Optimal design of thin walled I beams for extreme natural frequency of torsional vibrations

The optimal design of thin-walled I beams so as to extremize the natural frequency of torsional vibration is considered. It is assumed that only one dimension of the cross-section, except for the web height, may be variable in given limits, along the axis of the beam. The optimality condition for the variable dimension is settled by means of Pontryagin's maximum principle. The effect of the constant, axial loads is also included. the solution of the problem formulated is generally found in an iterative way. Some numerical examples of optimization of the I beam with variable widt of flanges are given.     

Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

Minimum weight design of beams in torsional vibration with several frequency constraints

The design of beams of cantilever form carrying and end inertia so as to minimize the total mass subject to the constraint that one, two or three of its torsional natural frequencies are fixed at specified values is considered. The problem is stated in variational form with the constraints introduced through Lagrange multipliers. The problem is taken analytically as far as possible and reduces to a set of first order non-linear differential equations. These are integrated numerically. The known solutions for less constrained problems are used as a basis from which to iterate to the required solutions. Optimum profiles and tables of numerical data computed for various examples are given.     

Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effects

An exact analytical method is presented for predicting the undamped natural frequencies of beams with thin-walled open cross-sections having no axis of symmetry. The governing differential equations give a characteristic equation of the 12th order with real coefficients. The roots are found numerically and the exact boundary conditions are considered especially for free ends to obtain natural frequencies. The simpler cases of neglecting cross-sectional warping and/or rotary inertia are also dealt with. It is seen that when the effect of rotary inertia is neglected significant errors incur for some boundary conditions, cross-section thicknesses and mode numbers. This is more profound when the warping effect is taken into account.     

Upper and lower bounds for fundamental natural frequency of beams

A better upper bound than the Rayleigh quotient is the Timoshenko quotient, the evaluation of which depends on a pair of compatible admissible moment and displacement functions. Based on both Rayleigh and Timoshenko quotients, a lower bound is readily computed. By means of an iteration procedure, both the upper and lower bound converge to the fundamental natural frequency.     

Optimal design of beams under flexural vibration

The minimum weight design of a cantilever beam in flexural vibration is considered. The aim is the maximization of a given natural bending frequency (usually the first) for a given beam weight or equivalently the minimization of beam weight for a specified value of a natural frequency. The beams considered are of rectangular section and are subject, in a range of cases presented, to a variety of constraints on lower and upper bounds on the cross-section dimensions or to the specification of a point mass at the end of the beam. Simple bending theory is regarded as applicable to the problem. A variational statement of the problem is made and the necessary conditions for a minimum are obtained as a system of non-linear equations which are solved numerically. Results are given in the form of tables and of figures showing computed optimum profiles. Some experiments on a sample set of beams of equal mass are described briefly. The optimum profile beam was found to have the greatest fundamental frequency, in support of the theoretical predictions.     

A potential model for torsional vibrations of molecules

The Schrödinger equation with a periodic potential which is the sum of two cosine functions admits simple analytic solutions for the ground and the first few excited states of a rotating body. This potential may be used to describe the torsional oscillations of a molecule with m and 2m-fold barriers.     

Finite difference analysis of torsional vibrations of pretwisted,rotating, cantilever beams with effects of warping

Theoretical natural frequencies of the first three modes of torsional vibration of pre-twisted, rotating cantilever beams are determined for various thickness and aspect ratios. Conclusions concerning individual and collective effects of warping, pretwist, tension-torsion coupling and tennis racket effect (twist-rotational coupling) terms on the natural frequencies are drawn from numerical results obtained by using a finite difference procedure with first order central differences. The relative importance of structural warping, inertial warping, pretwist, tension-torsion and twist-rotational coupling terms is discussed for various rotational speeds. The accuracy of results obtained by using the finite difference approach is verified by a comparison with the exact solution for specialized simple cases of the equation of motion used in this paper.     

Non-linear vibrations of three-layer beams with viscoelastic cores I. Theory

Approximate equations of motion are developed for large amplitude motions of three-layer axially restrained unsymmetrical beams with viscoelastic cores. The external force consists of a constant plus an oscillatory term. The combination of this form of forcing and the large amplitude motions cause the beam to respond at multiples of the forcing frequency. This can lead to difficulties in the complex modulus approach to viscoelasticity. These are overcome here through use of hereditary integrals and their relationships with complex moduli. Theoretical results on the frequency response of clamped, symmetrical beams are compared with earlier experimental work. On the whole, reasonable agreement is found.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

Optimal design of beams in torsion under periodic loading

The optimal design of beams in torsion under harmonically varying torques is discussed. The analysis covers the cases when the excitation frequency is either less than or greater than the fundamental frequency of the beam. The beams analyzed are in the main assumed to have rectangular cross-section but the theory is easily extended to other section shapes. In each case the problem is stated in variational form with the introduction of constraints through Lagrange multipliers. The mathematical analysis of the various problems presented results in a system of non-linear differential equations with associated boundary conditions. The solutions given for some of the cases provide expressions for the design variable and the response, along the length of the beam, in terms of the forcing frequency and some constants which can be determined for the particular problem. The computed results and data are given in tabular form and some optimum profiles are shown graphically.     

Analysis of torsional vibrations of an ethane molecule

Based on the solution for the Mathieu equation, we obtained wave functions for the internal rotation of an ethane molecule that satisfy the symmetry properties of the group G₃₆. We calculated the frequency of the principal torsional transition (273 cm⁻¹). The types of symmetry of the energy levels and transition probabilities in IR and Raman spectra are determined. We note drawbacks of divisible permutable-inverse nuclear groups and the groups of molecular symmetry associated with their construction, as well as difficulties that appear when equivalent rotations are used. The possibility of avoiding the application of an extended group is indicated. Belarusian State University, 4, F. Skorina Ave., Minsk, 220050, Belarus. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 64, No. 1, pp. 26–31, January–February, 1997.     

简明设计复合扭振变幅杆

传统设计复合扭振变幅杆的方法是利用各形状函数杆之间的角位移连续和力矩连续的边界条件,来确定方程中的待定系数,导出频率方程。本文利用各形状函数分界之间机械阻抗相等的方法来设计复合扭振变幅杆。可简化设计,物理意义明显。这种方法也适用于纵振复合变幅杆的设计。     

Non-linear free vibrations of inextensible beams

Non-linear free vibrations of inextensible clamped-free and free-free beams are analyzed by using Galerkin's method and the harmonic balance method.     

Transverse vibrations of pre-twisted sandwich beams

A set of equations of motion governing the bending and extensional displacements of a pre-twisted sandwich beam of rectangular cross-section are derived by using Hamilton's principle. The middle viscoelastic core is assumed to deform mainly through the classical shearing mechanism. The eigenvalues and loss factors of simply supported pre-twisted sandwich beams are computed by using the variational method. Analysis of the results revealed that pre-twisting the beam increases the real part of the eigenvalue by as much as 20% while reducing the loss factor by as much as 30 %. The loss factor of very soft, thickcored beams is especially sensitive to even small angles of pre-twist: e.g., a 22· 5° pre-twist may reduce the loss factor by as much as 80%. The effect of pre-twist is, however, shown not to be appreciable for soft, thin-cored beams. In any case, pre-twisting of the beam has a detrimental effect on the maximum loss factor that one can obtain for a specific size of the beam when only the shear parameter of the beam is changed.