Sen Yung Lee

Nonlinear Analysis on Hunting Stability for High-Speed Railway Vehicle Trucks on Curved Tracks

Based on the heuristic nonlinear creep model, the nonlinear coupled differential equations of the motion of a ten-degree-of-freedom truck system, considering the lateral displacement, the vertical displacement, the roll and yaw angles of the each wheelset, and the lateral displacement and yaw angle of the truck frame, moving on curved tracks, are derived in completeness. To illustrate the accuracy of the analysis, the limiting cases are examined. The influences of the suspension parameters, including those losing in the six-degree-of-freedom system, on the critical hunting speeds evaluated via the linear and nonlinear creep models, respectively, are studied and compared.

Exact Vibration Solutions for Nonuniform Timoshenko Beams with Attachments

The exact solution for the free vibration of a symmetric nonuniform Timoshenko beam with tip mass at one end and elastically restrained at the other end of the beam is derived. The two coupled governing characteristic differential equations are reduced into one complete fourth-order ordinary differential equation with variable coefficients in the angle of rotation due to bending. The frequency equation is derived in terms of the four normalized fundamental solutions of the differential equation. It can be shown that, if the coefficients of the reduced differential equation can be expressed in polynomial form, the exact fundamental solutions can be found by the method of Frobenius. Finally, several limiting cases are studied and the results are compared with those in the existing literature. A (x) E(x) G(x) I(x) J(x)

Stability of a Timoshenko beam resting on a Winkler elastic foundation

The influences of a Winkler elastic foundation modulus, slenderness ratio and elastically restrained boundary conditions on the critical load of a Timoshenko beam subjected to an end follower force are investigated. The characteristic equation for elastic stability is derived. It is found that the critical flutter load for the cantilever Timoshenko beam will first decrease as the elastic foundation modulus is increased and when the elastic foundation modulus is greater than the corresponding critical value, which corresponds to the lowest critical load, it will increase, instead. In particular, if the elastic foundation modulus is large enough, the critical flutter load for the cantilever Timoshenko beam can be greater than that of the Bernoulli-Euler beam. For a clamped-translational or clamped-rotational elastic spring supported beam resting on an elastic foundation, there exists a critical value of the spring constant for each beam. At this critical point, the critical load jumps and the type of instability mechanism changes. The jump mechanisms for beams resting on elastic foundations of different modulus values are different.

Elastic stability of non-uniform columns

Abstract A simple and efficient method is proposed to investigate the elastic stability of three different tapered columns subjected to uniformly distributed follower forces. The influences of the boundary conditions and taper ratio on critical buckling loads are investigated. The critical buckling loads of columns of rectangular cross section with constant depth and linearly varied width ( T 1 ), constant width and linearly varied depth ( T 2 ) and double taper ( T 3 ) are investigated. Among the three different non-uniform columns considered, taper ratio has the greatest influence on the critical buckling load of column T 3 and the lowest influence on that of column T 1 . The types of instability mechanisms for hinged-hinged and cantilever non-uniform columns are divergence and flutter respectively. However, for clamped-hinged and clamped-clamped non-uniform columns, the type of instability mechanism for column T 1 is divergence, while that for columns T 2 and T 3 is divergence only when the taper ratio of the columns is greater than certain critical values and flutter for the rest value of taper ratio. When the type of instability mechanism changes from divergence to flutter, there is a finite jump for the critical buckling load. The influence of taper ratio on the elastic stability of cantilever column T 3 is very sensitive for small values of the taper ratio and there also exist some discontinieties in the critical buckling loads of flutter instability. For a hinged-hinged non-uniform column ( T 2 or T 3 ) with a rotational spring at the left end of the column, when the taper ratio is less than the critical value the instability mechanism changes from divergence to flutter as the rotational spring constant is increased. For a clamped-elastically supported non-uniform column, when the taper ratio is greater than the critical value the instability mechanism changes from flutter to divergence as the translational spring constant is increased.

Deflection of nonuniform beams resting on a nonlinear elastic foundation

Abstract The static deflection of a general elastically end restrained non-uniform beam resting on a non-linear elastic foundation subjected to axial and transverse forces, governed by a non-linear fourth order non-homogeneous ordinary differential equation with variable coefficients, is examined. By using the method of perturbation, the governing differential equation is transformed into a set of self-adjoint linear fourth order ordinary differential equations with variable coefficients. It is shown that the deflection of the beam can be expressed in terms of the fundamental solutions of these linear ordinary differential equations. Especially if the coefficients of the linear fourth order ordinary differential equations are in an arbitrarily polynomial form, then the exact solution for the static deflection of the beam can be obtained.

Analysis of non-uniform beam vibration

Abstract In this paper a systematic development of the solution theory for the non-uniform Bernoulli-Euler beam vibration, including both forced and free vibrations, with general elastically restrained boundary conditions, is presented. The frequency equation and dynamic forced response, which is shown in closed integral form, are concisely expressed in terms of the fundamental solutions of the system. If the exact, closed form fundamental solutions are not available, then approximate fundamental solutions can be obtained through a simple and efficient numerical method. The present analysis can also be applied to the vibrational analysis of a beam with viscous and hysteretic damping.

Free vibrations of a non-uniform beam with general elastically restrained boundary conditions

Abstract A simple and efficient method is presented to study the problem of free vibration of a non-uniform Bernoulli-Euler beam with general elastic restraints at boundary points. With one set of particularly chosen fundamental solutions of the system, the frequency equation is derived and expressed in concise form. If the closed form fundamental solutions of the system are not available, the approximate fundamental solutions can be obtained through a newly developed algorithm which is shown to be efficient, convenient and accurate. Finally, several limiting cases of the general system are examined and some examples are presented to illustrate the validity and accuracy of the proposed method.

Dynamic Analysis of Nonuniform Beams With Time-Dependent Elastic Boundary Conditions

The dynamic response of a nonuniform beam with time-dependent elastic boundary conditions is studied by generalizing the method of Mindlin-Goodman and utilizing the exact solutions of general elastically restrained nonuniform beams given by Lee and Kuo. The time-dependent elastic boundary conditions for the beam are formulated. A general form of change of dependent variable is introduced and the shifting polynomials of the third-order degree, instead of the fifth-order degree polynomials taken by Mindlin-Goodman, are selected. The physical meaning of these shifting polynomial functions are explored. Finally, the limiting cases are discussed and several examples are given to illustrate the analysis.

The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

Abstract The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamiltons principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.

Free Vibrations of a Rotating Inclined Beam

By utilizing the Hamilton principle and the consistent linearization of the fully nonlinear beam theory, two coupled governing differential equations for a rotating inclined beam are derived. Both the extensional deformation and the Coriolis force effect are considered. It is shown that the vibration system can be considered as the superposition of a static subsystem and a dynamic subsystem. The method of Frobenius is used to establish the exact series solutions of the system. Several frequency relations that provide general qualitative relations between the natural frequencies and the physical parameters are revealed without numerical analysis. Finally, numerical results are given to illustrate the general qualitative relations and the influence of the physical parameters on the natural frequencies of the dynamic system.