5.1 Nonlinear SDOF Model

Figure 5.1.1 Nonlinear Stiffness

So far, we have discussed linear systems where the spring force is linearly proportioned with the displacement and the damping force is linearly proportioned with the velocity. In practice, material will stay linear until a certain limit (elastic limit) is reached depending on its mechanical properties. If the elastic limit is exceeded, the material will start to have a nonlinear relationship between the applied force and the response. the stiffness of the spring, for example, will not remain constant as illustrated in the graph below. In this section, we will introduce the methods used to analyze nonlinear systems.

Material nonlinearity

So far, we have discussed linear systems where Material nonlinearity occurs when the mechanical properties of a material, such as stiffness or damping, change as a function of the applied load or deformation.

Geometric nonlinearity

Geometric nonlinearity arises when large displacements or rotations significantly affect the system's response, leading to changes in stiffness or damping. External forces, such as impact or vibrations, can induce nonlinear behavior in the system.

Why it is Important to Consider Nonlinearities?

It is crucial to consider and account for nonlinearities in many cases involving structural dynamics. Ignoring nonlinear effects can lead to inaccurate predictions and flawed design decisions. Nonlinearities can introduce phenomena such as resonance, bifurcations, and chaotic behavior, which are not present in linear systems. These nonlinear phenomena can significantly affect the system's response, stability, and energy dissipation characteristics. Neglecting nonlinearities can result in unexpected failures, reduced performance, or compromised safety margins, particularly in systems subjected to large deformations, high loads, or dynamic loads. Furthermore, nonlinearities can impact the accuracy of simulation models and analytical solutions. Linear models might provide reasonable approximations for small-amplitude vibrations or low-stress conditions. However, as the amplitudes increase or loads become more significant, the linear assumptions break down, and nonlinear effects become more dominant. Neglecting these effects can lead to substantial errors and inaccuracies in predictions, making it essential to incorporate nonlinearities into the analysis and design of mechanical systems. Overall, considering and accounting for nonlinearities in structural dynamics is vital for accurately understanding and predicting the behavior of mechanical systems. It allows engineers to make informed decisions, enhance system performance, optimize designs, and ensure the reliability and safety of structures subjected to complex loading conditions.