Analysis of Tombstoning Phenomenon During Reflow

1998 ◽  
Vol 120 (1) ◽  
pp. 61-67 ◽  
Author(s):  
Q. Qi ◽  
R. L. Mahajan
Keyword(s):  
Dynamic Model ◽  
Contact Angles ◽  
Molten Solder ◽  
Dynamic Equations ◽  
Gap Size ◽  
Gravity Effect ◽  
Algebraic Equations ◽  
Solder Volume ◽  
Nonlinear Algebraic Equations

A dynamic model is established to investigate tombstoning phenomenon of passive SMT components during reflow. This model consists of three dynamic equations for the motion of the component and four nonlinear algebraic equations for area conservation, constant curvatures for solder fillets, and solder height-length relations. The formulation neglects the gravity effect in the solder. It is shown that formation of tombstoning depends on a combination of parameters such as component geometry, pad sizes, gap size between pads, molten solder volume and properties, and contact angles. For some values of these parameters encountered in practice, tombstoning can take place. For a given component, this model may be used to improve solder pad and stencil designs to avoid tombstoning.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

Periodic Solutions in Rotor Dynamic Systems With Nonlinear Supports: A General Approach

1989 ◽  
Vol 111 (2) ◽  
pp. 187-193 ◽  
Author(s):  
C. Nataraj ◽  
H. D. Nelson
Keyword(s):  
Dynamic Systems ◽  
Original Problem ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Trigonometric Collocation ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Future Work ◽  
Rotor Dynamic ◽  
Large Rotor

A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.

The Influence of Small Particles on the Structure of a Turbulent Shear Flow

2004 ◽  
Vol 126 (4) ◽  
pp. 613-619 ◽  
Author(s):  
David I. Graham
Keyword(s):  
Shear Flow ◽  
General Trend ◽  
Turbulent Shear ◽  
Mass Loading ◽  
Turbulent Shear Flow ◽  
Algebraic Equations ◽  
Reynolds Stresses ◽  
Temporal Correlations ◽  
Fluid Velocities ◽  
Nonlinear Algebraic Equations

In this paper, an analytical solution is found for the Reynolds equations describing a simple turbulent shear flow carrying small, wake-less particles. An algebraic stress model is used as the basis of the model, the particles leading to source terms in the equations for the turbulent stresses in the flow. The sources are proportional to the mass loading of the particles and depend on the temporal correlations of the fluid velocities seen by particles, Rijτ. The resulting set of equations is a system of nonlinear algebraic equations for the Reynolds stresses and the dissipation. The system is solved exactly and the influence of the particles can be quantified. The predictions are compared with DNS results and are shown to predict trends quite well. Different scenarios are investigated, including the effects of isotropic, anisotropic and non-equilibrium time scales and negative loops in Rijτ. The general trend is to increase anisotropy and attenuate turbulence with higher mass loadings. The occurrence of turbulence enhancement is investigated and shown to be theoretically possible, but physically unlikely.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

scholarly journals On the continuous analogue of the Seidel method

2018 ◽  
Vol 20 (4) ◽  
pp. 364-377
Author(s):  
I. V. Boikov ◽  
A. I. Boikova
Keyword(s):  
Differential Equations ◽  
Systems Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Seidel Method ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Continuous Analogue ◽  
Solution Methods ◽  
Linear And Nonlinear ◽  
Equations With Delay

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.

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