Although the small-scale effect and nonlinear damping on the nonlinear vibration of microbeam electrostatic resonators are important, they have been overlooked by researchers. We use the slender beam model including the small-scale effect and nonlinear damping to investigate the nonlinear vibrations of the electrostatic resonators in the present paper. We apply the Galerkin method on a nonlinear partial differential equation to obtain the nonlinear ordinary differential equations for the first and third modes. The two equations include constant terms. The multiple-scale method is used to obtain the approximate analytical solutions of the two equations. The approximate analytical solutions discover the effects of driving electric field, small-scale effect, and nonlinear damping on structural vibrations. The results suggest that the small-scale effect, the direct current (DC) voltage, and the alternating current (AC) voltage have some critical effects on the vibrations of microresonators.
Steel–concrete continuous composite beams are widely used in buildings and bridges and have many economic benefits. Slip has always existed in composite beams and will reduce the stiffness of composite beams. The effect of creep under a long-term load will also be harmful. Many scholars ignore the combined effects of slip and creep. In order to more accurately study the mechanical properties of steel–concrete continuous composite beams under long-term loads, this paper will consider the combined actions of slip and creep. By combining the elastic theory and the age-adjusted effective modulus method, the differential equation of the composite beam is derived via the energy variational method. The analytical solutions of axial force, deflection and slip under a uniform load are obtained by substituting the relevant boundary conditions. The creep equation is used to simulate the behavior of concrete with time in ANSYS. The analytical solution is verified by establishing a finite element model of continuous composite beams considering slip and creep. The results suggest the following: the analytical solution is consistent with the finite element simulation results, which verifies the correctness of the analytical solution. Considering the slip and creep effects will increase the deflection of the composite beam and the bending moment of the steel beam, reduce the bending moment of the concrete slab and have a significant impact on the structural performance of the continuous composite beam. The research results considering the coupling effect of slip and creep on continuous composite beams can provide a theoretical basis for related problems.
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