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The chapter focuses on fundamental theories, formulas, solution methods, and a set (toolbox) of MATLAB functions for the static analysis of linearly elastic bodies in two and three dimensions. The formulation and solution of an elasticity problem requires basic equations—equations of equilibrium, strain-displacement relations (geometric relations), conditions of compatibility, stress-strain relations (constitutive law), and boundary conditions. According to the strain-displacement relations, the six components of strain at a point are completely determined by the three displacement components u , v , and w . Therefore, the strain components cannot be arbitrary functions of x , y , and z . Depending on the geometric configuration of the elastic body in consideration, a two-dimensional elasticity problem usually falls into one of the two categories—plane stress and plane strain. A problem of plane strain is concerned with a long prismatic or cylindrical body. The body is subjected to external loads that are perpendicular to the longitudinal ( z ) axis and do not change along the length. It is assumed that frictionless constraints are imposed at the two ends of the body, which permit x , y deformation, but do not allow the displacement in the z direction.

The chapter reviews fundamental theories, formulas, solution methods, and a set (toolbox) of MATLAB functions for static analysis of Euler–Bernoulli beams. The system requirements for the MATLAB toolbox—a PC with Win 98SE/NT/2000 and XP or Mac with OS 9.x and up, and the software MATLAB (version 5.x and up) installed on the computer. A fundamental problem in the static analysis of Euler-Bernoulli beams is stated as the given external loads and boundary disturbances that determines the beam response (displacement, rotation, bending moment, and shear force), which is governed by a given equation and boundary conditions. There are many solution methods available for the static analysis of Euler-Bernoulli beams. The chapter presents three analytical solution methods for the beam problem—method of singularity functions, the boundary value approach, and distributed transfer function method (DTFM). The MATLAB functions are developed based on the distributed transfer function method, which is an exact analytical modeling and solution method for many one-dimensional distributed systems including bars, shafts, strings, and beams.

Presenting students with a comprehensive and efficient approach to the modelling, simulation, and analysis of dynamic systems, this textbook addresses mechanical, electrical, thermal and fluid systems, feedback control systems, and their combinations. It features a robust introduction to fundamental mathematical prerequisites, suitable for students from a range of backgrounds; clearly established three-key procedures – fundamental principles, basic elements, and ways of analysis – for students to build on in confidence as they explore new topics; over 300 end-of-chapter problems, with solutions available for instructors, to solidify a hands-on understanding; and clear and uncomplicated examples using MATLAB®/Simulink® and Mathematica®, to introduce students to computational approaches. With a capstone chapter focused on the application of these techniques to real-world engineering problems, this is an ideal resource for a single-semester course in dynamic systems for students in mechanical, aerospace and civil engineering.

The Inductrack system provides a novel way to achieve magnetic levitation by using Halbach arrays of permanent magnets (PMs). Due to the complexities of the nonlinear electro-magneto-mechanical coupling in the system, most previous analyses of the Inductrack system rely on steady-state results and consequently cannot fully capture the dynamic behaviors of the system in transient scenarios. In this article, a new three degrees-of-freedom (3DOF) transient model of the Inductrack system is proposed. This model describes the rigid-body motion of the Inductrack vehicle with axial (longitudinal) and vertical (transverse) displacements and pitch rotation, and it is derived without any assumption of steady-state quantities. Compared to a recently available 2DOF lumped-mass model developed by the authors, the inclusion of the pitch rotation in the new model results in a much more complicated mechanism of electro-magneto-mechanical coupling. Numerical results show that the pitch rotation can have a significant effect on the dynamic response and stability of the Inductrack system, which necessities vibration control for the safe operation of the Inductrack system.

A framework for modeling the transient response of percussive drilling systems is presented. The proposed approach is based on the distributed transfer function method (DTFM), which is a semi-analytical modeling technique. Experimental results obtained from a percussion testbed for the Regolith and ice drill for the exploration of new terrains (TRIDENT) were incorporated into this modeling technique. DTFM is shown to be a convenient, modular modeling approach, capable of handling complex boundary conditions and drill rod geometries. Moreover, this technique is computationally simple and allows for straightforward incorporation of experimentally measured boundary forcing via numerical convolution, as well as control of the frequency content in the transient response. An experimental study is used to demonstrate the ability of the proposed approach to characterize unknown boundary conditions.

A new structural design method is developed in this paper to improve the surface accuracy and enhance the performance of large deployable mesh reflectors. A self-standing truss with hard points (SS...

Mid-frequency transient vibration analysis of flexible structures plays an important role in a variety of engineering applications. In a mid-frequency region, neither low-frequency methods like the finite element analysis (FEA) nor high-frequency methods like the statistical energy analysis (SEA) are directly applicable to transient vibration analysis. For optimal design of multi-body structures, a mid-frequency transient vibration analysis tool with a good balance of accuracy and efficiency in computation is in demand. In this paper, to address the aforementioned issue, a model reduction method is developed for mid-frequency transient vibration analysis of beam structures. The method is based on the augmented distributed transfer function method (augmented DTFM). In this work, the augmented DTFM is modified for model reduction in mid-frequency analysis of beam structures, which is an extension of the authors’ previous effort. The idea behind this approach is to properly select several modes in the low-frequency region and a number of modes in a mid-frequency region that encompasses the excitation frequency spectrum, from the infinite series given by the augmented DTFM. This way, a reduced model of a beam structure for mid-frequency transient analysis is systematically obtained. The proposed model reduction method is validated in numerical examples, where the augmented method is compared with other methods, including the FEA. The accuracy and efficiency of the new method on the computation of transient displacement and shear force is demonstrated. As shown in the simulation results, a proper balance between accuracy and efficiency in model reduction can be achieved by the augmented DTFM. The computation savings by the proposed method, compared with the traditional numerical methods, can be of several orders of magnitude.

Quadrotors are a popular type of unmanned aerial vehicles (UAVs) that have been widely used and extensively researched in recent years. Quadrotors, while exhibiting high agility in maneuvering, generally lack stability. There exist several types of instability in modeling and control of quadrotors. During an aggressive maneuver process, a traditional quadrotor model developed based on Euler angles may encounter singularities due to the Euler sequence. This type of instability can be avoided via quaternion-based modeling. Although quaternion-based models can eliminate the singularities due to the Euler sequence, there still exist other types of singularities for maneuvers in head-direction and thrust-direction. A UAV usually equips with a camera or other type of sensing device, which is mounted on the front or top of the U AV and which points towards the forward direction. Keeping those sensing devices pointing at the desired orientation can be a difficult task because the definition of reference heading direction vector introduces singularity and renders the mixer of the quadrotor saturated. As a result, the UAV becomes unstable. In this article, a special strategy in dealing with the head-direction-induced singularity and saturation is proposed. In this strategy, pre-yawing is adopted when a quadrotor’s current heading direction is approaching singularity. As shall be shown, by pre-yawing before the quadrotor passes through the singularity, the quadrotor can converge toward the heading direction faster after exiting the singularity. Also, the pre-yawing strategy can reduce or avoid mixture saturation by reducing the feedback error. Consequently, the proposed strategy can also reduce the deviation from the desired trajectory. The benefits of this new control strategy in tracking are demonstrated through comparison with existing quadrotor models in the literature.

The concept of Inductrack refers to the magnetic levitation technology achieved by Halbach arrays of permanent magnets. In an Inductrack system, the dynamic behaviors involved with transient responses are difficult to capture due to the highly nonlinear, time-varying, electromagnetic-mechanical couplings. In the literature, dynamic modeling of Inductrack systems that aims to analyze the transient behaviors has been widely addressed. However, one common issue with the previous investigations is that most of the dynamic models either partly or completely adopted certain steady-state and ideal case assumptions. These assumptions are extremely difficult to maintain in a transient scenario, if not impossible. Therefore, while providing good understanding of Inductrack systems, the previous results in dynamic modeling have a limited utility in providing guidance for feedback control of Inductrack systems. Recently, a benchmark transient Inductrack model was created for characterizing the transient time response of the system with fidelity, which enables model-based feedback control design. In this work, based on the transient model, a new control method for the Inductrack dynamic system is developed. The proposed control method consists of a linear part and a nonlinear part. The linear part is devised based on a state feedback configuration; the nonlinear part is accomplished by fitting a nonlinear “force-current” mapping function. With this nonlinear feedback controller, the levitation gap of the Inductrack vehicle can be effectively stabilized at both constant and time-dependent traveling speed. The proposed control law is demonstrated in numerical examples. The nonlinear control design is potentially extensible to more complicated Inductrack systems with higher degrees of freedom.

Flexible structures carrying moving subsystems have various engineering applications, including cable transport, fast transit systems, and weapon systems. In some applications, the vibration of the supporting structure induced by successively moving subsystems can become significant and develop into parametric resonance. In study of the parametric resonance caused by moving subsystems, a conventional approach is to model subsystems as moving concentrated external loads, which leads to traditional resonance due to periodic excitation. In this paper, with consideration of the inertia effect and flexible coupling of subsystems, parametric resonance of a beam structure induced by groups of oscillators moving over it is investigated. Through a special formulation of sequential state equations, dynamic stability of the beam structure is predicted by eigenvalues of a time-domain mapping matrix. From numerical simulation, it shows that apart from the speed of oscillators that directly determines the characteristic period, the inertia and stiffness of the oscillators can also alter the parametric resonance conditions. This phenomenon cannot be captured with the conventional moving load assumption.

As a new strategy for magnetic levitation envisioned in 1990s, the Inductrack system with permanent magnets (PMs) aligned in Halbach arrays has been intensively studied and applied in many projects. Due to the nonlinear, time-varying electro-magneto-mechanical coupling in such a system, the dynamic behaviors are complicated with transient responses, which in most cases can hardly be predicted with fidelity by a steady-state Inductrack model. Presented in this paper is a benchmark 2-DOF transient Inductrack model, which is derived from the first laws of nature, without any assumed steady-state quantities. It is shown that the dynamic response of the Inductrack dynamic system is governed by a set of nonlinear integro-differential equations. As demonstrated in numerical simulations with the transient model, unstable vibrations in the levitation direction occur when the traveling speed of the vehicle exceeds a threshold. To resolve this instability issue, feedback control is implemented in the Inductrack system. In the development, an assembly of Halbach arrays and active coils that are wound on the PMs is proposed to achieve a controllable source magnetic field. In this preliminary investigation, the proposed control system design process takes two main steps. First, a PID controller is set and tuned based on a simple lumped-mass dynamic system. Second, the nonlinear force-current correlation is obtained from a lookup table that is pre-calculated by steady-state truncation of the full transient Inductrack model. With the implemented feedback control algorithms, numerical examples display that the motion of the vehicle in levitation direction can be effectively stabilized at different traveling speeds. Although only a 2-DOF transient model is used here, the modeling technique and the controller design approach developed in this work are potentially applicable to more complicated models of Inductrack Maglev systems.

In this paper, the authors represent the robustness capabilities of their previously implemented parallel feedforward compensator configuration, applicable to nonminimum-phase (NMP) systems. In the previous papers, it was shown how a paralleled feedforward compensator can stabilize a general purpose NMP system with ease. However, the robustness analysis was postponed for a later study. Thus, herein, the robustness capabilities of the proposed configuration in the presence of noise and disturbance are investigated. It will be shown that the main idea behind the stability of the parallel feedforward compensation with derivative (PFCD) methodology comes handy when trying to test for the robustness purposes. At the end, a simulation example is brought to represent the ballparks of the methodology.

To assure high pointing precision of the Tianma radio telescope (TMRT), new models based on a inclinometer measurement system are added to the classical pointing model. This involves four main tasks. Firstly, the inclinometer measurement system is set up and its precision is evaluated. Secondly, a feedback control strategy for pointing error for the turbulences of uneven azimuth track and thermal deformations of alidade is implemented. Thirdly, after removing the linear effect of uneven track, sine fitting inclinometer data obtains the model of nonlinear effect of azimuth-track-level unevenness. The nonlinear effect is less than ±5 arc-sec. The azimuth angles associated with large pointing error show a good agreement with those corresponding to poor track unevenness. Finally, the TMRT inclinometers monitor the effect of thermal deformations of alidade on the pointing precision of elevation from stow position, to observing almost motionless radio source, to arbitrary postures of radio telescope. The measured data after subtracting other affected factors of inclinometer including intrinsic drift, azimuth-track unevenness, variation of the center of gravity have similar tendency with the simulated results of finite element model (FEM). Applying established real-time modified model to observe the radio source 2334 + 8226, the checking results show that the pointing precision improved 65%.

A new form-finding method, namely the Fixed Nodal Position Method (FNPM), is developed for design of geometric configuration and internal force distribution for a class of truss structures with a requirement of high shape or surface accuracy. Different from existing form-finding methods, which usually follow a stress-first-and-displacement-later procedure, the FNPM first assigns nodal coordinates for a truss structure, and then determines the internal force distribution of the structure by an optimization process. The highlight of the FNPM is that the prescribed nodal coordinates are unchanged during the form finding process. This special feature of fixed nodal positions allows the proposed method to place the nodes of a truss structure at desired locations, satisfying complicated constrains and yielding high shape/surface accuracy as required. Moreover, because the assignment of geometric configuration (nodal coordinates) and the determination of internal force distribution are undertaken separately, the computational effort in the FNPM-based form finding is much less than that by a conventional method. The proposed method is applied to two large deployable mesh reflectors of 217 nodes and 865 nodes, respectively, and show high efficiency and accuracy in form finding.

A double-beam system, which is a structure composed of two parallel beams that are interconnected by a viscoelastic layer, is seen in many engineering applications. Vibration analysis is essentially important for the safe and reliable operation, and optimal design of such dynamic systems. This paper presents an analytical method, the distributed transfer function method (DTFM), for modeling and vibration analysis of double-beam systems with arbitrary beam linear densities and flexural rigidities , and general boundary conditions. Exact closed-form analytical solutions for natural frequencies, mode shapes, and steady-state responses to periodic excitations are determined. The proposed method is applicable to a double-beam system with lower beam being fully, partially, or not supported by an elastic foundation. Through numerical study, the accuracy and efficiency of the proposed method are validated, and the effects of the stiffness, length, and location of an elastic foundation are investigated. It is shown that the DTFM is a useful tool for optimal design of elastically connected double-beam systems.

In traditional form-finding of a deployable mesh reflector (DMR), the nodes of the DMR mesh are placed on the desired working surface and the surface accuracy of the DMR is measured either by the deviation of the nodes from the desired working surface or by the deviation of the mesh from its best-fit surface. Placement of nodes on working surface and inaccurate measures of surface accuracy cause non-negligible surface errors that cannot be further reduced. To deal with these issues and to further improve surface accuracy of DMRs, a new mesh geometry design method, called the Projecting Surface Method (PSM), is presented in this paper. The highlight of the PSM is that it purposely places the nodes of a DMR off its working surface, to achieve higher surface accuracy. To this end, a direct RMS error measuring the deviation of a DMR mesh from its desired working surface is introduced and a projecting surface for hosting the nodes of the DMR mesh is defined. By the direct RMS error and projecting surface, an optimization process produces a mesh geometry with its best-fit surface closest to the desired working surface, leading to significant surface error reduction. As shown in numerical examples of DMRs with 37, 271 and 817 nodes, the PSM can reduce surface errors by 50% or more. The proposed method is usable with existing form-finding methods for further improvement of surface accuracy of DMRs.

As a new strategy for magnetic levitation envisioned in the 1990s, the Inductrack system with Halbach arrays of permanent magnets has been intensively researched. The previous investigations discovered that an uncontrolled Inductrack system may be unstable even if the vehicle travels well below its operating speed and that instability can be persistent near and beyond the operating speed. It is therefore necessary to stabilize the system for safety and reliability. With strong nonlinearities and complicated electromagneto-mechanical coupling, however, the transient response of such a dynamic system is difficult to predict with fidelity. Because of this, model-based feedback control of Inductrack systems has not been well addressed. In this paper, by taking advantage of a recently available two degrees-of-freedom transient model, a new feedback control method for Inductrack systems is proposed. In the control system development, active Halbach arrays are used as an actuator, and a feedback control law, which combines a properly tuned proportional-integral-derivative controller and a nonlinear force-current mapping function, is created. The proposed control law is validated in numerical examples, where the transient motion of an Inductrack vehicle traveling at constant speeds is considered. As shown in the simulation, the control law efficiently stabilizes the Inductrack system in a wide range of operating speed, and in the meantime, it renders a smooth system output (real-time levitation gap) with fast convergence to any prescribed reference step input (desired levitation gap).

Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

In this paper, inspired by the geometric inversion transformation, a novel transformation of the z-plane root locus to a pseudo s-plane is proposed. In the z-plane, the stability of a discrete closed-loop system (a sampled-data control system) requires that all the system poles lie within the unit circle. In root locus analysis, the unit circle region seems congested, compared to the stability region of a continuous system, which is the left half of the s-plane. In the case of fast sampling, the poles of a discrete system can really be in a small neighborhood, thus making the control implementation difficult. The geometric transformation developed in this work helps widen or enlarge the space for the system poles and preserves most of the features of z-plane root loci, including marginal stability and root loci branching off at vertical angles. The usefulness of the new transformation in design of discrete control systems is demonstrated in a numerical example.

For an unmanned aerial vehicle (UAV), its navigation in terrains can be quite challenging. To reach the destination within the required time, the maneuver of the quadrotor must behave aggressively. During this aggressive maneuvering, the quadrotor can experience singularities in the yaw-direction rotation. Thus, it is essentially important to develop a mathematical model and control method that can avoid singularities while enabling such an aggressive maneuver. In our previous effort, we demonstrated a vertical loop aggressive maneuver performed by a quadrotor UAV, which utilizes the controlled loop path following (CLPF) method. As found in this work, conventional modeling and tracking control method may not be good enough if specific requirements, such as fast coasting speed and sharp turns, are imposed. The numerical simulation by singularity-free modeling and the CLPF method enables a quadrotor to be operated in aggressive maneuverability with features like automatic flipping and precise trajectory following. The current research extends the maneuverability of a quadrotor by using a different and more capable control approach. More complex trajectories are used to test this new control method. In this paper, a quadrotor is used to demonstrate the capability of the proposed control method in delivering an aggressive and singularity-free maneuver. A quaternion-based mathematical model of the quadrotor is derived to avoid the singularities of rotation during the aggressive maneuvers. At the same time, a new control method, namely the full quaternion differential flatness (FQDF) method, is developed for quadrotors to combat the requirement of a fast maneuver in three-dimensional space. The FQDF method, which makes use of full quaternion modeling and differential flatness, enables the quadrotor to react to the reference trajectory timely and to exhibit aggressive rotation without any singularity. Also, the singularities resulting from the heading direction can be resolved by a new algorithm. The FQDF method is compared with the reference literature’s methods and is tested in different trajectories from the ones in the previous studies. The numerical simulation demonstrates the aggressive maneuverability and computational efficiency of the proposed control method.

Transient analysis of medium-frequency (mid-frequency) and high-frequency vibrations plays an important role in the research and development of complex structures in aerospace, automobile, civil, mechanical, and ship engineering. Low-frequency analysis tools, like the finite element methods, do not work well for mid- and high-frequency problems because they require a huge number of degrees-of-freedom and consequently costly computation, and are sensitive to material properties and boundary conditions. High-frequency analysis tools, such as the statistical energy analysis (SEA) and its variations, are unsuitable for midfrequency problems because they describe the vibrational behaviors of multibody structures in a global manner and cannot provide detailed local information about displacements and internal forces. In this paper, a new method, which is called the augmented distributed transfer function method (DTFM), is proposed for transient vibration analysis of two-dimensional beam structures at medium and high frequencies. Without the need for discretization and numerical integration, the augmented DTFM consistently delivers analytical transient solutions from low to high-frequency regions. A unique feature of the proposed method is that it can provide local information about system response, such as the displacements and internal forces of a structure, at any point and in any frequency region. Additionally, the proposed method provides a platform for model reduction, by which, a balance of efficiency and accuracy in mid- and high-frequency analyses can be achieved. The proposed method is demonstrated in numerical examples.

In Part I of this two-part paper, a new benchmark transient model of Inductrack systems is developed. In this Part II, the proposed model, which is governed by a set of non-linear integro-differential governing equations, is used to predict the dynamic response of Inductrack systems. In the development, a state-space representation of the non-linear governing equations is established and a numerical procedure with a specific moving circuit window for transient solutions is designed. The dynamic analysis of Inductrack systems with the proposed model has two major tasks. First, the proposed model is validated through comparison with the noted steady-state results in the literature. Second, the transient response of an Inductrack system is simulated and analyzed in several typical dynamic scenarios. The steady-state response results predicted by the new model agree with those obtained in the previous studies. On the other hand, the transient response simulation results reveal that an ideal steady-state response can hardly exist in those investigated dynamic scenarios. It is believed that the newly developed transient model provides a useful tool for dynamic analysis of Inductrack systems and for in-depth understanding of the complicated electro-magneto-mechanical interactions in this type of dynamic systems.

In this work, the properties of the class of parallel feedforward compensators to stabilize linear closed-loop systems are studied. The characteristic equation and its root locus behavior, including its asymptotes, are investigated to leave out the compensators that will not result in a stable closed-loop system. Even though there have been numerous studies relevant to parallel feedforward compensation that result in the optimal integration of squared errors (ISE), the broader view of all possible compensators has not been of much interest in the literature. Nevertheless, this study is important because, in the presence of noise and disturbance, an optimal ISE control design for the nominal plant may perform poorly while a finite ISE design may have a robust and efficient performance. One of such class compensators is parallel feedforward compensator with derivative effort (PFCD) that for a vast number of processes can have impressive properties such as no branch comebacks to the right half plane (RHP) of the root locus plot (LHP black hole effect). The example in this paper shows how effectively PFCD can contract the root locus branches into the LHP.

In this work, a new parallel feedforward compensator for the feedback loop of a linear nonminimum-phase system is introduced. Then, analytical statistical arguments between the existing developed methods and the innovated method are brought. The compelling arguments suggest the parallel feedforward compensation with derivative (PFCD) method is a strong method even though at its first survey it seems to be optimistic and not pragmatic. While most of the existing methods offer an optimal integral of squared errors (ISE) for the closed-loop response of the nominal plant, the PFCD offers a finite ISE; in reality, typically, the nominal plant is not of main concern in the controller design and the performance in the presence of mismatch model, noise, and disturbance has priority. In this work, there are several arguments brought to bold the importance of the innovated PFCD design. Also, there is a closed-loop design example to show the PFCD effectiveness in action.

Large deployable mesh reflectors are found in many space applications. To warrant the performance of this type of space structure, it is desirable to have a methodology for systematic design of dep...

Flexible multistage rotor systems with water-lubricated rubber bearings (WLRBs) have a variety of engineering applications. Filling a technical gap in the literature, this effort proposes a method of optimal bearing placement that minimizes the vibration amplitude of a WLRB-supported flexible rotor system with a minimum number of bearings. In the development, a new model of WLRBs and a distributed transfer function formulation are used to define a mixed continuous-and-discrete optimization problem. To deal with the case of uncertain number of WLRBs in rotor design, a virtual bearing method is devised. Solution of the optimization problem by a real-coded genetic algorithm yields the locations and lengths of water-lubricated rubber bearings, by which the prescribed operational requirements for the rotor system are satisfied. The proposed method is applicable either to preliminary design of a new rotor system with the number of bearings unforeknown or to redesign of an existing rotor system with a given number of bearings. Numerical examples show that the proposed optimal bearing placement is efficient, accurate and versatile in different design cases.

Conventional analytical methods for transient heat conduction solutions, due to complicated derivations and formidable calculations, have been limited to composite bodies with two or three layers. Developed in this work is a new analytical solution method for transient heat conduction in hollow composite cylinders with an arbitrary number of layers and subject to general boundary conditions. In this method, a distributed transfer function formulation gives an s-domain solution of a composite cylinder and inverse Laplace transform of the s-domain solution via a newly derived residue formula yields the transient solution of the cylinder in an explicit and closed form. Unlike conventional analytical methods, the proposed method does not require different derivations for different cylinder configurations (such as number of layers, boundary conditions and thermal resistance at layer interfaces). Because it only involves two-by-two matrices in the solution process, the proposed method is highly efficient in computation, as demonstrated in numerical examples.

Developed in this work, is a simple and innovative control method, by which a nonminimum-phase (NMP) process can be easily stabilized in a closed-loop setting. The method is named as the parallel feed-forward compensation with derivative effort (PFCD). Through use of a high order process, the control system designed by the PFCD method is shown to be less influenced by noise, disturbance, and model mismatch, compared to other methods. Moreover, the necessary data required for implementing the PFCD method are discussed. The proposed control method is illustrated on tip position control in a slewing beam as a flexible robot arm, in which the effectiveness of the PFCD method is demonstrated. In addition, the proposed control method is compared with the existing methods in terms of stability and performance. The paper is concluded with notes about the advantages.

When an unmanned aerial vehicle (UAV) navigates reactively over an unknown land, it may encounter terrains that require aggressive maneuver to keep the designed coasting speed while staying on the path. During this execution of the aggressive maneuver, the UAV can experience singularity. In this article, a vertical loop aggressive maneuver performed by a quadrotor UAV is investigated. Due to the physical configuration of the quadrotor, the conventional modeling and tracking control method may not be desirable if certain requirements, such as fast coasting speed and sharp turns, are imposed. In this work, a new modeling and maneuver control method, which is called the controlled loop path following (CLPF) method is developed. As shown in numerical examples, the proposed singularity-free model and control method enables a quadrotor to be operated in aggressive maneuverability with features like automatic flipping and precise trajectory following.

Transient vibrations of flexible structures at mid- and high-frequencies have important applications in aerospace, civil, auto and ship engineering. In this paper, a new method is developed for the determination of the transient vibration solutions of two-dimensional beam frames in mid- and high-frequency regions. In the development, the governing equations of a beam frame structure are formulated by an augmented Distributed Transfer Function Method (DTFM), without the need for discretization and approximation. The augmented DTFM differs from the traditional DTFM in that it does not contain the singularities of subsystem transfer functions, which is crucially important in a mid- or high-frequency analysis. The proposed method delivers exact eigensolutions of a beam structure from low- to high-frequencies without numerical instability. With the platform provided by the augmented DTFM, the transient response of a beam structure can be conveniently estimated by either modal expansion or the residue formula for inverse Laplace transform. A highlight of the augmented DTFM lies in that detailed information at mid- and high-frequencies, such as local displacement, slope, bending moment and shear force at any point, can be obtained, which otherwise may be difficult with conventional methods for mid- and high-frequency analysis. The proposed method is illustrated on several examples and is computationally efficient and stable from low- to high-frequency regions. In the numerical simulation, the augmented DTFM is shown to produce more accurate results than traditional finite element analysis (FEA). The proposed method is extensible to three-dimensional beam structures.

Fast projection systems are seen in various engineering applications, including weaponry systems. This work is concerned with the vibration of coupled gun barrel-bullet systems. The vibration of the muzzle end of a gun barrel (launching structure) is critical to shooting accuracy and launching safety. Under a rapid and repeated launching process, the launching structure may experience parametric resonance that is induced by accelerating projectiles. In this paper, a mathematical model of the coupled gun barrel-bullet is developed. In the development, the gun barrel is modeled by a cantilever beam; the projectiles are modeled as moving rigid bodies with time-varying velocities; and the dynamic coupling between the gun barrel and projectiles are described by pairs of springs and dampers. With this model, the dynamic response of the coupled system is determined through use of an extended solution domain (ESD) technique, which facilitates systematic solution of the dynamic response of the coupled beam-rigid body system. Numerical results show that parametric resonance can be induced in the launching structure, which is highly dependent on system parameters and projectile launching rate.

In this paper, a mathematical model of a double beam structure carrying a high-pressure driven projectile is developed for investigation of the physical behaviors of gun barrels during firing. The dynamic response of such a weapon system is particularly interesting when reduction of muzzle vibrations and relevant dynamic stress in the structure is needed to improve the life cycle of the gun. In the model presented, the Timoshenko beam theory is implemented, and realistic characteristics of the physical system are considered. Numerical simulation results are presented for the displacement and rotation of the two beams, and the rigid-body projectile mass.