Fixed-free Boundary Conditions Research Articles

Morphing beam dynamics under moving force and moving moment with inertia effects

Morphing structures are re-configurable structures that can change its geometry to perform multiple functions in multiple operating conditions. Morphing beam structures have considerable applications in industrial robots, morphing aircraft, deployable space structures, etc. In this study, dynamic modelling and analysis of a telescopic type morphing beam, modelled as moving load problem with inertia effects, is performed. The moving loads are assumed to travel along the length of the beam, from fixed to free end and free to fixed end. The material and geometric parameters of the beam are assumed to be constant. The Rayleigh beam theory is used to model the beam, taking into account the rotating inertia effects. Dirac Delta function is used to model the moving loads in the governing equation. A hybrid analytical and numerical approach that couples eigenfunction expansions and Laplace transformation, along with the Crank–Nicholson numerical scheme, is developed to solve the coupled differential equations. The number of oscillations per unit travel time of the moving load and the Dynamic Amplification Factor (DAF) of the beam’s tip response are used to quantify the dynamic effects. Numerical results are investigated for the various non-dimensionalized speeds defined in terms of the moving loads’ critical speed. Numerical result shows that loads moving at low speeds have a more pronounced impact on the dynamic response compared to high speeds. Moving moment induces significant oscillatory behaviour for both (Fixed-free and Free-fixed) boundary conditions. In contrast, the moving mass induces oscillation only when it travels from free-end to fixed-end.

Static and dynamic analysis of silicone/polyurethane embedded GFRP composite for magnetorheological elastomer

The current study focuses on the static and dynamic properties of GFRP composites embedded with a silicone(Si)/polyurethane(PU) blend used in automobile vibration dampening. Epoxy resin, glass fiber, and Si/PU blends of various compositions were used to create the hybrid composite specimens. For the present study, five different Si/PU blend compositions were considered to make five Si/PU embedded GFRP composite laminates were fabricated by hand layup method. The fabricated laminateswere cut to the standard sizes using water jet machining for mechanical and vibration testing. All tests were carried out in accordance with ASTM standards. Free Vibration test of five elastomeric composite specimens was carried out by impulse hammer method under fixed-free boundary condition. The current study found that the hybrid composite specimen with 0% Si and 100% PU elastomer has higher tensile strength, impact energy, and natural frequency than the other Si/PU blend elastomeric composite specimens.The better interfacial adhesion of PU rubber helps to transfer stress across interfaces more effectively and thereby enhancing the overall mechanical integrity of the composite laminate.

Efficient computational homogenisation of 2D beams of heterogeneous elasticity using the patch scheme

Modern ‘smart’ materials have complex heterogeneous microscale structure, often with unknown macroscale closure but one we need to realise for large scale engineering and science. The multiscale Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately predict large scale, system level, solutions through computations on only small sparse patches of the given detailed microscale system. Here the microscale system is that of a 2D beam of heterogeneous elasticity, with either fixed–fixed, fixed–free, or periodic boundary conditions. We demonstrate that the described multiscale Patch Scheme simply, efficiently, and stably predicts the beam’s macroscale, with a controllable accuracy, at finite scale separation. Dynamical systems theory supports the scheme. This article points the way for others to use this innovative systematic non-intrusive approach, via a developing toolbox of functions, to model and compute accurately macroscale system-levels of general complex physical and engineering systems.

Effect of Vibration Control Mechanisms on the Vibrational Behavior of Al 6061/SiC Composite Beam

A machine in action in a plant mine or other mechanical system is a major source of vibrations that spread from foundation and across the surrounding area. There should be the least amount of vibrations possible. The vibrations must be isolated, absorbed, or dampened in order to do this. This study investigates how an oil damper and various types of vibration isolators affect vibration frequency of a stir cast composite beam made up of 90% Al6061 and 10% SiC to examine the influence of an oil damper well, as single and double absorbers on the vibration frequency of a stir cast composite beam. Investigating the frequencies on the Vibration Fundamental Trainer under fixed-hinged, hinged-hinged, and fixed-free boundary conditions. The experiments were carried out on the constructed composite beam. The results indicate that the absorbers vibration amplitudes decrease, along with the damper and isolators effects. Absorber and isolator frequencies observed were lower, than the frequency. Increased beam holes resulted in decreased stiffness and reduced frequencies. Compared to scenarios the frequencies were higher when the beam had fixed end conditions without holes. When compared to absorbers and isolators, oil dampers have more effectively decreased vibrations.

Lateral torsional buckling behaviour of tapered steel section with web opening – finite element analysis

PurposeRecently, this steel section has found increasing popularity in residential, industrial and commercial buildings with their high load-carrying capacity due to the nature of high strength to weight ratio properties. However, the rise on the price of steel section should be more emphasized; therefore, the optimization in steel section design is needed to overcome the issue of material cost. As such, tapered steel sections save on material use, while the introduction of web openings allows the placement of mechanical and electrical services, plumbing and also aesthetic design considerations.Design/methodology/approachThe purpose of this study is to investigate the lateral torsional buckling behavior of a tapered steel section with an ellipse-shaped opening by analyzing its structural parameters. To achieve this, the finite element analysis (FEA) of the section is modeled using LUSAS software, which allows for a detailed analysis of the section's behavior under varying loads and conditions. It involves the variation in web opening size, opening layout, opening rotation angle and the tapering ratio. Eigenvalue buckling analysis is adopted to know the parametric effects of each 108 model. The size of opening varies from 0.2 to 0.5 d of the total depth where the opening located. There are three type of layouts applied in this study, which are the layouts A, B and C. There are three types of rotation angles for the ellipse-shaped opening, including the non-rotated vertical opening and two additional types formed by rotating the opening 45 degrees clockwise and counterclockwise around the center-point of the ellipse. A fixed-free boundary condition was applied, resulting in a simulation of a cantilever beam. The models are fixed at one end with a larger depth, and free at the other end with a smaller depth. Loading condition is an application of 10 kN/m uniform distributed load in the direction of gravity along the mid-span of the top flange.FindingsIt is observed that the model 82 with Layout A, tapering ratio 0.3, opening size 0.5 d and opening rotated in 45 degree anti-clockwise direction results in the highest structural efficiency among the 108 models. Therefore, the buckling moment of model 82 is 1,013.08 kNm with structural efficiency of 481.26, which shows an increase of 3.17% compared to the controlled model.Originality/valueThe FEA results shows a significant increase in ductility and stiffness of the tapered steel section with elipse shape opening and consequently changes in the behaviour of yield point.

Parametric analysis to investigate the modal response of PZT cantilever beams

This study investigates the parametric influence of physical dimensions and material properties of a composite cantilever beam on its modal response under fixed-free boundary conditions. The considered beam was made of PZT and aluminium materials. Euler-Bernoulli beam theory was used to investigate the modal response of a simple beam first and later expanded to the composite beam. Different combinations of physical dimensions and material properties, including the moment of inertia, length, area, young modulus and density, were used to analyse their influence on the obtained modal response. The obtained results showed that the density, the area of the beam cross-section, and the length of the beam have an inverse effect on the natural frequency. While the moment of inertia and modulus of elasticity has a positive influence on the values of the natural frequency. The analysis provides valuable insights into the dynamic behaviour of the beam and the effects of different parameters on its resonance characteristics, paving the way for optimising energy harvesting systems and enhancing their performance.

Long-Wave Anti-Plane Motion in a Pre-Stressed Compressible Elastic Laminate with One Fixed and One Free Face

In this paper, long-wave anti-plane shear motion in a multilayered laminate composed of pre-stressed compressible elastic layers is investigated. The layers of the laminate are perfectly bonded, while a fixed-free boundary condition is prescribed on the outer faces of the laminate. The solution of the model is determined analytically via the propagator matrix and numerically through the asymptotic approach. Moreover, the numerical results featuring harmonic curves are presented graphically, together with an asymptotic long-wave analysis of the vibration modes. As a special case of materials, linear isotropic with one shear modulus is considered. A polynomial long-wave low-frequency approximation of the related dispersion relation is also studied. It governs dispersion curves including the lowest harmonic. It is revealed that a low-frequency mode exists in both the two- and three-layered laminates, which are taken as prototypical structures. Lastly, comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.

A uniform framework for the dynamic behavior of linearized anisotropic elastic rods

A uniform framework for the dynamic behavior of a rod composed of a linearized anisotropic material is constructed based on an asymptotic reduction method. Taylor–Young series expansions of the displacement vector and stress tensor are adopted at the beginning. By substituting the series expansions into the three-dimensional (3D) equilibrium equations and the lateral traction condition followed by proper manipulations, four scalar rod equations are obtained for the leading-order displacement vector and the twist angle of the central line. The associated one-dimensional (1D) boundary conditions are obtained through the virtual work principle. One key feature of the present theory is the establishment of recursive relations so that most of the unknowns from the expansion can be eliminated. Also, all the remainders are retained, so the pointwise error estimate of the equations is established. One important advantage of the present theory is that no ad hoc assumption on the forms of displacement and external loading is adopted. Therefore, the rod theory provides a uniform framework to study dynamic behaviors. As an illustrative example, the natural vibration of a linearized isotropic rod with fixed-free boundary conditions is studied. The natural frequency is calculated by applying this uniform framework for the bending, torsional, and longitudinal modes. The obtained results are compared with those of the 3D simulations, Euler–Bernoulli, and Timoshenko beam theories. It turns out that the present theory is accurate enough for moderate and long rods in the bending vibration, and provides explicit results with high accuracy for either long or short rods in torsional and longitudinal vibrations.

Finite-difference analysis of stresses of a non-uniform functionally graded material circular disk rotating in the magneto-thermal environment: An equal mass study

The stress field of a functionally graded material rotating disk is studied for different cases of non-uniform thickness variation in the magneto-thermal environment. Three different cases of thickness variation are considered by assuming the variation of non-uniform thickness profiles to be linear, rational, and exponential functions of radius. The mass of the functionally graded material disk is considered equal in all cases of uniform/non-uniform thickness variation. The finite-difference method is used to obtain the numerical results for an Al/Al2O3 functionally graded material disk of fixed-free boundary conditions. The resultant thermo-elastic analysis has shown that the decrease in outer end thickness significantly increases circumferential stress at that end. In the absence of a magnetic field, for the disk with thin outer end thickness minimum stress intensity can be found with linearly varying thickness profile, and high intensity of circumferential stress in case rational and exponential variation of thickness profiles can significantly be reduced with an optimum magnetic field. The transient stress fields and the effect of material properties are also analyzed in detail. All the analyses showed that along with affecting the magnitude, the presence of the magnetic field changes the location and nature of maximum stress in the disk. Finally, results are compared with the finite-element method and available analytical results to verify the analysis.

MODAL FREQUENCY ANALYSES OF THE VARIABLE STIFFNESS MECHANISM DESIGN OF THE SOFT ROBOTIC SYSTEM

In this study, mechanism design and numerical analysis are investigated and directly simulated through additively manufacturing materials of the thermoplastic polyurethane TPU and ABS. Systematic motion planning of humanoid arm systems improved concerning the designed soft robotic arm-like variable stiffness system through the completed novel methodology for the analysis. Soft robotics variable stiffness for mechanism design is a novel research area. Additionally, the humanoid arm-like variable stiffness mechanism herein is taken as a case study in this technology. The variable stiffness types for soft robotics are inflatable robotic technology, smart materials technology, mechanism technology, and a combination of them. The variable stiffness mechanism has a novel design opportunity via the boundary conditions and the orientation of the initial conditions for soft robotics. The relation between the boundary conditions and variable stiffness is not investigated sufficiently. The novel field of study completed herein, the soft robotics variable stiffness, is a fundamental investigation for further development in the mechanism design. Variable stiffness mechanisms can transmit the translational and rotational motions into required directions with required displacements and applied forces on the multibody systems. The stiffness for the fixed-free structural constraint boundary condition of the specified initial condition orientation is 8 Nm/rd compared to the stiffness value of the 65.6 Nm/rd fixed-fixed end boundary condition.

측정 위치에 대한 강건성을 가지는 구조 진동 신호 기반의 결함 있는 복합재 구조물의 분류

In the present work, a new method to classify healthy and damaged composite structures using experimentally obtained structural vibration data is proposed and evaluated. After fabricating healthy and damaged laminated composite beam specimens, structural vibration data for fixed-free boundary conditions is experimentally obtained via random excitation. The measured vibration signals are converted into images using a Short-Time Fourier Transform and used as input data for learning and testing. First, an autoencoder is used to detect the presence of damage. The autoencoder model is trained using the vibration data of the healthy composite structure. The vibration data of a healthy composite structure is input to the trained autoencoder model with the data of a damaged composite structure, and errors between the input and output data are compared to detect the presence of damage. Second, a convolutional neural network model is used to classify the healthy and damaged composite structures with two different damage locations. This study confirms that the proposed technique can effectively detect and locate damage in composite structures.

Bending of cross-ply laminated composite beams with various boundary conditions and loading

The classical laminate theory is used to create analytic solutions for laminated beams in this study. Several boundary conditions and loads are considered in analytic bending. The boundary conditions taken were hinged-hinged, clamped–clamped and clamped-free and the loading casas taken were center point load and uniformly distributed load. The lamination angles of cross-ply laminated beams were [00]s, [90 90]s, [090]s, and [900]s.The transverse deflection is maximum for lamination angles [90 90]s, [900]s, [090]s, and [00]s respectively for both types of loading, center point load and uniform load. The transverse deflection for center point load is higher than that for uniform for all boundary conditions and laminations.The transverse deflection for fixed-free boundary condition is higher than deflection of hinged-hinged, also deflection of hinged-hinged is higher than for clamped–clamped boundary condition. The results obtained were in excellent agreement.

Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation

In the present paper, nonlinear forced vibrations of an axial moving nanobeam which is vertically influenced by an external harmonic excitation and gravity is analyzed by considering the effects of linear damping. Considering certain assumptions, a nonlinear Euler-Bernoulli beam theory is developed. With the implementation of the nonlocal elasticity theory, the governing integro-partial-differential equation is obtained by using the Hamilton principle. The multiple scale method is employed to obtain a steady-state response for the size-dependent viscoelastic nanobeam with fixed-free boundary conditions. Subsequently, the trivial and non-trivial steady-state response and the bifurcation point types are examined. Finally, the effects of damping coefficient and nonlocal parameter on stability and bifurcation of trivial and non-trivial solutions are studied. It is found that the effect of nonlocal parameter on the steady-state response and the bifurcation point types is quite important.

Closed-form solutions of non-uniform axially loaded beams using Lie symmetry analysis

In this paper, the governing differential equation of a beam with axial force is studied using the Lie symmetry method. Considering the inhomogeneous beam and non-uniform axial load, the governing equation is a fourth-order linear partial differential equation with variable coefficients with no closed-form solution. We search for a favourable coordinate system where the governing equation has a simpler-form or a closed-form solution. A favourable coordinate transformation is found using the Lie transformation group method. The system of determining equations for the governing equation of a beam with non-uniform axial load is derived and then solved to find a favourable coordinate system dependent on the spatially variable stiffness, mass, and axial force. The class of non-uniform axially loaded beams which have a closed-form solution is determined. The fixed-free boundary condition is imposed to find the invariant closed-form solution. A comparison between the analytical solution derived by the Lie symmetry method and the numerical solution is presented. Lie symmetry analysis yields hitherto undiscovered closed-form solutions for non-uniform axially loaded beams.

Multibody Dynamics of Nonsymmetric Planar 3PRR Parallel Manipulator with Fully Flexible Links

This paper presents the implementation of the floating frame of reference formulation to model the flexible multibody dynamics of a nonsymmetric planar 3PRR parallel manipulator. All of the links, including the moving platform, of the manipulator under study are assumed flexible whereas the joints are assumed rigid. Using the Euler-Bernoulli beam, the flexibility of the links is modeled by using the Rayleigh-Ritz and finite element approximations. In both approximations, fixed-free boundary conditions are applied to the elastic coordinates of the links. These boundary conditions enable the evaluation of the elastic displacement at a link tip coincident with the end-effector of the manipulator which is of interest in the high precision robotics application. Both the approximations were compared by applying two different types of loads to the manipulator. It is shown that the elastic displacements obtained by using both the approximations have an agreement with a slight difference in the magnitude. In addition, the sensitivity analysis shows that the rigidity of the manipulator is much affected by the in-plane depth of the manipulator links’ cross section.

Transfer matrix formulations and single variable shear deformation theory for crack detection in beam-like structures

This study aims to estimate crack location and crack length in damaged beam structures using transfer matrix formulations, which are based on analytical solutions of governing equations of motion. A single variable shear deformation theory (SVSDT) that considers parabolic shear stress distribution along beam cross-section is used, as well as, Timoshenko beam theory (TBT). The cracks are modelled using massless rotational springs that divide beams into segments. In the forward problem, natural frequencies of intact and cracked beam models are calculated for different crack length and location combinations. In the inverse approach, which is the main concern of this paper, the natural frequency values obtained from experimental studies, finite element simulations and analytical solutions are used for crack identification via plots of rotational spring flexibilities against crack location. The estimated crack length and crack location values are tabulated with actual data. Three different beam models that have free-free, fixed-free and simple-simple boundary conditions are considered in the numerical analyses.

Analytical Solution for Free Vibration of Annular Mindlin Plate with a Circumferential Open Crack

Mindlin plate theory is employed to obtain the free vibration response of an annular moderately thick plate with a circumferential open crack with fixed-free boundary conditions. To model the crack, a set of continuously distributed rotational springs are employed at the crack location. The corresponding spring stiffness value is a function of the crack depth and is given as a closed-form function. To obtain the vibration behaviour, the eigenvalue problem is solved to obtain the natural frequencies and mode shapes. The current method is verified by comparing the results with those obtained from finite element analysis. Through a parametric study, the effects of the crack depth and its radial location on the natural frequencies and mode shapes are investigated. The results show that for a constant crack depth, the reduction in natural frequency is a strong function of the radial location of the crack.

Surface energy effect on nonlinear free axial vibration and internal resonances of nanoscale rods

This study aims to investigate effects of the surface energy parameters on the nonlinear axial vibration and internal resonances of nanoscale rods. The surface energy parameters considered are the surface density and the surface Lamé constants. The nonlinear governing equation of motion is derived using the Hamilton's principle which includes the strain energy and the kinetic energy of the nanorod surface and bulk. The strain and kinetic energies of the nanorod bulk are obtained using the classical theory of elasticity and those of the nanorod surface are obtained using the surface elasticity theory. Then, the nonlinear governing equation of motion is solved using the method of multiple scales and natural frequencies are obtained for fixed-fixed and fixed-free boundary conditions. Effects of the surface energy parameters on the frequencies are examined for various cases. It is observed that the type of effect of the surface energy parameters depends on their sign. It is also seen that the type of effect of the surface energy parameters on nonlinear frequencies may become different at higher frequency numbers. The observations of this study can be a useful reference for more accurate design and fabrication of nano-electro-mechanical systems in which rod-shaped nano-structures, carbon nanotubes, play a significant role.

Experimental Modal Analysis of Hand–Arm Vibration in Golf: Influence of Grip Strength

Interest in the design of products that link performance and comfort is rapidly growing in the field of sport. To this end, the equipment industry is progressively shifting towards customization and it is focusing on man-machine interaction. The notion itself remains insufficiently studied by the scientific community. With regard to golf, several works conclude that vibrations that are perceived in the handle may be harmful and they have significant influence on comfort as well as performance. In that respect, the present paper investigates the effects of grip strength on three indicators of club dynamics: modal characteristics, overall vibratory levels, and vibration dose perceived by the club user, according to ISO 5349 standard. The study can be broken down into three steps. First, the experimental modal characteristics of a golf club are identified while using free-free, fixed-free, and grip-free (with three levels of grip strength) boundary conditions. Subsequently, a numerical model is developed and updated using experimental results. Finally, the root mean squared values and vibration dose transmitted to the hand-arm system after ball contact are extracted from the validated numerical model.

Surface stress effects on the mechanical properties of silicon nanowires: A molecular dynamics simulation

A primary challenge to use silicon nanowires as a truly potential building block in nanoscale devices is the implementation of scale effects into operational performance. Therefore, surface stress effects—as a direct result of size reduction—on transport properties became a major field of study. Previous computational simulations have focused so far on geometrical parameters with symmetrical cross sections, while silicon nanowires with nonsymmetrical cross sections are the major result of top-down fabrication techniques. A recent study has drawn a new aspect on the role played by the surface stress with a torsional profile on silicon nanowires to address the existing controversy from experimental and computational studies. Motivated by its success, the implications of this surface stress profile on the tensile properties of silicon nanowires are studied through molecular dynamics simulations. Deformation associated with the surface stress is computed for different length-to-thickness and width-to-thickness ratios. Then, tensile properties are investigated for a constant strain rate. Atomic calculations are carried out on silicon nanowires along the ⟨100⟩ crystal orientation for fixed-fixed and fixed-free boundary conditions. A combination of compressive uniaxial surface stress and torsional surface stress contributes to the mechanical behavior of silicon nanowires. A transition on elastic properties is obtained through changing the cross section from square to rectangular configuration. Further to addressing the controversy regarding the contribution of the surface stress on the mechanical properties, limits associated with available analytical approaches are highlighted for silicon nanowires.