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Featured Application: A high fidelity aeroelastic model of a typical fin–actuator system is established, in which the friction model is more accurate. The influence of freeplay and friction on the system stability is analyzed using the time-domain method and frequency-domain method, and some suggestions for flutter suppression design are put forward. Both the dynamic characteristics and structural nonlinearities of an actuator will affect the flutter boundary of a fin–actuator system. The actuator models used in past research are not universal, the accuracy is difficult to guarantee, and the consideration of nonlinearity is not adequate. Based on modularization, a high-fidelity modeling method for an actuator is proposed in this paper. This model considers both freeplay and friction, which is easy to expand. It can be directly used to analyze actuator characteristics and perform aeroelastic analysis of fin–actuator systems. Friction can improve the aeroelastic stability, but the mechanism of its influence on the aeroelastic characteristics of the system has not been reported. In this paper, the LuGre model, which can better reflect the friction characteristics, was integrated into the actuator. The influence of the initial condition, freeplay, and friction on the aeroelastic characteristics of the system was analyzed. The comparison of the results with the previous research shows that oversimplified friction models are not accurate enough to reflect the mechanism of friction's influence. By changing the loads, material, and geometry of contact surfaces, flutter can be effectively suppressed, and the power loss caused by friction can be minimized. [ABSTRACT FROM AUTHOR]

The free vibration of plates and shallow shells with/without cutouts made of functionally graded materials (FGM) is investigated using variational FG shallow shells with temperature dependent mechanical characteristics of the constituent materials. First‐order shear deformation theory of shallow shells is employed. It is supposed that material properties vary through thickness according to a power‐law distribution of the constituent's volume fraction. They depend on both the temperature and the thickness. Temperature field is modeled by one‐dimensional heat transfer equation, since the temperature is varied only in thickness direction. Solution of this equation is determined by a polynomial power series expansion. Corresponding software was developed to implement the proposed approach. The vibration analysis was carried out for FG plates and shallow shells with cutout and various boundary conditions. [ABSTRACT FROM AUTHOR]

A novel asymptotical approach to integrate differential equations governing quasi-regular structures, particularly suitable on the stage of construction design, is proposed. [DOI: 10.1115/1.2202158]

Adaptive tracking control of the speed of a very elastically attached circular load driven by a direct current motor accompanied with an adaptive conventional and a fractional-order Proportional Integral Derivative (PID) controller is studied. It refers to improving the closed-loop control system response of elastically coupled components of drivelines. The motor and the load mechatronic models and the block diagrams are constructed. Parameters of the PID controller in the model reference control are both constant, as well as vary in time. The adaptive control method is improved by the application of a new closed-loop control structure canceling error dynamics. A few competing control strategies are tested based on the application of two types low and high frequency stepwise increasing variations of loading torque and damping coefficient of motion. Moreover, the performance of the control strategies is verified by Integral Time-Weighted Absolute Error (ITAE) index, since their robustness is evaluated by applying a sine modulated triangle waves of selected electric parameters. Therefore, a dynamic forcing and parameter uncertainty is applied. Simulation results are compared for checking the proposed methods. [ABSTRACT FROM AUTHOR]

In this work the chaotic dynamics of flexible curvilinear Euler–Bernoulli micro‐beams embedded into a stationary temperature field is investigated. The temperature field is modelled based on a the Duhamel–Neumann theory and is free from the restrictions on the temperature field distribution along beam thickness. The von Kármán geometric strain–stress relations are employed. The governing nonlinear PDEs are yielded by the Hamilton principle with an account of the modified couple stress theory. The finite dimension problem is truncated to a finite system of nonlinear ODEs using the finite difference method (FDM) and then the Cauchy problem is solved with a help of the Runge–Kutta method. Action of the 2D thermal field is defined by solution to the heat transfer PDE which is also solved by FDM of the second order of accuracy. The so‐called charts of vibration regimes (amplitude‐frequency planes) are constructed. In particular, novel features of nonlinear (chaotic) dynamics versus the change of the magnitude of the size (length) dependent parameter are reported. The carried out numerical analysis is supported by the monitoring of frequency power spectra based on the fast Fourier transform (FFT), phase space projections, Poincaré maps and LLEs (largest Lyapunov exponents). We have also analyzed system chaotic dynamics of the classical and size‐dependent beam models versus series of values of the two control parameters, i.e. beam curvature and its size‐dependent length. Moreover, we have detected and illustrated the novel scenarios of transition form regular to chaotic vibrations of the studied beams, governed by non‐linear PDEs. [ABSTRACT FROM AUTHOR]

Bone fracture, formation and adaptation are related to mechanical strains in bone. Assessing bone stiffness and strain distribution under different loading conditions may help predict diseases and improve surgical results by determining the best conditions for long-term functioning of bone-implant systems. In this study, an experimentally wide range of loading conditions (56) was used to cover the directional range spanned by the hip joint force. Loads for different stance configurations were applied to composite femurs and assessed in a material testing machine. The experimental analysis provides a better understanding of the influence of the bone inclination angle in the frontal and sagittal planes on strain distribution and stiffness. The results show that the surface strain magnitude and stiffness vary significantly under different loading conditions. For the axial compression, maximal bending is observed at the mid-shaft, and bone stiffness is also maximal. The increased inclination leads to decreased stiffness and increased magnitude of maximum strain at the distal end of the femur. For comparative analysis of results, a three-dimensional, finite element model of the femur was used. To validate the finite element model, strain gauges and digital image correlation system were employed. During validation of the model, regression analysis indicated robust agreement between the measured and predicted strains, with high correlation coefficient and low root-mean-square error of the estimate. The results of stiffnesses obtained from multi-loading conditions experiments were qualitatively compared with results obtained from a finite element analysis of the validated model of femur with the same multi-loading conditions. When the obtained numerical results are qualitatively compared with experimental ones, similarities can be noted. The developed finite element model of femur may be used as a promising tool to estimate proximal femur strength and identify the best conditions for long-term functioning of the bone-implant system in future study. [ABSTRACT FROM AUTHOR]

1) Background: The aim of this study is to present the results of experiments in which surface electromyography (sEMG) and thermal imaging were used to assess the muscle activation during gait and to verify the hypothesis that there is a relationship in the case of low fatigue level between sEMG-measured muscle activation, assessed in the frequency domain, and thermal factors calculated as minimum, maximum, kurtosis, mean, median, and mode from the area of interest. 2) Methods: Comparison of activity calculated from the recorded sEMG data for rectus femoris (RF), biceps femoris, tibialis anterior, and gastrocnemius medialis (GM) with thermal data obtained from the infrared vision. 3) Results: Data of 14 healthy volunteers obtained during 10 min of treadmill gait are presented and analyzed. The analysis revealed statistically significant linear correlations for RF (five moderate relationships) and GM (one good relationship) and moderate nonlinear correlations for all examined muscles. Also, a detailed protocol for precise, repeatable thermal examination is presented. 4) Conclusions: Estimated moderate linear and nonlinear correlations between thermal and electromyographic parameters are found for low level of muscle fatigue, which suggests that the presented method is useful in the analysis of muscle activation with the use of a thermal imaging as a complement to sEMG. [ABSTRACT FROM AUTHOR]

Free vibrations of microplates with non‐classical shape are considered in the paper. Governing equations are based on the modified couple stress theory and Kirchhoff–Love plate theory. It is assumed that the plate is isotropic or orthotropic and satisfy various boundary conditions. An analysis is performed by the variational‐structural method which is based on R‐functions theory and Ritz method. The testing of the proposed method is carried out by comparison with known in literature results and analytical solutions. The developed method is applied to the study of the influence of the material length scale parameter, boundary conditions, shape parameters, material characteristics on vibration frequencies. Also, size‐dependent analysis for buckling of the isotropic and orthotropic microplate subjected to uniaxial load is performed by the proposed approach. Critical loads for a plate with complex shape, different types of boundary conditions and material are calculated in order to study their effect on buckling. [ABSTRACT FROM AUTHOR]

Three-DOF system with a limited power supply (or non-ideal system) having the Mises girder as absorber is considered. Stationary resonance regimes of vibrations near stable equilibrium positions of the system are constructed in two approximations of the multiple scales method. Namely, vibrations near the resonance 1:1 between the motor rotation frequency and the linear sub-system frequency and vibrations near the resonance 1:1 between the motor and absorber frequencies are analyzed. This analysis near the first resonance is made as in supposition that elastic springs of the Mises girder are weak and have an order of the small parameter, as well as in supposition that these springs are not weak. It is obtained that the effective absorption of the elastic vibrations can be obtained in the second case. Additional numerical simulation permits to find regimes of absorption when it is possible to guarantee also the fast outcome from the first resonance region. The most appropriate for absorption and such outcome from the first resonance are vibrations near the second resonance and the snap-through motions. [ABSTRACT FROM AUTHOR]

This work focuses on analyzing the dynamical behavior of a mechanical system consisting of a double spatial pendulum in contact with a movable obstacle. The pendulum's ability to move in space is achieved by the use of special Cardan–Hook joints as the links of the pendulum. The mechanical system is equipped with a special movable obstacle, i.e. a rotating circular plate situated below the pendulum, which limits the space of admissible positions. A significant part of this work is devoted to the modeling of the contact between the pendulum and the obstacle. In this regard, a special class of reduced models is derived with the resultant friction force and moment acting on the finite size of the contact area along with a compliant model of impact based on the Hertz stiffness. Finally, the effective model suitable for fast and realistic numerical simulations is obtained. The contact model is tested numerically and the effect of its parameters on the system bifurcation dynamics is investigated. [ABSTRACT FROM AUTHOR]

Initial observations have indicated that each movement pattern or skill set has a fundamental mechanical structure. The purpose of this study was to examine biomechanical characteristics in the chasse step movement patterns between professional athletes (PA) and beginner players (BP). Large data sets were obtained for comparison by capturing kinematic and kinetic information of the dominant foot using the Oxford Foot Model (OFM) during table tennis strokes. Nine male PA and nine BP (all with dominant right feet) participated in a table tennis footwork test. A Vicon motion analysis system and a Novel Pedar insole plantar pressure measurement system were used to record kinematic and kinetic data, respectively. Findings from the study indicated that PA not only showed significantly larger forefoot and rear-foot dorsiflexion, but also demonstrated larger hallux plantarflexion. Also, PA showed significantly larger forefoot inversion and abduction at the forward-end of the step. Peak pressure values were higher under the lateral forefoot, and the medial and lateral rear-foot with faster changes in angular velocity recorded for PA during the chasse step phase. Greater peak pressures were also recorded under the other toes, and in the central and lateral forefoot during the forward swing phase when compared to BP. The results of the present study demonstrated that PA possessed greater foot drive technique. These findings may help coaches and beginners to comprehend the internal mechanisms of the chasse step technique and assist beginners in improving the mechanical efficiency of their performance. [ABSTRACT FROM AUTHOR]

In this paper, we studied both kinematic and dynamic models of a mammal-like octopod robot. To control robot legs, we employed a simple gait generator based on a sine function and we proposed modified generators of gait. The introduced generators allowed us to obtain better kinematic and dynamic properties of motion of the whole robot during walking. By changing the length and the height of a single robot step, it is possible to use one model to control the initial, regular, and final phases of the robot gait without the need of generating additional control signals. The simulation parameters were estimated based on the prototype designed in Inventor software. In turn, for numerical research, we used a simulation model implemented in Mathematica. The obtained results were presented in the form of time histories of basic kinematic and dynamic quantities of the robot gait as well as the configuration of the robot legs at the chosen time moments. The presented method allows us to precisely control the robot position in the vertical direction, which is also presented in this paper. As a result, we obtained better stability of the whole robot during walking and performing tasks, also on slippery terrains, i.e. on terrains with relatively low coefficient of friction between the ground and the robot feet. [ABSTRACT FROM AUTHOR]

A parametric pendulum excited by a discrete wave-modulated step function of length is subjected to a mathematical analysis and numerical modeling. We observe an existence of almost periodic solutions of ordinary differential equations with linear boundary value conditions. An exemplary oscillator subject to both an almost periodic step elongation and forcing synchronizes with the forcing, tending to almost periodic motions like stable limit cycles. Conditions for that synchronization as well as trajectories of numerical solutions on time history plots and phase planes are shown to confirm correctness of the analytical derivations and dedicated numerical modeling. [ABSTRACT FROM AUTHOR]

In this article, a two-degree-of-freedom system consisting of two pendulums with magnets embedded in a variable magnetic field is investigated experimentally and numerically. Pivots of the pendulums are coupled by an elastic element. The magnetic interaction originates from permanent magnets, mounted at free ends of the pendulums and current-powered air coils underneath. A novel model for the magnetic force is proposed and verified experimentally. Nonlinear dynamics of the system is examined by means of time series, bifurcation diagrams, phase portraits, and Poincaré sections. Regions of chaotic and regular motion are predicted numerically and justified experimentally. Multiperiodic motion and coexisting solutions are detected, and pictures in basins of their attraction are reported, among other. [ABSTRACT FROM AUTHOR]

In this article, a mechanical 1-degree-of-freedom oscillator with harmonic forcing and impacts was analysed both numerically and experimentally. A special attention was paid to the mathematical modelling and realistic numerical simulations of a real system. The developed experimental rig consists of a cart mounted on a guide, connected with springs to the support, and equipped with one-sided stiff limiter of motion. The cart was excited by an unbalanced disc mounted on the shaft of a stepper motor. Modelling was focused on the mathematical description of the impact process, where soft obstacle with Hertzian stiffness was assumed and different forms of non-linear damping were tested, including original modifications of the already used models. Model parameters are identified based on two experimental solutions corresponding to free vibrations – with and without impacts. Then, the model was validated by means of experimental and numerical analysis of bifurcation dynamics of the forced system. [ABSTRACT FROM AUTHOR]

The goal of the study was to perform both kinematic and dynamic simulation of an octopod robot walking on a flat and hard surface. To drive robot legs, different non-linear mechanical oscillators were employed as central pattern generators. Aside from using some well-known oscillators, a new model was proposed. Time series of robot's kinematic and dynamic locomotion parameters were computed and discussed. Displacement and velocity of the centre of gravity of the robot, ground reaction forces acting on the robot legs, as well as some aspects of energy consumption of a walking robot were analysed to assess the central pattern generators. The obtained kinematic and dynamic parameters showed some advantages of the applied generator. In particular, the gait of the robot was most stable when the robot was driven by the proposed central pattern generator model. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the effect of cartilage stiffness on inner foot biomechanics of Chinese bound foot while balanced standing using finite element method. A three-dimensional FE model of bound foot involving 28 bones, 72 ligaments, 5 plantar fascia, cartilages, and encapsulated soft tissue was constructed and validated. To conduct the sensitivity analysis of cartilage stiffness, the incremental Young's modulus of 1, 5, 10, and 15 MPa were assigned to the cartilage. 25% of the body weight was applied to the Achilles tendon to adjust the anterior- posterior displacement of center of pressure agreeable with the measured result. As the Young's modulus of cartilage increased, the peak von Mises stress in the fifth metatarsal increased obviously, while that in the calcaneus remains unchanged. The plantar fascia experienced reduced total tension with stiffer cartilage. The cartilage stiffening also caused a general increase of contact pressure at mid- and forefoot joints. Cartilage stiffening due to foot binding gave rise to risks of foot pain and longitude arch damage. Knowledge of this study contributes to the understanding of bound foot biomechanical behavior and demonstrating the mechanism of long-term injury and function damage in terms of weight-bearing due to foot binding. [ABSTRACT FROM AUTHOR]

The change in mechanical properties of the femoral bone tissue surrounding hip endoprosthesis stems during the post-operative period is one of the causes of implant instability, and the mathematical description of this phenomenon is the subject of much research. In the present study, a model of bone adaptation, based on isotropic Stanford theory, is created for further computer investigation. The results of implementation of such a mathematical model are presented regarding the choice of cement mantle rational thickness in cemented hip arthroplasties. The results show that for cement mantle thicknesses ranging from 1–1.5 mm, a peak stress value in the proximal part of the mantle exceeds the limit of durability of bone cement. Moreover, results show that high reduction in the bone density of distal and proximal regions was observed in cases of cement mantle thicknesses varying from 1–3 mm. No significant changes in bone density of the abovementioned regions were obtained for 4 mm and 5 mm. The outcome of numerical investigations can be treated as valuable and will lead to the improvement of cemented hip replacement surgery results. [ABSTRACT FROM AUTHOR]

The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations. [ABSTRACT FROM AUTHOR]

In this part of the paper, the theory of nonlinear dynamics of flexible Euler-Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied. [ABSTRACT FROM AUTHOR]

We consider a linear mechanical system under the action of potential, gyroscopic and dissipative (partial) forces. The classical Kelvin-Chetaev theorems are not applicable here, and another approach, which is based on Barbashin-Krasovskii theorem, is suggested. This approach is based on decomposition of the whole system and is convenient for systems of high dimension or with uncertain parameters. Some advantages of the proposed method are demonstrated by examples. [ABSTRACT FROM AUTHOR]

A new dynamic two-dimensional friction model is developed that is based on the bristle theory. Actually, it is the Reset Integrator Model converted into a two-dimensional space. Usually, two-dimensional friction models are in fact one-dimensional models that are rotated into a slip velocity direction. However, this common approach cannot be applied to the bristle model. That is why the idea of a two-dimensional bristle is presented. The bristle's deformation is described using polar coordinates. The carried-out numerical simulation of a planar oscillator has proved that the new model correctly captures the mechanism of smoothing dry friction by dither applied in both a perpendicular and co-linear way regarding body velocity. Furthermore, the introduced mathematical model captures two-dimensional stick-slip behaviour. Cartesian slip velocity components are the only inputs to the model. In addition, our proposed model allows one to describe friction anisotropy using bristle parameters. The paper contains the results of an experimental verification of the new friction model, conducted with a special laboratory rig employed to investigate a two-dimensional motion in the presence of dither as well as to validate our numerical results. [ABSTRACT FROM AUTHOR]

The (G′/G)-expansion scheme is used to execute many wave results for the partial differential equation, namely, the nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation which plays a significant role in mathematical physics, and this equation is accomplished in quantum electron–positron–ion magneto plasmas. The Lie approach is used to find the infinitesimal generators, and group invariant solutions, and help us to reduce the considered PDEs into ODEs. Then a planer dynamical system approach is used to see the existence of closed-form solutions. All possible phase portraits are obtained and the existence of electrostatic wave potentials is reported. Meantime, periodic and super-nonlinear electrostatic wave potentials are found for different initial value conditions with the help of the Runge–Kutta method. Moreover, new traveling wave patterns of the considered models are constructed and presented graphically. Then, the conserved vectors of the given physical model with the use of the multiplier scheme are presented. [ABSTRACT FROM AUTHOR]

It is shown by operational method that the boundary value problem of the theory of elasticity related to stresses, which can be reduced to three strains compatibility equations and to three equilibrium equations, in fact is of sixth order. Hence, it is not required to formulate additional boundary conditions. [DOI: 10.1115/1.1547681]

The paper is focused on the kinematic, dynamic and power consumption analysis of the constructed prototype of the hexapod robot walking with tripod gait on a flat and hard ground. The movements of the robot legs are controlled by different well known oscillators working as central pattern generators (CPGs). The mentioned models, as well as those proposed in our previous paper, are employed and compared from the viewpoint of fluctuations of the robot gravity center both in vertical and movement direction, contact forces between the robot legs and the ground as well as energy demand of the whole robot during walking process. Time histories of the key kinematic and dynamic quantities describing locomotion of the robot are numerically studied and experimentally verified. Power consumption of the whole robot is experimentally investigated based on the current consumption in the applied servo motors which drive the robot legs. We show that the proposed CPG model is more efficient regarding acceleration/deceleration of the whole robot, contact forces and overload acting on the robot legs as well the energy demand during walking of the robot. The appropriate choice of the robot leg movements, depending on the actual situation, can have a positive influence on the investigated issues, i.e. avoid unnecessary acceleration/deceleration of the robot, decrease the contact forces between the legs and the ground and reduce energy demand of the whole robot. Consequently, it also allows to improve structural stability of the robot during walking process. [ABSTRACT FROM AUTHOR]

Results of the simulations of vibrations of collecting electrode system of dry electrostatic precipitator (ESP) are reported. The system of nine electrodes 16m long, suspension beam (upper beam) and anvil beam (bottom beam) is analyzed. First, the variants regarding differ position of the anvil beam are analyzed. Second, the differ position of a point of the impulse force as well as direction of the impulse force are investigated (in this case the anvil beam is in a 'classic' position). Discretization of both beams and electrodes is carried out employing the modified rigid finite element method (RFEM). It allows to study an influence of the mass distribution and geometric properties of the analyzed system. Moreover, the method is convenient for the introduction of additional lumped mass parameters modeling the joints that fasten the electrodes to the suspension beam, distance-marking bushes, riveted and screw joints, etc. Position of the anvil beam and position/direction of the point of impulse force application can have essential influence on tangent and normal accelerations at different points of the plates, and thus on the effectiveness of the dust removal process. We aim to find optimal position of an anvil beam, which guaranties the maximum amplitudes of vibrations and their proper distribution in the plates. [ABSTRACT FROM AUTHOR]

Some special cases of a larger class of smooth models of the resultant friction force occurring on a planar contact area are developed and presented. These models are constructed under assumptions of classical Coulomb's law of friction on each element of contact and instantaneous transition between two modes: fully developed sliding and rigid stick state without local slips and deformations of the contact area. They are able to model cases of different values of static and kinetic friction factors. The considered models are very effective tools for fast and reliable numerical simulations of mechanical systems with frictional contacts. They are applied in modelling and numerical analysis of a special kind of mechanical system, i.e. a kind of a pendulum with special frictional driving. The rotational joint of the pendulum is elastically suspended in the motion plane. The pendulum is driven by circular frictional contact with a rotating disk. Examples of self-excited bifurcation and chaotic dynamics as well as stick-slip behaviour of the pendulum are presented. [ABSTRACT FROM AUTHOR]

This work is devoted to strain analysis and optimal design of a Functionally Graded (FG) rods and beams with small inclusions. The homogenization procedure plays a key role in our investigations. The method is illustrated using an example of the rod longitudinal deformation and bending of a beam. We consider the cases of FG inclusion sizes and FG steps between inclusions separately. Particular problems of optimal design are discussed in some details. The mathematical model of the bending beam, which adapts to the external load action, is proposed and an illustrative example of the adaptation process is given. [ABSTRACT FROM AUTHOR]

The externally excited and damped vibration of the double pendulum in the vertical plane are considered. The pendulum can collide many times during the motion with a motionless obstacle having the rough surface. The double pendulum colliding with this object has been modeled as a piecewise smooth system constrained by the unilateral constraint. In the relatively long time between the collisions, the differential equations govern the motion of the pendulum. When the contact with the obstacle appears, the pendulum exhibits a discontinuous behavior. An important element of the solving algorithm is aimed on the continuous tracking of the position of the pendulum in order to detect the collision with the unilateral constraints and to determine the state vector of the pendulum at the impact time instant. A single collision is described by the Euler's laws of motion in the integral form. The equations are supplemented by the Poisson's hypothesis and Coulomb's law of friction. The friction law is formulated for the instantaneous values of the reaction forces. The values of their impulses depend on the existence of a slip between the contacting bodies. Therefore, the Coulomb law cannot be generalized for the linear impulses of the forces in a simple way. We have applied the Routh method in order to solve the problem. The method has a simple geometrical interpretation in the impulse space. The angular velocities of both pendulum parts as well as the reaction forces at the joints of the system, which change in stepwise manner, have been presented in the paper. [ABSTRACT FROM AUTHOR]

The purpose of this study was to elaborate a two-dimensional approach for unipennate and bipennate striated skeletal muscle modelling. Behavior of chosen flat pennate muscle is modelled as a rheological system composed of serially linked passive and active fragments having different mechanical properties. Each fragment is composed of three elements: mass element, elastic element and viscous element. Each active fragment furthermore contains the contractile element. Proposed approach takes into consideration that muscle force depends on a planar arrangement of muscle fibers. Paper presents results of numerical simulations, conclusions deduced on the base of these results and a concept of experimental verification of proposed models. [ABSTRACT FROM AUTHOR]

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA). [ABSTRACT FROM AUTHOR]

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement. [ABSTRACT FROM AUTHOR]

The nonlinear dynamic problem of catenary risers is solved by means of the rigid finite element method. The method enables us to model slender systems such as lines, cables and risers undergoing large base motions. The formulation allows elimination of large values of stiffness coefficients: shear, longitudinal and torsional (dependent on permissibility of the system analyzed). The analysis presented in the paper is concerned with the dynamics of a riser initially bent and undergoing heave excitation. Comparison of the results with those obtained using the finite difference and finite element methods shows good compatibility and indicates the correctness of the models obtained by means of the rigid finite element method. Numerical effectiveness of the method enables it to be applied in solving the dynamic optimization problem. The problem described is the compensation of the horizontal vibrations of the vessel or a platform by vertical displacements of the upper end of the riser. Bending moments which arise during the motion of a platform or a vessel can seriously influence and in some cases damage the structure. Thus the aim of the optimization is to ensure that the maximum bending moment at a given point does not exceed its static counterpart. The results of numerical simulations show that the systems enabling heave compensation can be used to minimize the difference between the bending moment caused by displacements of the platform or a vessel and the static moment. [ABSTRACT FROM AUTHOR]

A method to study dynamical instability and non-linear parametric vibrations of symmetrically laminated plates of complex shapes and having different cutouts is proposed. The first-order shear deformation theory (FSDT) and the classical plate theory (CPT) are used to formulate a mathematical statement of the given problem. The presence of cutouts essentially complicates the solution of buckling problem, since the stress field is non-uniform. At first, a plane stress analysis is carried out using the variational Ritz method and the R-functions theory. The obtained results are applied to investigate buckling and parametric vibrations of laminated plates. The developed method uses the R-functions theory, and it may be directly employed to study laminated plates of arbitrary forms and different boundary conditions. Besides, the proposed method is numerical-analytical, what greatly facilitates a solution of similar-like non-linear problems. In order to show the advantage of the developed approach, instability zones and response curves for the layered cross- and angle-ply plates with external cutouts are constructed and discussed. [ABSTRACT FROM AUTHOR]

The dynamic response of a nonlinear system with three degrees of freedom in resonance that is loaded, inter alia, with a non-ideal excitation is investigated. A direct current motor (DC motor) with an eccentrically mounted rotor serves as a non-ideal source of energy. The general coordinate corresponding to the rotor dynamics steadily increases as a result of rotational motion. The decomposition of the equations of motion proposed in the paper allows us to separate the vibration of rotor from its rotations. The presented approach can be used to separate the vibration from rotations in many other mechanical and mechatronic systems. The behaviour of the considered non-ideal system near two simultaneously occurring resonances is examined using the Krylov-Bogolyubov averaging method. The stability analysis of the resonant response is also carried out. [ABSTRACT FROM AUTHOR]

In the process of modelling of the Celtic stone rotating and rolling on a plane surface, different versions of simplified models of contact forces between two contacting bodies are used. The considered contact models take into account coupled dry friction force and torque, as well as rolling resistance. They were developed using Padé approximations, their modifications, and polynomial functions. Before the use of these models, some kinds of regularisations have been made, allowing to avoid singularities in the differential equations. The models were tested both numerically and experimentally, giving some practical guesses of the most essential elements of contact modelling in the Celtic stone numerical simulations. Since the tested contact models do not require the space discretisation, they can find application in relatively fast numerical simulations of rigid bodies with friction contacts. [ABSTRACT FROM AUTHOR]

Nonlinear chains of atoms (NCA) are complex systems with rich dynamics, that influence various scientific disciplines. The lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. We have converted assumed PDE into nonlinear ODE by using the optimal system. The new auxiliary scheme is used to find the Travelling wave solutions, while graphical behaviour visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, and fundamental principles in physics that assert certain quantities remain constant over time. [ABSTRACT FROM AUTHOR]

This analysis investigates the occurrence of super nonlinear waves (SNWs) in a two-component electron-ion plasma containing Maxwell distributed electrons. The effect of Mach number (M) on the presence of SNWs is also discussed. Analytical findings for all conceivable dark-pitch solitons, singular solitons, associated hyperbolically, trigonometrically, and rational solitons, and super nonlinear periodic wave solutions are discussed in this study. A graphical representation of the research findings is also presented. This study successfully shows 2D- graphs to offer further insight into the physical properties of the solutions described for the various ranges of M that is, for M < 1 the velocity of the wave profile will be treated as subsonic, for M > 1 the velocity of the wave profile will be treated as supersonic and for M = 1 the velocity of the wave profile will be treated as sonic for super nonlinear waves in two-component Maxwellian plasma. We have further demonstrated the sensitivity assessments for the altered dynamical structural system's supersonic, subsonic, and sonic wave profiles, utilizing six unique initial conditions. That whole research justifies the appearance of SNWs in such a two-component Maxwellian plasma, which used to be formerly unidentified until about the study's start in 2011. The new analytical findings of the different categories and sensitivity analysis for the dynamical system with supersonic, subsonic, and sonic wave profiles are new things here. [ABSTRACT FROM AUTHOR]

This work is focused to analyze the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) by the Lie symmetry method. GCBS equation has been utilized to explain the wave profiles in soliton theory. GCBSE was constructed by Bogoyavlenskii and Schiff in different ways (explained in the introduction section). With the aid of Lie symmetry analysis, we have computed the symmetry generators of the GCBSE and commutation relation. We observed from the commutator table, translational symmetries make an Abelian algebra. Then by using the theory of Lie, we have discovered the similarity variables, which are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE). Using the new auxiliary method (NAM), we have to discover some new wave profiles of GCBSE in the type of few trigonometric functions. These exits some parameters which we give to some suitable values to attain the different diagrams of some obtained solutions. Further, the GCBSE is presented by non-linear self-adjointness, and conserved vectors are discovered corresponding to each generator. • Generalized Calogero-Bogoyavlenskii-Schiff equation (GCBSE) • GCBS equation has been utilized to explain the wave profiles in soliton theory • Computation of symmetry generators of the GCBSE and commutation relation • Similarity variables are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE) • New auxiliary method used to discover some new wave profiles of GCBSE in the type of few trigonometric functions • Nonlinear self-adjointness theory; • Conserved quantities. [ABSTRACT FROM AUTHOR]

In this current exploration, some new optical soliton structures of fractional nonlinear Schrodinger equation with the oscillating nonlinear coefficient are constructed with three different definitions of fractional operators beta, Riemann–Liouville, and M-Truncated derivatives. These structures are computed with the help of the new auxiliary equation method. This method gives the new analytical solutions of the considered model. The analysis is done by considering the different definitions of the derivatives like Beta, Riemann–Liouville (RL), and M-Truncated derivatives. The considered equation is converted to an ordinary differential equation (ODE) by the use of this complex transformation. The graphical explanation of some obtained results is also elaborated in detail. This work is new and does not exist in literature. • Nonlinear Schrodinger equation. • Beta derivative. • Riemann-Liouville (RL) fractional derivative. • M-truncated fractional derivative. • New auxiliary equation method (NAEM). • Optical soliton structures. • Graphical analysis. [ABSTRACT FROM AUTHOR]

A novel numerical/analytical approach to study geometrically nonlinear vibrations of shells with variable thickness of layers is proposed. It enables investigation of shallow shells with complex forms and different boundary conditions. The proposed method combines application of the R-functions theory, variational Ritz’s method, as well as hybrid Bubnov–Galerkin method and the fourth-order Runge–Kutta method. Mainly two approaches, classical and first-order shear deformation theories of shells are used. An original scheme of discretization regarding time reduces the initial problem to the solution of a sequence of linear problems including those related to linear vibrations with a special type of elasticity, as well as problems governed by non-linear system of ordinary differential equations. The proposed method is validated by the investigation of test problems for shallow shells with rectangular planform and applied to new vibration problems for shallow shells with complex planforms and variable thickness of layers. [ABSTRACT FROM AUTHOR]

We consider a mathematical model of two-layer beams coupled by boundary conditions in a stationary temperature field taking into account geometric nonlinearity. The stationary temperature field is defined by a 2D heat transfer equation with boundary conditions of the first kind. The geometric nonlinearity is introduced via von Kármán's relations for both beams. Equations of beam deflection are derived due to the Euler–Bernoulli hypothesis. The contact interaction is described using Winkler's model. Scenarios of a transition from regular to chaotic regimes are studied. Phase synchronization of beam vibrations versus both character and intensity of the applied temperature field is investigated. [ABSTRACT FROM AUTHOR]