# Deterministic and Random Response Evaluation of a Straight Beam with Nonlinear Boundary Conditions

## Abstract

### Purpose

Deterministic and random responses of one-dimensional continuous structures with nonlinear boundary conditions are evaluated in this article. Dynamic behaviors (i.e., the time-domain response for the deterministic case and mean-square response for the stochastic case) of a straight beam with a cubic nonlinear elastic boundary is investigated numerically.

### Methods

The nonlinear vibration problem is discretized by a new global spatial discretization method, and its responses are compared with those from the finite element method and assumed mode method.

### Results

The effects of selected various types of linear homogeneous boundary conditions and the number of expansion modes on the accuracy of the responses under harmonic excitation are discussed. Detailed discussions on the influence of nonlinear boundary conditions on dynamic behaviors uncover a counter-intuitive phenomenon: with the increase of the nonlinear spring stiffness, the stationary mean-square displacement of the mid-point descends first and then ascends slightly.

### Conclusion

Numerical results show that the present procedure possesses high precision for evaluating harmonic responses for both low-frequency and high-frequency excitations. Furthermore, it can be directly generalized to two-dimensional continuous systems and various types of nonlinear boundary conditions.

## Keywords

Straight beam Nonlinear boundary condition Deterministic and random responses New global spatial discretization method Assumed mode method## Notes

### Funding

This study was supported by the National Natural Science Foundation of China under Grant Nos. 11532011, 11872328, 11621062, and 11772100, and the Fundamental Research Funds for the Central Universities under Grant No. 2018FZA4025.

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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