The behavior of cambered belts transiting cylindrical rollers

2021 ◽  
pp. 095440892110644
Author(s):  
Jinxin Shi ◽  
Sheng Pan ◽  
Ron E. Markum ◽  
James K. Good
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Length Variation ◽  
Lateral Deformation ◽  
Roll To Roll ◽  
Manufactured Products ◽  
Lateral Behavior ◽  
The Web

The lateral deformations of webs in roll-to-roll process machines can affect the quality of the manufactured products. Webs with simple nonuniform length variation across their width (camber) will steer toward the long side, leading to the steady state lateral deformation and hence registration. Most previous studies have focused on a cambered web in a free span between two rollers. These studies assume some displacement and slope boundary conditions are known and seek the remaining conditions that would dictate the steady state lateral deformation of the web. This article focuses on the lateral behavior of a cambered web belt transiting between two aligned rollers as the simplest case of multiple span cambered web. Dynamic simulation has been conducted to better understand the response of a cambered web under tension that has been witnessed in tests. There are no boundary conditions enforced and no steady state deformation of the cambered webs. Thus there is no closed-form solution to the lateral movement of a cambered web transits over multiple rollers. This explained why the previous research focused more on the experimental exploration without few theoretical validations. The web travels toward the long side continually from one span to the next until a web guide attempts to return the web to an acceptable lateral location in the process machine.

System Dynamics of the Open-Draw With Web Adhesion: Particle Approach

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Sverker Edvardsson ◽  
Tetsu Uesaka
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Time Dependent ◽  
Temporal Fluctuation ◽  
Continuous Solutions ◽  
Continuous Growth ◽  
System Parameters ◽  
Paper Machines ◽  
Particle Approach ◽  
The Web

In the present work we propose a particle approach, which is designed to treat complex mechanics and dynamics of the open-draw sections that are still present in many of today’s paper machines. First, known steady-state continuous solutions are successfully reproduced. However, it is shown that since the boundary conditions depend on the solution itself, the solutions for web strain and web path in the open-draw section are generally time-dependent. With a certain set of system parameters, the nonsteady solutions are common. A temporal fluctuation of Young’s modulus, for example, destabilizes the system irreversibly, resulting in the continuous growth of web strain, i.e., break. Finally we exemplify with some strategic draw countermeasures how to prevent a dangerous evolution in the web strain.

Nonmodal Solution of Spherical Shells With Cutouts Excited by High-Frequency Axisymmetric Forces

1970 ◽  
Vol 37 (4) ◽  
pp. 977-983 ◽  
Author(s):  
M. C. Junger
Keyword(s):  
Boundary Conditions ◽  
High Frequency ◽  
Near Field ◽  
Closed Form Solution ◽  
Radial Force ◽  
Form Solution ◽  
Mode Shapes ◽  
High Frequency Response ◽  
Circular Cutout ◽  
Line Loads

A closed-form solution is obtained for the high-frequency response of a thin spherical shell embodying a circular cutout and excited axisymmetrically by a concentrated radial force. The solution is constructed by combining the shell response to the radial exciting force with its response to radial, tangential, and moment line loads applied along the cutout boundary, these line loads being selected to match the boundary conditions. Concise expressions for the shell response are obtained by applying the Sommerfeld-Watson transformation to the slowly converging high-frequency modal series which is thereby reduced to only two terms, viz., an exponentially decaying near-field and a standing or propagating-wave field. These two terms are in the nature of the creeping waves commonly used to formulate electromagnetic or acoustic diffracted wave fields in the short-wavelength limit. The method is illustrated for the simple case of a circular cutout with a clamped boundary, but lends itself to more complicated boundary conditions, viz., intersecting shells or wave guides. The natural frequencies and mode shapes are found from a single, characteristic equation involving trigonometric functions.

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 Solution of Dual-Phase Lag Heat Conduction Problem With Time Periodic Boundary Conditions

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Pranay Biswas ◽  
Suneet Singh ◽  
Hitesh Bindra
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Closed Form ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fourier Series Expansion ◽  
Phase Lag ◽  
Dual Phase ◽  
Time Periodic

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

Response of Pounding Dynamic Vibration Neutralizer Under Harmonic and Random Excitation

2018 ◽  
Vol 86 (2) ◽  
Author(s):  
Sami F. Masri ◽  
John P. Caffrey
Keyword(s):  
Steady State ◽  
Relative Motion ◽  
Closed Form Solution ◽  
Harmonic Excitation ◽  
Random Excitation ◽  
Form Solution ◽  
Primary System ◽  
Tuning Parameters ◽  
Dynamic Vibration ◽  
Highly Nonlinear

Exact steady-state solutions are obtained for the motion of an single-degree-of-freedom (SDOF) system that is provided with a highly nonlinear auxiliary mass damper (AMD), which resembles a conventional dynamic vibration neutralizer (DVN), whose relative motion with respect to the primary system is constrained to remain within a specified gap, thus operating as a “pounding DVN.” This configuration of a conventional DVN with motion-limiting stops could be quite useful when a primary structure with a linear DVN is subjected to transient loads (e.g., earthquakes) that may cause excessive relative motion between the auxiliary and primary systems. Under the assumption that the motion of the nonlinear system under harmonic excitation is undergoing steady-state motion with two impacts per period of the excitation, an exact, closed-form solution is obtained for the system motion. This solution is subsequently used to develop an approximate analytical solution for the stationary response of the pounding DVN when subjected to random excitation with white spectral density and Gaussian probability distribution. Comparison between the analytically estimated rms response of the primary system and its corresponding response obtained via numerical simulation shows that the analytical estimates are quite accurate when the coupling (tuning parameters) between the primary system and the damper are weak, but only moderately accurate when the linear components of the tuning parameters are optimized. It is also shown that under nonstationary, the pounding DVN provides slightly degraded performance compared to the linear one but simultaneously limits the damper-free motion to specified design constraints.

Application of Orthotropic Plate Theory to Windmill Blade Design

1981 ◽  
Vol 103 (4) ◽  
pp. 892-894 ◽  
Author(s):  
C. Rubin
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Closed Form ◽  
Orthotropic Plate ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Layered Structures ◽  
Form Solution ◽  
Blade Design ◽  
Free Boundary Conditions

The windmill blade is treated as a semi-infinite orthotropic wedge with free-free boundary conditions. A closed form solution for the deflections and stresses is obtained as a function of the loading. The loading may be quite general. Results for three different materials which are commonly used for windmill blades (aluminum, sitka spruce, and fiberglass) are obtained. Applications also include ribbed, corrugated, and layered structures. In addition, other types of boundary conditions may be used to obtain solutions to a wide variety of other orthotropic plate problems.

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.

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