Clamped-free Beam Research Articles

Frequency equation using new set of fundamental solutions with application on the free vibration of Timoshenko beams with intermediate rigid or elastic span

Stepped beams are crucial power transmission components in many mechanical engineering systems. These beams may be dynamically analyzed using the stepped Timoshenko model or the rigid mass model. In this paper, a new set of fundamental solutions is derived in order to normalize the Timoshenko beam equation at the origin of the coordinates. This set of solutions is used to derive the frequency equation of both stepped and rigid mass models. The validity ranges of these models were investigated by comparing the modal frequency results of both models. In addition, selected cases were compared using mode shape analysis. Three different models with classical end conditions are considered through this work. These are free–free, pinned–pinned and clamped–free beam configurations. The numerical results of the current work show that increasing the intermediate diameter ratio and decreasing the length ratio of the rigid mass results in decreasing the percentage deviation between the rigid mass model results and the elastic model results.

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

PurposeRecently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems.Design/methodology/approachIn this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition.FindingsThe theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio.Originality/valueA new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

A tangent stiffness–based approach to study free vibration of shear-deformable functionally graded material rotating beam through a geometrically non-linear analysis

An improved mathematical model to study the free vibration behavior of rotating functionally graded material beam is presented, considering non-linearity up to second order for the normal and transverse shear strains. The study is carried out considering thermal loading due to uniform temperature rise and using temperature-dependent material properties. Power law variation is assumed for through-thickness symmetric functional gradation of ceramic–metal functionally graded beam. The effects of shear deformation and rotary inertia are considered in the frame-work of Timoshenko beam theory. First, the rotating beam configuration under time-invariant centrifugal loading and thermal loading is obtained through a geometrically non-linear analysis, employing minimum total potential energy principle. Then, the free vibration analysis of the deformed beam is performed using the tangent stiffness of the deformed beam configuration, and employing Hamilton’s principle. The Coriolis effect is considered in the free vibration problem, and the governing equations are transformed to the state-space to obtain the eigenvalue problem. The solution of the governing equations is obtained following Ritz method. The validation is performed with the available results, and also with finite element software ANSYS. The analysis is carried out for clamped-free beam and for clamped–clamped beam with immovably clamped ends. The results for the first two modes of chord-wise and flap-wise vibration in non-dimensional speed-frequency plane are presented for different functionally graded material compositions, material profile parameters, root offset parameters and operating temperatures.

A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams

A systematic procedure is developed for studying the dynamic response of a rotating nonuniform Euler–Bernoulli beam with an elastically restrained root. To find the solution, a novel approach is used in that the fourth-order differential equation describing the vibration problem is first written as a first-order matrix differential equation, which is then solved using the power series method. The method can be used to obtain an approximate solution of vibration problems for nonuniform Euler–Bernoulli beams. Specifically, numerical examples are presented here to demonstrate the usefulness of the method in frequency analysis of nonuniform Euler–Bernoulli clamped-free cantilever beams. Results for mode shapes and frequency parameters were found to be in satisfactory agreement with previously published results. The effects of tapering, both equal and unequal, were investigated for both a cantilever wedge and cantilever cone.

Multiple internal resonances and nonplanar dynamics of a cruciform beam with low torsional stiffness

The three-dimensional motion of a slender clamped-free beam with a cruciform cross-section of low torsional stiffness, subject to lateral harmonic excitation, is investigated. Special attention is given to the nonlinear oscillations at multiple internal resonances, and to its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on the subject. The nonlinear integro-differential equations describing the flexural-flexural-torsional coupling of the beam are used, together with the Galerkin method, to obtain a minimal (three-degree-of-freedom) reduced order model, whose response is investigated through continuation of the relevant ordinary differential equations. Both inertial and geometric nonlinearities are considered. By varying the stiffness parameters of the beam and using tools of nonlinear dynamics, a complex dynamic behavior is observed around 1:1:1 resonance, where a variety of local bifurcations lead to multiple coexisting solutions which include planar and nonplanar motions. Notwithstanding the topologically involved portraits of the attractor-basin response, the analysis of system global dynamics may provide hints for safe structural design.

Multi-objective optimal design of hybrid viscoelastic/composite sandwich beams by using the generalized differential quadrature method and the non-dominated sorting genetic algorithm II

In this study, the multi-objective optimal design of hybrid viscoelastic/composite sandwich beams for minimum weight and minimum vibration response is aimed. The equation of motion for linear vibrations of a multi-layer beam is derived by using the principle of virtual work in the most general form. These governing equations together with the boundary conditions are discretized by the generalized differential quadrature method (GDQM) in the frequency domain for the first time. Also, the time and temperature dependent properties of the viscoelastic materials are taken into consideration by a novel ten-parameter fractional derivative model that can realistically capture the response of these materials. The material variability is accounted for by letting an optimization algorithm choose a material freely out of four fiber-reinforced composite materials and five viscoelastic damping polymers for each layer. The design parameters, i.e., the orientation angles of the composites, layer thicknesses and the layer materials that give the set of optimal solutions, namely the Pareto frontier, is obtained for the three and nine-layered clamped-free sandwich beams by using a variant of the non-dominated sorting genetic algorithms (NSGA II).

Broadband and tunable PZT energy harvesting utilizing local nonlinearity and tip mass effects

In this paper, a broadband piezoelectric energy harvester with an applied restoring force is studied. The restoring force is modeled as a spring in the boundary conditions of the system. The system is composed of a clamped-free beam with a tip mass which is supported by a spring at the free end. A piezoelectric layer is bounded to the upper surface of the beam and the system is exposed to an external harmonic base excitation. A nonlinear distributed parameter model of the harvester is derived using the Hamilton's principle utilizing the Euler–Bernoulli beam theory assumption. The nonlinear strain-displacement field is used to take the large transverse deflections effects into account. The reduced-order model is derived by implementing the Galerkin discretization method and the dimensionless form of the equations is used to analyze the system. The effect of the piezoelectric layer on the free vibrations of the structure is determined. Effects of tip mass and base acceleration on the frequency response of the system are investigated. The results show that, the tip mass can amplify the scavenged voltage and tune the resonance frequency. To study the influence of different types of the restoring force, linear, hardening, and pure nonlinear behavior are considered for the applied spring. The results demonstrate that by applying a pure nonlinear restoring force, the resonance bandwidth of the harvester increases which causes the harvester to generate energy in a larger frequency bandwidth and the output voltage increased remarkably in comparison to clamped-free energy harvester. This phenomenon enhances the harvester efficiency in the cases of the excitations with the time-varying frequency or the random excitations and the results in a broadband energy harvester. Finally, studying the frequency response curves for the different values of the spring location declares that, as the location of the attached spring changes, the nonlinear resonance frequency alters. The maximum increase in the bandwidth is accompanied by the case where spring is attached at the free end of the beam.

Damage detection in an uncertain nonlinear beam

The present paper proposes the use of discrete time Volterra series expanded with Kautz filters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic sti_ness e_ect. The Volterra kernels and Kautz functions are estimated in a stochastic way, where the limits of confidence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering di_erent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion.

State of art: Piezoelectric Vibration Energy Harvesters

This paper explains two novel piezoelectric vibration energy harvesters. One is lumped masssingle-axis impact lead ball piezoelectric vibration energy harvester(LMPVEH) and the other is clamped free beam single-axis impact lead ball piezoelectric vibration energy harvester(CMPVEH). Both energy harvesters works under the striking mass of lead ball and under the principle of direct piezoelectric effect and the piezoelectric material is lead zirconate titanate (PZT-5H) .Single-axis impact lead ball novel piezoelectric vibration energy harvesters are demonstrated along with the literature review in the table format. And also explained the effect of temperature in the performance of energy harvesters. The output voltage for LMPVEH is 25 Volts and the CMPVEH is 98 V.

Design and Performance of a Prototype Clamped Free Beam Novel Energy Harvester for low power applications

In this paper a prototype clamped free beam novel energy harvester (NEH) was designed and demonstrated in detail by using CATIA V5 tool. And also it was tested on 20g electromagnetic exciter at different striking loads and under the frequency of 60 Hz by using lead zirconate titanate (PZT-5H) discs or patches or transducers or circular diaphragms. The recorded output power is 5.264 mW at 0.49 N load. 5.264 mW output power can be used for operating the low power devices such MEMS devices and electronic microprocessors.

Assessment tools for numerical resolution of a contact dynamic problem with modal basis reduction.

This paper is devoted to the study of the modal basis reduction method in the framework of the dynamical behavior of a mechanical system with multiple joint clearances. The final objective is to estimate contact forces in a confined tube bundle during a dynamic loading for nuclear components sizing. The modal basis reduction combined with an explicit integration scheme is envisioned to deal with the large number of tubes and potential contact zones. In this paper, method is first applied to the example of a clamped-free beam impacting on a spring on its free edge. A semi-analytical resolution allows assessing the validity of the modal basis reduction method depending of some parameters, like frequency truncation and ratio between the bending stiffness and the spring stiffness. This study leads to criteria on numerical parameters which have to be respected to ensure stability and accuracy of results.

Vibration mitigation for clamped-free beam using time-delayed controller via Macro-fiber composite

Vibration mitigation for clamped-free beam using time-delayed controller via Macro-fiber composite

Nonlocal Timoshenko frequency analysis of single-walled carbon nanotube with attached mass: An alternative hamiltonian approach

Abstract In the present paper, the nonlocal vibration analysis of single-walled carbon nanotube as nanosized mass-sensor is examined. The nanotube is modeled as a clamped-free Timoshenko beam carrying an attached mass at its free end. Using the nonlocal Timoshenko beam theory, a re-formulation of Hamilton's principle has been presented and the equations of motion and the general corresponding boundary conditions have been derived. The main purpose of this paper is to show the sensitivity of the nanotube to the values of added mass and the in presence of nonlocal parameter on the fundamental frequencies values. Some numerical examples have been performed and discussed and the obtained results are compared with those of available works in literature and listed in bibliography.

Simulation of forced vibration in milling process considering gyroscopic moment and rotary inertia

Prevention of resonance in vibration of cutting tool is essential for achieving high quality and efficiency of the milling process. The resonance causes the cutting tool to oscillate with great amplitude and increases cutting tool wear and production costs. Using a 3-D nonlinear dynamic model of the milling process including both structural and cutting force nonlinearities, gyroscopic moment, and rotary inertia, different types of resonances in milling process are investigated in this article. The cutting tool is modeled as a rotating clamped-free beam which is excited by cutting forces. Using the method of multiple scales, frequency response function of the system in primary and super harmonic resonances is obtained. Using this model, the influences of axial depth of cut, cutting tool diameter, cutting tool length, and the number of cutter teeth on the frequency response of the tool tip vibrations are studied. The results showed that increase of axial depth of cut increases the steady state vibration response of the tool tip in all resonance cases.

Comparison analysis of energy loss between micro clamped–clamped and clamped-free beam in vertical motion flux modulation magnetic sensor

The 1/f noise is one of the main noise sources of giant magnetoresistive sensors, which will cause intrinsic detection limit at low frequency. To suppress this noise, a vertical motion flux modulation (VMFM) integrated with simple microelectromechanical-systems (MEMS) structure is proposed. However, the energy loss of VMFM structure is not considered yet. In this paper, energy loss between MEMS clamped---clamped beam (C---C beam) and clamped-free beam (C-F beam) used as VMFM structure are compared. In this comparison, quality factor (Q) is proposed to characterize the energy loss of the VMFM structure. Theory analysis and experimental results show that quality factor of a C---C beam is higher than that of a C-F beam, indicating a lower energy in VMFM. Finally, the sensitivity of a VMFM magnetic sensor modulated by a C---C beam is tested, and the sensitivity gets 3.85 mV/V/Gs.

Robust nonfragile observer-based H2/H∞ controller

A robust nonfragile observer-based controller for a linear time-invariant system with structured uncertainty is introduced. The robust stability of the closed-loop system is guaranteed by use of the Lyapunov theorem in the presence of undesirable disturbance. For the sake of addressing the fragility problem, independent sets of time-dependent gain-uncertainties are assumed to be existing for the controller and the observer elements. In order to satisfy the arbitrary H2-normed constraints for the control system and to enable automatic determination of the optimal bound of the performance functions in disturbance rejection control, additional necessary and sufficient conditions are presented in a linear matrix equality/inequality framework. The observer-based controller is then transformed into an optimization problem of coupled set of linear matrix equalities/inequality that can be solved iteratively by use of numerical software such as Scilab. Finally, concerning the evaluation of the performance of the controller, the control system is implemented in real time on a mechanical system, aiming at vibration suppression. The plant under study is a multi-input single-output clamped-free piezo-laminated smart beam. The nominal mathematical reduced-order model of the beam with piezo-actuators is used to design the proposed controller and then the control system is implemented experimentally on the full-order real-time system. The results show that the closed-loop system has a robust performance in rejecting the disturbance in the presence of the structured uncertainty and in the presence of the unmodeled dynamics.

Analytical solution for nonlinear oscillation of workpiece in turning process

Chatter phenomenon is one of the essential problems in metal machining processes. It causes cutting tool wear and increase of production costs. Chatter vibration is an unstable self-excited vibration that the regenerative chatter is its most common type. In this paper, regenerative chatter in turning process is investigated. A three-dimensional nonlinear dynamic model of the turning process including both structural and cutting force nonlinearities and gyroscopic effects is presented. The workpiece is modeled as a rotating clamped–free beam which is excited by cutting forces. Using the method of multiple-scales, analytical approximate response of the system is obtained. To validate the model its responses are compared with the experimental results. Using this model the influences of tool longitudinal position, workpiece diameter, depth of cut, and rotational speed of workpiece on the stability results are studied. Using these results, turning velocity intervals for stable and unstable cuts are determined.

Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm

In this paper, the forced nonlinear dynamic behavior of thin-walled beams with open cross section under external dynamic loads is analyzed by means of a high order implicit algorithm. This algorithm is developed using a 3D nonlinear model that takes into account the large torsion without any assumption on the torsion angle amplitude neither in the constitutive law nor in the derivation for governing dynamic equations. This algorithm is built by employing the following four steps: 1 – the space and time discretization procedures, 2 – a change of variable, 3 – a homotopy transformation, 4 – techniques used in the Asymptotic Numerical Method (ANM) (Cochelin et al., 2007; Mottaqui et al., 2010) [1,2]. The originality of this work reside in the fact that we use, for the first time, this algorithm for nonlinear analysis of thin-walled beams with open cross section under an arbitrary load. The space and time discretizations are performed respectively by the finite elements method and by the classical implicit Newmark scheme. The performance of the high order implicit algorithm is tested on four examples of nonlinear dynamic: a mono-symmetrical beam with a T cross section under external dynamic load, a mono-symmetrical beam with U cross-section under external dynamic load, a bi-symmetrical clamped-free beam IPE300 under harmonic loads and a bi-symmetrical simply supported beam with cruciform section under harmonic loads. A comparison of the obtained results with those computed by the Abaqus industrial code is given. This comparison confirms the robustness, accuracy and efficiency of this high order implicit algorithm.

Vibration analysis of a rectangular-hollow-sectional beam with an external crack

The vibration property of a rectangular-hollow-sectional beam with an external crack were theoretically and experimentally studied in this paper. The external part-through crack and through-thickness crack were considered. Linear elastic fracture mechanics were employed to establish the theoretical formula for computing the equivalent torsion spring stiffness of the two external cracks. Then, the experimental method of obtaining the equivalent torsion spring stiffness was developed. It is seen that the test data match fairly well with the analytical data. On this basis, the transfer matrix method was introduced to obtain the frequency equations of these cracked beams under three classic boundary conditions, and followed by the numerical method for calculating natural frequencies. Experimental tests on clamped–free beams were used to verify the accuracy. It should be noted that a part-through crack may lead to a more distinct natural frequency reduction than the through-thickness crack.

Investigation on Different Damping Mechanisms on the Q Factor of MEMS Resonators

It is the objective of this study to investigate the impact of different damping mechanisms occurring in micromachined clamped-clamped and clamped-free beams and evaluate the Q–factors determined experimentally against the predictions of analytical models. In detail, design parameters of the beams such as the length, width and thickness are varied to separate thermo-elastic, anchor, and surface related damping effects. Based on the analytical models, several sets of these simple structures which systematically vary in one geometrical parameter are manufactured and measured. By comparing the experimental data with analytical expressions, this study points out some limitations of current damping models especially with respect to anchor damping.