Raymond M. Brach

Rigid Body Collisions

The approach overcomes the difficulties encountered by others on the treatment of contact velocity reversals and negative energy losses. The interaction process between the two bodies is modeled using two coefficients. These are the classical coefficient of restitution, e, and the ratio, μ, of tangential to normal impulses. The paper reveals that these coefficients have a much broader intepretation than previously recognized in the solution of collision problems

Microparticle detachment from surfaces exposed to turbulent air flow: controlled experiments and modeling

Abstract This work presents the results of experiments conducted to characterize the detachment of microparticles from surfaces exposed to turbulent air during accelerated free-stream flow. Smooth glass plates used as substrates are scanned with an atomic force microscope to determine their roughness-height distributions. Microparticles of different sizes, materials and shapes (mostly microspheres) are deposited as sparse monolayers onto the substrates under controlled clean and dry conditions. The microparticles attach to the substrate in a condition of static equilibrium due to adhesion and reside completely within the viscous sublayer as the flow is accelerated. Microvideographic observations of individual microparticle detachment show that detachment occurs primarily as rolling motion along the surface and not as lift-off. Detachment is not necessarily followed by entrainment in the flow. Results are presented as detachment fractions as function of time. The experimental results reveal that detachment is governed by a balance of the moments of aerodynamic drag and rough-surface pull-off forces. This is substantiated using a recently developed attachment theory that takes into account surface roughness to determine the pull-off force of microparticles. The sensitivity of the free-stream threshold velocity for detachment to five factors contained in the experiments and the model is analyzed. Results indicate that the surface energy of adhesion and the microsphere radius have the most influence on the threshold velocity for detachment.

Impact dynamics with applications to solid particle erosion

Abstract A reformulation and solution is presented of the equations of impulse and momentum for an arbitrarily shaped rigid body striking a flat massive surface. The model makes use of the classical coefficient of restitution, e , and tangential coefficient, μ. The latter is defined as the ratio of the tangential to normal impulse components. It is shown that a distinction between μ and friction coefficients, f , is necessary for a proper interpretation of impact data. The expression of impact energy loss is placed into a particularly convenient form. Comparisons are made of this impact model to others from the field of solid particle erosion and wear.

SURFACE ROUGHNESS EFFECTS ON MICROPARTICLE ADHESION

A new self-consistent model is developed to treat the static contact of a microparticle with a flat barrier in the presence of molecular adhesion and surface roughness. Separation between their mean datum planes is modeled considering the elastic deformation of the microparticle and surface. The contact pressure is computed from the Lennard-Jones law following the Derjaguin approximation. The elastic deflection of the mean datum plane is calculated from the effective pressure by the half-space elastic theory. Roughness is modeled by introducing a Gaussian distribution to the gap between the surfaces. An effective pressure is defined as the statistical average of the contact pressure over the roughness heights. A solution satisfying all of the above conditions gives a self-consistent method of modeling adhesion between the microparticle and the flat barrier. Using collocation methods the equations are discretized as a large system of nonlinear algebraic equations. A continuation method is used to find the multiple numerical solutions for the mean separation and the effective contact pressure. Finally, adhesive contacts of both smooth and rough surfaces are simulated in a comparative manner to elucidate the features of surface roughness in the presence of molecular adhesion. The standard deviation of the Gaussian distribution is used as a parameter to assess the effects of roughness on the pull-off force. It is shown that increasing surface roughness significantly reduces the pull-off force and decreases the tendency for the microsphere to snap-on and snap-off.

Macrodynamics of Microparticles

A primary goal of this paper is to describe the development of two, independent engineering models for the oblique mechanical impact dynamics of solid aerosol particles, treated as microspheres, in the presence of adhesion forces. One model is algebraic and is based on rigid body impact theory* using coefficients such as the coefficient of restitution and the impulse ratio. This model is augmented by an energy conservation expression. Being algebraic and based directly on Newtons laws, the model offers a rigor and simplicity that makes it ideal for analyzing, displaying and interpreting experimental data. Dealing with impulses, this model does not require a detailed knowledge of the forces to analyze energy loss. The second model takes the form of a simulation using the differential equations of planar motion of a sphere in contact with a flat barrier. It uses Hertzian theory for the normal restoring force, an idealized tensile line force around the periphery of the contact region to represent adhesion a...

A Mathematical Model of the Impact and Adhesion of Microsphers

A model is presented for the low velocity planar impact of a micrometer-sized sphere (microsphere) having an arbitrary angle of approach to a surface in the presence of arbitrary contact and external forces. This model, based upon classical impact dynamics and Hertzian theories, analytically relates the velocity change of the microsphere to the physical parameters of the microsphere and the surface and to the microsphere-surface adhesion forces. The model is based upon two fundamental assumptions, namely, that the energy losses due to the process of material deformation and the process of adhesion are independent, and that the energy loss due to the adhesion process occurs only during the rebound phase of the impact. No assumptions are made about the nature of inelastic deformations in the formulation of the model, permitting it to apply equally well to viscoelastic, elastic-plastic, or other materials or combinations thereof. The utility and accuracy of the model is assessed by comparing its predictions ...

Experiments on the Low-Velocity Impact of Microspheres with Planar Surfaces

Measurements of individual normal and oblique impacts of microspheres with planar surfaces are described and analyzed. Incident velocities from ∼ 2 to 25 m/s and angles from 20° to 90° were controlled in the experiments for various combinations of microsphere and surface materials. For normal (90°) incidence, a single-component phase doppler particle analyzer system measured the incident and rebound normal velocities, particle diameter, and measurement volume crossing time. The resulting values of the kinematic coefficient of restitution revealed the effects of adhesion at lower incident velocities. In addition, the kinematic coefficient of restitution showed a direct dependency on surface material hardness. For oblique (<90°) incidence, a pulsed laser light sheet visualization technique was used to determine the particle incident and rebound, normal and tangential velocity components. The resulting impulse ratios variation with incidence angle helped delineate between rolling and sliding impacts. The sl...

Formulation of rigid body impact problems using generalized coefficients

Abstract The equations of motion of a rigid body expressed in terms of impulse and momentum are linear. When applied to rigid body collisions, it is known that the equations of motion are insufficient to provide a solution of the classical impact problem; an additional equation is needed for each unknown impulse component. Using a set of coefficients, a problem formulation is presented that extends Newtons approach for collinear impacts of particles to three-dimensional impact problems. Being linear and algebraic these equations can be solved, providing a set of solution equations in terms of the physical system parameters, initial conditions and the coefficients. A unique feature of these equations is that they are independent of the contact process(es) and apply to all collisions meeting the rigid body assumptions, whether energy is or is not conserved (contact processes may involve the release of stored energy). Certain solution behavior, including the energy change can be found by treating the coefficients as parameters. By imposing work-energy and/or kinematic constraints, coefficients can be bounded to insure realistic solutions. Coefficients are defined for couple-impulses so the approach is not limited to point contact. Examples are given of the collision of a sphere against a massive barrier (surface). In one, the sphere has an initial cross spin (about its roll-spin axis) and the tangential process is Coulomb friction. Another, including experimental data, is for microspheres ( ∼ 1–100 μm diameter), where the dynamic contact processes are not fully understood.

Models of Rebound and Capture for Oblique Microparticle Impacts

ABSTRACT Simple algebraic, rigid body impact models using coefficients of restitution and coefficient of friction have been used extensively in the study of the impact of microparticles with surfaces. Recent work by the authors has shown that rigid body impact theory can be extended to include an adhesion coefficient to model oblique impacts of microparticles with surfaces in the presence of molecular level forces. In this paper, the model is fully described and exploited for engineering applications. The behavior of coefficients is investigated both analytically and experimentally as initial impact velocity and angel of incidence vary. As the initial velocity of microparticles grows smaller and smaller, the significance of adhesion forces increases, eventually reaching a point where no rebound occurs and the particle is captured. Equations for this region of the growth of the influence of adhesion, designed for empirically fitting, are presented. They are capable of representing the behavior of the mater...

Classical planar impact theory and the tip impact of a slender rod

Abstract This paper covers the topic of the planar eccentric impact of a rigid body at a point. The tip impact of a slender rod against a massive surface or barrier is used as a means to illustrate the principles and solution techniques. Two issues are addressed; the first is how the planar impact problem should be, or at least can be, formulated and solved. The second is how close classical solutions match those obtained by methods used in the field of shock and vibration. In formulating the classical impact problem, three well-defined coefficients of restitution are available, the kinematic (defined as a ratio of velocities), kinetic (defined as a ratio of impulses) and energetic (defined as a ratio of energies). Their relationship, advantages and disadvantages are discussed. The best approach to treat tangential impulses in general has not yet reached agreement. For eccentric impacts with Coulomb friction, for example, various combinations of sliding and sticking including tangential velocity reversals can occur during contact. Several approaches have been proposed and all are related. A method using the impulse ratio as a basic parameter is covered and provides solutions for arbitrarily shaped lamina and arbitrary initial conditions. Solutions of the tip impact of a long slender rod using the classical approach are found for various combinations of coefficients of friction and restitution and initial velocities. The final velocities, impulses and energy losses from these solutions are compared with solutions obtained by integrating the differential equations of motion of a rigid rod striking a viscoelastic surface. The comparisons show excellent agreement for most of the cases considered.