scholarly journals A closed-form solution for steady-state coupled phloem/xylem flow using the Lambert-W function

2013 ◽  
Vol 36 (12) ◽  
pp. 2150-2162 ◽  
Author(s):  
A. J. HALL ◽  
P. E. H. MINCHIN
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Lambert W Function ◽  
Xylem Flow

Design and Analysis of Automotive Serpentine Belt Drive Systems for Steady State Performance

1997 ◽  
Vol 119 (2) ◽  
pp. 162-168 ◽  
Author(s):  
R. S. Beikmann ◽  
N. C. Perkins ◽  
A. G. Ulsoy
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Design Parameters ◽  
Belt Drive ◽  
Automotive Engine ◽  
Theoretical Predictions ◽  
Serpentine Belt ◽  
Drive Systems

Serpentine belt drive systems with spring-loaded tensioners are now widely used in automotive engine accessory drive design. The steady state tension in each belt span is a major factor affecting belt slip and vibration. These tensions are determined by the accessory loads, the accessory drive geometry, and the tensioner properties. This paper focuses on the design parameters that determine how effectively the tensioner maintains a constant tractive belt tension, despite belt stretch due to accessory loads and belt speed. A nonlinear model predicting the operating state of the belt/tensioner system is derived, and solved using (1) numerical, and (2) approximate, closed-form methods. Inspection of the closed-form solution reveals a single design parameter, referred to as the “tensioner constant,” that measures the effectiveness of the tensioner. Tension measurements on an experimental drive system confirm the theoretical predictions.

A Closed-Form Approach for Optimum Tolerance Allocation of Assemblies with General Tolerance-Cost Function

2011 ◽  
Vol 201-203 ◽  
pp. 1272-1278
Author(s):  
Kuo Ming Cheng ◽  
Jhy Cherng Tsai
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Equality Constraints ◽  
Allocation Problem ◽  
Tolerance Allocation ◽  
Lambert W Function ◽  
Manufacturing Cost ◽  
Worst Case ◽  
Constrained Minimization Problems

Tolerancing is one of the most crucial foundations for industry development and an index of product quality and cost. As tolerance allocation is based on manufacturing costs, this paper proposes a comprehensive method for optimal tolerance allocation with minimum manufacturing cost subject to constraints on dimensional chains and machining capabilities. The general reciprocal power and exponential cost-tolerance models with equality constraints as well as the worst-case and statistical tolerancings are employed in this method. A closed-form solution for the optimization problem by applying Lagrange multipliers is derived. The optimal tolerance allocation problem for reciprocal exponential cost-tolerance model by introducing Lambert W function is demonstrated. For constrained minimization problems with only equality constraints, the optimum design can be obtained by solving simultaneous equations without differentiating. An example is illustrated to demonstrate this approach. The result also shows that tolerance can be allocated economically and accurately using this method. The contribution of this paper is to solve the optimal tolerancing allocation problem by an efficient and robust method with simultaneous active constraints.

A Dishing Model for Chemical Mechanical Polishing of Plug Structures

2010 ◽  
Vol 126-128 ◽  
pp. 276-281
Author(s):  
Shih Hsiang Chang
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Chemical Mechanical Polishing ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mechanical Polishing ◽  
Quantitative Prediction ◽  
Wafer Surface ◽  
Maximum Value ◽  
Pattern Density

It is well known that dishing occurring in chemical mechanical polishing of plug structures leads to considerable wafer surface non-planarity and reduces the current/charge conduction. Thus, a closed-form solution for quantitative prediction of dishing is needed. A contact-mechanics-based approach to describe the steady-state dishing occurring in chemical mechanical polishing of plug structures is presented. The model is then applied to investigate the effect of pattern geometry on dishing in details. It was shown that plug dishing strongly depends on plug size, but minimally on pattern density. In addition, the maximum value of dishing occurs at a critical pattern density for fixed pitch.

Moving Load on a Fluid-Solid Interface: Supersonic Regime

1973 ◽  
Vol 40 (1) ◽  
pp. 137-142 ◽  
Author(s):  
T. C. Kennedy ◽  
G. Herrmann
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Moving Load ◽  
Closed Form Solution ◽  
Point Load ◽  
Form Solution ◽  
Steady State Response ◽  
Entire Space ◽  
State Response ◽  
Supersonic Regime

The steady-state response of a semi-infinite solid with an overlying semi-infinite fluid subjected at the plane interface to a moving point load is determined for supersonic load velocities. The exact, closed-form solution valid for the entire space is presented. Some numerical results for the displacements at the interface are calculated and compared to the results obtained when no fluid is present.

Closed-Form Steady-State Response Solution of the Twin Rotor Damper and Experimental Validation

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Richard Bäumer ◽  
Uwe Starossek
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Design Parameters ◽  
Control Force ◽  
Steady State Response ◽  
Active Mass ◽  
State Response ◽  
Twin Rotor

In previous research, the twin rotor damper (TRD), an active mass damper, was presented including control algorithms for monofrequent vibrations. In a preferred mode of operation, the continuous rotation mode, two eccentric masses rotate in opposite directions about two parallel axes with a mostly constant angular velocity. The resulting control force is harmonic. Within this paper, the steady-state response of a single-degree-of-freedom (SDOF) oscillator subjected to a harmonic excitation force with and without the TRD is studied. A closed-form solution is presented and validated experimentally. It is shown that the TRD provides damping to the SDOF oscillator until a certain frequency ratio is reached. The provided damping is not only dependent on the design parameters of the TRD but also depends on the steady-state vibration amplitude. The solution serves as a powerful design tool for dimensioning the TRD. The analytical closed-form solution is applicable for other active mass dampers.

scholarly journals Time-dependent solution of the NIMFA equations around the epidemic threshold

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1299-1355
Author(s):  
Bastian Prasse ◽  
Piet Van Mieghem
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Closed Form ◽  
State Vector ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mean Field Approximation ◽  
Contact Network ◽  
Epidemic Threshold ◽  
Linear Differential

AbstractThe majority of epidemic models are described by non-linear differential equations which do not have a closed-form solution. Due to the absence of a closed-form solution, the understanding of the precise dynamics of a virus is rather limited. We solve the differential equations of the N-intertwined mean-field approximation of the susceptible-infected-susceptible epidemic process with heterogeneous spreading parameters around the epidemic threshold for an arbitrary contact network, provided that the initial viral state vector is small or parallel to the steady-state vector. Numerical simulations demonstrate that the solution around the epidemic threshold is accurate, also above the epidemic threshold and for general initial viral states that are below the steady-state.

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