PSI - Issue 83

Mohamed Rjilatte et al. / Procedia Structural Integrity 83 (2026) 208–217

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3.2 Numerical Analysis of Nonlinear Free Vibrations In the following study, a beam clamped at both ends is considered, which implies that the spring stiffness coefficients at these points tend to infinity. The analyzed beam is suspended by elastic cables positioned at x=1/3 and x=2/3 of its total length. The beam is assumed to be homogeneous and of rectangular cross-section. The objective of this section is to investigate the influence of the maximum vibration amplitude on the beam’s behavior, illustrated below through the nonlinear mode shape and the corresponding curvature.

/ Wmax r is calculated as a function

In the present analysis of free vibrations, the dimensionless maximum amplitude (

/ Nl l   ) and is presented in the form of curves in Fig. 2.

of the ratio between the nonlinear and linear frequencies (

Fig. 2: Curves of the ratio between linear and nonlinear frequencies of the cable-stayed beam, near the first mode . Fig. 2 clearly shows that an increase in the dimensionless maximum amplitude results in an increase in the ratio of nonlinear to linear frequencies, which is a typical indication of hardening-type nonlinear behavior.

Fig. 3: First normalized nonlinear mode of a cable-stayed beam clamped at its ends, for different values of the maximum vibration amplitude Wmax.

Figure 3 shows the first nonlinear mode of a beam suspended by cables. In this figure, it should be noted that for different vibration amplitudes, the effect of geometric nonlinearity can be observed.