A Strain Energy Comparison of Discrete Modeling for Vibrating Continuous Systems

1972 ◽  
Vol 94 (1) ◽  
pp. 23-30
Author(s):  
S. K. Tolani ◽  
R. D. Rocke
Keyword(s):  
Strain Energy ◽  
Mass Matrix ◽  
Free Vibrations ◽  
Continuous Systems ◽  
Euler Beam ◽  
Closed Form Solutions ◽  
Lumped Parameter ◽  
Lumped Parameter Models ◽  
Vibration Problems ◽  
Consistent Mass Matrix

Lumped parameter models commonly used to describe continuous one-dimensional and Bernoulli-Euler beam vibration problems have been compared on the basis of maximum system strain energy. The consistent mass matrix approach has been included in the comparison. Standard matrix techniques have been employed to mathematically obtain desired solutions. Closed form solutions and solutions via the models to the system strain energy were obtained for all systems in three dynamic states: Free vibrations, constant base acceleration, and half sine base acceleration. Behavior of the strain energy errors, in general, were found to be similar to those of the frequency root errors.

Finite Element Analysis of Vibrating Frequency for Skew Slab Bridge Based on Finite Strip Thought

2010 ◽  
Vol 163-167 ◽  
pp. 1121-1127
Author(s):  
Mei Liang Yang ◽  
Gui Yun Xia ◽  
Jian Ren Zhang
Keyword(s):  
Mass Matrix ◽  
Maximum Error ◽  
Euler Beam ◽  
Element Analysis ◽  
Finite Strip ◽  
System A ◽  
Specification Method ◽  
Mesh Density ◽  
Cartesian Coordinate ◽  
Consistent Mass Matrix

Based on the finite strip thought and displacement interpolation function of Bernoulli-Euler beam element, using the transformation relationship between skew coordinate and Cartesian coordinate system, a new kind of thin parallel slab element was established, element stiffness matrix and consistent mass matrix were derived. The vibrating frequency of simply supported skewed slab was calculated. Computing results were compared with theoretical results and Ansys results. The maximum error was 2.68%. Changing the mesh density of skew slab, the convergence of present element was tested. Examples show that this element has the features of high precision and strong convergence. At last, the vibrating frequency coefficients of skew slab bridge with different ratios of span to width were provided, which can be adopted to compute the vehicle’s impact factor of skew slab bridge by specification method.

scholarly journals Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2288
Author(s):  
Hongming Luo ◽  
Guanhua Sun
Keyword(s):  
Structural Dynamics ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Time Integration ◽  
Partition Of Unity ◽  
Implicit Time ◽  
Lumped Mass ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Consistent Mass Matrix

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

Stability Criteria for Distributed Nonlinear Sampled-Data Systems Defined by Green’s Functions

1974 ◽  
Vol 96 (3) ◽  
pp. 315-321 ◽  
Author(s):  
G. Jumarie
Keyword(s):  
Stability Criterion ◽  
Feedback Gain ◽  
Continuous Systems ◽  
Mean Square ◽  
Data Systems ◽  
Sampled Data ◽  
Input Output ◽  
Output Stability ◽  
Lumped Parameter ◽  
The Mean

Sampled-data, nonlinear, distributed systems, which exhibit a structure similar to that of the standard closed loop with lumped parameter, are investigated from the viewpoint of their input-output stability. These systems are governed by operational equations involving discrete Laplace-Green kernels. Their feedback gains are bounded by upper and lower values which depend explicitly on the time and the distributed parameter. The main result is: an input-output stability theorem is given which applies both in L∞ (O, ∞) and L2 (O, ∞). This criterion, which may be considered as being an extension of the ≪circle criterion≫, involves the mean square value on the bounds of the feedback gain. Stability conditions for continuous systems are derived from this result. In the special case of systems with distributed periodical time-varying feedback gains, a stability criterion is given which applies in Marcinkiewicz space M2 (O, ∞). This result which involves the mean square value of the feedback gain is generally less restrictive than the L2 (O, ∞) stability criterion mentioned above.

Elastic Fields in a Polygon-Shaped Inclusion With Uniform Eigenstrains

1997 ◽  
Vol 64 (3) ◽  
pp. 495-502 ◽  
Author(s):  
H. Nozaki ◽  
M. Taya
Keyword(s):  
Composite Materials ◽  
Closed Form ◽  
Strain Field ◽  
Strain Energy ◽  
Convex Polygon ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Closed Form Solutions ◽  
Arbitrary Convex ◽  
Shaped Inclusion

In this paper the elastic fields in an arbitrary, convex polygon-shaped inclusion with uniform eigenstrains are investigated under the condition of plane strain. Closed-form solutions are obtained for the elastic fields in a polygon-shaped inclusion. The applications to the evaluation of the effective elastic properties of composite materials with polygon-shaped reinforcements are also investigated for both dilute and dense systems. Numerical examples are presented for the strain field, strain energy, and stiffness of the composites with polygon shaped fibers. The results are also compared with some existing solutions.

Lumped parameter models for two-ventricle and healthy and failing extracardiac Fontan circulations

2021 ◽  
Author(s):  
Matthew G Doyle ◽  
Marina Chugunova ◽  
S Lucy Roche ◽  
James P Keener
Keyword(s):  
Single Ventricle ◽  
Left Ventricular ◽  
Sensitivity Analyses ◽  
Surgical Strategies ◽  
Lumped Parameter ◽  
Lumped Parameter Models ◽  
Fontan Failure ◽  
Existence Of Periodic Solutions ◽  
Mathematical Definitions ◽  
Extracardiac Fontan

Abstract Fontan circulations are surgical strategies to treat infants born with single ventricle physiology. Clinical and mathematical definitions of Fontan failure are lacking, and understanding is needed of parameters indicative of declining physiologies. Our objective is to develop lumped parameter models of two-ventricle and single-ventricle circulations. These models, their mathematical formulations and a proof of existence of periodic solutions are presented. Sensitivity analyses are performed to identify key parameters. Systemic venous and systolic left ventricular compliances and systemic capillary and pulmonary venous resistances are identified as key parameters. Our models serve as a framework to study the differences between two-ventricle and single-ventricle physiologies and healthy and failing Fontan circulations.

Consistent mass matrix in fluid sloshing problems

AIAA Journal ◽  
1976 ◽  
Vol 14 (2) ◽  
pp. 245-247 ◽  
Author(s):  
Grant P. Steven
Keyword(s):  
Mass Matrix ◽  
Fluid Sloshing ◽  
Consistent Mass Matrix
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