Distributed Transfer Function Method Research Articles

Distributed Transfer Function Formulations for Vibration Analysis of Cracked Beam and Frame Structures

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

An Approach to Modeling Percussive Drilling Systems

Abstract 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.

Investigation of boundary flexibility on the performance of piezoelectric vibration energy harvesting beam systems by DTFM

The purpose of this paper is to investigate the effect of boundary flexibility on the performance of piezoelectric vibration energy harvester (PVEH) beam systems, which has not been studied comprehensively in the literature despite its importance. The coupled electromechanical equations of motion of a piezoelectric cantilever beam with a tip mass are established, with the base boundary constrained by translational and rotational springs. An exact closed-form solution of the frequency response function (FRF) of the PVEH is obtained by the distributed transfer function method (DTFM). The DTFM is a systematic powerful tool for the dynamic analysis of distributed parameter continua with non-classical boundary conditions, intermediate constraints, coupled fields, and non-proportional damping without adding much complexity to the solution formulation. Moreover, the DTFM computes the derivatives of the response, that is, the strains, which are required in the electromechanical coupling formulation, simultaneously without any differentiation. Numerical results showing the effects of boundary flexibility on energy harvesting efficiency are presented. A first-order rational function relating the boundary stiffness parameters and the harvesting efficiency is determined by nonlinear curve fitting of the calculated data. Physical insights and applicability of this analytical function for end-of-line quality check of the boundary of PVEH are discussed.

Medium-Frequency Vibration Analysis of Timoshenko Beam Structures

Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

Damping Mechanisms in Cable-Harnessed Structures for Space Applications: Analytical Modeling

Abstract Recent developments in the aerospace industry have driven focus toward accurately modeling the effects of the cables and electronic cords on space structures. In the past, researchers have modeled the mass and stiffness effects of these cables but primarily overlooked their damping effects through careful analytical model developments. The objective of the current work is to present analytical models for cable-harnessed structures that also include the damping effects in their vibration response. Obtaining simple, low-order and high-fidelity models are highly advantageous in designing robust vibration real-time control algorithms for structures. Additionally, the analytical models are useful tools in providing insight into and better understanding of the dynamics of space structures as they are often difficult to be tested prior to launch due to their large size and at best only a few components may be tested. Motivated by the space applications, this work considers beam structures wrapped with cables which are modeled using beam and string theory assumptions. Two different damping models namely Kelvin–Voigt and hysteretic damping are considered. The homogenization approach is used as a starting point for structures of periodic wrapping patterns. Using the variational principle, the governing partial differential equation for the transverse coordinate of vibrations is found for three cable patterns and the results are compared to those from the distributed transfer function method (DTFM). Finally, the effects of several structural parameters are studied on the overall system damping.

A New Approach to Transient Vibration Analysis of Two-Dimensional Beam Structures at Medium and High Frequencies

Abstract 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.

Effects of Varying Dynamics of Flexible Workpieces in Milling Operations

Abstract In this study, surface error calculations and stability conditions are presented for milling operations in case of slender parts. The dynamic behavior of the flexible beam-type workpiece is modeled by means of finite element method (FEM), while the varying dynamical properties related to the feed motion as well as the material removal process are incorporated in the model. The FEM-generated direct frequency response function is verified through a closed-form solution based on the distributed transfer function method. Relative errors and convergence of the FEM are investigated based on the analytical solutions of the continuum model, from which appropriate element size and mode number can be selected for modal coordinate transformations. The pattern in the variation of the natural frequencies is explored using the analytical model in case of high radial depth of cut relative to the original cross section of the beam-like workpiece. Both the stability conditions and the resulted surface errors are predicted as a function of the tool position. The presented approach and the results are validated by laboratory tests.

Parametric study of rocking cores-moment frames with supplemental viscous damping and self-centering devices using a distributed parameter model

A distributed parameter model is proposed for the rocking cores-moment frames (RCMFs) with supplemental viscous damping and self-centering devices. In this model, the moment frame and rocking cores are simplified as a shear beam and a flexural beam, respectively, while viscous damping and self-centering devices are substituted by rotational constraint with linear viscous damping and elastic stiffness. Three nondimensional parameters, frame-core stiffness parameter, base-rotation fixity parameter and damping parameter, are introduced to describe the dynamic behavior of the full physical structural system. Closed-form solutions for complex modal shapes are derived, and then parametric study is conducted to explore the effects of these three parameters on modal damping ratios, eigenvalues, complex modal shapes, and drift concentration factor. Explicit expressions of modal damping ratios with these three parameters are established through curve-fitting technique that can be conveniently used for engineering practice. Frequency responses of base shear forces and moments are obtained using the distributed transfer function method. It is concluded that the modal damping ratios are significantly affected by these three parameters, and their peak values will be remarkably reduced as the increasing of the frame-core stiffness and base-rotation fixity. For design purpose, appropriate ranges of these three parameters are recommended to achieve better drift uniformity control, response reduction, and higher modes effect mitigation. The proposed model and conclusions of the parametric study can be used as a tool for rapidly evaluating seismic performance in a preliminary design phase of the RCMFs.

Flutter and Divergence Instability of Axially-Moving Nanoplates Resting on a Viscoelastic Foundation

Moving nanosystems often rest on a medium exhibiting viscoelastic behavior in engineering applications. The moving velocity and viscoelastic parameters of the medium usually have an interacting impact on the mechanical properties of nanostructures. This paper investigates the dynamic stability of an axially-moving nanoplate resting on a viscoelastic foundation based on the nonlocal elasticity theory. Firstly, the governing partial equations subject to appropriate boundary conditions are derived through utilizing the Hamilton’s principle with the axial velocity, viscoelastic foundation, nonlocal effect and biaxial loadings taken into consideration. Subsequently, the characteristic equation describing the dynamic characteristics is obtained by employing the Galerkin strip distributed transfer function method. Then, complex frequency curves for the nanoplate are displayed graphically and the effects of viscoelastic foundation parameters, small-scale parameters and axial forces on divergence instability and coupled-mode flutter are analyzed, which show that these parameters play a crucial role in affecting nanostructural instability. The presented results benefit the designation of axially-moving graphene nanosheets or other plate-like nanostructures resting on a viscoelastic foundation.

On the phenomenon of natural-frequency splitting of a guitar string caused by a magnetic field

A magnet can affect the vibration of the metal strings in a musical instrument. We show that the magnetic field from a magnetic pickup can cause a frequency splitting of a metal guitar string’s normally degenerate transverse vibration modes, leading to a beat note in the resulting sound. The amount of frequency splitting induced by the magnet depends on the product of the induced magnetization in the ferromagnetic string, and the magnet’s spatial gradient at the position of the string, and is on the order of a few Hz. We apply free vibration theory to the string to obtain an eigenvalue problem, which we solve using a distributed-transfer function method. This method accurately predicts the natural frequencies of the vibrating guitar string in a non-uniform magnetic field. Videos of a high-E string vibrating with and without a nearby magnet will be shown.

A closed-form analytical solution method for vibration analysis of elastically connected double-beam systems

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.

Semi-Analytical Solution for Vibration of Nonlocal Piezoelectric Kirchhoff Plates Resting on Viscoelastic Foundation

Semi-analytical solutions for vibration analysis of nonlocal piezoelectric Kirchhoff plates resting on viscoelastic foundation with arbitrary boundary conditions are derived by developing Galerkin strip distributed transfer function method. Based on the nonlocal elasticity theory for piezoelectric materials and Hamilton's principle, the governing equations of motion and boundary conditions are first obtained, where external electric voltage, viscoelastic foundation, piezoelectric effect, and nonlocal effect are considered simultaneously. Subsequently, Galerkin strip distributed transfer function method is developed to solve the governing equations for the semi-analytical solutions of natural frequencies. Numerical results from the model are also presented to show the effects of nonlocal parameter, external electric voltages, boundary conditions, viscoelastic foundation, and geometric dimensions on vibration responses of the plate. The results demonstrate the efficiency of the proposed methods for vibration analysis of nonlocal piezoelectric Kirchhoff plates resting on viscoelastic foundation.

Transfer Function Analysis of Constrained, Distributed Piezoelectric Vibration Energy Harvesting Beam Systems

This paper presents a novel formulation and exact solution of the frequency response function (FRF) of vibration energy harvesting beam systems by the distributed transfer function method (TFM). The method is applicable for coupled electromechanical systems with nonproportional damping, intermediate constraints, and nonclassical boundary conditions, for which the system transfer functions are either very difficult or cumbersome to obtain using available methods. Such systems may offer new opportunities for optimized designs of energy harvesters via parameter tuning. The proposed formulation is also systematic and amenable to algorithmic numerical coding, allowing the system response and its derivatives to be computed by only simple modifications of the parameters in the system operators for different boundary conditions and the incorporation of feedback control principles. Examples of piezoelectric energy harvesters with nonclassical boundary conditions and intermediate constraints are presented to demonstrate the efficacy of the proposed method and its use as a design tool for vibration energy harvesters via tuning of system parameters. The results can also be used to provide benchmarks for assessing the accuracies of approximate techniques.

Optimal Vibration Reduction of Flexible Rotor Systems by the Virtual Bearing Method

This paper presents a new approach to optimal bearing placement that minimizes the vibration amplitude of a flexible rotor system with a minimum number of bearings. The thrust of the effort is the introduction of a virtual bearing method (VBM), by which a minimum number of bearings can be automatically determined in a rotor design without trial and error. This unique method is useful in dealing with the issue of undetermined number of bearings. In the development, the VBM and a distributed transfer function method (DTFM) for closed-form analytical solutions are integrated to formulate an optimization problem of mixed continuous-and-integer type, in which bearing locations and bearing index numbers (BINs) (specially defined integer variables representing the sizes and properties of all available bearings) are selected as design variables. Solution of the optimization problem by a real-coded genetic algorithm yields an optimal design that satisfies all the rotor design requirements with a minimum number of bearings. Filling a technical gap in the literature, the proposed optimal bearing placement approach is applicable to either redesign of an existing rotor system for improvement of system performance or preliminary design of a new rotor system with the number of bearings to be installed being unforeknown.

Thermo-electro-mechanical vibration analysis of piezoelectric nanoplates resting on viscoelastic foundation with various boundary conditions

Based on the nonlocal Kirchhoff plate theory, the thermo-electro-mechanical vibration responses are studied for a rectangular piezoelectric nanoplate resting on viscoelastic foundation with various boundary conditions. In doing this, the governing equations of motion are first derived for vibration analysis, where thermo-electro-mechanical loadings, nonlocal effect, piezoelectric effect and viscoelastic foundation have been taken into consideration. Subsequently, the Galerkin strip distributed transfer function method is proposed to solve the governing equations, which enables one to obtain the semi-analytical solutions of natural frequencies for piezoelectric nanoplates with arbitrary boundary conditions. Here, the developed mechanics model is first validated by comparing the obtained results with those available in the literature. Also, the effects of nonlocal parameter, boundary conditions, viscoelastic foundation, external electric voltage, increment temperature, biaxial force and geometric dimensions on the vibration behaviors are carefully examined for the piezoelectric nanoplate. The results demonstrate the efficiency and robustness of the developed model for vibration analysis of a complicated multi-physics system comprising piezoelectric nanoplates, viscoelastic foundation and thermo-electro-mechanical loadings.

A Discrete-Continuous Method of Mechanical System Modelling

Abstract The paper describes a discrete-continuous method of dynamic system modelling. The presented approach is hybrid in its nature, as it combines the advantages of spatial discretization methods with those of continuous system modelling methods. In the proposed method, a three-dimensional system is discretised in two directions only, with the third direction remaining continuous. The thus obtained discrete-continuous model is described by a set of coupled partial differential equations, derived using the rigid finite element method (RFEM). For this purpose, firstly the general differential equations are written. Then these equations are converted into difference equations. The derived equations, expressed in matrix form, allow to create a global matrix for the whole system. They are solved using the distributed transfer function method. The proposed approach is illustrated with the examples of a simple beam fixed at both ends and a simply supported plate.

A semi-analytical method for nonlocal buckling and vibration of a single-layered graphene sheet nanomechanical resonator subjected to initial in-plane loads

A semi-analytical method named Galerkin strip distributed transfer function method (GSDTFM) is employed to solve the buckling and vibration of a single-layered graphene sheet (SLGS) nanomechanical resonator subjected to initial in-plane loads within the framework of the nonlocal Kirchhoff plate theory. The buckling load and the frequency shift are obtained by using the GSDTFM. And the effects of the initial in-plane load, the nonlocal parameter as well as the attached nanoparticle location on the frequency shift are investigated. The simulated results show that the buckling load of the SLGS depends on not only the side length of the SLGS, but also the small scale effects. The impact of the initial in-plane load on the fundamental frequency is significant. The frequency shift at an initial tensile load is larger than that at an initial compressive load. These results are consistent with the previously reported ones.

Vibration analysis of a single-layered graphene sheet-based mass sensor using the Galerkin strip distributed transfer function method

The free vibration of a single-layered graphene sheet (SLGS)-based mass sensor is analyzed using the Galerkin strip distributed transfer function method (GSDTFM) based on the nonlocal Kirchhoff plate theory. The dynamic equations of the SLGS-based mass sensor are formulated, and the semi-analytical solutions of the frequency shift are computed with the GSDTFM. The effects of the nonlocal parameter, the attached nanoparticle locations, the plate side length, as well as the boundary conditions, on the frequency shift are studied. The simulated results show that the frequency shift of the SLGS-based mass sensor becomes smaller when the nonlocal parameter increases. The SLGS-based mass sensor is more sensitive when the attached nanoparticle is closer to the SLGS center or the side length of the SLGS becomes smaller. The boundary conditions strongly affect the frequency shift. Stiffer boundary condition causes larger frequency shift.

A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems

Flexible multistage rotating systems that are supported or guided by long water-lubricated rubber bearings (WLRBs) have a variety of engineering applications. Vibration analysis of this type of machinery for performance and duality requires accurate modeling of WLRBs and related rotor–bearing assemblies. This work presents a new model of WLRBs, with attention given to the determination of bearing dynamic coefficients. Due to its large length-to-diameter ratio, a WLRB cannot be described by conventional pointwise bearing models with good fidelity. The bearing model proposed in this paper considers spatially distributed bearing forces. For the first time in the literature, the current study addresses the issue of mixed lubrication in the operation of WLRBs, which involves interactions of shaft vibration, elastic deformation of rubber material and fluid film pressure, and validates the WLRB model in experiments. Additionally, with the new bearing model, vibration analysis of WLRB-supported flexible multistage rotating systems is performed through use of a distributed transfer function method, which delivers accurate and closed-form analytical solutions of steady-state responses without discretization.

Modeling vibration response and damping of cables and cabled structures

In an effort to model the vibration response of cabled structures, the distributed transfer function method is developed to model cables and a simple cabled structure. The model includes shear effects, tension, and hysteretic damping for modeling of helical stranded cables, and includes a method for modeling cable attachment points using both linear and rotational damping and stiffness. The damped cable model shows agreement with experimental data for four types of stranded cables, and the damped cabled beam model shows agreement with experimental data for the cables attached to a beam structure, as well as improvement over the distributed mass method for cabled structure modeling.