Dynamic buckling of simple two-bar frames using catastrophe theory

Non-linear static and dynamic elastic buckling of simple imperfect two-bar frames, treated as continuous systems, are analyzed with the aid of catastrophe theory using a comprehensive and readily employed procedure. Static catastrophes are extended to the corresponding dynamic catastrophes of undamped frames under step loading (autonomous systems) by properly determining the dynamic singularity and bifurcational sets. Attention is focused on fold and cusp catastrophes. A local analysis based on Taylor's expansion of the non-linear equilibrium equation of the frame allows us: (a) to classify the total potential energy function of the frames to the canonical form of the corresponding universal unfolding of the seven elementary Thom's catastrophes, and (b) to easily obtain static and dynamic buckling loads, critical points (singularity sets) and related imperfection sensitivities (bifurcational sets). An illustrative example associated with a static and dynamic fold catastrophe demonstrates the efficiency and reliability of the methodology proposed herein.     

Dynamic Stability Problems of Anisotropic Cylindrical Shells via a Simplified Analysis

An analytical–numerical method involving a small number of generalized coordinates is presented for the analysis of the nonlinear vibration and dynamic stability behaviour of imperfect anisotropic cylindrical shells. Donnell-type governing equations are used and classical lamination theory is employed. The assumed deflection modes approximately satisfy simply supported boundary conditions. The axisymmetric mode satisfying a relevant coupling condition with the linear, asymmetric mode is included in the assumed deflection function. The shell is statically loaded by axial compression, radial pressure and torsion. A two-mode imperfection model, consisting of an axisymmetric and an asymmetric mode, is used. The static-state response is assumed to be affine to the given imperfection. In order to find approximate solutions for the dynamic-state equations, Hamiltons principle is applied to derive a set of modal amplitude equations. The dynamic response is obtained via numerical time-integration of the set of nonlinear ordinary differential equations. The nonlinear behaviour under axial parametric excitation and the dynamic buckling under axial step loading of specific imperfect isotropic and anisotropic shells are simulated using this approach. Characteristic results are discussed. The softening behaviour of shells under parametric excitation and the decrease of the buckling load under step loading, as compared with the static case, are illustrated.     

Non-linear buckling of simple models with tilted cusp catastrophe

Non-linear buckling universal solutions of simple multiple-parameter discrete models are discussed via a comprehensive and readily employed procedure using Catastrophe Theory. Attention is focused on perfect models whose total potential energy (TPE) function, upon small disturbance breaking symmetry, reduces to the universal unfolding of the tilted cusp catastrophe. A local analysis based on simple approximations allows us to classify the TPE universal unfolding of any model to one of the seven elementary Thom's catastrophes by defining the corresponding control parameters. Subsequently, using global analyses one can obtain exact results for establishing the non-linear equilibrium paths: (a) of the “perfect” perturbed model (due to the small effect of an extra parameter) with the corresponding “imperfect” bifurcation and limit points(s), and (b) of the imperfect models (resulting after inclusion of the effect of normal imperfection parameters) together with the corresponding to each parameter limit points. Moreover, conditions for the direct evaluation of (non-degenerate) hysteresis points associated with a tilted cusp point in the control parameter plane, are established. Numerical results illustrate the methodology proposed herein.     

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k- model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of U _max/U _bar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.List of symbols C ₁ , C ₂ ,C turbulence model constants - F _x body force - k kinetic energy of turbulence - l _m mixing length - L length of the diffuser - u, v, w rms value of the fluctuating velocity - u, v, w turbulent component of the velocity - mean velocity in the x direction - _A average velocity at inlet - U _bar average velocity in any cross section - U _max maximum velocity in any cross section - V mean velocity in the y direction - W local width of the diffuser at any cross section - x, y coordinates - dissipation rate of turbulence - _m eddy diffusivity - Von Karman constant - mixing length constant - _l laminar viscosity - _eff effective viscosity - v kinematic viscosity - density - _k effective Schmidt number for k - effective Schmidt number for - stream function - non dimensional stream function     

Plastic buckling of cylindrical shells under biaxial loading

The predictions for plastic buckling of shells are significantly affected by the plasticity model employed, in particular in the case of nonproportional loading. A series of experiments on plastic buckling of cylindrical aluminum alloy shells under biaxial loading (external pressure and axial tension), with well-defined loading and boundary conditions, was therefore carried out to provide experimental data for evaluation of the suitability of different, plasticity models. In the experiments, initial imperfections and their growth under load were measured and special attention was paid to buckling detection and load path control. The Southwell plot was applied with success to smooth the results. The results show that axial tension decreases resistance to buckling under external pressure in the plastic region due to softening of the material behavior. Comparison with numerical calculations usingJ ₂ deformation and incremental theories indicate that both theories do not predict correctly plastic buckling under nonproportional loading.Babcock (SEM Member), deceased, was Professor of Aeronautics and Applied Mechanics, California Institute of Technology, Pasadena, CA 91125.     

Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences

We construct a suspension of Smale's horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-hòlder or template—a branched two-manifold with a semiflow—in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional quadratic like maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of one-dimensional bifurcations with a subsequence of area-preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos!     

Taylor–Couette flow with independently rotating end plates

Results are presented from a combined numerical and experimental study of steady bifurcation phenomena in a modified Taylor–Couette geometry where the end plates of the flow domain are allowed to rotate independently of the inner cylinder. The ends rotate synchronously and the ratio between the rate of rotation of the ends _e and the inner cylinder _i defines a control parameter :=_e/_i. Stationary ends favour inward motion along the end walls whereas rotating walls promote outward flow. We study the exchange between such states and focus on two-cell flows, which are found in the parameter range between =0 and =1 for =2. Hence is used as an unfolding parameter. A cusp bifurcation is uncovered as the organizing centre for the stability exchange between the two states. Symmetry breaking bifurcations, which lead to flows that break the mid-plane symmetry are also revealed. Overall, excellent agreement is found between numerical and experimental results. PACS 47.20, 47.11, 47.54     

A General Method for Estimating Dynamic Parameters of Spatial Mechanisms

Dynamic equations of motion require a large number of parameters for each element of the system. These can include for each part their mass, location of center of mass, moment of inertia, spring stiffnesses and damping coefficients. This paper presents a technique for estimating these parameters in spatial mechanisms using any joint type, based on measurements of displacements, velocities and accelerations and of external forces and torques, for the purpose of building accurate multibody models of mechanical systems. A form of the equations of spatial motion is derived, which is linear in the dynamic parameters and based on multibody simulation code methodologies. Singular value decomposition is used to find the essential parameter set, and minimum parameter set. It is shown that a simulation of a four-bar mechanism (with spherical, universal, and revolute joints) and based on the estimated parameters gives accurate response.     

Overall stability of pin-jointed frameworks after the onset of elastic buckling

Conclusions The total potential energy surfaces (V-surfaces) in the adjacent neighborhood of the initial position of the two perfect pin-jointed frames studied in examples 1 and 2 are confined to a three-dimensional space since only two of the members were assumed to be on the point of buckling. If however more than two members of a frame can contract flexurally, the representation of the V-surfaces is not so simple. The negative regions of the V-surface are of primary interest in the study of the post-buckling hehaviour of a frame since then an accelerated motion of the system away from the initial position of the frame may ensue. Therefore in the case of a multi-dimensional surface the location of the negative regions of the surface and the evaluation of the corresponding unstable modes of the given frame may be carried out more conveniently analytically using the expressions given by (2.8) and (2.9).In order to prove that the system is in unstable equilibrium, when the frame is on the point of buckling, it suffices to find at least one adjacent position for which V is negative. This may be a much easier task in the case of a complex frame than representing the entire shape of the surface in a multi-dimensional adjacent space so that the choice of a particular post-buckling mode or the possibility of a snap-through from one mode into another may be fully understood. Similarly, in the case of stable equilibrium of the system when a given frame is on the point of buckling, the result may be obtained quickly if the test for positive definiteness of the quadratic from in the expression for V is carried out.The effect of initial imperfections in the members becomes apparent on comparing the loaddisplacement characteristic of the frame in example 1 with the slope to the equilibrium path for the perfect frame. It is observed that adequate agreement between the two is established when the buckling mode is more fully developed.Then clearly, the initial imperfections in the members can be expected to affect the initial regions of the total potential energy surfaces, but agreement between such surfaces for perfect as well as in perfect frames may be adequate after the initial distortions of the imperfect frame have been overcome.Consequently the V surfaces as well as the stability criterion formulated for perfect frames by expressions 2.10 and 2.11 will in general give an indication of the post-buckling behavior of pin-jointed elastic frames when their members possess initial imperfections.This paper represents a part of the Investigation into the Post Buckling Behavior of Frames sponsored by the Aluminium Development Association of Great Britain and carried out by the author at Cambridge University from the years 1957 to 1960.     

Bifurcation sequences of a Coulomb friction oscillator

In some parameter ranges, the dynamics of a forced oscillator with Coulomb friction dependent on both displacement and velocity is reducible to the dynamics of a one-dimensional map. In numerical simulations, period-doubling bifurcations are observed for the oscillator. In this bifurcation procedure, the map arising from the Coulomb model may not have standard form. The bifurcation sequence of the Coulomb model is compared to that of the standard one-dimensional maps to see if it exhibits universal behavior. All observed components of the bifurcation sequence fit the universal sequence, although some universal events are not witnessed.     

On the nonlinear dynamic buckling mechanism of autonomous dissipative/nondissipative discrete structural systems

Summary Nonlinear dynamic buckling of nonlinearly elastic dissipative/nondissipative multi-mass systems, mainly under step load of infinite duration, is studied in detail. These systems, under the same loading applied statically, experience a limit point instability. The analysis can be readily extended to the case of dynamic buckling under impact loading. Energy, topological and geometrical aspects for the total potential energyV, which is constrained to lie in a region of phase-space whereV0, allow conclusions to be drawn directly regarding dynamic buckling. Criteria leading to very good, approximate and lower/upper bound dynamic buckling estimates are readily established without solving the highly nonlinear set of equations of motion. The theory is illustrated with several analyses of a two-degree-of-freedom model.     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

A comparative study on algorithms of boundary element solution for free torsional vibration of bodies of revolution

In this paper the functionu=rsin in cylindrical coordinates (r,,z) is introduced into the equation for free torsional vibration of bodies of revolution (where=v / r represents the angle of twist). With the static fundamental solution (–1 /R) a mixed BEM / FEM equation is derived. The domain integral term in the equation is discretized by Serendipity elements instead of commonly used constant value finite elements in the literature. The equation is an algebraic eigenvalue one. The dynamic fundamental solution (e ^1R /R) is also used for deriving the other mixed BEM / FEM equation. An appropriate iterative solution procedure is described. An algebraic eigenvalue equation can be obtained and its solution accuracy is almost interior meshing independent. A number of examples are studied. The results show the good economy and high accuracy of the algorithms proposed.The Project is Supported by National Natural Science Foundation of China.     

The moving, plane heat source in an elastic medium

Summary This note presents an exact solution for the stress and displacement field in an unbounded and transversely constrained elastic medium resulting from the motion of a plane heat source travelling through the medium at constant speed in the direction normal to the source plane.Nomenclature mass density - diffusivity - thermal conductivity - Q heat emitted by plane heat source per unit time per unit area - speed of propagation of plane heat source - shear modulus - Poisson's ratio - T temperature - _x, _y, _z normal stress components - u _x, u_y, u_z displacement components - c speed of irrotational waves - t time - x, y, z Cartesian coordinates - =x–vt moving coordinate     

The wedge subjected to tractions: a paradox re-examined

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2 satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for not equal but very close to 235-3, the classical solution can still be very large as approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2= and 2. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r ^- and ln r are replaced by R ^-gl and ln R where R=r/r ₀is the dimensionless radial distance and r ₀ is an arbitrary constant having the dimension of length.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Nonlinear dynamic buckling of a viscoelastic model under impact loading

A simple nonlinear buckling analysis is applied to a one-degree-of-freedom arch under impact loading in which viscous damping may also be included. Such a loading consists of a falling body striking centrally the joint mass of the arch in such a way that a completely plastic impact can be postulated. When there is no damping the exact dynamic buckling load for such a kind of loading-associated with an unbounded motion can be established by using a static criterion (approach). More specifically, it was shown that the dynamic buckling load corresponds to that unstable equilibrium state where the total potential energy of the system is zero. Furthermore, it was proved that the second variation of the total potential energy at the foregoing unstable equilibrium state is negative definite. This implies that the curve loading versus displacement resulting by the vanishing of the total potential energy has always a maximum on the afore mentioned unstable state. It was also found that the system may become sensitive to initial conditions. If damping is included the foregoing static criterion yields lower bound buckling estimates. These findings were verified by employing a highly efficient approximate technique as well as the numerical scheme of Runge-Kutta for solving any nonlinear initial-value problem.     

Calculation of transverse flow of a stream of rarefied gas past a plate

We give the results of a calculation by the Monte Carlo method of the coefficient of resistance and the field of flow past a plate placed perpendicular to a stream of rarefied gas at Mach numbers M = 2–20 and Reynolds numbers Re₀27. The calculations were carried out for two forms of the law governing the variation of the coefficient of viscosity as a function of temperature (T, T). The results are compared with available calculated and experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–112, July–August, 1976.     

Cardiovascular Mechanics: Investigation of Two Components,Tissue Heart Valves and Blood Cells

A fundamental anatomical composition of the heart valves is presented along with its relationship to the tissue mechanical behavior. During the loading and unloading phases of the tissue different stress strain pathways are followed with the curves composing the characteristic hysteresis loop, exhibiting the viscoelastic mechanical behavior of valvular tissue. The storage modulus and the phase shift (tan ) as well as the collagen modulus of human heart valves were measured in orthotropic directions using uniaxial dynamic tensile tests at 10 Hz. Viscoelastic properties of human erythrocytes are presented as calculated from micropipette aspiration experiments. Employing the hemorheometre, from filtration experiments an index of rigidity (IR) of erythrocytes is estimated. A relationship between the global parameter IR and the shear elastic modulus of erythrocyte membrane, , is established. The same two techniques adapted for leukocytes and their subpopulations have been used and a relationship between the rigidity index of leukocytes (ILR) and their apparent bulk viscosity (_app), has been found.     

Global bifurcations in the motion of parametrically excited thin plates

In this paper we investigate global bifurcations in the motion of parametrically excited, damped thin plates. Using new mathematical results by Kovai and Wiggins in finding homoclinic and heteroclinic orbits to fixed points that are created in a resonance resulting from perturbation, we are able to obtain explicit conditions under which Silnikov homoclinic orbits occur. Furthermore, we confirm our theoretical predictions by numerical simulations.