scholarly journals SOME PROBLEMS OF OPTIMIZATION AND CONTROL OF THE NATURAL FREQUENCIES OF AN ELASTICALLY SUPPORTED RIGID BODY

2021 ◽  
Vol 3 (2) ◽  
pp. 88-102
Author(s):  
S. Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Fundamental Frequency ◽  
Natural Frequencies ◽  
Optimization Problems ◽  
Free Vibrations ◽  
The Body ◽  
Natural Vibrations ◽  
Natural Oscillations ◽  
Optimal Position ◽  
Elastic Supports

The article analytically investigates the behavior of the frequencies and modes of natural vibrations of a rigid body, based on point elastic supports, when the position of the supports changes. It is assumed that the body is in plane motion and has two degrees of freedom. A linear description of body vibrations is accepted. The problems of determining such optimal positions of elastic supports at which the fundamental frequency of the structure reaches its maximum value are considered. Two groups of problems were studied. The first group concerns a body supported by only two supports. It was found that in the absence of restrictions on the position of the supports to maximize the fundamental natural frequency, these supports should be positioned so that the basic natural vibrations of the body are translational. Simple analytical conditions are formulated that must be satisfied by the corresponding positions of the supports. In real practical situations, these positions may be unreachable due to the presence of various kinds of restrictions due to design requirements. In this paper, optimization problems are considered taking into account a number of restrictions on the position of supports, typical for practice, expressed analytically by equations and inequalities. For each of the considered types of constraints, results are obtained that determine the optimal positions of the supports and the corresponding maximum values of the main natural frequencies. The approach applied allows us to consider other types of restrictions, which are not considered in the article. In the second group of problems for a body resting on an arbitrary number of supports, the optimal position of an additional elastic support introduced in order to maximize the fundamental frequency in fixed positions and the stiffness coefficients of the remaining supports was sought. It was found that this position depends on the value of the stiffness coefficient of the introduced support. Results are obtained that qualitatively and quantitatively characterize this position and the corresponding frequencies and modes of natural oscillations, including taking into account practically established limitations. The research method uses a qualitative approach, systematically based on the well-known Rayleigh theorem on the effect of imposing constraints on the free vibrations of an elastic structure.

scholarly journals QUALITATIVE PROPERTIES OF VIBRATIONS OF ELASTICALLY SUPPORTED RIGID BODY

2020 ◽  
Vol 2 (2) ◽  
pp. 85-94
Author(s):  
S Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Natural Vibrations ◽  
Qualitative Properties ◽  
Engineering Structures ◽  
Natural Oscillations ◽  
Geometric Characteristics ◽  
Operating Modes

The paper investigates free vibrations of an absolutely rigid body, supported by a set of linearly elastic springs and performing a plane-parallel motion. The proposed system has two degrees of freedom, which makes it elementary to determine the frequencies and modes of its natural oscillations by using exact analytical expressions. However, these expressions are rather cumbersome, which makes it difficult to study the behavior of frequencies and modes when the characteristics of the model change. Therefore, the aim of the work was to find out the qualitative properties of the modes of free vibrations depending on the elastic, inertial and geometric characteristics of the system, as well as to study the effect of changing the position of elastic supports on its natural frequencies. The main qualitative characteristic of the mode of natural vibrations of the system in consideration is the position of its node – a point that remains stationary during natural vibrations. For the practically important case of a system with two supports, it has been established in the work that, in the general case, of two modes corresponding to two different natural frequencies, one has a node located inside the gap between the supports, and the other – outside this gap. Analytical conditions are found that must be satisfied by the inertial and geometric characteristics of the system, which make it possible to determine which of the two modes corresponds to the internal position of the node. It is noted that these conditions do not depend on the stiffness of the supports. Analytical results were also obtained, allowing to determine a more accurate qualitative localization of the node. To clarify the behavior of natural frequencies when the position of the supports changes, an explicit expression is obtained for the derivative of the square of the natural frequency of the system with respect to the coordinate defining the position of the support. This expression can be used to solve a variety of problems related to the control and optimization of the operating modes of engineering structures subjected to dynamic, in particular periodic, effects. The results of the work were obtained using qualitative methods of the mathematical theory of oscillations. In particular, the theorem on the effect of imposing constraints on the natural frequencies of an elastic system is systematically used.

scholarly journals QUALITATIVE PROPERTIES OF VIBRATIONS OF ELASTICALLY SUPPORTED RIGID BODY

2020 ◽  
Vol 2 (2) ◽  
pp. 85-94
Author(s):  
S Bekshaev ◽  
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Natural Vibrations ◽  
Qualitative Properties ◽  
Engineering Structures ◽  
Natural Oscillations ◽  
Geometric Characteristics ◽  
Operating Modes

The paper investigates free vibrations of an absolutely rigid body, supported by a set of linearly elastic springs and performing a plane-parallel motion. The proposed system has two degrees of freedom, which makes it elementary to determine the frequencies and modes of its natural oscillations by using exact analytical expressions. However, these expressions are rather cumbersome, which makes it difficult to study the behavior of frequencies and modes when the characteristics of the model change. Therefore, the aim of the work was to find out the qualitative properties of the modes of free vibrations depending on the elastic, inertial and geometric characteristics of the system, as well as to study the effect of changing the position of elastic supports on its natural frequencies. The main qualitative characteristic of the mode of natural vibrations of the system in consideration is the position of its node – a point that remains stationary during natural vibrations. For the practically important case of a system with two supports, it has been established in the work that, in the general case, of two modes corresponding to two different natural frequencies, one has a node located inside the gap between the supports, and the other – outside this gap. Analytical conditions are found that must be satisfied by the inertial and geometric characteristics of the system, which make it possible to determine which of the two modes corresponds to the internal position of the node. It is noted that these conditions do not depend on the stiffness of the supports. Analytical results were also obtained, allowing to determine a more accurate qualitative localization of the node. To clarify the behavior of natural frequencies when the position of the supports changes, an explicit expression is obtained for the derivative of the square of the natural frequency of the system with respect to the coordinate defining the position of the support. This expression can be used to solve a variety of problems related to the control and optimization of the operating modes of engineering structures subjected to dynamic, in particular periodic, effects. The results of the work were obtained using qualitative methods of the mathematical theory of oscillations. In particular, the theorem on the effect of imposing constraints on the natural frequencies of an elastic system is systematically used.

Topology Optimization of Rigid-Body Systems Considering Collision Avoidance

2020 ◽  
Vol 142 (8) ◽  
Author(s):  
Fritz Stöckli ◽  
Kristina Shea
Keyword(s):  
Topology Optimization ◽  
Rigid Body ◽  
Collision Avoidance ◽  
Optimization Problems ◽  
Computational Design ◽  
Optimization Method ◽  
The Body ◽  
Graph Grammar ◽  
Automated Synthesis ◽  
Inertia Properties

Abstract Passive dynamic mechanisms can perform simple robotic tasks without requiring actuators and control. In previous research, a computational design method was introduced that integrates dynamic simulation to evaluate and evolve configurations of such mechanisms. It was shown to find multiple solutions of passive dynamic brachiating robots (Stöckli and Shea, 2017, “Automated Synthesis of Passive Dynamic Brachiating Robots Using a Simulation-Driven Graph Grammar Method,” J. Mech. Des. 139(9), p. 092301). However, these solutions are limited, since bodies are modeled only by their inertia properties and thus lack a shape embodiment. This paper presents a method to generate rigid-body topologies based on given inertia properties. The rule-based topology optimization method presented guarantees that the topology is manifold, meaning that it has no disconnected parts, while still connecting all joints that need to be part of the body. Furthermore, collisions with the environment, as well as with other bodies, during their predefined motion trajectories are avoided. A collision matrix enables efficient collision detection as well as the calculation of the swept area of one body in the design space of another body by only one matrix–vector multiplication. The presented collision avoidance method proves to be computationally efficient and can be adopted for other topology optimization problems. The method is shown to solve different tasks, including a reference problem as well as passive dynamic brachiating mechanisms. Combining the presented methods with the simulation-driven method from Stöckli and Shea (2017, “Automated Synthesis of Passive Dynamic Brachiating Robots Using a Simulation-Driven Graph Grammar Method,” J. Mech. Des. 139(9), p. 092301), the computational design-to-fabrication of passive dynamic systems is now possible and solutions are provided as STL files ready to be 3D-printed directly.

Calculation of the frequency of own oscillations of polyurethane shock absorbers used in shipbuilding

2020 ◽  
pp. 152-157
Author(s):  
Ю.Ф. Титова ◽  
С.Н. Яковлев
Keyword(s):  
Vibration Isolation ◽  
Shock Absorber ◽  
Free Vibrations ◽  
Empirical Dependence ◽  
Natural Vibrations ◽  
Natural Oscillations ◽  
Rubber Material ◽  
Shape Coefficient ◽  
Calculated Dependence ◽  
Shock Absorbers

В работе приведено обоснование необходимости применения амортизаторов и замены традиционного эластомерного материала резины на более совершенный материал – полиуретан. Целью работы является экспериментальное исследование деформативности полиуретанового виброизоляционного массива амортизатора и получение расчетной зависимости для определения собственной частоты колебаний системы «агрегат-амортизатор». В работе представлен подробный анализ зависимости коэффициента виброизоляции от соотношения частот вынужденных и свободных колебаний. Отмечено, что эффективность виброизоляции тем выше, чем больше это соотношение частот. Учитывая, что частота вынужденных колебаний - величина заданная, для повышения эффективности виброизоляции следует понижать частоту собственных колебаний амортизатора благодаря применению новых виброизоляционных материалов. В качестве эластомерного материала предложен полиуретан фирмы «Synair» (Великобритания) твердостью 40,45 и 50 ShA. Приведено описание нагружающего устройства для исследования деформативности виброизоляционного массива амортизатора серии КАС (корабельный амортизатор сварной). Получена эмпирическая зависимость коэффициента ужесточения в зависимости от коэффициента формы виброизоляционного массива амортизатора. Представлена эмпирическая зависимость по определению модуля упругости эластомерного материала виброизоляционного массива амортизатора в зависимости от условий контактирования на торцах и фактора формы массива. Представлена расчетная зависимость по определению частоты собственных колебаний амортизатора в зависимости от величины статической деформации. Представленная зависимость позволяет на стадии проектирования определить собственную частоту колебаний и предупредить возникновение такого опасного явления, как резонанс. The paper substantiates the need for the use of shock absorbers and the replacement of the traditional elastomeric rubber material with a more advanced material - polyurethane. The aim of the work is an experimental study of the deformability of a polyurethane vibration-isolating array of a shock absorber and obtaining a calculated dependence for determining the natural frequency of oscillations of the aggregate-shock absorber system. The paper presents a detailed analysis of the dependence of the vibration isolation coefficient on the ratio of the frequencies of forced and free vibrations. It is noted that the effectiveness of vibration isolation is higher, the greater the ratio of the frequencies of forced and natural vibrations. Taking into account the fact that the frequency of forced vibrations is a given value, to increase the efficiency of vibration isolation, it is necessary to reduce the frequency of natural vibrations of the shock absorber due to the use of new vibration insulation materials. The work describes the loading device for studying the deformability of the vibration-isolating array of the KAS series shock absorber (welded ship shock absorber). An empirical dependence of the tightening coefficient is obtained depending on the shape coefficient of the vibration-isolating array of the shock absorber. The paper presents a calculated dependence on determining the frequency of natural oscillations of the shock absorber depending on the magnitude of the static deformation.

scholarly journals Most important results of the scientific activity of the Seismological Division GS RAS in 2016-2020 (seismic research)

2021 ◽  
Vol 3 (1) ◽  
pp. 54-74
Author(s):  
Aleksei Liseikin ◽  
Viktor Seleznev
Keyword(s):  
Seismic Data ◽  
Natural Frequencies ◽  
Low Frequency ◽  
Fold Increase ◽  
Scientific Activity ◽  
Operating Conditions ◽  
Natural Vibrations ◽  
Natural Oscillations ◽  
Moho Boundary ◽  
Seismic Impacts

The article presents the most important results of seismic studies carried out in 2016-2020 at the Seismological Division GS RAS. Work at the Chirkey’s HPP, where natural oscillations of the dam and their seasonal changes were studied in detail and a method for monitoring the natural frequencies of the structure was developed. Research at the Sayano-Shushenskaya HPP, where the processes of interaction of operating hydroelectric units with surrounding structures were studied and it was found that under certain operating conditions of the equipment, there is a 10-20-fold increase in the natural oscillations of the dam, the source of which is natural oscillations (organ vibrations) in the penstocks. A method has been developed for assessing the response of structures to seismic impacts, based on the method of coherent reconstruction of standing wave fields and allowing to calculate the vibrations of an object arising from seismic impacts at its base. The possibilities of determining the natural vibrations of large objects based on low-frequency seismological records and their monitoring are demonstrated on the example of the results of the analysis of satellite images and seismological materials when determining the causes of the landslide on the Elbashinsky dump of the Kolyvan anthracite deposit in the Novosibirsk region. The possibility of using river seismic data to study the structure of the earth’s crust at all depth, including the Moho boundary, has been substantiated using the example of data obtained during the development of the CDP-2D profile in the lower reaches of the river Lena.

Measurement of Young’s Modulus and Shear Modulus of Some Structures Welded Using Flux Cored Wire by Vibration Tests

2014 ◽  
Vol 216 ◽  
pp. 151-156 ◽  
Author(s):  
Liviu Bereteu ◽  
Mircea Vodǎ ◽  
Gheorghe Drăgănescu
Keyword(s):  
Shear Modulus ◽  
Arc Welding ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Vibration Modes ◽  
Cored Wire ◽  
Flux Cored Arc Welding ◽  
Longitudinal Elastic Modulus ◽  
Vibration Tests ◽  
Non Destructive

The aim of this work was to determine by vibration tests the longitudinal elastic modulus and shear modulus of welded joints by flux cored arc welding. These two material properties are characteristic elastic constants of tensile stress respectively torsion stress and can be determined by several non-destructive methods. One of the latest non-destructive experimental techniques in this field is based on the analysis of the vibratory signal response from the welded sample. An algorithm based on Pronys series method is used for processing the acquired signal due to sample response of free vibrations. By the means of Finite Element Method (FEM), the natural frequencies and modes shapes of the same specimen of carbon steel were determined. These results help to interpret experimental measurements and the vibration modes identification, and Youngs modulus and shear modulus determination.

Determination of the Critical Operating Speeds of Planar Mechanisms by the Finite Element Method Using Planar Actual Line Elements and Lumped Mass Systems

1979 ◽  
Vol 101 (2) ◽  
pp. 210-223 ◽  
Author(s):  
S. Kalaycioglu ◽  
C. Bagci
Keyword(s):  
Dynamic Systems ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Large Error ◽  
Free Vibrations ◽  
Lumped Mass ◽  
Critical Speeds ◽  
Rotating Shafts ◽  
Line Elements ◽  
Kinematic Motion

It has been a well-established fact that dynamic systems in motion experience critical speeds, such as rotating shafts and geared systems whose undeformed reference geometry remain the same at all times. Their critical speeds are determined by their natural frequencies of considered type of free vibrations. Linkage mechanisms as dynamic systems in motion change their undeformed geometries as function of time during the cycle of kinematic motion. They do also experience critical operating speeds as rotating shafts and geared systems do, and their critical speeds are determined by the minima of their natural frequencies during a cycle of kinematic motion. Such a minimum occurs at the critical geometry of a mechanism, which is the position at which the maximum of the input power is required to maintain the instantaneous dynamic equilibrium of the mechanism. Actual finite line elements are used to form the global generalized coordinate flexibility matrix. The natural frequencies of the mechanism and the corresponding mode vectors (mode deflections) are determined as the eigen values and eigen vectors of the equations of instantaneous-position-free-motion of the mechanism. Method is formulated to include or exclude the link axial deformations, and apply to any number of loops having any type of planar pair. Critical speeds of planar four-bar, slider-crank, and Stephenson’s six-bar mechanisms are determined. Experimental results for the four-bar mechanism are given. Effect of axial deformations and link rotary inertias are investigated. Inclusion of link axial deformations in mechanisms having pairs with sliding freedoms is seen to predict critical speeds with large error.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

Does a rigid body limit maneuverability?

2000 ◽  
Vol 203 (22) ◽  
pp. 3391-3396 ◽  
Author(s):  
J.A. Walker
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Alternative Measure ◽  
Pectoral Fin ◽  
Caudal Fin ◽  
Minimum Radius ◽  
Current Definition ◽  
Theoretical Minimum

Whether a rigid body limits maneuverability depends on how maneuverability is defined. By the current definition, the minimum radius of the turn, a rigid-bodied, spotted boxfish Ostracion meleagris approaches maximum maneuverability, i.e. it can spin around with minimum turning radii near zero. The radius of the minimum space required to turn is an alternative measure of maneuverability. By this definition, O. meleagris is not very maneuverable. The observed space required by O. meleagris to turn is slightly greater than its theoretical minimum but much greater than that of highly flexible fish. Agility, the rate of turning, is related to maneuverability. The median- and pectoral-fin-powered turns of O. meleagris are slow relative to the body- and caudal-fin-powered turns of more flexible fish.

Sign in / Sign up

Export Citation Format

Share Document