The follower load problem and bifurcation to periodic motions

The problem of a follower load applied to a thin cantilevered elastic rod is presented. The stability of the straight static solution is considered, and a post-critical analysis is performed to describe non-trivial periodic motions. As the magnitude of the follower load is increased past a critical value, the real valued frequencies of the linearized system coalesce and then split apart with complex-conjugate values, rendering the straight state unstable. A Liapunov-Schmidt reduction is used to analyze the non-linear governing equations and to investigate the possibility of non-trivial periodic solutions. The rod is assumed to be incompressible and unshearable while only general requirements are made on the constitutive rule for bending. It is found that in the post-critical regime the higher order terms in the constitutive rule play an important role in determining whether non-trivial periodic solutions are possible.     

Direct method for analysis of flexible cantilever beam subjected to two follower forces

The static analysis of the flexible non-uniform cantilever beams under a tip-concentrated and intermediate follower forces is considered. The angles of inclination of the concentrated forces with respect to the deformed axis of the beam remain unchanged during deformation. The governing non-linear boundary-value problem is reduced to an initial-value problem by change of variables. The resulting problem can be solved without iterations. It is shown that there are no critical loads in the Euler sense (divergence) for any flexural-stiffness distribution and angles of inclination of the follower forces. In particular, if the follower forces are tangential, the rectilinear shape of the non-uniform cantilever beam is the only possible equilibrium configuration. In this paper some equilibrium configurations of the uniform cantilever under normal or tangential follower forces are presented using direct method.     

Non-conservative stability of multi-step non-uniform columns

The non-conservative stability of non-uniform columns under the combined action of concentrated and variably distributed forces is solved analytically. Two types of follower force system are considered: (i) concentrated follower forces and variably distributed follower forces, (ii) concentrated follower forces and variably distributed conservative forces. The exact solutions for stability of four kinds of one-step non-uniform columns subjected to the two types of follower force system are derived for the first time. Then a new exact approach, which combines the exact solutions of one-step columns and the transfer matrix method, is presented for the non-conservative stability analysis of multi-step non-uniform columns. The advantage of the proposed method is that the resulting eigenvalue equation for a multi-step non-uniform column with any kinds of two end support configurations, an arbitrary number of spring supports and concentrated masses can be conveniently determined from a second order determinant. The decrease in the determinant order, as compared with previously developed procedures, leads to significant savings in the computational effort. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM.     

A Simple Approach to the 1:1 Resonance Bifurcation in Follower-Load Problems

We consider the problem of 1:1 resonance in autonomous, timereversible systems. We first present an abstract treatment for n-dimensionalsecond-order systems, and then apply our method to two simplemechanical examples involving follower loads. As the magnitude of the follower load is increased past a criticalvalue, the trivial solution loses stability as the real-valuedfrequencies of the linearized system first coalesce and then splitapart with complex-conjugate values. In Hamiltonian systems this isusually referred to as the Hamiltonian–Hopf bifurcation. Some novelfeatures of our analysis are the direct exploitation of reversibilityand a Liapunov–Schmidt reduction of the second-order (Newtonian)equations of motion, the latter of which requires no complexification.The analysis of the resulting two-parameter, one-dimensionalbifurcation equation yields the possibility that families ofnontrivial periodic solutions may exist for load values in excess of the critical value.     

On the comparative optimal analysis and synthesis of four-bar function generating mechanism using different heuristic methods

In this research, optimal synthesis of function generation of four-bar linkages is investigated using different heuristic optimization methods. The novelty of this study is that we considered the five precision point’s distribution and both stroke angles of the follower and crank links as the optimization variables. Therefore, a non-linear optimization was done with seven variables. Different optimization techniques, including heuristic and gradient based methods were used for optimization and were completely compared. Finally, general solutions were found and distribution of precision points for some desired functions was rationalized. GA-Hybrid seems to be the best method in finding global optimums in cases that the problem is sorely nonlinear, and also it is fast. Even it seems that distribution of precision points is independent of stroke angles of the follower and crank links. Some precision points are out of the design domain, and they are sorted in a way that they can describe the behavior of desired function better.     

Three-dimensional equilibria of nonlinear pre-curved beams using an intrinsic formulation and shooting

This article describes a shooting method for computing three-dimensional equilibria of pre-curved nonlinear beams with axial and shear flexibility using the intrinsic beam formulation. For distributed and concentrated follower loads acting on a cantilevered beam, the method amounts to a direct solution approach requiring only a single shot (zero iterations) to compute the equilibria. This is possible since the system equations are defined in a local coordinate system that rotates and translates with the beam, akin to the follower loads themselves. A general procedure employing nonconservative follower loads, which invokes the Picard–Lindelöf theorem on uniqueness and existence of solutions, is also introduced for finding all solutions for three-dimensional pre-curved beam problems with conservative loading. This is particularly useful in beam buckling problems where multiple stable and unstable solutions exist. Three-dimensional equilibrium solutions are generated for many loading cases and boundary conditions, including three-dimensional helical beams, and are compared to similar solutions where available in the literature. Excellent agreement is documented in all comparison cases. For buckling examples, the stability of the computed solutions is assessed using a dynamic finite element code based on the same intrinsic beam equations. Due to the ability to avoid iteration, the presented approach may find application in model-based control for practical three-dimensional problems such as the control of manipulators utilized in endoscopic surgeries and the control of spacecraft with robotic arms and long cables.     

U型波纹管及相关结构环向屈曲的有限元分析(Ⅱ）--半圆环壳的屈曲

环壳不仅是U型波纹管的一个组成部分，更是一类重要的结构，在航天、核能和海洋工程中有重要的应用，其屈曲是人们关注的问题之一，其中对半圆环壳的分析还较为少见。本文采用文[7]的有限元法（考虑了屈曲前的弯曲和屈曲时载荷的转动并按线性化特征值问题处理）计算了正Gauss曲率半圆环壳在均匀外压作用下的屈曲，将所得结果与已知的近似解进行了对比、并讨论了其中的差异。本文除了给出临界载荷和子午线的屈曲模态外，还给出了前屈曲弯曲应力分布，以便仔细了解屈曲问题。     

非保守圆薄板的轴对称振动和稳定性

建立了受切向均布随从力作用的圆薄板在面内周边可移、不可移两种情况下的轴对称控制方程，用打靶法直接导出求解变系数常微分方程特征值问题数值解的计算式．通过数值计算，给出了周边可移、不可移的简支、固支圆板自振频率和临界载荷的特征曲线以及相应的临界发散载荷，并分析了泊松比对圆板自振频率和临界载荷的影响．     

FREE VIBRATION OF A SIMPLY SUPPORTED BEAM UNDER A NON-CONSERVATIVE DISTRIBUTED LOAD

对受非保守载荷的简支梁在后屈曲附近的自由振动进行了研究. 基于可伸长梁的大变形理论,建立了受沿轴线分布切向非保守力作用的简支梁后屈曲附近自由振动的几何非线性模型. 在小振幅和谐振动假设下,简化得到后屈曲梁线性振动的控制方程. 采用打靶法求解振动问题的控制方程,给出了前三阶固有频率与载荷之间的特征关系曲线. 结果表明：非保守载荷作用下梁的振动响应与保守载荷作用下梁的振动响应有着明显不同.     

FLUTTER INSTABILITY OF SUPPORTED PIPES CONVEYING FLUID SUBJECTED TO DISTRIBUTED FOLLOWER FORCES

In the past decades,it has been reported that divergence is the expected form of instability for fluid-conveying pipes with both ends supported.In this paper,the form of instability of supported pipes ...     

Surface Instability of Laminated Materials of Regular Structure Under Triaxial Compression

The surface instability of laminated composites of regular structure is analyzed in a three-dimensional linearized formulation. A case is considered where the material is compressed by a surface distributed dead or follower load. Numerical results are presented for a specific problem.     

Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force

The effects of a piezoelectric layer on the stability of viscoelastic plates subjected to the follower forces are evaluated. The differential equation of motion of the viscoelastic plate with the piezoelectric layer is formulated using the two-dimensional viscoelastic differential constitutive relation and the thin plate theory. The weak integral form of the differential equations and the force boundary conditions are obtained. Using the element-free Galerkin method, the governing equation of the viscoelastic rectangular plate with elastic dilatation and Kelvin–Voigt distortion is derived, subjected to the follower forces coupled with the piezoelectric effect. A generalized complex eigenvalue problem is solved, and the force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force; this force is used to improve the stability of the non-conservative viscoelastic plate. For the viscoelastic plate with various boundary conditions, the results for the instability type and the critical loads are presented to show the variations in these factors with respect to the location of the piezoelectric layers and the applied voltages. The stability of the viscoelastic plates can be effectively improved by the determination of the optimal location for the piezoelectric layers and the most favorable voltage assignment.     

Influence of boundary conditions on the stability of a column under non-conservative load

The cantilever follower force problem with external damping is extended to a three-parameter case, including a concentrated mass, a linear elastic spring, and a partial follower force at the free end. As a result of the study an unexpected, hitherto unrecognized feature of stability/vibration is identified.Normally, the boundary conditions have great influence on the stability limits. However, it is proved that there exists a force at which the critical frequency is independent of all three boundary parameters. The characteristics of this force/frequency combination are discussed in detail, especially in relation to the corresponding eigenfunctions.Also a direct study of the onset of flutter as a function of the boundary parameters is included.     

Post-buckling of a hinged-fixed beam under uniformly distributed follower forces

Based on geometrically non-linear theory for extensible elastic beams, governing equations of statically post-buckling of a beam with one end hinged and the other fixed, subjected to a uniformly distributed, tangentially compressing follower forces are established. They consist of a boundary-value problem of ordinary differential equations with a strong non-linearity, in which seven unknown functions are contained and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and in conjunction with analytical continuation, the non-linear governing equations are solved numerically and the equilibrium paths as well as the post-buckled configurations of the deformed beam are presented. A comparison between the results of conservative system and that of the non-conservative systems are given. The results show that the features of the equilibrium paths of the beams under follower loads are evidently different from that under conservative ones.     

Series solution for large deflections of a cantilever beam with variable flexural rigidity

In this paper large deflection and rotation of a nonlinear Bernoulli-Euler beam with variable flexural rigidity and subjected to a static co-planar follower loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytical technique for nonlinear problems, namely the Homotopy Analysis Method (HAM). The present solution can be used for the analysis of a wide range of loads, material/cross section properties and lengths for beams undergoing large deformations. The results obtained from HAM are compared with results reported in previous works. Finally, the load–displacement characteristics of a uniform cantilever beam with different material properties under a follower force applied normal to the deformed beam axis are presented.     

Elastic cables–rigid body coupled dynamics: asymptotic modeling and analysis

An elastic cables–rigid body coupled model is proposed for investigating dynamic interactions between cables’ nonlinear transversal vibrations and boundary tower’s torsional dynamics, arising in large transmission line–tower systems and suspended cable–bridge tower systems. By introducing a weak torsion assumption and a large moment of inertia for the tower, an asymptotic expansion of cables–tower coupled dynamics is conducted in a weakly nonlinear framework, and a cables–tower reduced coupled model is eventually established. After model’s validations using direct numerical simulations, two distinct kinds of coupled dynamics are fully investigated. The first is that an external torque is applied to the tower and the two cables would both be indirectly excited, asymmetrically, by the torsional/oscillating tower. The two cables’ responses are the same in this case. The second is that only one of the two cables, i.e., the leader cable, is directly excited, and the other cable, i.e., the follower one, is only indirectly excited through cables–tower dynamic interactions. In such kind of leader–follower dynamics, the leader cable is quite different from the follower one. Nonlinear coupled frequency response diagrams for both systems are constructed using numerical continuation algorithms, mainly focused on the coupled steady solutions’ stabilities and bifurcations. Furthermore, the dynamic effects of tower’s moment of inertia, wing span and damping are thoroughly investigated.

     

Autonomous and non-autonomous fixed-time leader–follower consensus for second-order multi-agent systems

Nonlinear Dynamics - This paper addresses the problem of consensus tracking with fixed-time convergence, for leader–follower multi-agent systems with double-integrator dynamics, where only a...     

Nonlinear finite element analysis of inflatable beams made from orthotropic woven fabric

This paper was devoted to the three-dimensional nonlinear finite element analysis of inflatable beams. The beams under consideration are made of modern textile materials and can be used as a load-bearing beams or arches when inflated. A 3D Timoshenko beam with a homogeneous orthotropic woven fabric (OWF) was proposed. The model took into account the geometric nonlinearities and the follower force resulting from the inflation pressure. The use was made of the usual total Lagrangian form of the virtual work principle to perform the nonlinear equilibrium equations which were discretized by the finite element method. Two kinds of solutions were then investigated: finite elements solutions for linearized problems which were obtained by the means of the linearization around the prestressed reference configuration of the nonlinear equations and nonlinear finite element solutions which were performed by the use of an optimization algorithm based on the Quasi-Newton method. As an example, the bending problem of a cantilever inflated beam under concentrated load was considered and the deflection results improve the existing theoretical models. As these beams are made from fabric, the beam models were validated through their comparison with a 3D thin-shell finite element model. The influence of the material effective properties and the inflation pressure on the beam response was also investigated through a parametric study. The finite elements solutions for linearized problems were found to be close to the theoretical results existing in the literature. On the other hand, the results for the nonlinear finite element model were shown to be close to the results for the linearized finite elements model in the case of high mechanical properties and the nonlinear finite element model was used to improve the linearized model when the mechanical properties of the fabric are low.     

Cooperative fault-tolerant tracking control of heterogeneous hybrid-order mechanical systems with actuator and amplifier faults

Nonlinear Dynamics - This paper studies the problem of leader–follower fault-tolerant tracking control for a class of heterogeneous multi-agent mechanical systems (MAMS) with actuator and...