Cantilever Beam System Research Articles

The applicability limits of the lowest-order substitute model for a cantilever beam system hard-impacting a moving base.

This paper examines the circumstances under which a one-degree-of-freedom approximate system can be employed to predict the dynamics of a cantilever beam comprising an elastic element with a significant mass and a concentrated mass embedded at its end, impacting a moving rigid base. A reference model of the system was constructed using the finite element method, and an approximate lowest-order model was proposed that could be useful in engineering practice for rapidly ascertaining the dynamics of the system, particularly for predicting both periodic and chaotic motions. The number of finite elements in the reference model was determined based on the calculated values of natural frequencies, which were found to correspond to the values of natural frequencies derived from the application of analytical formulas. The precision of the parameter identification and the outcomes yielded by the substitute model were validated through the calculation of the regions of stable periodic solutions using the analytical Peterka method. Subsequently, the qualitative and quantitative limits of the substitute model's applicability were determined. The quantitative limits were delineated through the utilization of Lyapunov exponents and characteristics associated with the energy dissipation due to impacts and the average number of impacts per excitation period. These characteristics provide a foundation for the introduction of global distance measures of the dynamic behavior of diverse systems within a specified range of the control parameter.

Stochastic nonlinear model updating in structural dynamics using a novel likelihood function within the Bayesian-MCMC framework

The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. The resonant decay method is employed to extract backbone curves, which capture the non-linear behaviour of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in estimating parameters of stochastic non-linear dynamical systems. Through the use of the proposed likelihood functions using backbone curves, the probability distributions of both linear and non-linear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and non-linear model updating is eliminated.

Torsional vibration suppression of cantilever beams system with the PNESI

Considering the intricate design of the conventional inerter structures, the compressive-torsional coupling effects of chiral metamaterial are used to achieve the torsional movement alongside linear movement, forming a simplified inerter mechanism. This simplified design is then integrated with the nonlinear energy sink (NES). In this study, a novel NES with the chiral metamaterial inerter structure is suggested to mitigate vibrations in the cantilever beam system. The introduction includes both the piecewise linear beam, providing nonlinear stiffness, and the chiral metamaterial inerter structure. These components are combined to form the piecewise nonlinear NES with inerter (PNESI) structure. Subsequently, the PNESI structure is integrated with the cantilever beam system, and a corresponding dynamic model is formulated to evaluate its vibration mitigation capabilities. The theoretical results are verified through experiments. The findings indicate that under steady-state vibration conditions, PNESI achieves peak vibration reductions of 88.6% in simulations and 62.7% in experiments for the cantilever beam system.

Secondary resonance of a cantilever beam with concentrated mass under time delay feedback control

The present work discusses the strongly nonlinear dynamical characteristics of the secondary resonance in a cantilever beam system with concentrated mass regulated through displacement and velocity time delay. Through the homotopy analysis method, the potential superharmonic and subharmonic resonance issues are resolved. A detailed analysis is conducted to investigate the influences of the time delay, feedback gain coefficient, excitation amplitude, and damping coefficient on the nonlinear response. The results show that the system exhibits hardening behavior for superharmonic resonance with one or two peaks and softening behavior for subharmonic resonance. In time delay control systems, increasing the damping coefficient occasionally causes an increase in amplitude. Due to the complexity of the system, it is challenging to definitively determine whether displacement or velocity time delay control is more suitable. Only when specific working conditions are given can it be compared which way is better. Prudently choosing the velocity and displacement time delay factors can restrain the amplitude of the system, adjust the response region, jump frequency and the number of solutions. The findings of this paper are thought to be useful in the design of time delay feedback controllers.

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Energy Analysis of a Damped Substructure Outrigger System

A damped outrigger system (DO) has been proposed to enhance the seismic performance, in which links between outriggers and perimeter columns are artificially disconnected and implemented by dampers. Such an operation essentially destroys the structural integrity and becomes a potential threat for structural safety. Moreover, its performance is very sensitive to the stiffness of perimeter columns. In this study, a mega-sub controlled system is employed to propose a novel outrigger system, i.e., a damped substructure outrigger system (DSO). The novel system has good structural integrity due to its main structure consisting of the core tube, outrigger, and perimeter column. To present further investigation, the govern equations of DSO are derived by the simplified model which is regard as a cantilever beam system with a multirotation spring and energy dissipation substructure. Then, the energy distribution and seismic performance are parametric investigated. Finally, the damping effects of DSO are discussed. The results indicate that DSO possesses the superiority of damping performance. Compared with DO, DSO is less sensitive to perimeter column stiffness. Moreover, the proposed system can obtain the high efficiency in energy dissipation but with less damping cost than that of the viscous damper. Also, the larger stroke of the viscous damper can be found for DSO.

Experimental Study on Performance of Impact Damper For Various Mass Ratios

Vibration plays a vital role in many structural elements. Excessive vibration can cause failure due to resonance and fatigue, malfunctioning of critical components, etc. Vibration can be controlled using passive, active and hybrid control systems. This paper presents the experimental studies carried out on the behaviour of impact dampers, which belong to the category of passive vibration control devices that are used to attenuate vibration of discrete and continuous systems. Impact damping is a technique for improving damping of a dynamic system by means of dissipation of energy due to collision of a free mass on the main mass.
 Experimental studies are carried out for forced vibration conditions by giving sinusoidal varying forcing function to the cantilever beam system using an exciter system and the time/frequency response function is determined. Impact mass is placed at free end of cantilever beam and the system is given constant base excitation and the damping of the system is studied by varying the frequency. Experiments are conducted for various mass ratios with different impact mass materials and the results are compared with theoretical predictions.

Multi-type dynamic load identification algorithm in continuous system: A numerical and experimental study based on SSM-Newmark-β

Dynamic load identification based on structural responses is an important problem in the field of engineering and plays an important role in the condition assessment of mechanical structures. Current popular load identification methods such as the state-space method (SSM) and the Green's kernel function method (GKFM) are implemented on discrete systems with outstanding performance, but when dealing with complex continuous systems, there are some limitations such as low efficiency and inaccuracy. In this paper, a continuous system load identification algorithm based on SSM and Newmark-β is proposed for the first time. Using modal coordinate transformation and modal truncation methods, the number of infinite vibration differential equations of a continuous system in physical space are converted to a finite number of vibration differential equations in modal space, where the modal truncation order and the optimal layout of the sensors are combined by modal strain energy. The Newmark-β method in modal space is derived and thus combined with SSM to obtain a load identification model Y=HF called SSM-Newmark-β method, which reduces the size of the transfer matrix H. The solution time of the proposed algorithm is 0.38 s for continuous systems, which is rather shorter than that for discrete systems. Furthermore, the simulations show that it has better accuracy and noise immunity than SSM and GKFM in the identification of sinusoidal load, impact load, and random load. The effect of time step is also discussed which reveals that the larger time step has less effect on the proposed algorithm. An experimental study is carried out on a cantilever beam system. The result verifies that the SSM-Newmark-β algorithm has better accuracy in load identification for the continuous system. This research provides a new sight for the real-time identification of dynamic loads for complex structures.

Geometric modifications to bluff body for enhanced performance in a wind-induced vibration energy harvester

Specialized sensors in wireless networks monitor the environment with minimal power, which is typically provided by batteries but it can also be powered with renewable sources using smart materials. One such solution is piezoelectric wind energy harvesting to power these sensors. However, the low output power of piezoelectric materials is still a concern. This research aims to improve piezoelectric wind energy harvesting by modifying the geometry of the bluff body. A square-shaped bluff body is used as a reference, and geometric changes are made by attaching square-shaped extensions. The study creates a mathematical model of a cantilever beam system with a piezoelectric material at fix end and a bluff body at the free end, using a multi-physics approach that combines electromechanical and aerodynamic models. The study uses MATLAB Simulink to solve the differential equations and verify the results with experiments. A parametric analysis examines the impact of external resistance, wind speed, and bluff body shape on circuit metrics. The study found that at a wind speed of 6 m/s and an optimal load resistance of 200 kΩ, the system generated 4.984 mW of power. This result demonstrates the potential of wind energy harvesting to power various distributed sensors for a wide range of applications.

Existence and stability of the periodic orbits induced by grazing bifurcation in a cantilever beam system with single rigid impacting constraint

Grazing, which can induce many nonclassical bifurcations, is a special dynamic phenomenon in some non-smooth dynamical systems such as vibro-impact systems with clearance. In this paper, the existence and stability of the periodic orbits induced by the grazing bifurcation in a cantilever beam system with impacts are uncovered. Firstly, the Poincaré mapping of the system is obtained by adopting the discontinuous mapping method. Then, by means of shooting method, the periodic orbits are determined, followed by the acquisition of Jacobian matrix in the case of non-impact is obtained subsequently. In addition, for various impacting patterns, a combination of inhomogeneous equations and inequations is obtained to determine the existence of period orbits after grazing, with the stability criterion of the grazing-induced periodic orbits given. Numerical results verify the effectiveness of theoretical analysis. What is more, we also give a conjecture about the relationship between eigenvalues and the type of periodic orbits when eigenvalues are imaginary numbers.

Global dynamics for impacting cantilever beam supported by oblique springs

The chaos and subharmonic bifurcation of a cantilever beam supported by oblique springs under bilateral asymmetric rigid constraints are investigated in this paper. It is difficult to investigate analytically the chaos and subharmonic bifurcation of the system because the stiffness term of the oblique spring support structure is a transcendental function. Firstly, the stiffness term of the system is fitted by the approximation method, and the homoclinic orbit and its internal orbits of the approximate system are compared with the orbits of the original system. Secondly, the threshold conditions for subharmonic bifurcation and homoclinic chaos are presented by applying the Melnikov method to the non-smooth impacting cantilever beam system. Moreover, the stability of impacting orbits is analyzed by combining characteristic multipliers of smooth manifolds with impact function, and the relationship between subharmonic bifurcation and chaos is investigated. Finally, the effects of damping, excitation frequency, excitation amplitude and impact coefficient of restitution on chaos and subharmonic bifurcation are investigated based on threshold conditions, which further verifies the theoretical analysis.

Performance improvement of NES based on eddy current damping

This paper investigates the application of an eddy current damping (ECD) to improve the performance of a nonlinear energy sink (NES). The effect of the damping on the vibration suppression performance of the NES is presented. Next, the principle of the ECD is introduced and combined with the NES, namely ECD enhanced NES (ECD-NES). On this basis, the effect of the ECD-NES parameters on its vibration reduction performance is analyzed. Then, the proposed ECD-NES is applied to a cantilever beam system, and the vibration suppression performance of the ECD-NES is compared with that of the NES without ECD. An optimization strategy for the ECD-NES is proposed under the condition of comprehensive consideration of displacement control and energy dissipation. Lastly, the effectiveness of the ECD-NES is verified experimentally on the cantilever beam system. The results show that adding the ECD is effective in improving the performance of the NES. After ECD enhanced NES is installed, the amplitude decay speed is increased by 24.18% in simulations and 18.85% in experiments; the energy dissipation rate is increased by 7.3% compared with the NES without ECD.

Method of measuring the stress of ferromagnetic materials based on EMAT and magnetic Barkhausen noise characteristic parameters

The magnetostrictive properties of ferromagnetic materials are related to their microstructure which varies between different stress states. This paper proposes a new method of stress detection based on EMAT (electromagnetic ultrasonic transducer) to better identify the early defect and stress state of materials. For experimental verification, an experimental platform consisting of cantilever beam, strain gauge and EMAT detection system was constructed. Compared with the traditional method of magnetostrictive effect detection, the transmitter and the receiver were deployed in 7 different ways to explore the relationship between electromagnetic ultrasonic signal and stress. According to the amplitude of EMAT signal and the intensity of magnetic field as obtained under different stress states, the relationship curves between them were drawn, with ten representative EMAT characteristic parameters selected. In addition, BP (back propagation) neural network was applied to establish the mapping relationship between the electromagnetic ultrasonic characteristic parameters and the stress state of the specimen, and the material stress was predicted with the maximum error of 8.10%. Moreover, BP neural network was adopted to combine EMAT characteristic parameters with Barkhausen characteristic parameters for establishing the mapping relationship between the characteristic parameters and stress state of specimen. The research results show that the accuracy of quantitative prediction was improved for material stress, with the maximum error reaching 6.27%.

Experimental Inter-Modal Targeted Energy Transfer in a cantilever beam undergoing Vibro-impacts

This work experimentally demonstrates inter-modal targeted energy transfer (IMTET) in a cantilever beam system with vibro-impacts. IMTET is known to achieve non-resonant rapid energy transfer from low-to-high frequency structural modes in mechanical systems. While previous works have computationally demonstrated the efficacy of IMTET, the theory and methodology supporting IMTET still lacks experimental verification. To address this task, an experiment is constructed whereby a steel beam is clamped at one end and excited at the other, and steel tips with clearances anchored to a host fixture serve as vibro-impacts (VIs) which induce non-resonant energy redistribution within the modal space of the beam. The system is modeled as an Euler–Bernoulli cantilever beam which is discretized by the finite element method, and the VI forces are simulated by a single-sided dissipative Hertzian contact model. The time-responses, modal dissipation ratios, and characteristic decay times measured in the experiment are compared to the computational models to confirm the validity of the theoretical predictions. It is found for both the experiment and computational model that the VIs transfer energy from low frequency structural modes to high frequency ones which, in turn, results in a substantial reduction in the characteristic decay time and thus confirms the efficacy of IMTET in practice. The diverse potential applications of IMTET in engineering practice are then emphasized.

Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method

In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass.

One-to-one internal resonance of a cable-beam structure subjected to a concentrated load

Cable-beam composite structure, such as a cable-stayed bridge in cantilever construction and a tower crane hoisting load, will sustain dynamic excitation. In order to investigate the dynamic behaviors at this stage, a cable-supported cantilever beam model subjected to a concentrated harmonic excitation at the cantilever end of the beam is established. The interactions including the second-order effect of the axial compressive load on the beam and the dynamic interaction between the beam and the cable due to cable's dynamic elongation are considered. A reduced two-degree-of-freedom system is obtained by using Galerkin discretization. Multiple time scales method is applied to solve the ordinary differential equations. Based on the derived modulation equations and the ordinary differential equations, frequency response, amplitude response, phase plane and time history curves are provided to explore the dynamic behaviors of the system. Three different lengths of the cable-stayed cantilever beam system are considered to study the changes in dynamic behaviors. In order to verify the validity in each case, Runge-Kutta method is used to directly solve the original ordinary differential equations and two results have a satisfying consistence. The analysis results show that frequency responses of both the beam and the cable exhibit hardening spring properties; the cable has more rich and complicated dynamic behaviors than the beam, for example the cable has more significant jump phenomenon.

Influence analysis of control signal phase on the vibration reduction effect of active constrained layer damping

The phase of the control signal applied on the piezoelectric constrained layer will have a direct influence on the vibration reduction effect in the process of implementing active constrained layer damping (ACLD) to the structure. Taking the ACLD cantilever beam system as the object, this paper focused on the study of the effect of control signal phase on the vibration suppression using the ACLD by establishing the experimental system and creating the relevant dynamic model. By analyzing the shear deformation of viscoelastic layer and combining piezoelectric equation, dynamic viscoelastic theory and mechanical analysis, a dynamic model of ACLD cantilever beam was created, which can effectively simulate two damping mechanisms and include the influence of control signal phase. The rationality of the dynamic model was verified by comparing the natural frequency and the time-domain vibration response corresponding to different control signal phases obtained by simulation and experiment. Using the presented dynamic model and the experimental results, the influence of the control signal phase on the vibration reduction effect was determined. Specifically, the control signal phase and the vibration response amplitude of the cantilever beam show a harmonic relationship centered on the response amplitude without control. The phase value corresponding to the best vibration reduction effect is not 180° like pure piezoelectric active vibration reduction and it is about 140° because of the hysteresis effect of viscoelastic material (for ZN-1 viscoelastic material selected in this study, the loss angle is about 40°).

Existence and Stability of the Periodic Orbits Induced by Grazing Bifurcation in a Cantilever Beam System with Single Rigid Impacting Constraint

Existence and Stability of the Periodic Orbits Induced by Grazing Bifurcation in a Cantilever Beam System with Single Rigid Impacting Constraint

Nonlinear saturation controller simulation for reducing the high vibrations of a dynamical system.

This paper studies the nonlinear vibrating behaviour of a nonlinear cantilever beam system (primary system) using a nonlinear absorber (the secondary system). The nonlinear vibrating behavior for the present dynamical system is considered with the effect of the external force. The one controller type, nonlinear saturation controller (NSC), is introduced to decrease the vibration of this system. Perturbation method treatment is produced to get the mathematical solution of the equations for the dynamical modeling with NSC. The perturbation technique is used to obtain the approximate solution of the dynamical system. This research focuses on resonance case with primary and 1:2 internal resonance. Time histories of the primary system and the controller are shown to demonstrate the reaction with and without control. The time-history response, as well as the impacts of the parameters on the system and controller, are simulated numerically using the MATLAB program. Routh-Hurwitz criterion is used to examine the stability of the system under primary resonance. A numerical simulation, using the MATLAB program, is obtained to show the time-history response, the effect of the parameters on the system and the controller. The effects of system parameters on the performance of the primary system and the controller are investigated. A comparison between all the obtained solutions made to confirm the results. Validation curves are provided to show how closely the perturbation and numerical solutions are related. A comparison is made with recently released papers.

Coriolis Force Sliding Mode Control Method for the Rotary Motion of the Central Rigid Body-Flexible Cantilever Beam System in TBM

Beam slab structure is often encountered in a complex tunnel boring machine. Beam slab structure is subject to dynamic load, which is easy to cause fatigue damage and affect its service life. Therefore, it is necessary to control the vibration of this kind of beam slab structure. In this study, the central rigid body-flexible beam model is established for the rotating beam and plate rotating around the y-axis. Based on the Hamilton variational principle, the dynamic equation of the central rigid body-flexible beam system is established, and the dynamic model of the central rigid body-flexible beam system considering the influence of Coriolis force and centrifugal force is given. The vibration control of the central rigid body-flexible beam system is studied. The vibration mode of the rotating Euler Bernoulli beam is determined by using the elastic wave and vibration mode theory. The influence of the rotating motion on the beam vibration is analyzed, and the variable structure control law is designed to suppress the beam vibration. Numerical simulation results show that the control method can effectively suppress the first-order and second-order vibration of the beam and verify the effectiveness of the control strategy.