scholarly journals Nonlinear vibrations of a beam with non-ideal boundary conditions and stochastic excitations - experiments, modeling and simulations

Author(s):  
T. Roncen ◽  
J-P. Lambelin ◽  
J-J. Sinou
Author(s):  
Amit K Rai ◽  
Shakti S Gupta

Here, we have studied the linear and nonlinear vibrations of a thin circular plate subjected to circularly, radially, and spirally moving transverse point loads. We follow Kirchoff’s theory and then incorporate von Kármán nonlinearity and employ Hamilton’s principle to obtain the governing equations and the associated boundary conditions. We solve the governing equations for the simply-supported and clamped boundary conditions using the mode summation method. Using the harmonic balance method for frequency response and Runge-Kutta method for time response, we solve the resulting coupled and cubic nonlinear ordinary differential equations. We show that the resonance instability due to a circularly moving load can be avoided by splitting it into multiple loads rotating at the same radius and angular speed. With the increasing magnitude of the rotating load, the frequency response of the transverse displacement shows jumps and modal interaction. The transverse response collected at the centre of the plate shows subharmonics of the axisymmetric frequencies only. The spectrum of the linear response due to spirally moving load contains axisymmetric frequencies, the angular speed of the load, their combination, and superharmonics of axisymmetric frequencies.


Author(s):  
Dumitru I. Caruntu ◽  
Ion Stroe

This papers deals with nonlinear vibrations of non-uniform beams with geometrical nonlinearities such as moderately large curvatures, and inertia nonlinearities such as longitudinal and rotary inertia forces. The nonlinear fourth-order partial-differential equation describing the above nonlinear effects is presented. Using the method of multiple scales, each effect is found by reducing the nonlinear partial-differential equation of motion to two simpler linear partial-differential equations, homogeneous and nonhomogeneous. These equations along with given boundary conditions are analytically solved obtaining so-called zero-and first-order approximations of the beam’s nonlinear frequencies. Since the effect of mid-plane stretching is ignored, any boundary conditions could be considered as long as the supports are not fixed a constant distance apart. Analytical expressions showing the influence of these three nonlinearities on beam’s frequencies are presented up to some constant coefficients. These coefficients depend on the geometry of the beam. This paper can be used to study these influences on frequencies of different classes of beams. However, numerical results are presented for uniform beams. These results show that as beam slenderness increases the effect of these nonlinearities decreases. Also, they show that the most important nonlinear effect is due to moderately large curvature for slender beams.


Author(s):  
Francesco Pellicano

In the present paper vibrations of circular cylindrical shells having different boundary conditions are analyzed. Sanders-Koiter theory is considered for shell modeling: both linear and nonlinear vibrations are analyzed. An energy approach based on Lagrange equations is considered; a mixed expansion of displacement fields, based on harmonic functions and Tchebyshev polynomials, is applied. Several boundary conditions are analyzed: simply supported, clamped-clamped, connection with rigid bodies. Comparisons with experiments and finite element analyses show that the technique is capable to model several and complex boundary conditions. Applications to geometrically nonlinear shells show that the technique is effective also in the case of nonlinear vibration: comparisons with the literature confirm the accuracy of the approach.


Author(s):  
Giovanni Ferrari ◽  
Giulio Franchini ◽  
Prabakaran Balasubramanian ◽  
Kostas Karazis ◽  
Marco Amabili

Abstract Nuclear fuel bundles in PWR reactors present vibrations due to coolant flow, which may result in fretting at the interface between the fuel rods and the retaining elements, named spacer grids (SGs). Seismic excitation may also occur during accidental events, such as earthquakes. In this perspective, forced vibration experiments were performed on a reduced-scale nuclear fuel bundle provided by Framatome Canada. The presence of partially loose fuel pellets inside the fuel rods was provided in the experiments. A maximum coolant flow of 5 meters per second was reached inside a water tunnel. The identification of vibration parameters was attempted in the linear regime, through modal analysis, and in the nonlinear regime, through a single-degree-of-freedom method based on harmonic balance. The value of the equivalent damping parameter was shown to increase strongly with the amplitude of the excitation, thus acting in the direction of safety. The fuel bundle presents a peculiar softening vibration behavior in the nonlinear regime, with a marked decrease of the peak vibration frequency. The comparison with other recent experiments shows that the boundary conditions constituted by SGs have a predominant effect on stiffness and damping during nonlinear vibrations. Therefore, the characterization of the boundary conditions at the SGs was attempted by means of dedicated experiments. Bending oscillations were tested in the frequency range between DC and 50 Hertz. Tests were repeated in presence and in absence of water. The resulting force-displacement loops clearly show the presence of hysteresis and of bilinear stiffness. The availability of a mathematical model for the stiffness and the damping at the boundary conditions will be indispensable for the future development of reduced-order models describing the vibrations of PWR fuel bundles.


Author(s):  
Sami A. Alkharabsheh ◽  
Mohammad I. Younis

We present an investigation into the dynamics of MEMS arches when actuated electrically including the effect of their flexible (non-ideal) supports. First, the eigenvalue problem of a nonlinear Euler-Bernoulli shallow arch with torsional and transversal springs at the boundaries is solved analytically. Several results are shown to demonstrate the possibility of tuning the theoretically obtained natural frequencies of an arch to match the experimentally measured. Then, simulation results are shown for the forced vibration response of an arch when excited by a DC electrostatic force superimposed to an AC harmonic load. Shooting technique is utilized to find periodic motions. The stability of the captured periodic motion is examined using the Floquet theory. The results show several jumps in the response during snap-through motion and pull-in. Theoretical and experimental investigations are conducted on a microfabricated curved beam actuated electrically. Results show softening behavior and superharmonic resonances. It is demonstrated that non-ideal boundary conditions can have significant effect on the qualitative dynamical behavior of the MEMS arch, including its natural frequencies, snap-through behavior, and dynamic pull-in.


2013 ◽  
Vol 361-363 ◽  
pp. 1115-1118
Author(s):  
Peng Liu ◽  
Jie Rui ◽  
Bo Lei ◽  
Fei Zheng

This paper establishes the shape function of high-pier with non-ideal boundary conditions in the top and uses the energy method to calculate its critical load. Then its effective length factor is achieved by using Euler's formula. Further, the FEM and energy method are respect used to calculate the effective length factor of the engineering example and comparative analysis is carried on. Results show that: The non-ideal boundary conditions have great influence on the effective length factor and should be considered in the calculation. The result from the formula of energy method is nearly the same as the one from the FEM which demonstrates this method is of good accuracy to calculate the effective length factor of high-pier. In addition, it is also of great convenience in the design of high-piers.


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