Microstructure Theory for a Composite Beam

1971 ◽  
Vol 38 (4) ◽  
pp. 947-954 ◽  
Author(s):  
C.-T. Sun
Keyword(s):  
Composite Beam ◽  
Continuum Model ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Present Theory ◽  
Flexural Waves ◽  
Subsequent Application ◽  
External Forces ◽  
Work Done ◽  
Constituent Layer

A continuum model with microstructure is constructed for laminated beams. In deriving equations, each constituent layer is considered as a Timoshenko beam. With a certain kinematical assumption regarding the deformations of the composite beam the kinetic and strain energies, as well as the variation of the work done by external forces, are computed. The energy and work expressions are “smoothed out” by a smoothing operation, thus transforming the laminated structuring into microstructure of a macro-homogeneous beam. Subsequent application of Hamilton’s principle yields the equations of motion and the boundary conditions. The equations thus obtained are employed to investigate free flexural waves in a composite beam. It is found that the dispersion curves according to the present theory agree very well with the curves obtained according to an exact analysis. Results according to the effective modulus theory are presented for comparison. Two simplified versions of the microstructure beam theory are also developed and discussed.

Energy Conserving Equations of Motion for Gear Systems

2004 ◽  
Vol 127 (2) ◽  
pp. 208-212 ◽  
Author(s):  
Sejoong Oh ◽  
Karl Grosh ◽  
James R. Barber
Keyword(s):  
Equations Of Motion ◽  
Interaction Force ◽  
Fundamental Principle ◽  
Step Change ◽  
Conservation Of Energy ◽  
Gear Design ◽  
Contact Points ◽  
Energy Conserving ◽  
External Forces ◽  
Work Done

A system of two meshing gears exhibits a stiffness that varies with the number of teeth in instantaneous contact and the location of the corresponding contact points. A classical Newtonian statement of the equations of motion leads to a solution that contradicts the fundamental principle of mechanics that the change in total energy in the system is equal to the work done by the external forces, unless the deformation of the teeth is taken into account in defining the direction of the instantaneous tooth interaction force. This paradox is avoided by using a Lagrange’s equations to derive the equations of motion, thus ensuring conservation of energy. This introduces nonlinear terms that are absent in the classical equations of motion. In particular, the step change in stiffness associated with the introduction of an additional tooth to contact implies a step change in strain energy and hence a corresponding step change in kinetic energy and rotational speed. The effect of these additional terms is examined by dynamic simulation, using a system of two involute spur gears as an example. It is shown that the two systems of equations give similar predictions at high rotational speeds, but they differ considerably at lower speeds. The results have implications for gear design, particularly for low speed gear sets.

Dynamics of Timoshenko Beams Conveying Fluid

1976 ◽  
Vol 18 (4) ◽  
pp. 210-220 ◽  
Author(s):  
M. P. Paidoussis ◽  
B. E. Laithier
Keyword(s):  
Equations Of Motion ◽  
Beam Theory ◽  
Present Theory ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Finite Difference Technique ◽  
Observed Behaviour ◽  
Pipes Conveying Fluid ◽  
Euler Bernoulli Beam ◽  
Conveying Fluid

The dynamics of pipes conveying fluid is described by means of the Timoshenko beam theory. The equations of motion are derived and solved ( a) by a finite-difference technique, and ( b) by a variational method. It is shown that the latter is the more efficient method. The eigenfrequencies of the system and its stability characteristics are compared with results obtained previously using the Euler-Bernoulli beam theory, and it is shown that in certain cases (e.g. short pipes) the two sets of results diverge. Experiments indicate that the present theory is more successful in predicting the observed behaviour. Furthermore, the present theory shows that, in some cases, cantilevered pipes may lose stability by buckling, whereas previous theories indicate that the system always loses stability by flutter.

scholarly journals Energy Considerations in Systems With Varying Stiffness

2003 ◽  
Vol 70 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. R. Barber ◽  
K. Grosh ◽  
S. Oh
Keyword(s):  
Energy Conservation ◽  
Total Energy ◽  
Dynamic Behavior ◽  
Contact Force ◽  
Elastic System ◽  
Equations Of Motion ◽  
Local Deformation ◽  
Elastic Systems ◽  
External Forces ◽  
Work Done

If the stiffness of an elastic system changes with time, a conventional Newtonian statement of the equations of motion will generally lead to solutions that violate the fundamental mechanics principle that the work done by the external forces be equal to the increase in total energy of the system. Timoshenko’s discussion of the problem of a vehicle driven across an elastic bridge is generalized to show that energy conservation can be restored only if the local deformation of the components is taken into account in determining the direction of the contact force. This result has important consequences for the interaction of elastic systems in general, including, for example, the dynamic behavior of meshing gears.

Solution of Impact-Induced Flexural Waves in a Circular Ring by the Method of Characteristics

1998 ◽  
Vol 65 (3) ◽  
pp. 569-579 ◽  
Author(s):  
V. P. W. Shim ◽  
S. E. Quah
Keyword(s):  
Numerical Scheme ◽  
Method Of Characteristics ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Circular Ring ◽  
Higher Order ◽  
Flexural Waves ◽  
The Method Of Characteristics ◽  
Finite Difference Equations ◽  
Generalized Displacements

A study of elastic wave propagation in a curved beam (circular ring) is presented. The governing equations of motion are formulated in two forms based on Timoshenko beam theory. Solutions are obtained using the method of characteristics, whereby a numerical scheme employing higher-order interpolation is proposed for the finite difference equations. Results obtained are verified by experiments; it is found that use of the higher-order numerical scheme improves correlation with experimental results. Comparison of the relative accuracy between the two mathematical formulations—one in terms of generalized forces and velocities and the other in terms of generalized displacements—shows the former to be mathematically simpler and to yield more accurate results.

How success enhances self-serving bias: A multinomial process model of the Implicit Association Test

2019 ◽  
Vol 47 (7) ◽  
pp. 1-9 ◽  
Author(s):  
Zhongyang Wang ◽  
Yi Li ◽  
Zheng Jin ◽  
Timothy Tamunang Tamutana
Keyword(s):  
Undergraduate Students ◽  
Implicit Association Test ◽  
Process Model ◽  
Self Esteem ◽  
Regulation Mechanism ◽  
Implicit Association ◽  
Desirable Outcome ◽  
Subsequent Application ◽  
External Forces ◽  
Better Than

Self-serving bias is individuals' belief that leads them to blame external forces when bad things happen and to give themselves credit when good things happen. To evaluate how underlying evaluative associations toward the self or others differ between individuals, and/or how the regulation mechanism of the influence of such associations differs, we used a multinomial process model to measure the underlying implicit self-esteem in these processes with 56 Chinese undergraduate students. The results indicated that participants assessed themselves as being better than others when their performance was followed by a desirable outcome. Subsequent application of the quadruple processes showed that both activation of positive associations toward self and regulation of the associations played important roles in attitudinal responses. Our findings may provide a supplementary explanation to that of previous results, promoting understanding of the mechanism underlying self-serving bias.

scholarly journals Numerical methods for General Relativistic particles

2018 ◽  
Vol 14 (S342) ◽  
pp. 19-23
Author(s):  
Fabio Bacchini ◽  
Bart Ripperda ◽  
Alexander Y. Chen ◽  
Lorenzo Sironi
Keyword(s):  
Numerical Methods ◽  
Gravitational Lensing ◽  
Complex Dynamics ◽  
Equations Of Motion ◽  
Numerical Algorithms ◽  
Relativistic Effects ◽  
Compact Objects ◽  
Light Rays ◽  
Massive Particles ◽  
External Forces

AbstractWe present recent developments on numerical algorithms for computing photon and particle trajectories in the surrounding of compact objects. Strong gravity around neutron stars or black holes causes relativistic effects on the motion of massive particles and distorts light rays due to gravitational lensing. Efficient numerical methods are required for solving the equations of motion and compute i) the black hole shadow obtained by tracing light rays from the object to a distant observer, and ii) obtain information on the dynamics of the plasma at the microscopic scale. Here, we present generalized algorithms capable of simulating ensembles of photons or massive particles in any spacetime, with the option of including external forces. The coupling of these tools with GRMHD simulations is the key point for obtaining insight on the complex dynamics of accretion disks and jets and for comparing simulations with upcoming observational results from the Event Horizon Telescope.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Free vibration analysis and post-critical free vibrations of nanocomposite rotating beams reinforced with graphene platelet

2021 ◽  
pp. 107754632110511
Author(s):  
Arameh Eyvazian ◽  
Chunwei Zhang ◽  
Farayi Musharavati ◽  
Afrasyab Khan ◽  
Mohammad Alkhedher
Keyword(s):  
Natural Frequency ◽  
Thermal Environment ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Weight Fraction ◽  
Free Vibrations ◽  
Graphene Platelet ◽  
The Impact

Treatment of the first natural frequency of a rotating nanocomposite beam reinforced with graphene platelet is discussed here. In regard of the Timoshenko beam theory hypothesis, the motion equations are acquired. The effective elasticity modulus of the rotating nanocomposite beam is specified resorting to the Halpin–Tsai micro mechanical model. The Ritz technique is utilized for the sake of discretization of the nonlinear equations of motion. The first natural frequency of the rotating nanocomposite beam prior to the buckling instability and the associated post-critical natural frequency is computed by means of a powerful iteration scheme in reliance on the Newton–Raphson method alongside the iteration strategy. The impact of adding the graphene platelet to a rotating isotropic beam in thermal ambient is discussed in detail. The impression of support conditions, and the weight fraction and the dispersion type of the graphene platelet on the acquired outcomes are studied. It is elucidated that when a beam has not undergone a temperature increment, by reinforcing the beam with graphene platelet, the natural frequency is enhanced. However, when the beam is in a thermal environment, at low-to-medium range of rotational velocity, adding the graphene platelet diminishes the first natural frequency of a rotating O-GPL nanocomposite beam. Depending on the temperature, the post-critical natural frequency of a rotating X-GPL nanocomposite beam may be enhanced or reduced by the growth of the graphene platelet weight fraction.

Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

1953 ◽  
Vol 20 (4) ◽  
pp. 469-474
Author(s):  
W. A. Nash
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Middle Surface ◽  
Elastic Buckling ◽  
Elastic Instability ◽  
Total Potential ◽  
External Forces ◽  
Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

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