The ability of a structural assembly to carry loads and forces determines how stable it will be over time. Viewing structural assemblages as comprising columns, beams, arches, rings, and plates, this book will introduce the student to both a classical and advanced understanding of the mechanical behavior of such structural systems under load and how modeling the resulting strains can predict the overall future performance?the stability?of that structure. While covering traditional beam theory, the book is more focused on elastica theory in keeping with modern approaches. This text will be an expanded and updated version a similar, previously published book, but with pedagogical improvements and updated analytical methods. This engineering textbook will provide a focused treatment on the study of how structures behave and perform when under stress loading, including plastic deformation and buckling. All advanced engineering students studying engineering mechanics, structural analysis and design, fatigue and failure, and other related subjects need to have this knowledge, and this book will provide it in a thorough and coherent fashion. Written by two of the world?s leading engineering professors in this subject area, the pedagogy has been classroom-tested over many years and should find a receptive readership among both students and instructors.

Fundamentals of Structural Stability

On transverse vibrations of a vertical Timoshenko beam

In this study the stability of a rotor system, which is comprised of a simply supported nonlinear spinning shaft with multi-rigid disk, near to the major critical speeds is investigated. The nonlinearity is due to the stretching and large amplitude. The influence of rotary inertia and gyroscopic effects are included, however, shear deformation is ignored. To analyze the nonlinear equations of motion, the method of multiple scales is applied to the ordinary differential equations of motion. The influences of different parameter such as number of disks, disk mass moment of inertia, rotational speed, external damping, and position of disks on the forward and backward linear frequencies, steady state response, stability and bifurcations of the rotor system are investigated. It is seen that in the higher rotational speeds, the backward frequency is increasing with an increase of number of disks, and in the lower rotational speeds, the backward frequency is decreasing with an increase of number of disks. By increasing number of disks, bifurcations occur in the lower speeds therefore, the instability occurrence for large number of disk is at speeds lower than that regarding to the low number of disks. By an increase of disk mass moment of inertia, the amplitude and the hardening effect decrease.

Nonlinear vibration analysis of a spinning shaft with multi-disks

A historical account of the contributions of the Aeroelasticity Branch (AB) and the Langley Transonic Dynamics Tunnel (TDT) to rotorcraft technology and development since the tunnel's inception in 1960 is presented. The paper begins with a summary of the major characteristics of the TDT and a description of the unique capability offered by the TDT for testing aeroelastic models by virtue of its heavy gas test medium. This is followed by some remarks on the role played by scale models in the design and development of rotorcraft vehicles and a review of the basic scaling relationships important for designing and building dynamic aeroelastic models of rotorcraft vehicles for testing in the TDT. Chronological accounts of helicopter and tiltrotor research conducted in AB/TDT are then described in separate sections. Both experimental and analytical studies are reported and include a description of the various physical and mathematical models employed, the specific objectives of the investigations, and illustrative experimental and analytical results.

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A Historical Overview of Aeroelasticity Branch and Transonic Dynamics Tunnel Contributions to Rotorcraft Technology and Development

In this paper, we investigate the nonlinear vibration of a metamaterial structure that consists of a rotating cantilever beam attached to a periodic array of spring–mass–damper subsystems deployed for vibration suppression. The full nonlinear model of the system is developed. The nonlinear response due to a primary resonance excitation is investigated, and the capability of the metastructure to suppress vibration is examined. The mass of the resonators (absorbers) comes at the expense of the host structure’s mass itself, which makes the total mass of the system conservative. Free and forced vibration analyses are performed. We first use the method of multiple scales to analyze the nonlinear behavior of rotating beams. The perturbation solutions are validated against their numerical counterparts. Results show the presence of a critical rotational speed at which the beam undergoes bifurcation and starts to flutter. The addition of the absorbers is observed to slightly reduce this critical speed. Nevertheless, the amplitude of limit-cycle oscillations beyond bifurcation is found to decrease when equipping the rotating beam with local absorbers. The results demonstrate the capability of the metamaterial structure as an efficient damping treatment. Furthermore, we show that careful placement of the absorbers along the cantilever beam (close to the tip) enables further vibration mitigation.

Vibration suppression of nonlinear rotating metamaterial beams

We consider the connection between vibration and buckling problems for a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The rod is assumed off-clamped, i.e. the axis of rotation does not pass through the rod's clamped end. The resulting fourth-order boundary value problem with a turning point for the free vibrations is solved using uniform approximations in a transitional parameter range where high rotation rates balance small off-clampings. Second approximations to the vibration eigenvalues are used to determine critical buckling rotation rates for the slightly off-clamped rods. Buckling is unexpected in this situation as the rod is wholly under tension.

Vibration and buckling of rotating flexible rods at transitional parameter values

This work considers a group of problems associated with rotating Timoshenko beams. The beam is not assumed to be hubclamped, i.e. the axis of rotation does not necessarily pass through the beam's clamped end. Cases of physical interest involving off-clamped beams include wobbling rotors, impellor blades, and turbine blades. For clamped-free boundary conditions, we seek solutions of the governing equations which correspond to transverse buckling. For the rotor, it is known that Euler-Bernoulli beams do not have buckled modes. By contrast, the Timoshenko beam will have an infinite number of buckled modes. In the impellor blade case, both Euler-Bernoulli and Timoshenko beams will have an infinite number of buckled modes. However, the Timoshenko beam will buckle at a lower eigenrotation speed. This is also true for the case of a rotating Timoshenko beam with clamped-clamped boundary conditions, e.g. a turbine blade clamped at both the rim and hub of a rotating platform. Analytic results for both the clamped-free and clamped-clamped cases are augmented by results obtained from numerical solution of the corresponding boundary value problems.

Transverse buckling of a rotating Timoshenko beam

Integrating and differentiating matrices allow the numerical integration and differentiation of functions whose values are known at points of a discrete grid. Previous derivations of these matrices have been restricted to one-dimensional grids or to rectangular grids with uniform spacing in at least one direction. The present work develops integrating and differentiating matrices for rectangular grids with non-uniform spacing in both directions. The use of these matrices as operators to reformulate boundary-value problems on rectangular domains as matrix problems for a finite-dimensional solution vector is considered. The method requires non-uniform grids which include “near-boundary” points. An eigenvalue problem for the transverse vibrations of a simply-supported rectangular plate is solved to illustrate the method.

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W. D. Lakin

Papers

Vibration and buckling of rotating flexible rods at transitional parameter values

Transverse buckling of a rotating Timoshenko beam

A combined integrating and differentiating matrix formulation for boundary value problems on rectangular domains