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.

## Tutorial

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

## Subject Review

### 1. System Description 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, Fd(t) is the force by a damping element, and Fs(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 x0 and v0 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.

### 2. Numerical Integration 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 zk = z(tk), tk = 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.

### References

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.