Intermediate Spring Support Research Articles

BUCKLING OF BEAMS WITH A BOUNDARY ELEMENT TECHNIQUE

The present work is devoted to the buckling study of non-homogeneous fixed- fixed beams with intermediate spring support. The stability issue of these beams leads to three-point boundary value problems. If the Green functions of these boundary value problems are known, the differential equations of the stability problems that contain the critical load sought can be turned into eigenvalue problems given by homogeneous Fredholm integral equations. The kernel function of these equations can be calculated from the associated Green functions. The eigenvalue issues can be reduced to algebraic eigenvalue problems, which are subsequently solvable numerically with the use of an efficient algorithm from the boundary element method. Within this article, the critical load findings of these beams are compared to those obtained using commercial finite element software, and the results are in excellent correlation.

Large-amplitude dynamics of a functionally graded microcantilever with an intermediate spring-support and a point-mass

Numerical modelling and simulations are conducted on the large-amplitude dynamics of a functionally graded microcantilever with a tip-mass, additionally supported by an intermediate spring; the functionally graded microsystem is subject to a base excitation. Since one end of the microsystem is free to move, it undergoes large deformation; curvature-related nonlinearities play an important role. Taking into account this type of nonlinearity, using the Mori–Tanaka homogenisation scheme, as well as the modified couple stress theory, an energy technique is employed to derive the nonlinearly coupled equations for the longitudinal and transverse motions. An inextensibility assumption is applied for the functionally graded microcantilever, and hence, a nonlinear equation of motion for the transverse motion involving inertial (apart from stiffness) nonlinearity is obtained. For the functionally graded microsystem considered, effects of the length-scale parameter, the material gradient index, the tip-mass, and the stiffness of the spring-support on the nonlinear resonant responses are highlighted by means of a Houbolt’s finite difference scheme together with Newton–Raphson method.

Modal interactions and energy transfers in large-amplitude vibrations of functionally graded microcantilevers

Modal interactions and internal energy transfers are investigated in the large-amplitude oscillations of a functionally graded microcantilever with an intermediate spring-support. Based on the Mori–Tanaka homogenization technique and the modified couple stress theory, the energy terms of the functionally graded microsystem (kinetic and size-dependent potential energies) are developed and dynamically balanced. Large-amplitude deformations, due to having one end free, are modeled taking into account curvature-related nonlinearities and assuming an inextensibility condition. The continuous model of the functionally graded microsystem is reduced, by means of the Galerkin method, yielding an inertial- and stiffness-wise nonlinear model. Numerical simulations on this highly nonlinear reduced-order model of the functionally graded microcantilever are performed using a continuation method; a possible case of modal interactions is determined by obtaining the natural frequencies of the microsystem. The nonlinear oscillations of the microcantilever are examined, and it is shown how the energy fed to the functionally graded microsystem (from the base excitation) is transferred between different modes of oscillation.

Size-dependent internal resonances and modal interactions in nonlinear dynamics of microcantilevers

The size-dependent internal energy transfer in the nonlinear dynamical behaviour of a microcantilever with an intermediate spring-support is investigated. A geometric size-dependent nonlinearity due to large changes in the curvature is taken into account in the longitudinal and transverse motions. Based on the modified couple stress theory, the potential energy of the system is developed; the kinetic energy is also constructed in term of the displacement field. The energy terms are balanced with the potential energy stored in the intermediate spring-support. The centreline-inextensibility assumption is applied leading to the continuous model of the system involving nonlinear inertial components as well as size-dependent nonlinear curvature components. Based on a weighted-residual technique, the continuous model is reduced and the resultant truncated model is solved via use of a continuation technique. The linear component of the truncated model is solved through an eigenvalue extraction method in order to verify the occurrence of internal energy transfer and modal interaction mechanisms. For the system tuned to internal resonances, the highly nonlinear dynamical response is obtained, taking into account both inertial and geometric (due to large rotations) nonlinearities. It is shown that taking into account the length-scale parameter changes the internal energy transfer mechanisms significantly.

Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support

The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.

Nonlinear vibrations and stability of an axially moving Timoshenko beam with an intermediate spring support

In this paper, the nonlinear forced vibrations and stability of an axially moving Timoshenko beam with an intra-span spring-support are investigated numerically. Taking into account the shear deformation and rotary inertia, three coupled nonlinear partial differential equations of motion are obtained using Hamilton's principle along with stress–strain relations. These equations are discretized into a set of coupled nonlinear ordinary differential equations via the Galerkin method. The pseudo-arclength continuation technique is used to solve the governing set of equations. The frequency–response curves of the system in the subcritical regime are obtained and examined. In addition, direct time integration is employed to investigate the global dynamics of the system through constructing the bifurcation diagrams of Poincaré maps.

Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis

The nonlinear coupled longitudinal-transverse vibrations and stability of an axially moving beam, subjected to a distributed harmonic external force, which is supported by an intermediate spring, are investigated. A case of three-to-one internal resonance as well as that of non-resonance is considered. The equations of motion are obtained via Hamilton’s principle and discretized into a set of coupled nonlinear ordinary differential equations using Galerkin’s method. The resulting equations are solved via two different techniques: the pseudo-arclength continuation method and direct time integration. The frequency-response curves of the system and the bifurcation diagrams of Poincare maps are analyzed.

Stability and bifurcations of an axially moving beam with an intermediate spring support

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton’s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincare maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.

크랙을 가진 탄성지지된 유체유동 외팔파이프의 동적 안정성

The dynamic stability of elastically restrained cantilever pipe conveying fluid with crack is investigated in this paper. The pipe, which is fixed at one end, is assumed to rest on an intermediate spring support. Based on the Euler-Bernoulli beam theory, the equation of motion is derived by the energy expressions using extended Hamilton's Principle. The crack section is represented by a local flexibility matrix connecting two undamaged pipe segments. The influence of a crack severity and position, mass ratio and the velocity of fluid flow on the stability of a cantilever pipe by the numerical method are studied. Also, the critical flow velocity for the flutter and divergence due to variation in the support location and the stiffness of the spring support is presented. The stability maps of the pipe system are obtained as a function of mass ratios and effect of crack.

Dynamics of cantilevered pipes conveying fluid. Part 2: Dynamics of the system with intermediate spring support

This paper, the second in a three-part series, deals with the three-dimensional (3-D) nonlinear dynamics of a vertical cantilevered pipe conveying fluid, additionally constrained by arrays of four or two springs or a single spring at a point along its length. Theoretical calculations are presented for the same pipe but different spring configurations, points of attachment and stiffnesses, the main generic difference being this: in some cases, the system loses stability by planar flutter, and thereafter performs two-dimensional (2-D) or 3-D periodic, quasiperiodic and chaotic oscillations; in other cases, the system loses stability by divergence, followed at higher flows by oscillations in the plane of divergence or perpendicular to it, again periodic, quasiperiodic or chaotic. Experiments were conducted for some of the systems studied theoretically, and agreement is found to be generally good, although some open questions remain.

Effects of damping on the dynamic stability of a rod with an intermediate spring support subjected to follower forces

The equation of motion in matrix form of an Euler beam, including the effect of internal damping subjected to non-conservative follower forces, is formulated based on Lagrangian approach and the assumed mode method. The beam is assumed to rest on an intermediate spring support of large stiffness modeling a rigid support. The effect of slight damping on the jump phenomenon for the critical follower load with respect to variation in the support location is examined for a specific example of a simply supported-free rod. For some cases, the modes of instability in the form of flutter or divergence without damping are found to be unaffected by the presence of slight damping, although there may be a sharp decrease in the first critical load for instability by flutter. However, for specific locations of the intermediate spring support, the presence of a small amount of damping may change the mode of instability from divergence to flutter with a sharp decrease in the critical follower load. An interesting finding is that this mode of instability by flutter only occurs for an isolated and narrow range of follower loads.

Dynamic stability of a rod with an intermediate spring support subject to subtangential follower forces

The equation of motion in matrix form of an Euler beam subject to a tip-concentrated subtangential follower force is formulated based on Lagrangian approach and the assumed mode method. The beam is assumed to rest on an intermediate spring support. The critical flutter load is first computed for a simply supported-free rod on an intermediate spring support of large stiffness modeling a rigid support. Convergence of the smallest critical load is found to be fast with just five terms for the assumed deflection function. Jump phenomenon for the critical follower load is found to occur for both variations in the support location, the stiffness of the spring support, as well as the parameter for subtangentiality. Detailed history of the changes in the load-frequency curves due to variation in these parameters is presented to explain the occurrence of the jump phenomenon. In respect of changes in the parameter for subtangentiality for a rod on an intermediate spring support, it is found that the rod can undergo multiple transition from divergence to flutter, and then back to divergence when the subtangentiality changes from zero to unity.

Divergence and flutter of a cantilever rod with an intermediate spring support

The equation of motion in matrix form of a cantilever Euler beam subject to a tipconcentrated follower force at the free end is formulated based on the Lagrangian approach and the assumed mode method. The non-conservative nature of the system is identified by the non-symmetric matrix in the equation of motion. The beam is assumed to rest on an intermediate spring support. Jump phenomenon for the critical follower load is found to occur for both variations in the support location as well as the stiffness of the spring support. Detailed history of the changes in the load-frequency diagram due to variation in the support location and the stiffness of the spring support is presented to explain the occurrence of the jump phenomenon. An interesting finding is that the rod is found to be extremely unstable for a certain combination of spring stiffness and support location for a cantilever rod.

Nonlinear Dynamics of a Fluid-Conveying Cantilevered Pipe with an Intermediate Spring Support

In this paper, the nonlinear planar dynamics of a vertical cantilevered pipe conveying fluid are explored, in the presence of an intermediate spring support, by means of a two-degree-of-freedom Galerkin discretization of the flexible system. The stability of the original equilibrium is examined first, and the regions in the parameter space where the system is stable or loses stability by divergence or flutter are determined. Then, by examining the nonlinear equations of motion, the stability of the other fixed points that emerge with increasing flow velocity is studied, for various system parameters, revealing a very rich bifurcational behaviour. The nonlinear dynamics is also studied in the vicinity of various bifurcations by means of centre manifold theory and normal form reduction, as well by numerical simulation, in the vicinity of picthfork, Hopf and double degeneracy bifurcations; local and global behaviour are explored. Finally, the dynamics in the presence of harmonic perturbations in the flow is investigated numerically in the neighbourhood of the double degeneracy, where heteroclinic orbits arise, and chaotic regions are shown to exist.

Non-conservative stability of intermediate spring supported columns with an elastically restrained end support

The non-conservative stability of an intermediate spring supported uniform column clastically restrained at one end and subjected to a follower force at the other unsupported end is studied. It is found that when the intermediate spring support is far from the unsupported end, the instability mechanism is flutter. As the intermediate spring support approaches the unsupported end, the instability mechanism is changed from flutter to divergence with the increase of intermediate spring stiffness. For the hinged-intermediate and guided-intermediatc spring supported columns, the critical buckling load of flutter instability will first decrease, then increase as the intermediate spring stiffness is increased. Nevertheless, when the instability mechanism is divergence, the critical buckling load depends on the location of the intermediate spring support only, whereas for the clamped-intermediate spring supported column the critical buckling load of divergence instability decreases monotonically to a fixed value as the intermediate spring stiffness is increased. Finally, the influence of elastic end restraints on the stability of the column is also investigated.

Buckling of Bars with Variably Positioned Supports

In considering the instability of members subjected to axial compressive loads, several classical examples of support conditions are usually mentioned, hinged-hinged, clamped-free, clamped-clamped, and clamped-hinged. The corresponding problems of members that are transversely supported at arbitrary points and that are evidently encountered in practice, while not representing any particular analytic difficulty, appear infrequently in the literature. A typical case, presented is that of a simply supported rod with an intermediate elastic spring support at an arbitrary point.