Dynamic coefficient of an elastically supported,pre-stressed beam

Summary An analysis is made of the problem of vibrations of a beam with an axial force resting on eltstic foundation, when the beam is uniform and of finite length and is subjected to an impulsive load. The solution is presented within the framework of a beam theory which includes the effects of shear deformation and rotary inertia. An example is provided where the dynamic coefficient for the bending moment is calculated. From the results of theoretical analysis, it becomes evident that the axial force in the beam and the stiffness of the foundation have considerable effect upon the dynamical behaviour of the system.

---

Übersicht Die Schwingungen eines auf elastischer Grundlage liegenden Balkens, der durch eine Axialkraft beansprucht ist, werden für den Fall analysiert, daß der Balken gleichförmig und von endlicher Länge ist, und daß eine Schlagbelastung an dem Balken angreift. Die Lösung wird innerhalb der Balkentheorie gegeben, die die Wirkung der Schubverformung und die Drehträgheit berücksichtigt. In einem Beispiel wird der Schwing-beiwert für das Biegemoment ermittelt. Die Ergebnisse zeigen, da\ die Axialkraft und die Steifigkeit der elastischen Grundlage auf das dynamische Verhalten des Systems erheblichen Einfluß haben.

---

---

The author wishes to acknowledge the consistent guidance and encouragent of Professor H, Saito of Tohoku University.     

On the dynamic behaviour of elastic plastic beams simply supported and axially loaded

Summary An elasto-plastic beam-column model withn degrees of freedom statically loaded with an axial compressive force is considered. Its dynamic behaviour in the elasto-plastic range, resulting from the action of a transversal distributed force periodically varying in time in condition of resonance with the structure in the elastic range, is examined. The principal differences with previous results obtained with one degree of freedom model are showed.

---

Sommario Si considera un modello adn gradi di libertà di un'asta elasto plastica sollecitata staticamente da una forza assiale di compressione. Se ne esamina la risposta, in campo elasto plastico, ad un carico trasversale distribuito, variabile con legge armonica ed in condizioni di risonanza con la struttura in campo elastico, evidenziando le principali differenze rispetto alle conclusioni tratte in precedenza utilizzando schemi ad un grado di libertà.

---

---

The present investigation has been promoted and financed by the Consiglio Nazionale Ricerche (C.N.R.) at the Istituto di Scienza e Tecnica delle Costruzioni del Politecnico di Milano.     

On the dynamic behaviour of a beam with an accelerating mass

Summary The dynamic behaviour of an Euler beam traversed by a moving concentrated mass, is analyzed for the general case of a mass moving with a varying speed. The equation of motion in a matrix form is formulated using the Lagrangian approach and the assumed mode method. The dimensionless form of the equation enables the numerical results to be applicable for a wide range of system parameters. The possibility of the mass separating from the beam is analyzed by examining the contact forces between the mass and the beam during the motion.     

The action of a concentrated force and a concentrated electric charge in an infinite transversally isotropic piezoceramic medium

An explicit solution of the problem of electroelasticity is constructed for a transversally isotropic medium under the action of a concentrated force and a concentrated electric charge. The cases where the force acts along the anisotropy axis and along the isotropy axis are considered. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 80–87, April, 2000.     

On the buckling of a girder, elastically supported and elastically clamped in a number of equidistant points

Summary Lately the authors have been engaged in an inquiry with respect to the safety of scaffolding constructions, as used in the erection and the repair of large buildings. A question of detail was presented by the determination of the buckling load of a prismatic girder, elastically supported and elastically clampled in a number of equidistant points. The present paper gives an answer to this problem with the understanding that the endpoints of the girder — though elastically clamped — are fixedly supported.     

Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. In order to highlight the symmetry breaking phenomenon and the approach to its analysis, the subsidiary simplified problem of a curved beam attached to a linearly elastic foundation, and subjected to uniformly distributed “mechanical” load, which is independent of deflections, is addressed first. The analysis is based on a two degrees of freedom reduced order (RO) model resulting from the Galerkin decomposition with linear undamped eigenmodes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.     

On the dynamic behaviour of a flexible beam carrying a moving mass

The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes.The ordinary differential equations of motion using the improved shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the improved model to the un-improved equations.     

Solution of a viscoelastic boundary-value problem on the action of a concentrated force in an infinite plane

We formulate a theorem containing the solution of a boundary-value problem of isotropic linear viscoelasticity on the action of a concentrated force in an infinite plane. The two creep functions that are used in the constitutive relations and correspond to the shear and bulk expansion states and assumed to be independent; the general forms of these functions are not specified. Formulas are presented for the stress, strain, and displacement components.     

Symmetry breaking in an initially curved pre-stressed micro beam loaded by a distributed electrostatic force

The symmetric and asymmetric buckling of an initially curved micro beam subjected to an axial pre-stressing load and transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam used as the base functions. The criteria of symmetric limit point buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beam and the axial load. Two symmetry breaking conditions, defining the relations between the axial load and the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one, are obtained. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric response curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric response curve. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict symmetric and non-symmetric buckling in electrostatically actuated curved pre-stressed micro beams. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. The nature of the latter and the relations between it and the symmetry breaking criteria are found to go through a prominent qualitative change as the initial distance between the beam and the electrode, characterizing the electrostatic force, changes.     

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The nonlinear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge–Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance (10⁻⁵). The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.     

The stress state of a viscoelastic orthotropic half-plane loaded with a concentrated force

The operator continued-fraction method is used to solve the problem on the stress state of a viscoelastic orthotropic half-plane loaded at a momentt=0 with a force normal to the boundary of the half-plane. The stress field in the half-plane is determined based on the Volterra principle and the solution of the corresponding elastic problem. The influence of the rheological parameters on the stress state of the half-plane is shown by an example of a specific material. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 124–130, February, 2000.     

Radiation emitted by a constant load moving uniformly in a circle on an elastically supported membrane

Wave radiation is studied which is due a constant load moving with a constant speed along a circular path over an unbounded membrane on a elastic foundation. The steady-state solution of the problem is obtained, showing that the radiation occurs for all load velocities. It is shown that the elastic field radiated by the supercritically moving load is confined in a spiral-like apex. The membrane displacements at the boundaries of this apex are discontinuous. The radiated energy per period of load rotation is calculated showing a discrete energy spectrum. For increasing load velocities, the total amount of radiated energy becomes larger. It also turns out that the major part of the radiated energy follows the direction of the load motion.     

Nonlinear dynamic force transmissibility of a flywheel rotor supported by angular contact ball bearings

The dynamic force transmissibility (DFT) of aerospace flywheel rotor system (FRS) supported by angular contact ball bearings (ACBBs) is examined in this paper. The influence of combined loads and contact angle variation is considered in the Sjovall formula to accurately solve the load distribution and nonlinear stiffness of ACBB. Subsequently, the lateral vibration model of FRS is established by considering the nonlinear stiffness characteristics of ACBB. The DFT of the system is solved via harmonic balance method and arc length continuation, and the stability of the results is determined. Numerical integration and dynamic tests are utilized to verify the accuracy of harmonic balance results. Based on the proposed model, the effects of rotor unbalance excitation, axial preload, and rotor damping on the DFT of the system are discussed. The soft-stiff transition phenomenon is observed in terms of the varying supporting stiffness of ACBB wherein deformation is measured under axial preload. The value of rotor unbalanced mass determines the nonlinear characteristics of FRS. The results provide an important reference for dynamic performance evaluation and vibration isolation device design of aerospace FRS.

     

On the dynamic stability of an elastically fixed high-drag airfoil under vortical parametric excitations

Conditions of dynamic instability at parametric resonance in an oscillatory system with three degrees of freedom, which models the motion of an elastically fixed high-drag airfoil under unsteady aerohydrodynamic loads due to the vortex trail separation, are obtained in the linear approximation.     

Energy consumed by a bearing supported spindle in the presence of a dynamic cutting force and of defects

A machine tool spindle system model is proposed in this paper to investigate the non-linear face-milling cutting forces behavior, which are neglected in the literature, in order to predict the total mechanical power of a spindle. A simulation of the structure of the spindle based on the finite-element method is elaborated to estimate the variable cutting forces and then the variable power loss generated by bearings, considering the angular position and contact angles of the variable balls. Experiments are elaborated to compare the experimental power values with the predicted results. Particular attention is paid to different types of defects (inner ring spalling, outer ring spalling, eccentricity, and unbalance) in order to study their impact on the power consumed by the spindle during the approach and cutting phases under different rotating conditions.