Temperature Effects on Nonlinear Vibration Behaviors of Euler‐Bernoulli Beams with Different Boundary Conditions

Abstract

This paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams with symmetric and nonsymmetric boundary conditions. It is assumed that in the considered model the temperature increases/decreases instantly, and the temperature variation is uniformly distributed along the length and the cross-section. By using the extended Hamilton’s principle, the mathematical model which takes into account thermal and mechanical loadings, represented by partial differential equations (PDEs), is established. The PDEs of the planar motion are discretized to a set of second-order ordinary differential equations by using the Galerkin method. As to three different boundary conditions, eigenvalue analyses are performed to obtain the close-form eigenvalue solutions. First four natural frequencies with thermal effects are investigated. By using the Lindstedt-Poincaré method and multiple scales method, the approximate solutions of the nonlinear free and forced vibrations (primary, super, and subharmonic resonances) are obtained. The influences of temperature variations on response amplitudes, the localisation of the resonance zones, and the stability of the steady-state solutions are investigated, through examining frequency response curves and excitation response curves. Numerical results show that response amplitudes, the number and the stability of nontrivial solutions, and the hardening-spring characteristics are all closely related to temperature changes. As to temperature effects on vibration behaviors of structures, different boundary conditions should be paid more attention.

Thermal Effects on the Nonlinear Forced Responses of Hinged-Clamped Beam with Multimodal interaction

“…It is globally observed a shift in frequency along the frequency axis, to the left for positive temperature difference and to the right for negative one in comparison to the temperature-free curve (∆𝑇 = 0 °𝐶). These results agree with those obtained by authors of papers [10]- [13]. The shift increases with the absolute value of temperature difference.…”

“…In order to eliminate the secular terms from 𝑢 𝑛1 , the 𝐴 𝑛 must be chosen so that the coefficient of exp(𝑖𝜔 𝑛 𝑇 0 ) is zero. This coefficient will contain 𝐹 𝑛 when Ω is near 𝜔 𝑛 as well as the nonlinear terms associated with any combinations of the form 𝜔 𝑛 ≈ ±𝜔 𝑚 ± 𝜔 𝑃 ± 𝜔 𝑞 (20) The eigenfunction of the eigenvalue problem (13) is…”

“…Moreover, it is widely known that structures are permanently exposed to the thermal field from accidental or environmental origins that induce temperature variation of their members [10]- [24]. Yaobing et al [13] have paid attention to the temperature effects on the nonlinear vibration behavior of Euler-Bernoulli beams with different boundary conditions. Effects of uniform temperature rise on the structural responses have been investigated through the authors limited their study to singlemode analysis for the sake of simplicity.…”

“…Introducing the quasistatic assumptions, the acceleration and velocity terms in the 𝑢 direction are neglected [13]. This leads to:…”

Vibration control for the nonlinear resonant response of a piezoelectric elastic beam via time-delayed feedback

“…As the efficient structure component, the elastic non-rigid support beam structure is very common in the engineering field, such as the main beam of cable-stayed bridge [1][2][3], beam of large stadiums or factories, underground structures in the pile [4]. The importance of a high coefficient of large and complex structures, and therefore its reliability and stability and higher accuracy requirements.…”