A Simplified Variation of Parameters Approach to Euler’s Equations

1999 ◽  
Vol 66 (1) ◽  
pp. 273-276 ◽  
Author(s):  
D. L. Richardson ◽  
J. W. Mitchell
Keyword(s):  
Euler Equations ◽  
Angular Displacement ◽  
Variation Of Parameters ◽  
Euler’S Equations ◽  
Euler's Equations ◽  
New Form

A normalized form of Euler’s equations is rewritten in a variation of parameters approach using amplitudes and angular displacement as parameters. This new form is compact and yields a more accurate numerically integrated solution over longer simulation times than does a conventional integration of the Euler equations.

scholarly journals Convergence of Vortex Methods for Euler's Equations

1978 ◽  
Vol 32 (143) ◽  
pp. 791 ◽  
Author(s):  
Ole Hald ◽  
Vincenza Mauceri Del Prete
Keyword(s):  
Vortex Methods ◽  
Euler’S Equations ◽  
Euler's Equations

A physics-based turbocharger model for automotive diesel engine control applications

2018 ◽  
Vol 233 (7) ◽  
pp. 1667-1686 ◽  
Author(s):  
Kang Song ◽  
Devesh Upadhyay ◽  
Hui Xie
Keyword(s):  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Turbine Rotor ◽  
Variable Geometry ◽  
Gas Velocity ◽  
Euler’S Equations ◽  
Variable Geometry Turbine ◽  
Euler's Equations ◽  
Compressor Power

Control-oriented models of turbocharger processes such as the compressor mass flow rate, the compressor power, and the variable geometry turbine power are presented. In a departure from approaches that rely on ad hoc empirical relationships and/or supplier provided performance maps, models based on turbomachinery physics and known geometries are attempted. The compressor power model is developed using Euler’s equations of turbomachinery, where the gas velocity exiting the rotor is estimated from an empirically identified correlation for the ratio between the radial and tangential components of the gas velocity. The compressor mass flow rate is modeled based on mass conservation, by approximating the compressor as an adiabatic converging-diverging nozzle with compressible fluid driven by external work input from the compressor wheel. The variable geometry turbine power is developed with Euler’s equations, where the turbine exit swirl and the gas acceleration in the vaneless space are neglected. The gas flow direction into the turbine rotor is assumed to align with the orientation of the variable geometry turbine vane. The gas exit velocity is calculated, similar to the compressor, based on an empirical model for the ratio between the turbine rotor inlet and exit velocities. A power loss model is also proposed that allows proper accounting of power transfer between the turbine and compressor. Model validation against experimental data is presented.

GRAPHIC TREATMENT OF EULER'S EQUATIONS

Science ◽  
1927 ◽  
Vol 66 (1700) ◽  
pp. 114-115
Author(s):  
C. Barus
Keyword(s):  
Euler’S Equations ◽  
Euler's Equations

Complete Six-Degrees-of-Freedom Nonlinear Ship Rolling Motion

1994 ◽  
Vol 116 (4) ◽  
pp. 191-201 ◽  
Author(s):  
M. Taz Ul Mulk ◽  
J. Falzarano
Keyword(s):  
Degrees Of Freedom ◽  
Second Harmonic ◽  
Equations Of Motion ◽  
Path Following ◽  
Euler Angle ◽  
Moment Curve ◽  
Six Degrees Of Freedom ◽  
Euler’S Equations ◽  
Euler's Equations ◽  
Asymmetric Nonlinearity

The emphasis of this paper is on nonlinear ship roll motion, because roll is the most critical ship motion of all six modes of motion. However, coupling between roll and the other modes of motion may be important and substantially affect the roll. Therefore, the complete six-degrees-of-freedom Euler’s equations of motion are studied. In previous work (Falzarano et al., 1990, 1991), roll linearly coupled to sway and yaw was studied. Continuing in this direction, this work extends that analysis to consider the dynamically more exact six-degrees-of-freedom Euler’s equations of motion and associated Euler angle kinematics. A combination of numerical path-following techniques and numerical integrations are utilized to study the steady-state response determined using this more exact modeling. The hydrodynamic forces are: linear frequency-dependent added-mass, damping, and wave-exciting, which are varied on a frequency-by-frequency basis. The linearized GM approximation to the roll-restoring moment is replaced with the nonlinear roll-restoring moment curve GZ(φ), and the linear roll wave damping is supplemented by an empirically derived linear and nonlinear viscous damping. A particularly interesting aspect of this modeling is the asymmetric nonlinearity associated with the heave and pitch hydrostatics. This asymmetric nonlinearity results in distinctive “dynamic bias,” i.e., a nonzero mean in heave and pitch time histories for a zero mean periodic forcing, and a substantial second harmonic. A Fourier analysis of the nonlinear response indicates that the harmonic response is similar to the linear motion response.

Translation of Euler's equations

1969 ◽  
Vol 9 (6) ◽  
pp. 37-64 ◽  
Author(s):  
A.Ya. Dubovitskii ◽  
A.A. Milyutin
Keyword(s):  
Euler’S Equations ◽  
Euler's Equations
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