Fixed-free Beam Research Articles

Numerical Simulation of Fixed-Free End Beam’s Modal Behaviour using Two-Way Coupled Fluid-Structure Interaction Approach

The fluid flow velocity stands as a pivotal parameter with a great influence on the mutual interaction between the fluid domain and structure. This paper focuses on structural deformation, structural velocity, von-Mises stress and frequency response of a fixed-free end beam using two-way coupled fluid-structure interaction within ANSYS Workbench. The parameters such as deformed fluid pressure and velocity were analysed across three diverse flow regimes: laminar, transitional and turbulent – each aligned with distinct fluid velocities. The outcomes show that the total deformation, velocity and von-Mises stress distribution of the fixed-free end beam increase as the inlet fluid velocity increases. As a result, the pressure and velocity distribution in the fluid flow which receiving the resultant or leftover structure deflection also increases as the incoming fluid velocity increases. Furthermore, the analysis probes the frequency response in turbulent flow conditions reveals higher values compared to laminar and transitional flows, albeit within the same natural frequency domain. This observation marks significant vibrational characteristics found in the presence of turbulent flow dynamics.

Vibration suppression of smart composite beam using model predictive controller

Abstract This work presents an adaptive model predictive control (MPC) strategy to suppress the vibration in a laminated composite beam. The control method incorporates a system identification algorithm to estimate the system parameters online, which provides a precise simulation of system dynamics. A fixed-free cantilever composite beam equipped with piezoelectric actuators was used to evaluate the efficacy of the control method. The sensors and actuators are securely bonded to the upper and lower surfaces at arbitrary locations along the beam’s length. A unified mechanical displacement field is applied to all layers, while displacements are considered independently for each layer. The beam is composed of eight layers of material, each with a thickness of 0.2 mm and orientations specified as (90°/0°/90°/0°). To achieve the best performance, the parameters of the MPC were adjusted numerically. The numerical analysis revealed that placing the actuator near the clamped end at the fixed end resulted in superior control outcomes, with a settling time of approximately 1.8 s. Conversely, the longest settling time occurred when the actuator was positioned at the free end, taking around 4 s. This model could potentially be expanded to address vibration in more intricate beams exhibiting nonlinear characteristics. The deflection readings measured at the end of the beam have been utilized as feedback control signals for predicting future behavior over a predetermined control horizon. The subsequent cost function is minimized through a quadratic equation to determine the sequence of optimal yet constrained control inputs. The suggested active vibration control system is then implemented and assessed numerically to examine the effectiveness of the control method.

Accelerometer Mass Loading Study Based on a Damage Identification Method using Fundamental Laws in Closed Systems

Many civil engineering structures can be evaluated as closed systems in which additional mass spatial distribution can change within the system in terms of the fundamental continuity and energy equations. Considering the heavy hours of traffic, it can be assumed that the position of the vehicle mass on the bridge is variable, but the total mass remains constant. This study aims to show that the damage to the closed systems can be successfully estimated by using a damage indicator that is valid for systems with constant mass distribution. In the analytical study, a 100-element 200 degree of freedom fixed-free beam is investigated and the assessment of the damage position is verified numerically and the validity of the parameter is examined in an experimental study. Except for the piece at the free end of the beam; It has been determined that the damage indicator calculated on the damaged elements is 60 times larger than the damage indicator calculated at the undamaged elements. In the elements at the free end of the beam; it was observed that this ratio is between 6 and 40 depending on the mass of the accelerometer. Therefore, a criterion for accelerometer mass and damage indicator is proposed.

Determining local modulus and strength of heterogeneous films by force-deflection mapping of microcantilevers.

Estimating the elastic modulus and strength of heterogeneous films requires local measurement techniques. For local mechanical film testing, microcantilevers were cut into suspended many-layer graphene using a focused ion beam. An optical transmittance technique was used to map thickness near the cantilevers, and multipoint force-deflection mapping with an atomic force microscope was used to record the compliance of the cantilevers. These data were used to estimate the elastic modulus of the film by fitting the compliance at multiple locations along the cantilever to a fixed-free Euler-Bernoulli beam model. This method resulted in a lower uncertainty than is possible from analyzing only a single force-deflection. The breaking strength of the film was also found by deflecting cantilevers until fracture. The average modulus and strength of the many-layer graphene films are 300 and 12GPa, respectively. The multipoint force-deflection method is well suited to analyze films that are heterogeneous in thickness or wrinkled.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

Analisa Getaran Pada Balok Jepit Bebas dan Jepit-Jepit dengan Variasi Posisi Motor Penggetar

This study aims to determine the difference in vibration produced when the position of the vibration source is placed in a different position on the cantilever beam which is supported by fixed-fixed and free-fixed supports. This research used 6061 series aluminum beam with a size of 32 mm x 32 mm x 600 mm. Variations the placement of the exciter of fixed-fixed beams was in the 15 cm, 30 cm, and 45 cm positions, while the fixed-free beams was in the 10 cm, 20 cm, 30 cm, 40 cm, and 50 cm positions. The results of the research on fixed-fixed beams showed that the maximum displacement value was at a position of 30 cm with a value of 0.097 mm, the minimum displacement at 45 cm with a value of 0.04 mm. The maximum velocity value was at a position of 15 cm with a value of 0.653 cm/s, the minimum velocity was at a position of 45 cm with a value of 0.397 cm/s. The maximum acceleration value was at a position of 15 cm with a value of 12.16 m/s², the minimum acceleration was at a position of 45 cm with a value of 8.33 m/s². While the fixed-free beam showed that the maximum displacement value was at a position of 50 cm with a value of 0.197 mm, the minimum displacement at a position of 10 cm with a value of 0.097 mm. The maximum velocity value was at a position of 30 cm with a value of 1.138 cm/s, the minimum velocity at a position of 10 cm with a value of 0.670 cm/s. The maximum acceleration value was at a position of 30 cm with a value of 21,833 m/s², the minimum acceleration was at a position of 10 cm with a value of 9.95 m/s².Keywords: vibration, displacement, velocity, acceleration, aluminum

Impact of the ground clearance on the annual energy production and tower cost of an offshore wind turbine

This article analyzes the impact of the ground clearance on the Annual Energy Production (AEP) and tower cost of a 20 MW offshore wind turbine. In addition, the influence of the rated wind speed on the analysis result will be considered. The AEP is computed by considering wind speed variation over the swept area of the rotor blades. The tapered tubular steel tower is considered for mass and cost calculation. The tower is considered as a fixed-free cantilever beam with concentrated mass at the free end. The analysis shows that the ground clearance only has a minor impact on the AEP but it has a remarkable impact on the tower mass. Specifically, when the ground clearance reaches 50 meters, the AEP only increases by roughly 3% while tower mass is nearly doubled compared to the case with no ground clearance. The results also reveal the significant impact of the rated speed on both the AEP and tower mass.

The natural frequency of drum-deployed thin-walled open tubular booms

Thin-walled tape springs are extended from cylindrical deployment drums to support instruments and sensors in a cantilevered manner on spacecraft. Attaching tape springs onto the deployment drums results in a partially flattened and partially restrained cross section, which is far from the ideal case of a fixed-free beam. The consequence is a more compliant root condition that has the potential to couple and amplify on-board microvibrations with the natural frequency of the extended instrumentation. In this paper it is shown that an Euler–Bernoulli beam model can be used to calculate the natural frequency of drum-deployed tape springs using elastic boundary conditions to represent the root condition. A Finite Element (FE) model and experimental data are used to validate the beam model’s correctly predicted relationships between the natural frequency and tape spring length, f1,Res.∝L−1.5, and its rotational stiffness, f1,Res.∝krot0.5. For the investigated beryllium copper tape springs the FE model and beam model are in excellent agreement with experiment, with the error <10%. For carbon fibre epoxy tape springs there is also strong agreement between the FE model and experiment, and approximately 10%–20% error with the beam model. On a scale from a hinged beam to a fixed-free beam, the non-dimensionalised beam equation reveals that the drum-deployed tape springs are close to the hinged beam end of the scale. Two stiffening methods are proposed to increase the stiffness of the tape springs, and hence move the tape springs towards being a fixed-free beam.

Studying the Critical Buckling Load of FG Beam Using ANSYS

In this article, the critical buckling load of functionally graded beam is calculated using ANSYS APDL Software (version 17.2) under mechanical and thermal load. In mechanical load, the effects of length to thickness ratio, power law index and mode number on the non-dimension critical buckling load of fixed-fixed and fixed-free FG beam. The results show that the length to thickness ratio is not effect on the non-dimension critical buckling load while the power law index and mode number effect on the non-dimension critical buckling load. In thermal load, the critical buckling load for fixed-fixed and pinned-pinned FG beam depend on length to thickness ratio, power law index and mode number. The results show that the critical buckling load increases with decreasing length to thickness ratio.

In-Situ Dynamic Response Measurement for Damage Quantification of 3D Printed ABS Cantilever Beam under Thermomechanical Load.

Acrylonitrile butadiene styrene (ABS) offers good mechanical properties and is effective in use to make polymeric structures for industrial applications. It is one of the most common raw material used for printing structures with fused deposition modeling (FDM). However, most of its properties and behavior are known under quasi-static loading conditions. These are suitable to design ABS structures for applications that are operated under static or dead loads. Still, comprehensive research is required to determine the properties and behavior of ABS structures under dynamic loads, especially in the presence of temperature more than the ambient. The presented research was an effort mainly to provide any evidence about the structural behavior and damage resistance of ABS material if operated under dynamic load conditions coupled with relatively high-temperature values. A non-prismatic fixed-free cantilever ABS beam was used in this study. The beam specimens were manufactured with a 3D printer based on FDM. A total of 190 specimens were tested with a combination of different temperatures, initial seeded damage or crack, and crack location values. The structural dynamic response, crack propagation, crack depth quantification, and their changes due to applied temperature were investigated by using analytical, numerical, and experimental approaches. In experiments, a combination of the modal exciter and heat mats was used to apply the dynamic loads on the beam structure with different temperature values. The response measurement and crack propagation behavior were monitored with the instrumentation, including a 200× microscope, accelerometer, and a laser vibrometer. The obtained findings could be used as an in-situ damage assessment tool to predict crack depth in an ABS beam as a function of dynamic response and applied temperature.

Demonstration of vibration control using a compliant parallel arm mechanism

There is still a demand for compact, portable, low-cost laboratory equipment for undergraduate engineering students, since turn-key systems are expensive and require substantial lab space. We propose a novel compliant mechanism demonstrating the fundamentals of vibration control to undergraduate engineering students. The mechanism consists of a translational two-degrees-of-freedom system using initially straight fixed-free flexible beams connected to primary and secondary masses. A linear actuator is connected to the primary mass and two laser displacement sensors are utilized to record displacement of each mass. Primary and secondary mass displacements are controlled by adjusting the forcing frequency while keeping the amplitude constant. All parts of the mechanism are three-dimensional printed to reduce cost. The cost of the device itself is less than $5, and with the actuator and data acquisition the total cost is $1300. The mechanism was demonstrated to 32 undergraduate Machine Dynamics and Vibrations students divided into two course sections. Feedback on the student learning experience was collected using the Student Assessment of Learning Gains survey instrument. Results indicate positive overall feedback, with students highlighting the value of using MATLAB to control the system and analyze results, the connections to real-world applications incorporated into the learning activity, and the hands-on nature of the learning experience.

Simulations, fabrication, and characterization of d31 mode piezoelectric vibration energy harvester

This paper presents the development work on d31 mode piezoelectric vibration energy harvester. The device structure consists of a fixed-free type cantilever beam with a seismic mass attached at the free end of the beam. On top of the cantilever beam, a ZnO piezoelectric layer is sandwiched between two metal electrodes. The harvester is designed using an FEM tool CoventorWare. The simulations are carried out to estimate the resonance frequency, mises stress, optimal load resistance, and generated power. The optimized design is then implemented using a five mask SOI bulk micromachining process. The fabricated harvester is characterized for frequency response using Polytec MSA-500 Micro System Analyzer. The experimental resonance frequency is found to be 235.38 Hz. The harvester is also evaluated for generated open-circuit voltage when subjected to harmonic acceleration. The open-circuit peak-to-peak voltage for 0.1 g acceleration is found to be 306 mV.

Performance Analysis of Cantilever Beam for Tunable Fabry Perot Filter

The design and simulation of the cantilever beam for tunable fabry perot filter is discussed in this paper. The fabry perot filter is tuned to operate in three frequency ranges namely short wave Infrared (SWIR), mid wave Infrared (MWIR) and long wave Infrared (LWIR) are operated in the frequency range of 1.6–2.5 μm, 3–5 μm and 8–12 μm respectively. The light that is emitted from the array of vertical cavity surface emitting laser (VCSEL) is tuned to the desired wavelength using this fabry perot filter. The fabry perot cavity length is varied using the MEMS cantilever. By applying the reflective coating to the arms of the cantilever, the fabry perot cavity is formed. The actuation given to the MEMS cantilever is electrostatic actuation. The potential difference between the two arms of the cantilever creates the electrostatic force, which attracts the free end of the cantilever. Due to the impact of the electrostatic force the arms get pulled in, as a result the separation between the mirrors varies, which in turn filters the desired wavelength. The array of cantilever is also simulated to get a desired pattern of the wavelength. The fabrication process flow for the simulated cantilever is explained. The fixed-free beam approach leads to the analysis of capacitance, stress, strain and displacement.

Design and simulation of a MEMS analog micro-mirror with improved rotation angle

This paper presents design and simulation of a MEMS analog torsional micro-mirror with improved rotation angle. The working range and consequently the rotation angle limitation of electrostatically actuated torsional micro-mirror due to pull-in instability are solved. Pedestal and suspension beams are presented to achieve electrostatically actuated torsional micro-mirror with extended working range. In the electrostatically actuated torsional micro-mirror, increasing the applied voltage decreases the equivalent stiffness of the structure and leads the system to an unstable condition by undergoing to a saddle node bifurcation. In the proposed structure to eliminate pull-in instability and increase the rotation angle range, mechanical stiffness of the structure is increased by adding a pedestal and locating auxiliary fixed-free beams in the free ends of the micro-mirror. The governing equations in relation to the proposed structure are derived. To verify, the numerical results are compared with simulated results.

Receptance coupling model for variable dynamics in fixed-free thin rib machining

An analytical approach for predicting thin rib, fixed-free beam dynamics with varying geometries is presented. The proposed method uses receptance coupling substructure analysis (RCSA) to predict the stepped thickness beam receptances (or frequency response functions) at the fixed-free beam’s free end and at the change in thickness. The assembly receptances are calculated by rigidly connecting receptances that describe the individual stepped beam sections, where the section receptances are computed using the Timoshenko beam model. Comparisons with finite element calculations are completed to verify the technique. It is observed that the RCSA predictions agree closely with finite element results. Experiments are also performed, where the stepped beam thickness is changed by multiple machining passes, and receptance measurements are carried out between passes. The RCSA predictions are compared to experimental results for natural frequency and stiffness. Agreement in natural frequency to within two percent is reported.

A modified pseudo-rigid-body modeling approach for compliant mechanisms with fixed-guided beam flexures

Abstract. In the present paper, we investigate a modified pseudo-rigid-body (MPRB) modeling approach for compliant mechanisms with fixed-guided beam flexures by considering the nonlinear effects of center-shifting and load-stiffening. In particular, a fixed-guided compliant beam is modeled as a pair of fixed-free compliant beams jointed at the inflection point, where each fixed-free beam flexure is further modeled by a rigid link connected with an extension spring by a torsion spring, based on the beam constraint model (BCM). Meanwhile, the characteristic parameters of the proposed MPRB model are no longer constant values, but affected by the applied general tip load, especially the axial force. The developed MPRB modeling method is then applied to the analysis of three common compliant mechanisms (i.e. compound parallelogram mechanisms, bistable mechanisms and 1-DOF translational mechanisms), which is further verified by the finite element analysis (FEA) results. The proposed MPRB model provides a more accurate method to predict the performance characteristics such as deformation capability, stiffness variation, as well as error motions of complaint mechanisms with fixed-guided beam flexures, and offers a new look into the design and optimization of beam-based compliant mechanisms.

Design and development of 3-stage determination of damage location using Mamdani-adaptive genetic-Sugeno model

Damage detection in structural elements like beams is one of important research areas for health monitoring. Initiation of a fault in the form of a crack or any damage puts a limitation on the service life of a structural member. So, in this paper, a method is proposed which uses the advantages of soft computing techniques like Fuzzy Inference Systems (Mamdani and Sugeno) and Adaptive Genetic Algorithm for three stage refinement of the data base generated using dynamic responses from a cracked fixed-free aluminum alloy beam element. For the crack element reference, a finite element model of a single transverse crack has been considered. The proposed method describes both Mamdani and Sugeno Fuzzy Inference Systems for training of damage parameters. In the Adaptive Genetic Algorithm, a statistics based method has been incorporated to limit the randomness of the search process. Finally, the results from the Mamdani-Adaptive Genetic-Sugeno model (MAS) are validated with the results from the experimental analysis.

Analytical solutions for fixed-free beam dynamics in thin rib machining

Two different analytical approaches for predicting thin rib, fixed-free beam dynamics with varying geometries are presented. The first approach uses the Rayleigh method to determine the effective mass for the fundamental bending mode of the stepped thickness beams and Castigliano’s theorem to calculate the stiffness both at the beam’s free end and at the change in thickness. The second method uses receptance coupling substructure analysis (RCSA) to predict the beam receptances (or frequency response functions) at the same two locations by rigidly connecting receptances that describe the individual stepped beam sections, where the receptances are derived from the Timoshenko beam model. Comparisons with finite element calculations are completed to verify the two techniques. It is observed that the RCSA predictions agree more closely with finite element results. Experiments are also performed, where the stepped beam thickness is changed by multiple machining passes, and receptance measurements are carried out between passes. The RCSA predictions are compared to experimental results for natural frequency and stiffness. Agreement in natural frequency to within a few percent is reported.

Output variance constrained bending control of rotating Euler–Bernoulli beam

In this article bending control of rotating Euler–Bernoulli beam is considered. It is assumed that the fixed-free elastic beam is attached to a servomotor using a variance constrained controller, specifically output variance constrained controller for vibration suppression. Equations of motion of the system obtained via Hamilton's principle and Galerkin method are used. The resulting linearized state-space models obtained considering just one or three modes are used for control system design. Output variance constrained controllers are designed in order to control bending at the beam tip and beam rotation angle with different variance constraint magnitudes. Closed-loop responses are analyzed when they experience white noise perturbations. Comparisons between system having tighter variance constraint and weaker variance constraint are also performed. Finally, robustness of Output variance constrained controllers with respect to the modeling uncertainty (i.e. variation of number of modes) is examined.

Influence of tuned mass damper on fixed-free 3D nonlinear beam embedded in nonlinear elastic foundation

This study investigated the damping effects of a tuned mass damper (TMD) on a fixed-free 3D nonlinear beam resting on a nonlinear elastic foundation with existing internal resonances. The nonlinear elastic foundation below (y-direction) and beside (z-direction) the nonlinear beam was simulated using cubic nonlinear springs via the method of multiple scales for the analysis of this problem. We first examined this system to ensure if the internal resonance exists. The 1:3 internal resonance was found in the 1st and 2nd modes of the system. This prompted us to add a TMD on the elastic beam in order to suppress internal resonance and vibrations. We also examined the influence of the mass and location of the TMD as well as damping and spring coefficients on the damping effects. Analysis data is presented in graphs, including 3D maximum amplitude plots of the modes and 3D maximum amplitude contour plots. Our results show that placing the TMD in the y-direction also suppresses vibrations in the z-direction. The locations of the maximum amplitudes between node points in the mode shapes and near the free end of the beam indicate where the TMD should be placed with regard to damping.