An Accurate Method for First and Second Derivatives of Dynamic Responses

2011 ◽  
Vol 27 (3) ◽  
pp. 389-398 ◽  
Author(s):  
Q. Liu ◽  
J. Zhang ◽  
L. Gu ◽  
L. Yan
Keyword(s):  
Analytical Method ◽  
Efficient Algorithm ◽  
Accurate Method ◽  
Dynamic Responses ◽  
New Method ◽  
Dynamic Loads ◽  
Dynamics Analysis ◽  
Design Variables ◽  
Second Derivatives ◽  
Derivatives Of

ABSTRACTThis paper has developed an accurate method for calculating the first and second derivatives of dynamic responses with respect to the design variables of structures subjected to dynamic loads. An efficient algorithm to calculate the dynamic responses, their first and second derivatives with respect to the design variables is formulated based on the Newmark-β method. The algorithm is achieved by direct differentiation and only a single dynamics analysis is required. An example is demonstrated with the new method proposed in this paper and the analytical method. The comparative numerical results show the new method is highly accurate compared to the analytical method.

Computation of finite strains from moire displacement patterns

1968 ◽  
Vol 3 (1) ◽  
pp. 65-75 ◽  
Author(s):  
W Bossaert ◽  
R Dechaene ◽  
A Vinckier
Keyword(s):  
Accurate Method ◽  
Finite Strains ◽  
New Method ◽  
Algebraic Functions ◽  
Partial Derivatives ◽  
Moiré Fringes ◽  
Practical Necessity ◽  
Displacement Patterns ◽  
Derivatives Of

A new and accurate method is described for deriving finite strains from moiré displacement patterns. The components u and v of the displacements at the mesh points of a Lagrangian co-ordinate network are estimated from the moiré fringes. In the neighbourhood of a point, where the strains are to be calculated, two algebraic functions are determined, approximating as closely as possible to the two sets of estimated u and v values. The strains are then calculated from the partial derivatives of these two functions. The procedure is repeated for all points at which the strains are required. When this procedure is applied to a great number of points the use of a computer becomes a practical necessity. Once the formulae have been programmed, the operator has only to estimate the u and v values at the mesh points of the co-ordinate network and to feed this set of numbers to the computer. This new method thus allows a considerable gain in time and brings about an improvement in accuracy compared with other methods.

Vibration and dynamics analysis of electric vehicle drivetrains

2021 ◽  
pp. 146134842097920
Author(s):  
Xia Hua ◽  
Eric Gandee
Keyword(s):  
Electric Vehicle ◽  
Dynamic Mesh ◽  
Dynamic Responses ◽  
Dynamic Loads ◽  
Dynamics Analysis ◽  
Motor Speed ◽  
Motor Torque ◽  
Gear Teeth ◽  
Gear Mesh ◽  
Bearing Loads

The importance of the vibration and dynamics of electric vehicle drivetrains has increased because of noise and durability concerns. In this study, the important dynamic responses of drivetrains, including the dynamic mesh force acting at the gear teeth, dynamic loads acting at the bearings, and torsional fluctuation of the tire or load under major vibration excitations, such as motor torque fluctuation excitation and spiral bevel gear mesh excitation, were investigated. The results demonstrate that at a lower motor speed, dynamic responses such as the dynamic mesh force, dynamic bearing loads, and dynamic torsional displacement of the tire or load under motor torque fluctuation are dominant. At a higher motor speed, however, the dynamic responses under the gear mesh excitation are dominant. In addition, increasing the pinion-motor torsional compliance is an effective approach for suppressing the dynamic responses of drivetrains under motor torque fluctuation.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

scholarly journals Advanced Graphical–Analytical Method of Pipe Tank Design Integrated with Sensitivity Analysis for Sustainable Stormwater Management in Urbanized Catchments

Water ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1035
Author(s):  
Bartosz Szeląg ◽  
Adam Kiczko ◽  
Anna Musz-Pomorska ◽  
Marcin K. Widomski ◽  
Jacek Zaburko ◽  
...  
Keyword(s):  
Sensitivity Analysis ◽  
Analytical Method ◽  
Design Method ◽  
Accurate Method ◽  
Urban Catchment ◽  
Storm Water Management Model ◽  
Catchment Areas ◽  
Calculation Methodology ◽  
Tank Design ◽  
Spatial Plan

Pipe tanks represent important runoff retention elements of urban stormwater systems. They enable us to reduce and retain runoff as well as to mitigate peak flows in the network. Pipe tanks are often taken into account while designing the spatial plan of urban catchment areas. Hence, there is a need to develop a relatively quick and accurate method for pipe tank dimensioning. A graphical–analytical method of designing a pipe tank is presented in the paper. In the assumed methodology, the possibility of employing machine learning for obtaining a more precise error prediction of the proposed pipe tank design method (compared with the tank volume simulations using the storm water management model (SWMM)) are considered. Thus far, this aspect has not been discussed in the literature. In the adopted calculation methodology, sensitivity analysis constitutes an important element, enabling us to assess the influence of the input data assumed for tank design on the dimensions of the outflow devices and the length of the retention chamber.

A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

1990 ◽  
Vol 112 (1) ◽  
pp. 83-87 ◽  
Author(s):  
R. H. Fries ◽  
B. M. Coffey
Keyword(s):  
Numerical Simulation ◽  
White Noise ◽  
State Space ◽  
Spectral Characteristics ◽  
The State ◽  
Second Derivatives ◽  
Time Histories ◽  
Sample Interval ◽  
Rail Irregularities ◽  
Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

Investigation of a Technique for the Differentiation of Empirical Data

1983 ◽  
Vol 105 (3) ◽  
pp. 200-202 ◽  
Author(s):  
D. M. Trujillo ◽  
H. R. Busby
Keyword(s):  
Dynamic Programming ◽  
Differentiable Function ◽  
Empirical Data ◽  
Numerical Experiments ◽  
Random Noise ◽  
Second Derivatives ◽  
Derivatives Of

A dynamic programming filter is derived to estimate the first and second derivatives of empirical data. A series of numerical experiments are conducted using a known differentiable function with various amounts of added random noise.

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