Lumped Inertia Research Articles

Research of a novel stereoscopic symmetrical quadruple hair gyroscope

This paper presents a novel stereoscopic symmetrical quadruple hair gyroscope (SSQHG) which is distinguished from the conventional flat structures to achieve better angular rate measurement performance. A symmetrical device architecture is designed to realize the differential detection of Coriolis force, thereby effectively eliminate the common mode interference. Four stereoscopic hair posts are adopted to increase the quality of the lumped inertia masses, so as to enhance the measurement sensitivity. A simplified mass-spring-damper model of idealized SSQHG is established and verified by a modal simulation to synthetic analysis the motion modal. A set of finite element method simulations including modal distribution simulation and harmonic response simulation are implemented to acquire accurate vibration information and to identify the structure parameters quantitatively. A micromachining procedure based on the standard deep dry silicon on glass is adopted to fabricate the proposed micro-gyroscope. The frequency response experiments indicate that the vacuum packaged prototype has a drive mode frequency of 536.2 Hz with a quality factor of 1319.7 and a sense mode frequency of 535.7 Hz with a quality factor of 1334.5. An analog closed-loop driving circuit is designed to realize the self-excited oscillation of SSQHG and an open-loop sensing circuit is designed to extract the Coriolis force signal. The preliminary experimental results demonstrate that the fabricated SSQHG prototype exhibits an angular rate sensitivity of 16.03 mV/°/s and a bias instability of 16.26°/h at room temperature.

Free vibration analysis of a rigid bar supported by arbitrary elastic beams

For convenience, a two-node conventional elastic beam element (C beam element) with the displacements of its 2nd node replaced by those of center of gravity (c.g.) of the joined rigid bar is called the modified beam element (M beam element). The objective of this paper is to present a modified finite element method (modified FEM) such that the free vibration characteristics of a rigid bar supported by a number of elastic beams can be easily determined. First of all, the displacements for the 2nd node of a C beam element joined with the rigid bar are determined in terms of those for the c.g. of the joined rigid bar to establish the M beam element. Next, the mass and stiffness matrices for the M beam element are derived based on the displacements for the 1st node of the C beam element and those for the c.g. of the joined rigid bar. Then, the overall property matrices of the entire unconstrained vibrating system (i.e. a rigid bar supported by a number of elastic beams) can be determined by using the assembly technique of the conventional FEM and considering the effects of lumped mass and rotary inertia of the rigid bar. Finally, the boundary (supporting) conditions are imposed to produce the effective property matrices of the constrained vibrating system and then the free vibration characteristics are determined with the standard approach. In order to confirm the presented theory and the developed computer program, the rigid bar is modeled by a number of C beam elements with bigger Young’s modulus (ER) and the conventional FEM is used to determine the natural frequencies and associated mode shapes of the vibrating system. It is found that the latter will converge to the corresponding ones obtained from the presented modified FEM when the magnitude of ER increases to certain values.

The analysis of free vibration of conservative oscillators with inertia and static type cubic non-linearities by variational-Padé technique

In this study, a new method based on the variational iteration method and the Pade approximation is introduced to obtain accurate analytical solutions for some special types of non-linear differential equations, arising in modelling the dynamic behaviour of oscillators with inertia together with cubic stiffness non-linearities. The comparison of the results of a new proposed variational Pade technique for analysing large-amplitude free vibrations of cantilever beams carrying a lumped mass and rotary inertia, with those of other conventional methods, ensures the excellent accuracy of the proposed method.

Investigation of synchroniser engagement in dual clutch transmission equipped powertrains

Transient response of a dual clutch transmission (DCT) powertrain to synchroniser mechanism engagements is investigated using a lumped inertia model of the powertrain. Original research integrates lumped inertia powertrain models for the DCT with a detailed synchroniser mechanism model and two separate engine models, comprising of a mean torque model and a harmonic torque model, using torque derived from piston firing. Simulations are used to investigate the synchroniser mechanism engagement process in a previously unscrutinised operating environment. Simulations are performed using both engine torque models, with the mean torque model demonstrates the highly nonlinear nature of synchroniser mechanism engagement, and the powertrain response to the engagement process. Through the introduction of harmonic engine torques, additional excitation is present in the mechanism during engagement, and increased vibration of the synchroniser sleeve results. The impact of vibrations is particularly important to the increased wear of indexing chamfer contact surfaces.

Using a Characteristic Force Approach to Determine the Eigensolutions of an Arbitrarily Supported Linear Structure Carrying Lumped Attachments

In this paper a characteristic force approach is developed that can be used to determine the eigenvalues and mode shapes of any arbitrarily supported linear structure carrying various lumped attachments, including a lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, and an undamped or damped oscillator with or without a rigid body degree of freedom. Using the proposed approach, each lumped element is first replaced by the load, either a force or moment, that it exerts on the linear structure, thus transforming the free vibration problem into a forced vibration one. By expressing the deflection of the linear structure in terms of these forces or moments and enforcing the compatibility conditions at each attachment points, the roots of the characteristic determinant of these loads can be graphically or numerically solved to find the eigenvalues of the combined system. Once the eigenvalues have been found, the corresponding mode shapes can be readily obtained. The proposed method is easy to code, systematic to apply, and can be easily modified to accommodate any arbitrarily supported one- or two-dimensional linear structure carrying various lumped attachments.

Modelling and control of a bio-inspired microgripper

A biologically inspired microgripper has been considered with its kinematic model. The dynamic model of the gripper has been developed considering the system with its equivalent lumped inertia, stiffness and damping parameters. The biomimetic gripper developed is conceptually similar to an agonist-antagonistic mechanism. This mechanism is most commonly used by almost all crustaceans not only for gripping, but also for cutting, crushing, walking, climbing, etc., where two muscles called agonist and antagonist are working against each other to allow movement in both directions and to stabilise the joint. The mathematical model is then simulated using MATLAB/SIMULINK. Based on the SIMULINK model developed, a PID controller is used for precise position control.

Impulse Response Method for Pipeline Systems Equipped with Water Hammer Protection Devices

Surge protection devices, such as surge tanks and air chambers, have been modeled with the impulse response method for transient analysis of water distribution systems. The lumped inertia model and continuity equation are used to represent nonpipe hydraulic elements. Results of pressure or discharge variations obtained by using the impulse response method and the method of characteristics are in good agreement. The impulse response method provides total pressure and discharge along any pipeline segment by direct integration of the ratio of complex head or complex discharge to a complex downstream discharge, respectively. A modification is proposed so that transition between turbulent and laminar flows can be considered. The representation of hydraulic devices has been incorporated into the impedance matrix method, which was developed for heterogeneous and multilooped pipe network systems. The potential advantages of the proposed method over other conventional approaches were investigated by applying the proposed method to hypothetical pipe network systems.

New Digital Multispeed Method for Indirect Measurement of Transient Electrical and Shaft Torques

This paper presents characteristics and solutions to some practical problems concerning the new digital Generalized Indirect Multispeed Method (GIMSM), for measurement of transient electrical and shaft torques occurring in a complex serial multimachine AC drive system. The method indirectly includes torsion and internal machine and shaft damping. It is based on the measurement of the transient speeds, using high resolution digital tachometers, of every lumped inertia component. Obtaining the torque values requires digital filtering, followed by numerical differentiation and integration. This procedure depends on the type of the transients and the electrical and mechanical characteristics of the drive. The measurement of transient torques using the proposed method is realized in the laboratory on atwo-machine induction motor (IM) drive. The results of the measurements of the electrical and shaft torques during the acceleration and speed reversal are presented in this article. The article further recommends the choice of optimum sampling rate and the algorithm for digital filter selection based on time–frequency analysis.

Optimal inertia lumping from modal mass matrices for structural dynamics

An optimal inertia lumping formulation using modal mass matrices obtained from structural analysis is presented, enabling recovery of lumped inertia information even when substructuring has been employed. The method is implemented with a commercial optimization program and uses reduced mass and stiffness formulations employed in commercial finite element analysis programs. The result is lumped inertia models of complex flexible bodies for use in structural dynamics, including flexible multibody dynamic simulation. A method for dealing with nondifferentiability of repeated eigenvalues that arise in the optimization process is derived and implemented in the proposed formulation. Numerical examples are presented to validate the method, relative to conventional mass lumping approaches.

Synthesis and Steady-State Analysis of High-Speed Elastic Cam-Actuated Linkages With Fluctuated Speeds by a Finite Element Method

A cam driving a lumped inertia through an elastic slider-crank follower linkage with its actuator, a D. C. motor, has been considered in this work. An iterative procedure utilizing the finite element method developed by Midha et al. (1978) is used to synthesize the cam profile to produce a desired output motion at a given design speed and damping ratio. The elastic deflections induced by the large inertia of the linkage operating at a fluctuated high speed of the driving cam are taken into account in synthesizing the cam profile. Responses of the synthesized linkages are simulated and found to be satisfactory at the design conditions.

Efficient Valve Representation in Fixed-Grid Characteristics Method

This work formulates a unified set of boundary conditions to efficiently represent the majority of valve and orifice devices found in water supply, transmission, and distribution systems. A particularly useful combination of mathematical components results when a lumped inertia model is linked with a throttling device. This combination of elements, termed a pipe replacement element/valve-in-line (PREVIL), has been constructed to permit a wide range of control-valve/short-pipe combinations to be conveniently modeled with the method of characteristics. The solution is quadratic in form and explicit, regardless of the number of pipes that are connected to the boundary condition. A variety of on-off and modulating valves can be accurately handled within this framework. An additional feature developed in this treatment of pressure-reducing an pressure-sustaining values, flow controllers, and other similar devices permits a more realistic representation of these important control devices. The response time of the regulating valve on opening or closing can be readily approximated so that a greater range of control behavior can be simulated. Application of the combined boundary condition is illustrated in a pipe network.

Study on Design Technology of Multiblade Fan. (2nd Report. Basic Study on Simple Numerical Model for Dynamic Analysis of Rotating Fan Shaft System).

This study aims to clarify deformation and stress generation mechanisms in a multiblade fan under static and dynamic torsional moments, and to establish a simple lumped inertia model for the dynamic analysis. Static and dynamic torsional tests were carried out to investigate the deformation and stress of the blades. Numerical calculations of the deformation and stress using the finite element method were also carried out under the same loading conditions. By comparing the experimental results with the numerical ones, the generation mechanisms of the deformation and stress of the blade were clarified. The results of these investigations are as follows. The relative displacement between the main and side plates causes the bending deformation in the flatwise and edgewise directions and the torsional deformation of the blade under the static and dynamic torsional moments. The stress caused by the deformation in the flatwise direction mainly governs the strength of the blade. Good coincidence is found between the static stress distribution and the dynamic one in the first mode. This means that the simple lumped inertia model is suitable for further dynamic analysis of the rotating fan shaft system.

Analysis and testing of propellant feed system priming process

This paper presents the analytical and the experimental results pertaining to the priming process of the propellant feed system with initial line pressures starting from 0 psia and greater. The analytical methods employ the method of characteristics to solve for the one-dimensional liquid transients in the liquid-full segments and the lumped inertia technique to model the dynamics of the partially filled segments or the two-phase segments. The advantages of these methods are (1) fluid compressibility and piping flexibility are accounted for in the solution; and (2) the characteristics method can be used in the solution of a complex system. The analytical results were compared with test results. Excellent correlation was obtained between predictions and test results verifying the essential aspects of the analytical modeling techniques and providing the foundation for analyzing a complicated network system.

Rigorous Theory of Vibration Analysis of Rotating Blade Based on Transfer Matrix Method. 2nd Report. Blade Model and Transfer Matrix.

This is a research on rigorous theory for the natural vibration analysis of blade, a geometrically and environmentally complicated beam. The reason why the term "rigorous" is emphasized here is that (1) all the factors related to beam deformation are taken into consideration, and (2) those factors are combined to form an organically unified theory of non-contradiction, which is fully confirmed. In the 1st report, such a unified theory has been formulated in the form of a set of state equations. In this 2nd report, they are converted into a set of transfer matrices, assuming lumped inertia. The reason of acceptability of this assumption is that the order of major blade vibrations of engineering importance is much lower than the number of lumped inertia. The reason of beneficialness of this assumption is that it separates inertia, Coriolis and centrifugal-force matrices from stiffness matrix to enable theoritical development of good perspective.

Synthesis and Steady State Analysis of High-Speed Elastic Cam-Actuated Linkages by a Finite Element Method

A cam driving a lumped inertia through an elastic slider-crank follower linkage with a curved beam coupler is considered. An iterative procedure utilizing the finite element method developed by Midha et al. (1978) is used to synthesize the cam profile to produce a desired output motion at a given design speed and damping coefficient. Nonlinear terms are neglected producing inhomogeneous, periodic, linear, ordinary differential equations. Responses of the synthesized linkages are simulated and found to be satisfactory at the design conditions.

Viscoplasticity of respiratory tissues

Low-frequency mechanical behavior of various respiratory tissues shows certain similarities. In this study we test the hypothesis that rate-independent plastic processes along with rate-dependent viscoelastic processes are responsible. We considered oscillatory responses of several respiratory tissues measured over prescribed ranges of frequency (up to 6 Hz) and amplitude of forcing. These included the excised cat lung, the human chest wall in vivo, and two components of the chest wall: the excised dog rib cage and the excised rabbit abdominal viscera; some data were previously reported and some are new. We analyzed these data using the viscoplastic model of Hildebrandt (J. Appl. Physiol. 28: 365-372, 1970). It consists of three compartments: a plastoelastic compartment mechanically in parallel with a viscoelastic compartment, both in series with a lumped inertia. We fitted oscillatory data of the above respiratory tissues to the model by a least-squares technique. The fit was qualitatively consistent with the observations and exhibited moderately good to very good quantitative correspondence. As an independent verification of this approach, we obtained the stress relaxation after a step-volume change. Based on the oscillatory response of cat lungs, the calculated stress relaxation function was found to be generally consistent with corresponding observations. This study indicates that both plasticity and viscoelasticity appear to be important determinants of mechanical behavior of respiratory tissues at low frequencies and that inertial effects are negligible.

Vibrations of Tapered Cantilever Beams and Shafts

SummaryVibration characteristics of a linearly tapered cantilever beam and a shaft have been studied by using the lumped inertia force method; a linear displacement distribution is considered over each element. The results are compared with some of those in the literature and with experimental observations. These comparisons indicate that, even using a few elements, a reasonable degree of accuracy can be obtained in the natural frequency, although it is essential to consider more elements in order to determine mode shape accurately.