Cantilever Rod Research Articles

Bending of magnetic filaments under a magnetic field

Magnetic beads and superparamagnetic (SP) colloid particles have successfully been employed for micromechanical manipulation of soft material, in situ probing of elastic properties, and design of smart materials (ferrogels). Here we derive analytical expressions for the equilibrium shape of magnetic fibers, considering two end-member cases, (a) SP or single-domain particles concentrated at the free end of cantilevered rods or tubes, and (b) filaments consisting of SP particles, with this case being mathematically equivalent to tubes containing SP particles. Our analysis yields also metastable equilibrium states (MES's), which only exist above a critical filament length, but become more stable with increasing magnetic field. The MES's for case (a) are, like the ground state, circular arcs, but more strongly bent. The multiform MES's in case (b), which comprise hairpin, sinuous, or even closed shapes, have recently been observed in experiments, too. We also study the effect of gravity on the balance between bending and magnetic energy, which leads to curves with inflection point if the influence of gravity is stronger than that of the magnetic field. Because of their simple experimental realization, case (a) magnetic filaments are deemed highly suitable for micromechanical experiments on long chains of polymer molecules. Another potential application of cantilevered magnetic filaments with magnetic material attached to the free end is in scanning probe microscopes. Because the magnetic field due to the magnetic tip is comparatively weak, the magnetization structure of the sample to be investigated would not be affected by the probe. Thus, for the examination of magnetically soft materials, probes in the form of magnetic filaments may hold advantages over tips usually employed in magnetic force microscopy.

Evaluation of the Critical Following Force for a Cantilever Rod

The authors use the methods of initial parameters in the matrix form to study the stability of rods under the action of the following forces. They consider rods with various fastening of the ends. The obtained theoretical results have been verified by an experiment. It is shown that the difference between the theoretical and experimental values of the critical following forces does not exceed 5%.

Post-buckling of a cantilever rod with variable cross-sections under combined load

Based on the geometrically nonlinear theory of axially extensible elastic rods, the governing equations of post-buckling of a clamped-free rod with variable cross-sections, subjected to a combined load, a concentrated axial load P at the free end and a non-uniformly distributed axial load q, are established. By using shooting method, the strong nonlinear boundary value problems are numerically solved. The secondary equilibrium paths and the post-buckling configurations of the rod with linearly varied cross-sections are presented.

ON COUPLED LONGITUDINAL AND LATERAL VIBRATIONS OF ELASTIC RODS

In this paper, the coupled longitudinal and lateral vibrations of a rectangular parallelepiped composed of a uniform, isotropic, elastic body are examined. The one-dimensional model for the body was developed by Green, Naghdi and several of their co-workers. It is based on a Cosserat or directed rod theory. The frequency spectrum is found to divide naturally into three regions which are separated by degenerate cases. When the cross-section of the body is square, a natural splitting of the modes occurs. One-half of this split involves motions coupling the axial extension and symmetric lateral deformation of the body. The other half involves asymmetric lateral deformations. The cases of a free–free rod and a cantilevered rod are discussed and the results compared to existing works in the literature which use the three-dimensional theory of elasticity to model the vibrations.

Stability Control of Active Compressed Rods with Shape Memory Polymers

In structural systems, applied compressive force fields cause second-order effects that may not only influence the buckling behavior, but also enhance the internal stress and the deformation field in the structure. Specifically, slender and thin-walled structural elements subjected to compressive force fields may experience buckling instability, which may affect their service life. In the case of imperfection-sensitive systems, loading and geometric imperfections may drastically affect the buckling response and reduce the buckling resistance. This behavior may be positively influenced by a structural adaptation to bending effects. In this connection, an attempt is made to use the potentials of the adaptive systems to influence the second-order as well as the buckling response of the slender compression elements by means of the actuator components. To this end, the stability control problem of adaptive slender rods with embedded Shape Memory Polymer (SMP) actuators is theoretically treated. The main goal of this investigation was to explore the possibility of minimizing the end deflection of a cantilever rod under eccentrically applied force via actuation of an SMP fiber. First, the stability theory of spatial rods is outlined, and the non-linear governing equations of straight rods are derived. The specific case treated is the problem of a cantilever rod with a single eccentric SMP fiber under eccentrically applied end compressive force. Numerical calculations were carried out with the help of the Runge-Kutta method. It is shown that the response of the rod can be effectively controlled by the appropriate action of the SMP actuator. Specifically, the goal of minimizing the end deflection of the eccentrically loaded cantilever rod through the action of the SMP actuator is achieved. In carrying out the computations, the second-order effects arising from the action of SMP fiber as well as the axial force were taken into consideration. Moreover, the follower action of the SMP force was considered.

Spectral Finite Element and Genetic Algorithm for Crack Detection in Cantilever Rod

Spectral Finite Element and Genetic Algorithm for Crack Detection in Cantilever Rod

Non-Local Periodic Motions of a Thin Cantilevered Rod

In this work we consider the existence of unforced periodic motionsfor a thin cantilevered elastic rod. In particular, largeamplitude motions that combine bending and twisting of the rodare studied. We use a lumped mass model consisting of a cantilevered planar rod whose base is attached to a torsional spring. The stability of the torsional vibration is studied and then the existence of a oneparameter family of large amplitude periodic motions is demonstrated.

Oscillations of a Prismatic Rod

Equations are obtained for the longitudinal and transverse oscillations of a prismatic rod that differ from the standard equations of structural mechanics (strength of materials). The difference lies both in the appearance of new terms in the equations of motion corresponding to the Timoshenko theory, and in a refinement of the boundary conditions and the conditions on the rigidity of the rod for transverse oscillations. The oscillations of a vertical cantilevered rod when its base moves horizontally from a state of rest are examined and a comparison is made with the classical theory of transverse (bending) oscillations of prismatic rods.

Flutter of a cantilever rod with a relocatable lumped mass

The dynamic stability of a cantilever rod modeled as an Euler beam with an arbitrarily located lumped mass is investigated in this work. The rod is subjected to a follower thrust at the free end. The effect of the presence of slight internal damping on the critical flutter load with respect to the variation in the location of the lumped mass is examined. It is found that an abrupt variation in the critical flutter load may occur with a slight variation in the location of the lumped mass for a rod without damping. Such a variation is less pronounced with the presence of slight internal damping.

A new formulation for the torsional vibration analysis of rotating cantilever rods

This paper presents a new modeling method for the analysis of torsional vibration of rotating cantilever rods. The natural frequency and mode shape variations due to the rotational motion could be accurately estimated with the modeling method, and the coefficient for the well-known Southwell equation could be obtained, as well. This method has couple of advantages compared to previous conventional modeling methods. Different from the previous modeling methods, the equations of motion of the rotating cantilever rod were derived with consistent linearization in a rigorous way. An eigenvalue problem to obtain the coefficients of the Southwell equation of rotating rods were derived from the original eigenvalue problem.

A general nonlinear structural model of a multirod (multibeam) system—II. Results

A numerical model for the non-linear structural analysis of multirod systems was developed in Part I of the paper. In the second part this model is used in order to investigate various examples of multirod systems. At first the model is applied to investigate the plane (Elastica) and nonplane behavior of straight cantilevered rods under various loads. In order to cope with increasing non-linearity, the physical rod is divided into subrods, thus enabling the accurate analysis at any level of nonlinearity. Good agreement is obtained between the present results and other theoretical and experimental results from the literature. Considering a real multirod system, the model is used to analyze the non-linear structural behavior of a two-rods system. The numerical results are compared with experimental results that were obtained during the course of the present research. Good agreement is presented between the numerical and experimental results. At last the influence of a flexible joint between the rods of a two-rods system is investigated.

試料共振型磁力計による微小磁気モーメント測定

A new type of highly sensitive magnetometer called a resonating sample magnetometer (RSM) is described. A magnetic sample is mounted on the end of a cantilever rod that incorporates a flat spring with a pick-up coil. The sample is magnetized in a de field, -5 kOeHa-7 emu, and a complete hysteresis loop can be measured in about 15 min.

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.

Chaotic non-planar vibrations of the thin elastica: Part I: Experimental observation of planar instability

In this two-part paper, the results of an investigation into the non-linear dynamics of a flexible cantilevered rod (the elastica) with a thin rectangular cross-section are presented. An experimental examination of the dynamics of the elastica over a broad parameter range forms the core of Part I. In Part II, the experimental work is related to a theoretical study of the mechanics of the elastica, and the study of a two-degree-of-freedom model obtained by modal projection. The experimental system used in this investigation is a rod with clamped-free boundary conditions, forced by sinusoidally displacing the clamped end. Planar periodic motions of the driven elastica are shown to lose stability at distinct resonant wedges, and the resulting motions are shown in general to be non-planar, chaotic, bending-torsion oscillations. Non-planar motions in all resonances exhibit energy cascading and dynamic two-well phenomena, and a family of asymmetric, bending-torsion non-linear modes is discovered. Correlation dimension calculations are used to estimate the number of active degrees of freedom in the system.

Large Rotatory Oscillations of Transversely Isotropic Rods: Spatio-Temporal Symmetry-Breaking Bifurcation

Time-periodic motions of an initially straight, transversely isotropic, cantilever rod, modeled as a general nonlinearly elastic Cosserat curve, are considered. By inspection, the problem possesses $O(2)$ spatial symmetry and, due to the requirement of time periodicity, $O(2)$ temporal symmetry (invariance under time shifts, modulo the period, and time reversal). First, the mathematical realization of these symmetries is investigated in terms of transformation groups acting on the governing partial differential equations (PDEs), which ultimately simplifies the task of finding (special classes of) solutions. The main point of this paper is twofold: The process just described (i) is difficult to implement in this problem (we provide an unconventional reformulation of the Cosserat model that mitigates the difficulty), and (ii) uncovers new solutions. More specifically, the existence of solutions, invariant under a certain “twisted” subgroup of the spatio-temporal symmetry group is established. These solution...

Toward the problem of static deformations in curvilinear rods

The article presents expressions for deformation tensor components, potential energy, differential equations, and boundary conditions as applied to a curvilinear rod with axis curved along a circle of given radius and subject to the effect of concentrated tensile and bending loads. Analytical solutions of three boundary value problems on bending and tension of a cantilever rod with axially constant cross section are given under some simplifying assumptions. Characteristics of the stress-strain state of curvilinear rods are noted for various methods of loading.

Transverse vibrations of prismatic rods by accounting for shears

Derivations of a differential equation of eight order and boundary conditions of flexural vibrations of a straight rod by accounting for shear strains caused by a transverse load are presented. Unlike existing approaches, use of coefficients of shear and angles of displacements is excluded in all relationships. The article gives the solution of the boundary-value problem on determining the function of plane displacement of sections with transverse shear for a round profile without the traditional introduction and additional determination of the shear function. An exact analytical solution of the boundary-value problem on free flexural vibrations of a cantilever rod by accounting for shears is presented. A numerical example of calculating flexural vibration parameters of a rod with round cross-sectional profile is examined along with evaluation of the effect of shear strain factor on frequency of the principal tone.

Cantilever Rod in Cross Wind

A thin elastic rod is held at one end in a strong cross wind. The nonlinear large deformation equations are formulated and solved by perturbation and numerical integration. The problem is governed by a nondimensional parameter K representing the relative importance of aerodynamic drag to flexural rigidity. For large K, phenomena such as nonuniqueness, instability, and hysteresis may occur.

Magnetoelastic buckling of two nearby ferromagnetic rods in a magnetic field

The interactive effect of two nearby soft ferromagnetic rods on the magnetoelastic buckling is studied. The cantilevered rods are parallel to each other. The buckling is due to a uniform magnetic field normal to the axes of the rods, making an arbitrary angle with the plane through the axes of the rods. The buckling analysis is based upon a perturbation theory in which the final (or buckled) state of the system is considered as a small perturbation of an intermediate equilibrium state, for which the rigid-body state is taken. The unknown rigid-body and perturbed fields are solved with use of the theory of complex functions. The forces of magnetic origin on the deflected rods are calculated and the buckling problem is reduced to an eigenvalue problem. It is found that the interaction between the two rods considerably reduces the buckling magnetic field in case of two nearby rods. The buckling mode turns out to be symmetric. These theoretically predicted results are confirmed experimentally. Finally, the influences of the direction of the basic magnetic field and of the predeflection (before buckling) are discussed.

An alternating-gradient magnetometer (invited)

This paper describes a type of vibrating-sample magnetometer capable of sensitivity exceeding 10−8 emu. The instrument is 1000 times more sensitive than a conventional VSM with comparable working space, and is much quicker to use than a SQUID magnetometer, which generates point-by-point data. The magnetic sample is mounted on the end of a cantilevered rod that incorporates a piezoelectric element. The sample is magnetized by a dc field (variable in magnitude), and is simultaneously subjected to a small alternating field gradient. The alternating field gradient exerts an alternating force on the sample, proportional to the magnitude of the field gradient and to the magnetic moment of the sample. The resulting deflection of the cantilever rod is measured by the voltage output of the piezoelectric element. By operating at or near a mechanical resonance frequency of the cantilever, the output signal is greatly amplified. In practice, the operating frequency is 100–1000 Hz, with mechanical Q values of 25–250. Mechanical and acoustic noise in the environment limits the sensitivity. Measurements have been made with a signal-to-noise ratio of about 500 on a 25-μm sphere with a moment of 3.7×10−6 emu; this corresponds to a sensitivity of at least 10−8 emu. A complete hysteresis loop over ±10 kOe can be made in about 100 s. Measurements have been made over a temperature range from 77 to 400 K.