Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions

2013 ◽  
Vol 714 ◽  
pp. 312-335 ◽  
Author(s):  
J. B. Bostwick ◽  
P. H. Steen
Keyword(s):  
Contact Angles ◽  
Solid Support ◽  
Linear Operators ◽  
Harmonic Oscillators ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Spherical Cap ◽  
Spherical Drop ◽  
Coupled Oscillations ◽  
Latitudinal Belt

AbstractA spherical drop is constrained by a solid support arranged as a latitudinal belt. This belt support splits the drop into two deformable spherical caps. The edges of the support are given by lower and upper latitudes yielding a ‘spherical belt’ of prescribed extent and position: a two-parameter family of constraints. This is a belt-constrained Rayleigh drop. In this paper we study the linear oscillations of the two coupled spherical-cap surfaces in the inviscid case, and the viscous case is studied in Part 2 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 336–360), restricting to deformations symmetric about the axis of constraint symmetry. The integro-differential boundary-value problem governing the interface deformation is formulated as a functional eigenvalue problem on linear operators and reduced to a truncated set of algebraic equations using a Rayleigh–Ritz procedure on a constrained function space. This formalism allows mode shapes with different contact angles at the edges of the solid support, as observed in experiment, and readily generalizes to accommodate viscous motions (Part 2). Eigenvalues are mapped in the plane of constraints to reveal where near-multiplicities occur. The full problem is then approximated as two coupled harmonic oscillators by introducing a volume-exchange constraint. The approximation yields eigenvalue crossings and allows post-identification of mass and spring constants for the oscillators.

Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous motions

2013 ◽  
Vol 714 ◽  
pp. 336-360 ◽  
Author(s):  
J. B. Bostwick ◽  
P. H. Steen
Keyword(s):  
Slip Boundary Condition ◽  
Boundary Integral ◽  
Integral Approach ◽  
Support Surface ◽  
Algebraic Equations ◽  
Viscous Effects ◽  
Spherical Cap ◽  
Spherical Drop ◽  
Coupled Oscillations ◽  
Geometrical Constraints

AbstractA spherical drop is constrained by a solid support arranged as a latitudinal belt. The spherical belt splits the drop into two deformable spherical caps. The edges of the belt support are given by lower and upper latitudes, yielding a support of prescribed extent and position: a two-parameter family of geometrical constraints. In this paper we study the linear oscillations of the two coupled surfaces in the viscous case, the inviscid case having been dealt with in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 312–335), restricting to axisymmetric disturbances. For the viscous case, limiting geometries are the spherical-bowl constraint of Strani & Sabetta (J. Fluid Mech., vol. 189, 1988, pp. 397–421) and free viscous drop of Prosperetti (J. Méc., vol. 19, 1980b, pp. 149–182). In this paper, a boundary-integral approach leads to an integro-differential boundary-value problem governing the interface disturbances, where the constraint is incorporated into the function space. Viscous effects arise due to relative internal motions and to the no-slip boundary condition on the support surface. No-slip is incorporated using a modified set of shear boundary conditions. The eigenvalue problem is then reduced to a truncated set of algebraic equations using a spectral method in the standard way. Limiting cases recover literature results to validate the proposed modification. Complex frequencies, as they depend upon the viscosity parameter and the support geometry, are reported for both the drop and bubble cases. Finally, for the drop, an approximate boundary between over- and under-damped motions is mapped over the constraint parameter plane.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

scholarly journals Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

2020 ◽  
Vol 10 (2) ◽  
pp. 493 ◽  
Author(s):  
Ma’en S. Sari ◽  
Wael G. Al-Kouz ◽  
Anas M. Atieh
Keyword(s):  
Boundary Conditions ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Shear Stiffness ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Small Scale ◽  
Algebraic Equations ◽  
Chebyshev Spectral Collocation ◽  
The Matrix

The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

scholarly journals The general theory of relaxation methods applied to linear systems

1939 ◽  
Vol 169 (939) ◽  
pp. 476-500 ◽  
Keyword(s):  
General Theory ◽  
Sufficient Conditions ◽  
Formal Proof ◽  
Linear Operators ◽  
Successive Approximations ◽  
Continuous Systems ◽  
Algebraic Equations ◽  
Relaxation Methods ◽  
Linear Differential ◽  
Homogeneous Linear

During the last few years Southwell and his fellow-workers have developed a new method for the numerical solution of a very general type of problem in mathematical physics and engineering. The method was originally devised for the determination of stresses in frameworks, but it has proved to be directly applicable to any problem which is reducible to the solution of a system of non-homogeneous, linear, simultaneous algebraic equations in a finite number of unknown variables.* Southwell’s “ relaxation m ethod” is one of successive approximation and, in order to complete the previous investigations of this method, it is necessary to prove that the successive approximations do actually converge towards the exact solutions. This formal proof is given in § 4. Southwell’s “ relaxation” methods are not directly applicable to continuous systems, where the number of unknown variables is infinite, but it is shown here that simple extensions and modifications of the relaxation method render it suitable for application to either discrete or continuous systems (§ 5). The general theory of relaxation methods is then developed in terms of the theory of linear operators (§7) and sufficient conditions are prescribed for the convergence of the process of approximation (§ 8). These general methods are then applied to the solution of non-homogeneous, linear integral equations (§ 10) and to the solution of nonhomogeneous, linear differential equations (§§ 11, 12).

Free vibration analysis of two parallel beams connected together through variable stiffness elastic layer with elastically restrained ends

2016 ◽  
Vol 20 (3) ◽  
pp. 275-287 ◽  
Author(s):  
Alborz Mirzabeigy ◽  
Reza Madoliat ◽  
Mehdi Vahabi
Keyword(s):  
Differential Equations ◽  
Elastic Layer ◽  
Flexural Rigidity ◽  
Free Vibration Analysis ◽  
Variable Stiffness ◽  
Differential Transform Method ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Transform Method ◽  
Differential Transform

In this study, free transverse vibration of two parallel beams connected together through variable stiffness Winkler-type elastic layer is investigated. Euler–Bernoulli beam hypothesis has been applied and the support is considered to be translational and rotational elastic springs in each ends. Linear and parabolic variation has been considered for connecting layer. The equations of motion have been derived in the form of coupled differential equations with variable coefficients. The differential transform method has been applied to obtain natural frequencies and normalized mode shapes of system. Differential transform method is a semi-analytical approach based on Taylor expansion series which converts differential equations to recursive algebraic equations and does not need domain discretization. The results obtained from differential transform method have been validated with the results reported by well-known references in the case of two parallel beams connected through uniform elastic layer. The effects of variation type and total stiffness of connecting layer, flexural rigidity ratio of beams, and boundary conditions on behavior of system are investigated and discussed in detail.

Development of a Dynamic Model for Predicting the Rigid and Flexible Motions of the Crank Slider Mechanism

1998 ◽  
Vol 120 (3) ◽  
pp. 678-686 ◽  
Author(s):  
H. Nehme ◽  
N. G. Chalhoub ◽  
N. Henein
Keyword(s):  
Equations Of Motion ◽  
Continuous Model ◽  
Differential Algebraic Equations ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Transverse Deformation ◽  
Connecting Rod ◽  
Lagrange Principle ◽  
Plane Transverse ◽  
Out Of Plane

A continuous model is developed to predict the rigid and flexible motions of the piston assembly/connecting rod/crankshaft mechanism for a single cylinder engine. The model accounts for the torsional vibration and the out-of-plane transverse deformation of the crankshaft along with the out-of-plane transverse deformation of the connecting rod. The eigenvalue problem of the crankshaft, including the counterweights, the flywheel, and the crank gear, is solved to obtain the analytical expressions for the elastic modes of the crankshaft. The resulting mode shapes are then used in the assumed modes method to approximate the structural flexibility terms. The differential-algebraic equations of motion are obtained by implementing the Lagrange principle. The digital simulation results illustrate the role played by the topological nonlinearities inherent in the system and reveal the relationships with which the rigid and flexible motions of the crank-slider mechanism would interact.

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 Non-isothermal droplet spreading/dewetting and its reversal

2015 ◽  
Vol 776 ◽  
pp. 74-95 ◽  
Author(s):  
Yi Sui ◽  
Peter D. M. Spelt
Keyword(s):  
Slip Length ◽  
Spreading Rate ◽  
Contact Angles ◽  
Lubrication Theory ◽  
Drop Diameter ◽  
Droplet Spreading ◽  
Spherical Cap ◽  
Drop Shape ◽  
Rate Versus ◽  
O 10

Axisymmetric non-isothermal spreading/dewetting of droplets on a substrate is studied, wherein the surface tension is a function of temperature, resulting in Marangoni stresses. A lubrication theory is first extended to determine the drop shape for spreading/dewetting limited by slip. It is demonstrated that an apparent angle inferred from a fitted spherical cap shape does not relate to the contact-line speed as it would under isothermal conditions. Also, a power law for the thermocapillary spreading rate versus time is derived. Results obtained with direct numerical simulations (DNS), using a slip length down to $O(10^{-4})$ times the drop diameter, confirm predictions from lubrication theory. The DNS results further show that the behaviour predicted by the lubrication theory – that a cold wall promotes spreading, and a hot wall promotes dewetting – is reversed at sufficiently large contact angles and/or viscosity of the surrounding fluid. This behaviour is summarized in a phase diagram, and a simple model that supports this finding is presented. Although the key results are found to be robust when accounting for heat conduction in the substrate, a critical thickness of the substrate is identified above which wall conduction significantly modifies wetting behaviour.

Dynamics of sessile drops. Part 1. Inviscid theory

2014 ◽  
Vol 760 ◽  
pp. 5-38 ◽  
Author(s):  
J. B. Bostwick ◽  
P. H. Steen
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Dynamic Contact ◽  
Linear Operators ◽  
Sessile Droplet ◽  
Physical Reason ◽  
Spherical Cap ◽  
Static Contact ◽  
Dynamic Contact Line ◽  
Line Condition

AbstractA sessile droplet partially wets a planar solid support. We study the linear stability of this spherical-cap base state to disturbances whose three-phase contact line is (i) pinned, (ii) moves with fixed contact angle and (iii) moves with a contact angle that is a smooth function of the contact-line speed. The governing hydrodynamic equations for inviscid motions are reduced to a functional eigenvalue problem on linear operators, which are parameterized by the base-state volume through the static contact angle and contact-line mobility via a spreading parameter. A solution is facilitated using inverse operators for disturbances (i) and (ii) to report frequencies and modal shapes identified by a polar $k$ and azimuthal $l$ wavenumber. For the dynamic contact-line condition (iii), we show that the disturbance energy balance takes the form of a damped-harmonic oscillator with ‘Davis dissipation’ that encompasses the dynamic effects associated with (iii). The effect of the contact-line motion on the dissipation mechanism is illustrated. We report an instability of the super-hemispherical base states with mobile contact lines (ii) that correlates with horizontal motion of the centre-of-mass, called the ‘walking’ instability. Davis dissipation from the dynamic contact-line condition (iii) can suppress the instability. The remainder of the spectrum exhibits oscillatory behaviour. For the hemispherical base state with mobile contact line (ii), the spectrum is degenerate with respect to the azimuthal wavenumber. We show that varying either the base-state volume or contact-line mobility lifts this degeneracy. For most values of these symmetry-breaking parameters, a certain spectral ordering of frequencies is maintained. However, because certain modes are more strongly influenced by the support than others, there are instances of additional modal degeneracies. We explain the physical reason for these and show how to locate them.

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