scholarly journals Energy analysis of Duffing oscillators with quadratic damping: exact solutions

2021 ◽  
Vol 2090 (1) ◽  
pp. 012008
Author(s):  
Z Rakaric ◽  
I Kovacic
Keyword(s):  
Energy Loss ◽  
Exact Solutions ◽  
Closed Form ◽  
Gamma Function ◽  
Energy Analysis ◽  
Incomplete Gamma Function ◽  
Restoring Force ◽  
Phase Trajectories ◽  
Quadratic Damping ◽  
Insight Into

Abstract Oscillators with a Duffing-type restoring force and quadratic damping are dealt with in this paper. Four characteristic cases of this restoring force are analysed: hardening, softening, bistable and a pure cubic one. Their energy-displacement relationships are considered, and the corresponding closed-form exact solutions are obtained in terms of the incomplete Gamma function, which represent new results. Such results provide insight into damped dynamics of the class of system, including finding the phase trajectories as well as the comparison between these cases from the viewpoint of the energy loss per cycle.

ON THE INCOMPLETE GAMMA FUNCTION AND ITS NEUTRIX CONVOLUTION FOR NEGATIVE INTEGERS

2019 ◽  
Vol 10 (1) ◽  
pp. 30-51
Author(s):  
Mongkolsery Lin ◽  
◽  
Brian Fisher ◽  
Somsak Orankitjaroen ◽  
◽  
...  
Keyword(s):  
Gamma Function ◽  
Incomplete Gamma Function

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

Ward’s Solitons II: Exact Solutions

1998 ◽  
Vol 50 (6) ◽  
pp. 1119-1137 ◽  
Author(s):  
Christopher Kumar Anand
Keyword(s):  
Algebraic Geometry ◽  
Exact Solutions ◽  
Closed Form ◽  
Solution Space ◽  
Compact Surface ◽  
Closed Form Expression ◽  
Chiral Model ◽  
Form Expression ◽  
Holomorphic Bundles

AbstractIn a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.

A Closed Form Solution for Flow During the Vacuum Assisted Resin Transfer Molding Process

1999 ◽  
Vol 122 (3) ◽  
pp. 463-475 ◽  
Author(s):  
K-T. Hsiao ◽  
R. Mathur ◽  
S. G. Advani ◽  
J. W. Gillespie, ◽  
B. K. Fink
Keyword(s):  
Closed Form ◽  
Scaling Laws ◽  
Resin Transfer Molding ◽  
Closed Form Solution ◽  
Form Solution ◽  
Flow Front ◽  
Molding Process ◽  
Transfer Molding ◽  
Front Region ◽  
Insight Into

A closed form solution to the flow of resin in vacuum assisted resin transfer molding process (VARTM) has been derived. VARTM is used extensively for affordable manufacturing of large composite structures. During the VARTM process, a highly permeable distribution medium is incorporated into the preform as a surface layer. During infusion, the resin flows preferentially across the surface and simultaneously through the preform giving rise to a complex flow front. The analytical solution presented here provides insight into the scaling laws governing fill times and resin inlet placement as a function of the properties of the preform, distribution media and resin. The formulation assumes that the flow is fully developed and is divided into two regimes: a saturated region with no crossflow and a flow front region where the resin is infiltrating into the preform from the distribution medium. The flow front region moves with a uniform velocity. The law of conservation of mass and Darcy’s Law for flow through porous media are applied in each region. The resulting equations are nondimensionalized and are solved to yield the flow front shape and the development of the saturated region. It is found that the flow front is parabolic in shape and the length of the saturated region is proportional to the square root of the time elapsed. The results thus obtained are compared to data from full scale simulations and an error analysis of the solution was carried out. It was found that the time to fill is determined with a high degree of accuracy while the error in estimating the flow front length, d, increases with a dimensionless parameter ε=K2xxh22/K2yyd2. The solution allows greater insight into the process physics, enables parametric and optimization studies and can reduce the computational cost of full-scale 3-dimensional simulations. A parametric study is conducted to establish the sensitivity of flow front velocity to the distribution media/preform thickness ratio and permeabilities and preform porosity. The results provide insight into the scaling laws for manufacturing of large scale structures by VARTM. [S1087-1357(00)02002-5]

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