Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2020 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems, a general algorithm has been developed to compute natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam’s cross section.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2021 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam's cross section.

scholarly journals Transverse Vibrations and Stability of Axially Traveling Sandwich Beam With Soft Core

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Xiao-Dong Yang ◽  
Wei Zhang ◽  
Li-Qun Chen
Keyword(s):  
Critical Speed ◽  
Natural Frequencies ◽  
Sandwich Beam ◽  
Axial Loading ◽  
Core Layer ◽  
Soft Core ◽  
Axially Moving Beam ◽  
Transverse Vibrations ◽  
Axially Moving ◽  
The Galerkin Method

The transverse vibrations and stability of an axially moving sandwich beam are studied in this investigation. The face layers are assumed to be in the membrane state, which bears only axial loading but no bending. Only shear deformation is considered for the soft core layer. The governing partial equation is derived using Newton's second law and then transferred into a dimensionless form. The Galerkin method and the complex mode method are employed to study the natural frequencies. In comparison with the classical homogenous axially moving beam, the gyroscopic matrix is no longer skew-symmetric because of the introduction of the soft core. The critical speed for the divergence of the axially moving sandwich beam is analytically obtained. The contribution of the core layer shear modulus to the natural frequencies and critical speed is discussed.

Lateral Vibration of Two Axially Translating Beams Interconnected by a Winkler Foundation

2006 ◽  
Vol 129 (3) ◽  
pp. 380-385 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Müftü
Keyword(s):  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Mode Shapes ◽  
Winkler Foundation ◽  
Axial Tension ◽  
Winkler Elastic Foundation ◽  
Critical Tension ◽  
Divergence Instability ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. The natural frequencies and associated mode shapes are obtained. The natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at a critical velocity and a critical tension; and, divergence and flutter instabilities coexist at postcritical speeds, and divergence instability takes place precritical tensions. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams are presented.

Nonlinear Dynamics and Bifurcations of an Axially Moving Beam

1999 ◽  
Vol 122 (1) ◽  
pp. 21-30 ◽  
Author(s):  
F. Pellicano ◽  
F. Vestroni
Keyword(s):  
High Dimensional ◽  
Axially Moving Beam ◽  
Dimensional System ◽  
Global Dynamics ◽  
Dimensional Phase Space ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method ◽  
Sample Case

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem: a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied. [S0739-3717(00)00501-8]

Support-Condition Analyses of Rotating Beams Under Distributed Imbalance Loading

2011 ◽  
Author(s):  
Kai Jokinen ◽  
Erno Keskinen ◽  
Marko Jorkama ◽  
Wolfgang Seemann
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Cross Section ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Element Model ◽  
Mode Shapes ◽  
Constant Cross Section ◽  
Support Condition ◽  
Beam Elements

In roll balancing the behaviour of the roll can be studied either experimentally with trial weights or, if the roll dimensions are known, analytically by forming a model of the roll to solve response to imbalance. Essential focus in roll balancing is to find the correct amount and placing for the balancing mass or masses. If this selection is done analytically the roll model used in calculations has significant effect to the balancing result. In this paper three different analytic methods are compared. In first method the mode shapes of the roll are defined piece wisely. The roll is divided in to five parts having different cross sections, two shafts, two roll ends and a shell tube of the roll. Two boundary conditions are found for both supports of the roll and four combining equations are written to the interfaces of different roll parts. Totally 20 equations are established to solve the natural frequencies and to form the mode shapes of the non-uniform roll. In second model the flexibility of shafts and the stiffness of the roll ends are added to the support stiffness as serial springs and the roll is modelled as a one flexibly supported beam having constant cross section. Finally the responses to imbalance of previous models are compared to finite element model using beam elements. Benefits and limitations of each three model are then discussed.

Vibration Response of Functionally Graded Rotating Disk With a Circumferential Crack

2010 ◽  
Author(s):  
Rui Liu ◽  
Hamid Nayeb-Hashemi
Keyword(s):  
Critical Speed ◽  
Crack Depth ◽  
Rotating Disk ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Campbell Diagram ◽  
Circumferential Crack ◽  
Inner Radius ◽  
The Galerkin Method

In this study, the vibration characteristics of a functionally graded rotating hollow disk with the circumferential surface crack are investigated. In order to simplify the problem, the circumferential crack of the rotating hollow disk is modeled as circumferential step indentation. The Galerkin Method is used to obtain the radial and hoop stresses for disks with clamped edge at the inner radius. Finite Difference scheme is adopted to solve the partial differential equation of motion of the rotating hollow disk to obtain the mode shapes and the Campbell Diagram. The first critical speed, which is one of the important parameters limiting the performance of the rotating disk, was obtained from the Campbell Diagram. The results show that the crack will reduce the stiffness and the critical speed of the rotating disk. Critical speed increases with decreasing the distance from inner radius to the crack and decreases with increasing crack depth. Furthermore, considering the functionally graded disk, the distribution of elastic modulus does not change significantly the effects of circumferential cracks on the vibration characteristics of the rotating.

Finite Width Effects on the Critical Speed of Axially Moving Materials

1998 ◽  
Vol 120 (2) ◽  
pp. 633-634 ◽  
Author(s):  
C. C. Lin
Keyword(s):  
Critical Speed ◽  
Beam Theory ◽  
Axially Moving Beam ◽  
Finite Width ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Simply Supported ◽  
Moving Beam ◽  
Two Dimensional Model ◽  
Axially Moving

An exact solution is provided to determine the effects of the free end boundary conditions and the slenderness ratios on the critical speed of two dimensional, axially moving materials. The axially moving beam theory and the mathematical, two-dimensional model with all edges simply supported are also considered for comparison.

scholarly journals Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Erasmo Viola ◽  
Marco Miniaci ◽  
Nicholas Fantuzzi ◽  
Alessandro Marzani
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Internal Boundary ◽  
Radius Of Curvature ◽  
Inertia Effects ◽  
Circular Arch

AbstractThis paper investigates the in-plane free vibrations of multi-stepped and multi-damaged parabolic arches, for various boundary conditions. The axial extension, transverse shear deformation and rotatory inertia effects are taken into account. The constitutive equations relating the stress resultants to the corresponding deformation components refer to an isotropic and linear elastic material. Starting from the kinematic hypothesis for the in-plane displacement of the shear-deformable arch, the equations of motion are deduced by using Hamilton’s principle. Natural frequencies and mode shapes are computed using the Generalized Differential Quadrature (GDQ) method. The variable radius of curvature along the axis of the parabolic arch requires, compared to the circular arch, a more complex formulation and numerical implementation of the motion equations as well as the external and internal boundary conditions. Each damage is modelled as a combination of one rotational and two translational elastic springs. A parametric study is performed to illustrate the influence of the damage parameters on the natural frequencies of parabolic arches for different boundary conditions and cross-sections with localizeddamage.Results for the circular arch, derived from the proposed parabolic model with the derivatives of some parameters set to zero, agree well with those published over the past years.

A New Approach to Compute Natural Frequencies and Mode Shapes of Non-Uniform Continuous Bar, Circular Shaft and Beam Vibration

2019 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Mode Shapes ◽  
Engineering Structures ◽  
Cross Sectional ◽  
Closed Form Solutions ◽  
Continuous Structure ◽  
Circular Shaft ◽  
Spatial Parameters

Abstract The wave equation governing longitudinal vibration of a bar and torsional vibration of a circular shaft, and the Euler-Bernoulli equation governing transverse vibration of a beam were developed in the eighteenth century. Natural frequencies and mode shapes are easily obtained for uniform or constant spatial parameters (cross sectional area, material property and mass distribution). But, real engineering structures seldom have constant parameters. For non-uniform continuous structure, a large number of papers have been written for more than 100 years since the publication of Kirchhoff’s memoir in 1882. There are analytical solutions only in few cases, and there are approximate numerical methods to deal with other (almost all) cases, most notably Stodola, Holzer and Myklestad methods in addition to Rayleigh-Ritz and finite element methods. This paper presents a novel approach to compute natural frequencies and mode shapes for arbitrary variations of spatial parameters on the basis of linear time-varying system theory. The advantage of this approach is that now it can be claimed that “almost” closed-form solutions are available to find natural frequencies and mode shapes of any non-uniform, linear and one-dimensional continuous structure.

scholarly journals Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

2018 ◽  
Vol 2018 ◽  
pp. 1-25 ◽  
Author(s):  
Blake Martin ◽  
Armaghan Salehian
Keyword(s):  
Reference Values ◽  
Natural Frequencies ◽  
Unit Length ◽  
Reference Value ◽  
Bending Stiffness ◽  
Mode Shapes ◽  
Beam Model ◽  
Continuous Bending ◽  
Spatially Varying ◽  
Euler Bernoulli Beam

The Lindstedt-Poincaré method is applied to a nonuniform Euler-Bernoulli beam model for the free transverse vibrations of the system. The nonuniformities in the system include spatially varying and piecewise continuous bending stiffness and mass per unit length. The expression for the natural frequencies is obtained up to second-order and the expression for the mode shapes is obtained up to first-order. The explicit dependence of the natural frequencies and mode shapes on reference values for the bending stiffness and the mass per unit length of the system is determined. Multiple methods for choosing these reference values are presented and are compared using numerical examples.

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