Torsion Function Research Articles

Analytic torque model for two-axis microelectromechanical system mirrors

The availability of recently developed microelectromechanical system (MEMS) micro-mirror technology provides an opportunity to replace macroscale actuators for free-space laser beam steering in light detection and ranging and communication systems. Precision modeling of mirror pointing and its dynamics are critical to the design and control of MEMS beam steerers. Beginning with Hornbeck's torque approach, we present a first-principles, analytically closed-form torque model for an electrostatically actuated two-axis (tip-tilt) MEMS mirror structure. The torque expression is a function of the mirror's physical parameters, such as angles, voltages, and size. An Euler dynamic equation formulation describes the gimballed motion as a pair of damped harmonic oscillators with a coupled torsion function. Static physical parameters such as MEMS mirror dimensions and voltages are inputs to the model as well as dynamic harmonic oscillator parameters, such as damping and restoring constants, which are calculated or fitted to measurements. A Taylor series expansion of the torque function provides insights into MEMS behavior, including operational sensitivities near "pull-in." MATLAB and SIMULINK simulations illustrate performance sensitivities, controllability, physical limitations, and other important considerations in the design of precise pointing systems. Commercial-off-the-shelf micromirror measurements confirm the model's validity in steady state and dynamic scanning operations.

Contribution of the Pericardium to Left Ventricular Torsion and Regional Myocardial Function in Patients with Total Absence of the Left Pericardium

The relationship between left ventricular (LV) torsional deformation and myocardial function has recently been recognized. However, little is known about whether the pericardium affects this relationship. Our aim was to identify the contribution of the pericardium to LV torsion and regional myocardial function in the clinical setting. We examined LV torsion in basal and apical LV short-axis views, and regional LV myocardial function, such as longitudinal strain in apical 4-chamber view, and circumferential and radial strains in parasternal LV short-axis views using 2-dimensional speckle-tracking imaging method in 5 patients with congenital total absence of the left pericardium and systolic paradoxical ventricular septal motion on M-mode echocardiogram and in 10 control subjects. Diagnosis of the pericardial defect was based on chest radiograph, computed tomography, jugular phlebogram, and M-mode and 2-dimensional echocardiogram. LV torsion was defined as the net difference in LV rotation in the basal and apical planes. There was no significant difference in LV ejection fraction determined by 2-dimensional echocardiography between the pericardial defect and control groups. LV torsion was markedly decreased in the pericardial defect group compared with the control group. There were no significant differences in longitudinal, radial, and circumferential systolic strains and systolic and early diastolic strain rates in the LV walls and in longitudinal systolic strains and systolic and early diastolic strain rates in the left atrial walls between the two groups. Pericardial defects cause a lack of LV torsion while maintaining LV regional myocardial function in patients with systolic paradoxical ventricular septal motion. Therefore, pericardium plays an important role in LV torsion.

Surface waves created by low-frequency magnetic fields

This paper analyses the effects of a low frequency A.C. magnetic field on the free surface of a liquid metal. The action of the vertical and uniform magnetic field is twofold. First it creates forced standing surface waves which generally exhibit symmetry related to that of the container; second it triggers non-symmetric free surface instabilities superimposed on the forced regime. A previous paper considered the case of a circular cylindrical tank where axisymmetric forced standing waves caused an electric current perturbation which then excited non-axisymmetric waves at a critical A.C. field intensity. Nonlinear interaction between the symmetric and non-symmetric modes was not taken into account. The present work treats the problem from a more general standpoint. Equilibrium perturbations are developed systematically to order N 2 (where N is the magnetic interaction parameter) and at this level of approximation we also need to consider nonlinear mode interactions and electromagnetic damping. The theory applies to tanks of arbitrary shape and the O( N) irrotational motion may be described by the torsion function for the particular pool cross-section. For circular and annular tanks we then derive a system of coupled Mathieu–Hill equations for the time-development of non-symmetric surface modes. Two main types of parametric resonance are predicted, namely the single or combination mode, and the particular type observed may depend on the geometry of the tank. Results of the stability analysis are confirmed by experimental work carried out in mercury pools.

Torsion of uniform bars with polygon cross-section

The solution of the torsion problem of uniform bars with polygon cross-section by means of the Trefftz integral for the complex torsion function is presented. The governing equations consist of the first-order integral of the Schwarz-Christoffel mapping, and another third-order integral, known as the Trefftz integral. The known series solution of the Schwarz-Christoffel integral is used. The solution of Trefftz's integral is carried out in series form according to the method of von Koppenfels and Stallmann, by the separation into regular and singular corner functions. The result is in the form of explicit corner functions, Frobenius and Taylor series. The series solutions are computed with convolution number algebra. An additional Fourier analysis allows one to compute the shear in a polygon cross-section with a round hole. Torsion stiffness is computed by integrals along the boundary. An estimate of the effect of rounding of corners is given.

Old and New on the Bass Note, the Torsion Function and the Hyperbolic Metric

Old and New on the Bass Note, the Torsion Function and the Hyperbolic Metric

A Hadamard-jensen inequality and an application to the elastic torsion problem

The (generalised) torsion function u of a domain Ω Rn is a function which is zero on the boundary of the domain and whose Laplacian is minus one at every point in the interior of the domain. Denote by │Ω│ the measure of Ω xc its centroid. We establish, for convex Ω Various improvements, generalisations and possible applications are discussed

Torsional rigidity of reinforced concrete bars with arbitrary sectional shape

In this paper, the torsional rigidity of arbitrarily shape bar made of different materials is studied on the basis of theory of elasticity and finite element approach. With additional boundary conditions for the common boundaries of different materials from the continuous conditions of deformation and traction across the interior boundary, the torsion function can be solved numerically from the second boundary-value problem of potential theory. The traction jump boundary conditions across the interior surfaces are enforced in the alternate finite element approach. Several examples are shown to check the computational approach proposed, and the approach, at last, is applied to calculate the torsional rigidity of reinforced concrete bar and some multiply connected cross sections such as tower leg section of the Tsing Ma Bridge and other engineering structures.

Peculiarities of the vibrations of shrouded blades in power turbines

1. The necessity to account for restraining the torsion of the blade section for vibrations of shrouded blades irrespective of how long they are made has been shown. The experimental and calculated data presented testify to the fact that the law of distribution of the normal stresses in the peripheral section of a shrouded blade vibrating mainly torsionally coincides with the law of variation of the torsion function along the periphery of the blade section. 2. The experimental and calculated data confirm that, in the computations of the natural frequencies and modes of vibrations, it is necessary to consider that there is a complete absence of warping of the peripheral section of the blade. The condition of equality to zero of the bimoment at the peripheral section holds good for the vibrations of unshrouded blades. In the case of shrouded blades, this condition leads to a significant error in the determination of the stressed state. 3. It is shown that relatively high, mainly bending, stresses occur in the peripheral section of rotating shrouded blades particularly with interunit vibrations of the rim. In view of this, it is very advisable to take steps in the design stage itself to reduce the stress levels in the peripheral section of a shrouded blade.

Determining the residual stresses in plastically deformed tubes subjected to combined bending and torsion

i. The necessity to account for restraining the torsion of the blade section for vibrations of shrouded blades irrespective of how long they are made has been shown. The experimental and calculated data presented testify to the fact that the law of distribution of the normal stresses in the peripheral section of a shrouded blade vibrating mainly torsionally coincides with the law of variation of the torsion function along the periphery of the blade section.

On the torsion of a cylinder with several cracks

In this paper, based on paper [1], the analytic expression of the torsion function for a cylinder containing arbitrary oriented cracks is obtained. The problem is reduced to solve a system of singular integral equations for the unknown dislocation density functions. Using the numerical method of the singular integral equations[2,7] the torsional rigidities and stress intensity factors are evaluated for several multicracked cylinders. Next, the creak-cutting method[5] is firstly extended to lve the torsion problem for a rectangular prism. The numerical results show that the method presented here is successful.

Torsion of Composite Bars by Boundary Element Method

A boundary element (B.E.) solution for the Saint‐Venant torsion problem for composite cylindrical bars of arbitrary cross section is presented. The composite bar consists of a cylindrical matrix material surrounding a finite number of inclusions with different shear moduli firmly bonded to it. The problem is formulated in terms of the torsion function, which is established by solving a Neumann‐type boundary value problem. Moreover, the torsional rigidity of the composite cross section and the shear stress components are evaluated. Several numerical examples are worked out and the results are compared with those available from analytical or other numerical solutions. The case of the homogeneous cross section with or without holes is obtained as a special case. The efficiency of the B.E. method is also demonstrated and examined.

Isoperimetric Inequalities in the Torsion and Clamped Membrane Problems for Convex Plane Domains

Bounds for the curvature of the level curve of the torsion function through an arbitrary point in a convex region D, are used to derive improved isoperimetric inequalities for maximum stress, the torsional rigidity and other functionals. These inequalities are exact if D is either a circle or an infinite strip. A similar procedure is used in the clamped membrane problem, and again improved isoperimetric inequalities are derived.

Flexure of beams with certain curvilinear cross sections

Special complex variables techniques are used to obtain the six flexure functions (one of which is the torsion function) of a certain isotropic cylinder under flexure. The cross section is bounded by the closed curver=a sin4(θ/4) (−π 1).

Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function

SynopsisIn this paper maximum principles are employed to relate solutions of certain classes of nonlinear elliptic problems to solutions of the associated torsion problem. By this method a number of new isoperimetric inequalities are derived. In special cases solutions of the nonlinear problems are also related to solutions of the clamped membrane problem.

On the torsion function, Green's function and conformal radius: An isoperimetric inequality of Pólya and Szegö, some extensions and applications

On the torsion function, Green's function and conformal radius: An isoperimetric inequality of Pólya and Szegö, some extensions and applications

Axially Symmetric Stress Distributions in Elastic Solids Containing Penny-Shaped Cracks Under Torsion

We present the axially symmetric stress distributions in elastic solids containing a pair of axially symmetric penny shaped cracks when the infinite elastic medium is kept under torsion. We derive the integral representation formula for the torsion function and the expressions for the stress-intensity factors.

Cauchy–Schwarz Inequality and the Evaluation of Torsional Rigidity of Anisotropic Bars

Bounds for torsional rigidity of anisotropic bars with one plane of elastic symmetry perpendicular to the axis of the bar are derived making use of the Cauchy–Schwarz inequality. The lower bound is obtained by formulating the problem in terms of the Prandtl stress function, while the formulation in terms of the torsion function supplies the upper bound. An illustrative example shows that even a single term first approximation leads to relatively close bounds.

A Theoretical Method for Solving an Elasto-Plastic Bending Problem

On the elasto-plastic torsion problem, many investigations have been made by following approaches. In the case of perfect-plastic body, the relaxation methods have been used and numerical calculations have been fulfilled adopting the non-linear stress-strain relation.In this paper, authers suggest and analytical method to solve theoretically the elasto-plastic torsion problem. At first, we employ the new parameters (τ', α), where τ' and α are the resultant shear stress derived from Saint-Venant's torsion function and the argument of τ' respectively. Then the basic equation for the elastic torsion problem is reduced to Laplace's equation in term of these parameters (τ', α). Developing the above (τ', α)-theory, a solution for elastoplastic torsin is obtained using the total strain theory of plasticity. At last, we can describe the torsion function (ω) in the form of the hypergeometric series, provided that the Ramberg-Osgood's law is employed as the non-linear stress-strain relation.

Elastic-viscoplastic torsion for a Sokolovsky's oval section

An approximate method of solution is used for the quasi-static torsion problem of a viscoplastic bar. After linearization, the governing equation is integrated and reduces to aPoisson equation. If the curvilinear coordinate system, made of the contours and the generators of the corresponding torsion function for a perfectly plastic material, is used, a closed form solution for the time-dependent torsion function can be found. The smoothness condition for the torsion function can also be established. As an example, theSokolovsky's oval is solved under the time-dependent twist. A variational equation can be seen similar to the case of perfectly plastic material.

The classical torsion problem for sections with curvilinear boundaries

This paper is concerned with the application of complex variable analysis to obtain solutions for the Saint-Venant torsion problem corresponding to certain sections with regular curvilinear boundaries. The area of the cross-section is conformally mapped on the unit circle by a transformation involving some parameters. By varying these parameters various shapes having several axes of symmetry are obtained. Sections bounded by n (> 2) equal and very approximately circular arcs are included as particular cases. Closed and exact expressions are derived for the complex torsion function, the torsional rigidity, and the shearing stresses at any point of the section. The distribution of shearing stress on the boundary is investigated in the general case and graphs showing its variation in four particular examples are plotted.