Free Response of Twisted Plates with Fixed Support Separation

2000 ◽  
Vol 123 (2) ◽  
pp. 175-180 ◽  
Author(s):  
Eric M. Mockensturm ◽  
C. D. Mote,
Keyword(s):  
Aspect Ratio ◽  
Natural Frequencies ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Free Response ◽  
Moving Plate ◽  
Discrete Eigenvalue ◽  
Initial Tension ◽  
Linear Partial Differential Equations ◽  
Axially Moving

In previous work, Mockensturm and Mote investigated the effects of twist on steady motions of an axially moving plate. It was found that twisting produces compressive stresses that increase with twist, aspect ratio, and initial, longitudinal tension. In this work, we study the effects of the stresses and non-flat equilibrium produced by twist on the free response. To accomplish this, the equations of motion are linearized about the equilibrium configuration, yielding a set of three, coupled, linear partial differential equations. The equations are discretized and the free response is predicted from the resulting discrete eigenvalue problem. As a function of twist angle, the natural frequencies first increase and then decrease rapidly to zero as the compressive lateral stresses become sufficiently large to cause wrinkling. The effects of thickness, aspect ratio, and initial tension on natural frequencies are also studied.

Free Response of Twisted Plates

1999 ◽  
Author(s):  
Eric M. Mockensturm ◽  
C. D. Mote
Keyword(s):  
Aspect Ratio ◽  
Natural Frequencies ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Free Response ◽  
Moving Plate ◽  
Discrete Eigenvalue ◽  
Initial Tension ◽  
Linear Partial Differential Equations ◽  
Axially Moving

Abstract In previous work, Mockensturm and Mote (1999) investigated the effects of twist on steady motions of an axially moving plate. It was found that twisting produces compressive stresses that increase with twist, aspect ratio, and initial, longitudinal tension. In this work, we study the effects of the stresses and non-flat equilibrium produced by twist on the free response. To accomplish this, the equations of motion are linearized about the equilibrium configuration, yielding a set of three, coupled, linear partial differential equations. The equations are discretized and the free response is predicted from the resulting discrete eigenvalue problem. As a function of twist angle, the natural frequencies first increase and then decrease rapidly to zero as the compressive lateral stresses become sufficiently large to cause wrinkling. The effects of thickness, aspect ratio, and initial tension on natural frequencies are also studied.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

1971 ◽  
Vol 13 (1) ◽  
pp. 51-59 ◽  
Author(s):  
B. Dawson ◽  
N. G. Ghosh ◽  
W. Carnegie
Keyword(s):  
Differential Equations ◽  
Shear Deformation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Slenderness Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Rotary Inertia ◽  
Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

scholarly journals Parametric Vibrations of Axially Moving Beams with Multiple Edge Cracks

2019 ◽  
Vol 24 (2) ◽  
pp. 241-252 ◽  
Author(s):  
Murat Sarıgül
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Multiple Cracks ◽  
Mean Velocity ◽  
Beam System ◽  
Crack Tips ◽  
Axially Moving Beams ◽  
Edge Cracks ◽  
Axially Moving

Nonlinear transverse vibrations of axially moving beams with multiple cracks is handled studied. Assuming that the beam moves with mean velocity having harmonically variation, influence of the edge crack on the moving continua are investigated in this study. Due to existence of the crack in the transverse direction, the healthily beam is divided into parts. The translational and rotational springs are replaced between these parts so that high stressed regions around the crack tips are redefined with the springs' energies. Thus, the problem is converted to an axially moving spring-beam system. The equations of motion and its corresponding conditions are obtained by means of the Hamilton Principle. In numerical analysis, the natural frequencies and responses of the spring-beam system are investigated for principal parametric resonance in detail. Some important results are obtained; the natural frequencies decreases with increasing crack depth. In case of the beam travelling with high velocities, the effects of crack's depth on natural frequencies seems to be vanished.

Dynamic Stability of Axially Moving Plates With Periodically Varying Speeds Under Partially Distributed Edge Tensions

2012 ◽  
Author(s):  
T. H. Young ◽  
S. J. Huang ◽  
A. C. Liu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Plane Elasticity ◽  
Moving Plate ◽  
System Equations ◽  
Axially Moving ◽  
Axially Moving Web ◽  
Moving Speed

This paper investigates the dynamic stability of an axially moving web which translates with periodically varying speeds and is subjected to partially distributed tensions on two opposite edges. The web is modeled as a rectangular plate simply supported at two opposite edges where the tension is applied, and free at the other two edges. The plate is assumed to possess internal damping, which obeys the Kelvin-Voigt model. The moving speed of the plate is expressed as the sum of a constant speed and a periodical perturbation with a zero mean. Due to the periodically varying speed of the moving plate, terms with time-dependent coefficients appear in the equations of motion, which may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the partially distributed edge tensions is determined exactly by the theory of plane elasticity. Then, the dependence on the spatial coordinates in the equations of motion is eliminated by the Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve this set of system equations analytically if the periodical perturbation of the moving speed is much smaller as compared with the average speed of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the sum-type appear between modes having the same symmetry class in the transverse direction. Unstable regions of main resonances are generally larger than those of sum-type resonances.

An Analytical Method for Free Vibration Analysis of Composite Beams Subjected to Initial Thermal Stresses

2011 ◽  
Vol 482 ◽  
pp. 1-9
Author(s):  
A. Mahi ◽  
E.A. Adda-Bedia ◽  
A. Benkhedda
Keyword(s):  
Aspect Ratio ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Thermal Stresses ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Ply Orientation

The purpose of this paper is to present exact solutions for the free vibration of symmetrically laminated composite beams. The present analysis includes the first shear deformation theory and the rotary inertia. The analytical solutions take into account the thermal effect on the free vibration characteristics of the composite beams. In particular, the aim of this work is to derive the exact closed-form characteristic equations for common boundary conditions. The different parameters that could affect the natural frequencies are included as factors (aspect ratio, thermal load-to-shear coefficient, ply orientation) to better perform dynamic analysis to have a good understanding of dynamic behavior of composite beams. In order to derive the governing set of equations of motion, the Hamilton’s principle is used. The system of ordinary differential equations of the laminated beams is then solved and the natural frequencies’ equations are obtained analytically for different boundary conditions. Numerical results are presented to show the influence of temperature rise, aspect ratio, boundary conditions and ply orientation on the natural frequencies of composite beams.

Effect of Fluid–Structure Interaction on Vibration of Moving Sandwich Plate With Balsa Wood Core and Nanocomposite Face Sheets

2020 ◽  
Vol 12 (07) ◽  
pp. 2050078
Author(s):  
Elham Haghparast ◽  
Amirabbas Ghorbanpour-Arani ◽  
Ali Ghorbanpour Arani
Keyword(s):  
Sandwich Plate ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Fluid Structure Interaction ◽  
Two Phase ◽  
Moving Plate ◽  
Fluid Structure ◽  
Balsa Wood ◽  
Structure Interaction ◽  
Axially Moving

This research presents theoretical investigation to analyze vibration of axially moving sandwich plate floating on fluid. This plate is composed of balsa wood core and two nanocomposite face sheets where the three layers vibrated as an integrated sandwich. The fluid–structure interaction (FSI) effects on the stability of moving plate are considered for both ideal and viscous fluid. Halpin–Tsai model is utilized to determine the material properties of two-phase composite consist of uniformly distributed and randomly oriented carbon nanotubes (CNTs) through the PmPV (poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene)vinylene]}) matrix. The governing equations are derived based on sinusoidal shear deformation plate theory (SSDT) which is more accurate than the conventional theories, and significantly, it does not require a shear correction factor. Employing Hamilton’s principle, the equations of motion are obtained and solved by the semi-analytical method. Results indicated that the dimensionless frequencies of moving sandwich plate decrease rapidly with increasing the water level and they are almost independent of fluid level when it is higher than 50% of the plate length. The results of this investigation can be used in design and manufacturing of marine vessels and aircrafts.

Dynamic Properties of a Moving Thread Line

1976 ◽  
Vol 98 (3) ◽  
pp. 868-875 ◽  
Author(s):  
M. A. Moustafa ◽  
F. K. Salman
Keyword(s):  
Fundamental Frequency ◽  
Dynamic Properties ◽  
Unit Length ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Initial Tension ◽  
Tension Distribution ◽  
Elastic Strings ◽  
Axially Moving ◽  
Mathematical Relations

A mathematical model representing the transverse vibration of axially moving elastic strings is presented considering tension and mass variation. A suggested numerical scheme was successfully used to solve the nonlinear partial differential equations of motion. For axially nonmoving strings, the effect of initial amplitudes, and consequently the tension variation on the fundamental frequency is obtained. Also, the effect of the initial tension and the mass of the string per unit length on the fundamental frequency and their corresponding mathematical relations are presented. For axially moving strings, the effect of the axial velocity on the fundamental frequency as well as the tension distribution along the thread is given. Also the behavior of the string at velocities equal and greater than the wave speed is shown.

Effects of Structural Properties on Natural Frequencies of Axially Moving Beams with an Intermediate Support

2015 ◽  
Vol 801 ◽  
pp. 129-135
Author(s):  
Şafak Aksoy ◽  
Ferid Kostekci
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Flexural Rigidity ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Axially Moving Beam ◽  
Intermediate Support ◽  
Linear Vibrations ◽  
Axially Moving ◽  
Non Linear System

In this study, linear vibrations of axially moving beam simply supported between the guides are examined and natural frequencies are calculated numerically. The vibrations of axially tensioned Euler-Bernoulli beam are investigated under clamped-clamped end conditions. Governing differential equations of motion are derived using Hamilton’s Principle for two regions of the beam. The boundaries at the outer ends of the beam are assumed immovable. Non-dimensional equations of motion are derived and the solutions of the linear problem are obtained. Assuming a weakly non-linear system, linear equations are obtained using the Method of Multiple Scales. First seven natural frequencies are calculated numerically based on the flexural rigidity, axial velocity and locations of the intermediate support.

Size-Dependent Vibration of Axially Moving Viscoelastic Micro-Plates Based on Sinusoidal Shear Deformation Theory

2017 ◽  
Vol 09 (02) ◽  
pp. 1750026 ◽  
Author(s):  
Ali Ghorbanpour Arani ◽  
Elham Haghparast
Keyword(s):  
Aspect Ratio ◽  
Size Effect ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Modified Couple Stress Theory ◽  
Axially Moving ◽  
Sinusoidal Shear Deformation Theory

In the present research, vibration and instability of axially moving viscoelastic micro-plate is investigated. Sinusoidal shear deformation theory (SSDT) is utilized due to its accuracy of polynomial functions than other plate theories. Based on Kelvin’s model, the viscoelastic structural properties of micro-plate are taken into consideration. The modified couple stress theory (MCST) is employed because of its capability to interpret the size effect. Using Hamilton’s principle, equations of motion are obtained and solved by hybrid analytical–numerical solution at different boundary conditions. Influences of various parameters such as size effect, axially moving speed, viscoelastic structural damping coefficient, thickness and aspect ratio on the vibration characteristics of moving viscoelastic micro-plate are discussed in detail. The results indicated that the critical speed of moving micro-plate is strongly dependent on the aspect ratio, therefore, the low aspect ratio should be considered for optimum design of this kind of moving micro-devices. The results of this investigation can be used in design and manufacturing of axially moving systems at the micro-scale such as micro-magnetic tapes.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

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