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Beams involving intermediate axially and transversely eccentric masses have many industrial applications such as plane wings with an intermediately positioned heavy gas turbine. In most of these applications, the beams undergo large deflections which give rise to geometric nonlinearities. So, the objective of the current research is to provide a no...
Contexts in source publication
Context 1
... mentioned in the introduction, variable cross section beams with eccentric intermediate mass find applications in many mechanical structures such as the wings of a plane with an intermediately positioned gas turbine, or a beam used for supporting an eccentric motor-pump system. The schematic view of a variable cross section beam with an intermediate eccentric mass í µí± ̂ is considered as shown in Figure (1). This í µí± ̂ may contain a small un-balance mass of í µí± ̂ rotating at a distance í µí±̂ from its center of rotation. ...
Context 2
... what follows, we will present a nonlinear dynamic model for this system. The nonlinear strain energy of the beam shown in Figure (1 (1) ...
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Citations
... The first approach considers the discretization of a beam to a sufficiently large number of rigid segments, so the beam is modelled as a system with a finite number of degrees of freedom. The method of rigid segments is explained in [26] with a detailed analysis of the relevant literature concerning the application of Euler-Bernoulli beam theory. ...
... The analysis of coupled axialbending vibration caused by the transverse eccentricity in multibody systems with uniform homogeneous beams can be found in [21] where both Timoshenko and Euler-Bernoulli beams were analysed as well as in [22][23][24] for the case of Euler-Bernoulli beams only and finally in [25] for non-uniform Euler-Bernoulli beams. Note that the corresponding results related to the problems of non-linear coupled axial-bending vibration can be encountered in [26,27] but only for Euler-Bernoulli cantilever beams. Of course, coupling between axial and bending vibration can occur for other reasons too. ...
Purpose
This study aims to obtain and present the closed-form solution of coupled axial-bending vibration problem for the general case of axially functionally graded (AFG) beams, to create the frequency equation and to propose numerical method for computing natural frequencies.
Methods
The general model of a beam is introduced using two cylindrical springs, one rotational spring, and a rigid body at each beam end. Mass centres of bodies are placed eccentrically in axial and transverse direction with respect to end points of the beam. The general boundary conditions are modelled by linearized Newton–Euler differential equations and the general case of the in-plane axial-bending vibrations of AFG beams is covered. It is assumed that the beam is made of a functionally graded material whose material and cross-sectional characteristics change along its longitudinal axis without any restrictions. The Euler–Bernoulli constitutive theory is applied for modelling. Partial differential equations of motion are transformed into a system of ordinary differential equations with variable coefficients, the form suitable for the implementation of the symbolic-numeric methods of initial parameters. Natural frequencies of the beam are computed as numerical solutions of the exact frequency equation.
Results and Conclusions
The closed-form solution of coupled axial-bending vibrations is derived for general case of AFG beams. Orthogonality conditions of mode shapes are derived, and constants in time function. Also, the frequency equation is derived and solved numerically to obtain natural frequencies. Obtained results of natural frequencies and mode shapes are compared to those available in open literature.
The coupled axial-bending vibration of an axially functionally graded Timoshenko cantilever beam of non-uniform cross-section is considered. The coupling phenomenon appears due to the presence of the transverse eccentricity of the mass centre of a rigid body attached to the beam free end. To study the coupled phenomenon, an approach based on the symbolic–numeric method of initial parameters is proposed. The influence of the transverse eccentricity on the cantilever beam natural frequencies and mode shapes is examined. The approach has the potential to be also applied to vibration problems of the systems composed of rigid bodies interconnected by non-uniform non-homogeneous beams.
In this paper, a novel vibrating beam gyroscope with exponential variable cross-section is designed. Relationship between the width and length of the variable cross-section exponential beam is in the form of an exponential function change. The exponential shape factor (ESF) of beam is less than or equal to zero, and the thickness of the beam remains constant with the beam length. The effects of curvature nonlinearity and inertia nonlinearity on the system are considered. The vibration control equations, boundary conditions, and nonlinear discretization model of the exponential beam micro-gyroscope (EBMG) are developed by using the extended Hamiltonian principle, the single-mode approximation method, and the Lagrange differential equations. The effects of direct current (DC) and alternating current (AC) voltages on the system response in both the drive and sense directions of the gyroscope are analyzed. The static response of the gyroscope system under the different ESF is solved by the Adomian decomposition method (ADM). The nonlinear discretization model is solved by the multi-scale method to analyze the influence of each parameter on the dynamic response of the gyroscope. The results show that with the increase of the ESF, the pull-in voltage and the first-order natural frequency of the EBMG increase gradually, and have a linear change pattern approximately. By adjusting the ESF, the difference between the peak frequency of Coriolis force response and the sense peak frequency can be controlled. Utilizing the nonlinear harden characteristics of the EBMG system, when the AC voltage is applied in the thickness direction of exponential beam, the EBMG system can obtain better bandwidth performance and linear measurable range by choosing the appropriate ESF, damping ratio, and AC voltage; when the AC voltage is applied in the width direction of the exponential beam, the EBMG can not only obtain the higher sensitivity performance, but also increase the linear detectable range by choosing an appropriate ESF.
Leaf spring is widely used in many large-stroke compliant mechanisms, such as positioning stage and flexural pivot, for its distributed compliance. However, the uniform cross-section rejects the possibility of improving the stress concentration and center shift of leaf springs. To this end, a generalized model is proposed to contain multiple types of leaf springs with variable crosssections for large deflection applications. An element integral method based on the beam constraint model (EIMBCM) is provided to model the nonlinear properties of elliptical-arc-fillet leaf springs (EAFLS), such as nonlinear deflection and center shift. Properties comparison of EAFLSs with specific parameters indicates that the generalized model can provide a comprehensive parametric method for mechanism design and avoid the empirical choice. The accuracy of the proposed nonlinear model is verified by nonlinear finite element results. The efficiency of the proposed model is illustrated by the design of a guiding mechanism.
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