The tool VB-09 plots the time response of a one-degree-of-freedom vibrating system with several types of nonlinear damping and spring elements. Forced responses are computed by numerical integration for five types of external loads: impulse, step, ramp, exponential and sinusoidal forces.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

**Example: A one-degree-of-freedom system with a nonlinear spring**

The input in the tool is

After clicking "Run", the results will be shown in another page. The time response of the displacement and velocity of the system are

The maximum displacement and velocity in absolute value are also shown as follows

Maximum Displacement = 1.00850664 at time t = 4.12912913

Maximum Velocity = 1.96596902 at time t = 3.67867868

**Figure 1. ** A nonlinear 1-DOF vibrating system

Shown in Fig. 1 is the free-body diagram of a one-degree-of-freedom vibrating system,
where x(t) is the displacement of mass m, F(t) is an external force, F_{d}(t) is the force by a damping element,
and F_{s}(t) is the force by a spring element. The motion of such a system is described by the differential equation

subject to the initial conditions

where x_{0} and v_{0} are the initial displacement and velocity of mass m, respectively. The damping force and spring force in general
are nonlinear functions of the displacement and velocity of the mass. Table 1 lists several commonly used models of damping and spring forces.

**Table 1. ** A nonlinear 1-DOF vibrating system

When both the damping and spring forces are linear, Eq. (1) is reduced to the commonly used governing equation for a spring-mass-damper system

Refer to the subject review of tool VB-01 for analysis and time response solution of 1-DOF linear systems.

With the state variables, Eq. (1) is converted into an equivalent state-space form

where

The nonlinear state equation (5) is solved by a fixed-step Runge-Kutta method of order four as given below:

where

with z_{k} = z(t_{k}), t_{k} = kh, k = 0,1,2,…, and h being the step size. For convergent results, the step size must be small enough. As a rule of thumb,

One may try several step sizes to compare the accuracy and convergence of numerical solutions.

1. Den Hartog, J. P., 1985, Mechanical Vibrations, Dover Publications, New York.

2. Timoshenko, S., 1990, Vibration Problems in Engineering, John Wiley & Sons; 5th edition.

3. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Elsevier Science, Boston.