibration. External energy can be supplied through either an applied force or an imposed displacement excitation. The applied force or displacement excitation may be harmonic, non-harmonic but periodic, non-periodic, or random in nature. The response of a system to a harmonic excitation is called harmonic response. The non-periodic excitation may have a long or short duration. The response of a dynamic system to suddenly applied non-periodic excitations is called transient response.

Equations of motion

Let’s examine the system which consists of a block with mass m, spring with stiffens k and the viscous damper with damping coefficient c. The force F(t) acts on the body and pulls it from its equilibrium position.

FIGURE 3.1 - A spring-mass-damper system

FIGURE 3.2 - Free Body Diagram a) External forces b) Effective forces

So now you can apply the Free Body Diagram Method or the Equivalent System Method. In this case we will apply the Free Body Diagram Method and by applying the Newton’s second law we will get the differential equation which describes the motion (vibrations) of the system.

$$m\ddot{x}+c\dot{x}+kx=F(t)$$

Since this equation is nonhomogeneous, its general solution x(t) is given by the sum of the homogeneous solution, and the particular solution. The homogeneous solution, which is the solution of the homogeneous equation

$$m\ddot{x}+c\dot{x}+kx=0$$

represents the free vibration of the system. As previously discussed the free vibrations dies out with time under each of three possible conditions of damping (under damping, critical damping and over damping) and under all possible initial conditions.

With time the general solution of the non-homogenous differential equation reduces to the particular solution which represents the steady-state vibration. The steady-state motion is present as long as the force or forcing function is present.