scholarly journals Discussion: “An Improved Method for Calculating Free Vibrations in Systems of Several Degrees of Freedom” (Dudley, Winston M., 1938, ASME J. Appl. Mech., 5, pp. A61–A66)

1939 ◽  
Vol 6 (1) ◽  
pp. A36
Author(s):  
C. Concordia
Keyword(s):  
Degrees Of Freedom ◽  
Free Vibrations ◽  
Improved Method

An Improved Method for Calculating Free Vibrations in Systems of Several Degrees of Freedom

1938 ◽  
Vol 5 (2) ◽  
pp. A61-A66
Author(s):  
Winston M. Dudley
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Matrix Theory ◽  
Free Vibrations ◽  
Initial Disturbance ◽  
Improved Method ◽  
Matrix Methods ◽  
Computing Machine ◽  
The Matrix ◽  
Time Required

Abstract In 1934 two English investigators (1) published a method for calculating the various modes and frequencies of vibration of a system having several degrees of freedom. Their method, which is based on matrices, greatly shortens the time spent in obtaining numerical solutions in many important problems, notably those with immovable foundations. In this paper is presented a new theorem which (a) makes possible a further reduction of nearly one half in the time required, so that solutions up to 20 deg or more of freedom are now practical and (b) makes it then possible to determine the motion of the system after any initial disturbance in a few minutes, instead of several hours as required by older methods. It is useful in the latter respect whether the modes have been determined by matrix methods, or not. Although the paper gives simpler proofs than any previously published, knowledge of the matrix theory is not required in using the method. Problems are analyzed by a tabular process, in which an ordinary computing machine helps greatly. Comments based on computing experience are given. A simple numerical example has been given elsewhere (1).

Best Structural Theories for Free Vibrations of Sandwich Composites via Machine Learning

2019 ◽  
Author(s):  
Marco Petrolo ◽  
Erasmo Carrera
Keyword(s):  
Degrees Of Freedom ◽  
Computational Cost ◽  
Free Vibrations ◽  
Numerical Results ◽  
Design Parameters ◽  
Sandwich Composites ◽  
Accuracy Requirement ◽  
Minimum Number ◽  
Carrera Unified Formulation ◽  
Structural Theories

Abstract This work presents a novel methodology for the development of refined structural theories for the modal analysis of sandwich composites. Such a methodology combines three well-established techniques, namely, the Carrera Unified Formulation (CUF), the Axiomatic/Asymptotic Method (AAM), and Artificial Neural Networks (NN). CUF generates structural theories and finite element arrays hierarchically. CUF provides the training set for the NN in which the structural theories are inputs and the natural frequencies targets. AAM evaluates the influence of each generalized displacement variable, and NN provides Best Theory Diagrams (BTD), i.e., curves providing the minimum number of nodal degrees of freedom required to satisfy a given accuracy requirement. The aim is to build BTD with far less computational cost than in previous works. The numerical results consider sandwich spherical shells with soft cores and different features, such as thickness and curvature to investigate their influence on the choice of generalized displacement variables. The numerical results show the importance of third-order generalized displacement variables and prove that the present framework can be of interest to evaluate the performance of any structural theory as typical design parameters change and provide guidelines to the analysts on the most convenient computational model to save computational cost without accuracy penalties.

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.

Modelling Articulated Vehicles with a Flexible Semi-Trailer

2013 ◽  
Vol 60 (3) ◽  
pp. 389-407
Author(s):  
Kornel Warwas ◽  
Iwona Adamiec-Wójcik
Keyword(s):  
Numerical Simulation ◽  
Genetic Algorithm ◽  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Free Vibrations ◽  
Reduced Model ◽  
Rigid Finite Element Method ◽  
Articulated Vehicles ◽  
Articulated Vehicle

Abstract The paper presents a model of an articulated vehicle with a flexible frame of a semi-trailer. The rigid finite element method in a modified formulation is used for discretisation of the frame. In order to carry out effective numerical simulation, a reduced model with a considerably smaller number of degrees of freedom is proposed. The parameters of the reduced model are chosen in an optimization process by using a genetic algorithm. To this end, it is assumed that the full and reduced model have to be similar in the range of static deflections and frequencies of free vibrations. Numerical simulations are concerned with the influence of the flexibility of the frame on the motion of the articulated vehicle during an overtaking maneuver. Results are presented and discussed

On normal mode vibrations

1964 ◽  
Vol 60 (3) ◽  
pp. 595-611 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Finite Number ◽  
Linear Systems ◽  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Free Vibrations ◽  
Non Linear ◽  
Non Linear Systems

1. Introduction. In linear systems, the concept of ‘free vibrations in normal modes’ is well defined and fully understood. The meaning of this phrase is far less clear when it is applied to non-linear systems. It is the purpose here to define and examine the free vibrations in normal modes (and their stability) in certain non-linear systems composed of masses and springs and having a finite number of degrees of freedom. Of necessity, such a paper is in some degree conceptual in nature.

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 Static and Vibrational Analysis of Partially Composite Beams Using the Weak-Form Quadrature Element Method

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Zhiqiang Shen ◽  
Hongzhi Zhong
Keyword(s):  
Degrees Of Freedom ◽  
Vibrational Analysis ◽  
Weak Form ◽  
Composite Beams ◽  
Free Vibrations ◽  
Computational Accuracy ◽  
Direct Evaluation ◽  
Quadrature Element Method ◽  
Partially Composite ◽  
Element Method

Deformation of partially composite beams under distributed loading and free vibrations of partially composite beams under various boundary conditions are examined in this paper. The weak-form quadrature element method, which is characterized by direct evaluation of the integrals involved in the variational description of a problem, is used. One quadrature element is normally sufficient for a partially composite beam regardless of the magnitude of the shear connection stiffness. The number of integration points in a quadrature element is adjustable in accordance with convergence requirement. Results are compared with those of various finite element formulations. It is shown that the weak form quadrature element solution for partially composite beams is free of slip locking, and high computational accuracy is achieved with smaller number of degrees of freedom. Besides, it is found that longitudinal inertia of motion cannot be simply neglected in assessment of dynamic behavior of partially composite beams.

An Improved Method of Matrix Displacement Analysis in Vibration Problems

1969 ◽  
Vol 20 (4) ◽  
pp. 321-332 ◽  
Author(s):  
W. Carnegie ◽  
J. Thomas ◽  
E. Dokumaci
Keyword(s):  
Strong Convergence ◽  
Degrees Of Freedom ◽  
Continuous Medium ◽  
Continuous Systems ◽  
Improved Method ◽  
Eigenvalues And Eigenvectors ◽  
Displacement Method ◽  
Vibration Problems ◽  
Number Of Segments

SummaryThis paper presents a method with strong convergence characteristics for the determination of eigenvalues and eigenvectors of continuous systems. The limitation on the number of undetermined constants in the displacement functions introduced by the conditions at the ends of a segment is removed by the introduction of points of freedom within the segment.This improves the convergence of eigenvalues and eigenvectors very rapidly with the number of segments, especially in torsional vibration problems where the convergence with the usual Matrix Displacement method is very poor. The continuous medium is successively approximated by the use of sub-systems with finite numbers of degrees of freedom. The principles upon which the method is based and the convergence of the results are discussed and illustrated by a series of examples.

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