Cantilever Beam Equation Research Articles

An analytical model of lateral condylar plate working length

BackgroundThe locking plate is a common device to treat distal femur fractures. Healing is affected by construct stiffness, thus many surgeon-controlled variables such as working length have been examined for their effects on strain at the fracture. No convenient analytical model which aids surgeons in determining working length has yet been described. We propose an analytical model and compare it to finite element analysis and cadaveric biomechanical testing. MethodsFirst, an analytical model based on a cantilever beam equation was derived. Next, a finite element model was developed based on a CT scan of a “fresh-frozen” cadaveric femur. Third, biomechanical testing in single-leg stance loading was performed on the cadaver. In all methods, strain at the fracture was recorded. An ANCOVA test was conducted to compare the strains. FindingsIn all models, as the working length increased so did strain. For strain at the fracture, the shortest working length (35 mm) had a strain of 8% in the analytical model, 9% in the finite element model, and 7% for the cadaver. The longest working length (140 mm) demonstrated strain of 15% in the analytical model, and the finite element and biomechanical tests both demonstrated strain of 14%. InterpretationThe strain predicted by the analytical model was consistent with the strain observed in both the finite element and biomechanical models. As demonstrated in existing literature, increasing the working length increases strain at the fracture site. Additional work is required to refine and establish validity and reliability of the analytical model.

Monotonically Iterative Method for the Cantilever Beam Equations

In this paper, we consider the existence of extremal solutions for the nonlinear fourth-order differential equation. By use of a new comparison result, some sufficient conditions for the existence of extremal solutions are established by combining the monotone iterative technique and the methods of lower and upper solutions. Finally, an example is given to illustrate the validity of our main results.

Analytical and numerical study of shape memory polymers as intervertebral discs under pure bending

Rupture of the elastic nucleus is one of the most critical challenges in total disc replacement (TDR). Smart polymers (SMPs) could be good candidates for TDR due to their unique properties such as self-healing, shape memory, adhesion control, adhesive capability and self-assembly. Herein, we investigate the behavior of SMPs as intervertebral discs (IVDs) under pure bending loading using analytical and numerical methods. For this purpose, the lumbar disc was considered as a cantilever beam and constitutive equation for pure bending mode using Euler-Bernoulli beam theory were developed in a full cycle of stress-free strain recovery. Numerical evaluation was performed to understand the effect of frozen volume fraction to determine the appropriate SMP for clinical evaluations. We conduct all analyses for the L4-L5 vertebral section because it is the most susceptible lumbar disc to damage. The numerical results show that SMPs with rescued frozen volume fraction are better representative for natural lumbar disc. Our results also indicate that SMPs that replace IVDs should be able to withstand at least 7.8° and 8.93° angle changes for pure bending and flexion-extension loads without any damage.

Modeling and analysis of a metal rubber vibration isolation system considering the nonlinear stiffness characteristics.

In this study, the complete design process of a metal rubber isolator using a numerical method applied in automobile underwater gliders (AUGs) is researched. A ring-like metal rubber isolator that has the potential to reduce the vibration of the AUGs is proposed. In the numerical design method, the equivalent cantilever beam model is used to identify the mechanical properties of the metal rubber isolator, whose accuracy is verified by experiment. The static stiffness and dynamic stiffness are gained through a fitting method, considering different mass loadings. Different material parameters of a metal rubber isolator, such as relative densities, wire diameters, wire spiral pitch diameters, and mass loading from the power system, greatly influence the mechanical properties and stiffness characteristic. The vibration isolation performance of a metal rubber isolator applied in the AUG is evaluated by a nonlinear single degree of freedom dynamic model, which is solved by the harmonic balance method. The linear stiffness component of the metal rubber isolator is the main influence factor for the vibration isolation performance. The nonlinear stiffness component could shift the isolation frequency.

Dynamic analysis of variable fractional order cantilever beam based on shifted Legendre polynomials algorithm

The shifted Legendre algorithm is used to solve the governing differential equation of variable order cantilever beam in time domain. According to the variable fractional model, the governing equations of viscoelastic polymer cantilever beam are built. The integer order and variable fractional order differential operator matrices are obtained based on the properties of shifted Legendre polynomials. The variable fractional differential equations are converted into algebraic equations and solved in time domain by operator matrix. The convergence analysis confirmed the high efficiency and accuracy of the proposed algorithm for solving differential equations. The deflection and stress of the polymer cantilever beams (PET and HDPE) under various external load (uniformly distributed and harmonic) are studied.

A novel damage identification method for flue gas turbine blades based on tip timing

Tip timing signal analysis has been applied to the online condition monitoring of high-speed blades. However, traditional tip timing analysis methods are not suitable for low-speed flue gas turbines. Therefore, this paper proposes a novel blade tip timing signal analysis method based on an investigation of the dynamic response characteristics of low-speed blades. First, the finite element modal theory is introduced to analyze the characteristics of blade damage. Second, an equivalent cantilever beam analysis model of flue gas turbine blades is established under complex environment and working conditions. In order to monitor the variation of local stiffness, a damage identification method based on the variation of the free end deflection of the equivalent cantilever beam is proposed. Finally, a rotating blade tip timing monitoring testing rig is established to verify the feasibility of the proposed method. The results show that the cracks originating at about 80% of the blade height have the greatest influence on blade stiffness, followed by blade root. The calculated blade damage parameters are 4.8464 mm and 3.7588 mm, and the crack influencing factors are 4.7476 and 3.6822, respectively, indicating that the change trend is consistent with the blade damage rules.

On the use of nonlinear impact oscillators in vibrating electromagnetic based energy harvesters

This work investigates the use of impact-oscillators in nonlinear electro-magnetic cantilever based broad bandwidth frequency energy harvester. The electro-mechanical model includes the dynamical equations of the nonlinear impact-oscillators, the nonlinear frequency resonant cantilever beam equation of the harvester, and the Lorenz magnetic force resulting from the surrounding nonlinear magnetic field. The simulated results of the harvested power show a band of resonant frequencies of the vibrating cantilever beam extended beyond its fundamental natural frequency. Due to the strong nonlinearity of both the impact oscillators’ forces and the magnetic flux, the harvested power of the enhanced system shows wider bands in its respective frequency response. Moreover, the harvested power of the cantilever beam shows that considering two magnets in repulsion generates higher values mainly due to the strong nonlinear transient magnetic flux, resulting into few times more power as compared to the case of considering only one magnet in the system.

Lower and upper solutions method to the fully elastic cantilever beam equation with support

The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, \t\t\t{u(4)(t)=f(t,u(t),u′(t),u″(t),u‴(t)),t∈[0,1],u(0)=u′(0)=0,u″(1)=0,u‴(1)=g(u(1)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \\quad t\\in [0,1], \\\\ u(0)=u'(0)=0, \\\\ u''(1)=0,\\qquad u'''(1)=g(u(1)), \\end{cases} $$\\end{document} where f:[0,1]times mathbb{R}^{4}rightarrow mathbb{R}, g: mathbb{R}rightarrow mathbb{R} are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.

Vibration Suppression for Beam-Like Repeating Lattice Structure Based on Equivalent Model by a Nonlinear Energy Sink

Based on the fully deployed space beam-like truss, the vibration reduction of the lattice structure is studied by using the local NES (nonlinear energy sink) attachment in this paper. The beam-like lattice structure is modeled as an equivalent linear continuous system (a finite length beam) by the equivalent method and validated with the finite element results. The dynamic vibration equations for the equivalent cantilever beam are established and the governing equations for the equivalent beam with NES are approximated by the Galerkin method. The displacement responses of the beam with and without NES attached under shock excitation are obtained. With NES at different positions, the amplitude responses of the coupled system under the external excitation at different positions are calculated to evaluate the suppression effect of the NES attachment to the structure. And with different masses of the NES, the amplitude responses of the coupled structure subject to the external excitation at different positions are also investigated to get the influence of the mass of the NES attachment to the vibration reduction. It can be seen from the results that the NES attachment can attenuate the response of the beam-like truss under transient excitation efficiently. And with the mass of NES attachment increasing, the vibration amplitude of the coupled system declines more rapidly, and the energy consumption efficiency of the NES attachment is higher. Moreover, the attenuation effect of the NES with different masses is experimentally analyzed. The experimental results are in good accord with the theoretical calculation.

An analytical formulation to evaluate natural frequencies and mode shapes of high-rise buildings

Abstract In this paper, an original analytical formulation to evaluate the natural frequencies and mode shapes of high-rise buildings is proposed. The methodology is intended to be used by engineers in the preliminary design phases as it allows the evaluation of the dynamic response of high-rise buildings consisting of thin-walled closed- or open-section shear walls, frames, framed tubes, and dia-grid systems. If thin-walled open-section shear walls are present, the stiffness matrix of the element is evaluated considering Vlasov’s theory. Using the procedure called General Algorithm, which allows to assemble the stiffness matrices of the individual vertical bracing elements, it is possible to model the structure as a single equivalent cantilever beam. Furthermore, the degrees of freedom of the structural system are reduced to only three per floor: two translations in the x and y directions and a rigid rotation of the floor around the vertical axis of the building. This results in a drastic reduction in calculation times compared to those necessary to carry out the same analysis using commercial software that implements Finite Element models. The potential of the proposed method is confirmed by a numerical example, which demonstrates the benefits of this procedure.

Numerical investigations of the transient cavitating vortical flow structures over a flexible NACA66 hydrofoil

In this paper, the cavitating flow over a flexible NACA66 hydrofoil is studied numerically by a modified fluid-structure interaction strategy with particular emphasis on understanding the flow-induced vibration and the cavitating vortical flow structures. The modified coupling approaches include (1) the hydrodynamic solution obtained by the large eddy simulation (LES) together with a homogenous cavitation model, (2) the structural deformation solved with a cantilever beam equation, (3) fluid-structural interpolation and volume mesh motion based on the radial basis functions and greedy algorithm. For the flexible hydrofoil, the dominant flow-induced vibration frequency is twice of the cavity shedding frequency. The cavity shedding frequency is same for the rigid and flexible hydrofoils, demonstrating that the structure vibration is not large enough to affect the cavitation evolution. The predicted cavitating behaviors are strongly three-dimensional, that is, the cavity is (a) of a triangular shape near the hydrofoil tip, (b) of a rectangular shape near the hydrofoil root, and (c) with a strong unsteadiness in the middle of the span, including the attached cavity growth, oscillation and shrinkage, break-off and collapse downstream. The unsteady hydroelastic response would strongly affect the cavitation shedding process with small-scale fragments at the cavity rear part. Furthermore, three vortex identification methods (i.e., the vorticity, the Q- criteria and the Ω method) are adopted to investigate the cavitating vortex structures around the flexible hydrofoil. It is indicated that the cavity variation trend is consistent with the vortex evolution. The vortex structures are distributed near the foil trailing edge and in the cavitation region, especially at the cavity-liquid interface. With the transporting downstream the shedding cavities, the vortices gradually increase in the wake flows.

A Note on the Eigenvalue Criteria for Positive Solutions of a Cantilever Beam Equation with Free End

In this paper, we study the existence of at least one positive solution to the fourth-order two-point boundary value problem (BVP) which models a cantilever beam equation, where one end is kept free. Here $$f \in {\mathcal {C}}\left( [0,1] \times {\mathbb {R}}_{+}, {\mathbb {R}}_{+}\right) $$ , $$g \in {\mathcal {C}}\left( [0,1] , {\mathbb {R}}_{+}\right) $$ and $$\lambda $$ is a positive parameter. The sufficient conditions are interesting, new and easy to verify. We have used some inequalities on the nonlinear function f and eigenvalues of a linear integral operator as bounds for the parameter $$\lambda $$ to obtain our results. Our approach is based on a revised version of a fixed point theorem due to Gustafson and Schmitt.

Analysis and calculation of bending fatigue life of curve-face gear

The curve-face gear pair is a new type of gear pair, and it has not been widely applied in many occasions as other kinds of gear pairs. In order to solve the bending fatigue life problem of this new gear pair with time-varying characteristics, a new theoretical calculation model was established. According to the loading state in actual operation, the single-tooth force model of curve-face gear pair was built, the effective loading of curve-face gear was analyzed, and loading spectrum in a period tooth was obtained. By building tooth equivalent cantilever beam model, the single-tooth bending stress calculation formula was derived, and the bending stress spectrums in tooth different sections were analyzed, meanwhile stress spectrum of bending fatigue hazard section in a period tooth was obtained. Combining the Miner’s rule and bending stress spectrum of dangerous section position, the bending fatigue life calculation method of curve-face gear was proposed. Through concrete example analysis, the bending fatigue life of curve-face gear was calculated. According to dynamic load spectrum extracted by ADAMS and bending stress calculated by ANSYS Workbench, the fatigue damage distribution of curve-face gear was obtained by nCode DesignLife. By constructing curve-face gear pair running experimental platform, the dynamic tooth root bending stress was measured. Comparing the theoretical calculation, simulation, and experimental results, the error between experiment and theory is within 10%, so the theoretical calculation method of bending fatigue life of curve-face gear was verified.

Vibratory Response Characteristics of High-Frequency Shape Memory Alloy Actuators

Abstract This paper presents a shape memory alloy actuator design using a bimorph structure capable of high-speed actuation and low power consumption. Two active layers of shape memory alloy wires are separated by a passive layer of thermoplastic polyurethane. This structure results in a bending actuator when current is alternated between the two active shape memory alloy layers. Actuators of lengths 20, 25, 30, 35, and 40 mm were tested at peak current input of 110, 120, 130, and 140 mA. The 40-mm actuator was shown to have a natural frequency of 11.4 Hz with a bending displacement of 26.4 mm perpendicular to the neutral position and a power input of 0.78 W (140 mA peak current input). A relationship between the input current and the resulting vibratory characteristics was found. As the current increases, the natural frequency decreases and the damping ratio increases. The experimental results are compared with a finite element method (FEM) vibration analysis and an Euler–Bernoulli cantilever beam equations.

Dynamic Testing and Analysis of Poisson’s Ratio of Lumbers Based on the Cantilever-Plate Bending Mode Shape Method

Abstract In this work, a method for dynamic testing of Poisson’s ratio of dimension lumber was proposed, based on the strain oscillation curve of first-order bending vibration of a cantilever plate. The method is efficient and convenient with good repeatability and high accuracy. Poisson’s ratios along grain and across grain (μLR and μRL, respectively) on radial sections (LR section) and across grain (μRT) on transverse sections of Sitka Spruce specimens were measured. The results show that the Poisson’s ratio along grain on the LR section μLR is one order higher than that across grain on the LR section μRL. Additionally, the accuracy of the elastic modulus of cantilever plate specimens calculated by theoretical equations of cantilever beam was analyzed. For cantilever plate of Sitka Spruce with an aspect ratio of four, the error of elastic modulus calculated with the cantilever formula is 0.7 %.

The method of lower and upper solutions for the cantilever beam equations with fully nonlinear terms

In this paper we discuss the existence of solutions of the fully fourth-order boundary value problem \t\t\t{u(4)=f(t,u,u′,u″,u‴),t∈[0,1],u(0)=u′(0)=u″(1)=u‴(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}=f(t, u, u', u'', u'''), \\quad t\\in [0, 1], \\\\ u(0)=u'(0)=u''(1)=u'''(1)=0 , \\end{cases} $$\\end{document} which models the deformations of an elastic cantilever beam in equilibrium state, where f:[0, 1]times {mathbb{R}}^{4}to mathbb{R} is continuous. Using the method of lower and upper solutions and the monotone iterative technique, we obtain some existence results under monotonicity assumptions on nonlinearity.

Solvability for fully cantilever beam equations with superlinear nonlinearities

This paper deals with the existence of solution for the fully fourth-order boundary value problem \t\t\t{u(4)(x)=f(x,u(x),u′(x),u″(1),u‴(x)),x∈[0,1],u(0)=u′(0)=u″(1)=u‴(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(x)=f(x, u(x), u'(x), u''(1), u'''(x)),\\quad x\\in [0, 1], \\\\ u(0)=u'(0)=u''(1)=u'''(1)=0 , \\end{cases} $$\\end{document} which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where f:[0, 1]times {{mathbb{R}}}^{4}to {mathbb{R}} is continuous. Some inequality conditions on f guaranteeing the existence and uniqueness of solutions are presented. The inequality conditions allow f(x, y_{0}, y_{1}, y_{2}, y_{3}) to grow superlinearly on y_{0}, y_{1}, y_{2}, and y_{3}.

Optimized Design of an Anti-Rotation and Anti-Overload Structure Based on Missile-Borne Semi-Strap-Down Inertial Navigation System

The Semi-Strapdown Inertial Navigation System (SSINS) provides a new solution to attitude measurement of a high-speed rotation missile. However, high dynamic and high overload test environments present significant challenges for attitude measurement systems. As a key component inside the SSINS, the Top-to-top Hemisphere Structure (THS), the rationality of its design greatly affects the navigation accuracy of the SSINS. First of all, this paper abstracts the new optimized THS into the equivalent cantilever beam structure stress model. For the first time, the relationship between the critical condition of plastic deformation at the top of the cavity and the structure parameters of the THS is obtained. Then, starting from the structural design angle of the lower hemisphere, the parameter extraction principle for effectively protecting the top hemisphere and improving the anti-rotation ability of the SSINS is obtained for the first time; furthermore, the uniform sampling method is used to obtain structural parameters satisfying the set conditions. Finally, the THS before and after optimization will be tested for anti-overload and anti-rotation in the same environment. The test results show that the optimized THS can increase the anti-rotation ability of the system to twice that before optimization. In addition, the first proposed THS' s structural optimization method opens up a new idea for the better application of the sensor in SSINS and the improvement of the positioning accuracy in the missile-borne environment.

Frequency lock-in during vortex induced vibration of a rotating blade

The vortex-induced vibration (VIV) of a rotating blade is studied in this paper. Uniform Euler–Bernoulli cantilever beam equation is used to model the rotating blade, while the nonlinear oscillator satisfying Van der Pol equation is used to represent the vortex shedding. The fluctuating lift force on the blade due to the wake dynamics is expressed such that it is proportional to the square of the magnitude of the relative velocity at which the blade interacts with the fluid. An unsteady lift due to vortex shedding acts on the blade, where as the blade is coupled with the fluid through a linear inertial coupling, resulting in a fluid–structure interaction problem. The coupled equations are discretized by using modal functions which satisfy the Eigenvalue problem. The work attempts to understand the interaction between the blade modes and the wake modes. The influence of the mass ratio and the reduced flow velocity on the coupling of the modes is explored. It is concluded form the study that the first mode is dominant and the effect of higher modes is less significant. Therefore, the single mode approximation is considered to study the instabilities associated with the frequency lock-in phenomenon of a rotating blade. The influence of fluid and blade damping, mass ratio and the coupling parameter on the lock-in phenomenon is investigated in this study.

Damage Identification of Beam with Breathing Crack under Fractal Dimension

Damage identification has been more urgent in engineering, especially beam structure which is often damaged by ambient excitation. It is based on time-frequency domain analysis of beam with breathing crack under fractal dimension in this paper. An equivalent cantilever beam model is constructed. The stiffness cosine model with time as parameter is used to simulate the opening and closing process of crack. The most widely used box dimension in fractal dimension is selected to calculate and analyze the displacement response of beam structure. The results show that the changes of box dimensional of beam structure displacement response can determine whether there is damage or not and identify the damage degree intuitively. It is more practical for damage identification in actual engineering.