Cantilever Bar Research Articles

Instrument for Measuring Milk Coagulation in Cheese Vats

A rugged and sensitive instrument measured properties of milk coagulation during cheese manufacture. The instrument consisted of a probe made from a stainless steel rod, 20cm long and .62cm in diameter, attached perpendicularly to a 15cm diameter, stainless steel disc. The probe was attached to a stainless steel line, an adjustment reel, and the end of a cantilever bar. A strain gauge, mounted on the cantilever bar, measured the resistance of the milk to movement of the probe. Forces on the probe varied with standing-wave motion of milk, coagulation, curd firming, curd syneresis, and clean-in-place operations. The curd adhered to the disc as continuous measurements were made, even in direct-acid cottage cheese curd. The instrument was applied successfully to Swiss, Cheddar, and cottage cheese manufacturing processes in both open and closed cheese vats. Curd strength at cutting varied three to four times in commercial operations. The instrument provided continuous curd firmness measures. A 10-cm diameter disc was preferred in 8-liter containers where proximity of side wall prevented adequate force dispersion from the 15-cm disc. A 5-cm diameter disc detected coagulation of ultrafiltered milk retentate.

Inelastic post-buckling of tapered flexible bars

Large deformations of buckled elasto-plastic flexible cantilever bars of variable stiffness with rectangular and circular cross-sections are considered. An improved Euler method is used to solve the resulting first-order, nonlinear differential equations. Deflections at the free end, and hence the values of critical loads are presented for the tapered bars.

Two fundamental cases in the non-linear bending of a straight bar

A general analytical approach of the non-linear problem of a linear elastic and isotropic straight bar of variable stiffness is indicated. The linear equivalence method, introduced by one of the authors, is applied to two fundamental cases for the isostatic straight bar, i.e. the cantilever bar (a Cauchy type problem) and the simply supported bar (a bilocal problem). Some numerical examples concerning moderate deformations and rotations are presented.

Diffraction constants for pressure gradient transducers

Conventionally the diffraction constant D of a receiving transducer (microphone or hydrophone) is defined as the ratio of the average sound pressure acting on the blocked diaphragm to the free-field sound pressure. This definition is inadequate for transducers that have complex vibration patterns, and, in the absence of ad hoc modifications, it will fail to give the correct results for pressure gradient transducers. A new definition based on the general theory of transducers with fixed velocity distributions is therefore introduced. The diffraction constant is assigned not only the function of accounting for scattered wave effects but also the function of extracting from the surface pressure distribution the component that drives the velocity distribution (or vibration pattern) of the transducer. The diffraction constant depends on the choice of the reference velocity that is associated with the acoustic port of the transducer. A spatial rms reference velocity is defined, and it is shown to be a suitable reference for pressure gradient transducers. Calculation of the diffraction constant is carried out for the following piezoelectric pressure gradient hydrophones: a flexural disk supported inside an annular mass, an end supported flexural bar, a cantilever bar, an oscillating rigid cylinder, and an oscillating rigid sphere. The use of the diffraction constant in conjunction with the electroacoustic circuit of the hydrophone to calculate voltage receiving response is described, and the circuit information needed to illustrate the procedure is given for the flexural disk hydrophone.

Parametric resonance by combined frequencies of cantilever bars under periodic axial load by the elastic-joint-method

Parametric resonance by combined frequencies of cantilever bars under periodic axial load by the elastic-joint-method

On the extensional vibrations of rotating bars

The extensional equations of motion for a cantilever bar rotating about an axis fixed in space are derived. It is shown that the form of the non-linear strain-displacement relation is important in determining the nature of the relationship between the frequency of extensional oscillations and the rotational speed. In particular, the frequency may or may not increase monotonically with rotational speed, depending on the degree of hardening in the effective extensional spring. The determination whether an instability occurs as the rotational speed increases is beyond the limits of engineering beam theory.

Optimization of Continuous One-Dimensional Structures under Steady Harmonic Excitation

The paper examines some questions relating to the optimal design of continuous one-dimensional structures driven by harmonically oscillating loads. Optimal-control methods are applied to a cantilever bar driven sinusoidally by an axial force at its tip to illustrate the minimum-weight design of one-dimensional structures under dynamic excitation. Realistic constraints are imposed during the optimizations, including a maximum allowable stress amplitude at any point along the bar and a minimum cross-sectional area. It is shown that in the absence of damping, the design space may contain many disjoint feasible regions, and multiple optima can exist. Detailed solutions are obtained for continuous bars with an excitation frequency less than, and then greater than, the fundamental free-vibration frequency. It is found that above a certain excitation frequency, two or more arcs with different constraints characterize the optimal designs. It is concluded that when more than two different constrained arcs characterize the optimal solution, the continuum approach may be impractical, and finite-element approximations may offer the only alternative.

Buckling of Cantilever Bars on Elastic Foundations

Legendre polynomials in conjunction with the Galerkin procedure are used to determine the buckling loads of cantilever bars supported laterally by an elastic foundation. The numerical results of example problems converge rapidly, and the use of the Legendre polynomials to represent the buckled deformation is found to be very desirable. Although the example problem considers only the axial compression to vary along the length of the bar, the general procedure can be extended to account for cases having variable cross sections or foundation stiffness, or both.

Stability of a rotating cantilever subjected to dissipative, aerodynamic, and transverse follower forces

The problem of determining the state of stability of a slender cantilever rotating at a uniform speed and subjected to aerodynamic, dissipative, and transverse follower forces is investigated. The equations of motion for the flexural-torsional deformations of a slender rotating cantilever bar are derived from a conservation law that embodies the field equations for the disturbed state of stress in a solid rotating about a fixed axis relative to the initial state of stress associated with the undisturbed form of equilibrium of the rotating system. The associated non-self-adjoint boundary value problem is determined, and a suitable adjoint variational principle is derived. This variational principle serves as the basis for determining approximately the complex eigenvalues and hence the critical flutter values of the transverse follower force for prescribed values of the geometric and material properties of the system. Several graphs which illustrate the variation of the critical flutter load with the speed of rotation, hub radius, the internal damping parameters, aerodynamic load parameters, and the warping rigidity are presented.

Snap-buckling of nonlinearly-elastic finite element bars

In this study, the material nonlinearities are added to the classical elastica problem of large deflections of buckled bars. This class of problems is treated by the finite-element matrix displacement method. The formulations developed are appropriate for a midpoint-tangent incremental procedure and coordinate transformation. The material nonlinearity is incorporated in the stiffness matrix explicitly for any nonlinearly-elastic material if its stress-strain or moment-curvature relation is known. An example of the postbuckling of a nonlinearly-elastic cantilever bar is illustrated with results in agreement with an alternative numerical solution obtained by Oden and Childs [1]. The example is further illustrated with the inclusion of various initial deflections to the bar.

Corrosion of 18Ni 180 and 18Ni 200 Maraging Steels in Chloride Environments

Abstract 18Ni 180 and 18Ni 200 maraging steels have shown high resistance to stress corrosion cracking in sea water. U-bends had no cracks after exposure for over three years. 18Ni 180 welds also had good resistance, but welds in the 18Ni 200 grade suffered stress corrosion cracking. Cathodic protection of U-bends was beneficial, although a sensitivity to hydrogen was observed at very negative potentials. Fracture toughness measurements in sea water and 3-1/2% NaCl gave KISCC values of 130 ksi (in)1/2 for the 18 Ni 180 steel and 94 ksi (in)1/2 for 18Ni 200. The corresponding values for welds were 122 ksi (in)1/2 and 70 ksi (in)1/2. Fracture toughness decreased when the precracked cantilever bars were coupled to zinc or, in the case of welds, held at −0.85 volt vs SCE. General corrosion rates of maraging steels in sea water and marine atmospheres were approximately half those of HY-80 and 4340 low alloy steels.

Stability of a cantilevered bar subjected to a transverse follower force of fluid jet

The present study is concerned with the theoretical and experimental investigations of the stability of a cantilevered bar subjected to the influence of a fluid jet that acts transversely at the free end. Fluid is carried along the length of the bar by means of a rubber hose placed suitably to avoid asymmetry and an attachment at the free end changes the direction of the fluid flow to achieve the desired result. Thus, in addition to having a transverse follower force at the end, the system is also subjected to the force of Coriolis acceleration.

Effects of damping on stability of elastic systems subjected to nonconservative forces

A question of the correlation between stability and quasistability regions of elastic and viscoelastic systems subjected to nonconservative forces is discussed. On the base of the method of expansion in fractional powers of parameters the more rigorous considerations are presented than the considerations used in the earlier papers where the semiintuitive assumptions, the arguments by analogy and the incomplete induction methods were applied widely. In the first part of the paper a number of general statements concerning both continuous and discrete systems are proved. It is shown that for real laws of damping a considerable part of the quasistability region belongs in fact to instability region. From this point of view a number of papers dealing with non-conservative stability problems (including panel flutter problems) must be reconsidered. To illustrate the general statements, in the second part of the paper a numerical examination of stability of cantilever bar made of the standard viscoelastic solid and subjected to follower and dead forces is presented. Some phenomena inherent to the nonconservative viscoelastic systems are discussed.

Stability of a Bar Under Eccentric Follower Force

Uniform cantilevered bar subject to eccentric compressive follower force, considering warping rigidity, bending-torsional flutter and stability

Instability modes of cantilevered bars induced by fluid flow through attached pipes

A cantilevered bar of uniform cross-section with flexible pipes conveying fluid attached to it is considered. It is shown that for certain cross-sections of the bar stability may be lost by either torsional divergence (torsional buckling) or torsional flutter, depending upon the location of the pipes with respect to the center of gravity of the cross-section. In addition, transverse flutter can also occur, but not transverse buckling. It is shown further that the Coriolis forces, which are present due to the motion of the fluid in vibrating pipes, may have either a stabilizing or a destabilizing effect, depending upon the parameters of the system.

Upper and Lower Bounds of Frequencies for Cantilever Bars of Variable Cross Sections

Upper and lower bounds of frequencies of transverse vibrations of cantilever bars of variable cross sections are presented, taking into account the effects of transverse shear and of rotatory inertia. Numerical results for the first four natural frequencies are presented in tabular form for different inertia characteristics of the bars.

Lateral buckling of a cantilever subjected to a transverse follower force

In this note we study the stability of bending-torsional equilibrium of a cantilevered bar subjected at its end section to a follower force; this case may represent a wing subjected to the jet of a turbine-engine.Making use of the dynamic analysis, we can determine the critical thrust.

Transverse Vibrations of Cantilever Bars of Variable Cross Section

Approximate formulas are derived for upper and lower values of the three-lowest natural frequencies for transverse vibrations of cantilever bars of variable cross section. For cones, truncated cones, wedges, and truncated wedges, as well as bars of constant cross section, the effects of rotatory inertia and shear are taken into account also and compared with the results obtained by neglecting these effects according to the elementary Bernoulli-Euler's theory of bending of beams. Numerical results for the first three natural frequencies are presented in tabular form for different inertia characteristics.

Response of Beams and Plates to Random Loads

Abstract With the use of generalized harmonic analysis the problem of vibrating damped beams and plates under stochastic loading is solved. The resulting equations give the cross-correlation functions for displacements, stresses, moments, and so on, in terms of the cross-correlation function of external pressure. Mean square values of these functions are special cases of these results. Using a method due to Rice, we also calculate the probable number of times per unit time the random displacements or stresses will exceed a given value. The case of simply supported bars, cantilever bars, clamped circular plates, and simply supported rectangular plates are worked out in detail.

Apparatus for the Production and Measurement of Small Motions

A cantilever bar for producing small motions of the order of 10−5 cm and greater is described together with an interferometer for measuring these motions. Two methods of using the interferometer are discussed. The first uses an uninterrupted monochromatic light source in which the interference fringes are observed to be visible and stationary for particular amplitudes of oscillation. The second method uses a stroboscopic light source synchronized with the oscillating bar. Motions of the cantilever bar measured by these two methods are compared with each other and with the theory of the cantilever bar.