part dyn micro forces mr v4[1]

8/2/2019 Part Dyn Micro Forces Mr v4[1]

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Particle Dynamics with Micro-scale Forces

Feasibility of Motion

Abstract:

The dynamics of micro sized particle has been modeled considering all the dominant forces. At

macro scale, gravity and inertia are the dominant forces. Friction coefficient is considered as

constant. On micro scale there are additional forces need to be considered. Magnitude of

electrostatic force of attraction by contact-electrification and Van der Waals force is also

comparable to that of the earlier mentioned ones. Friction coefficient, at micro scale is no longer

a constant but is a function of applied normal load. Surface forces of attraction being the

dominant one at micro scale, the friction factor becomes a strong function of surface forces at

static condition. Moreover, surface forces are a function of surface roughness. So friction factor

is indirectly a function of surface roughness. This manuscript is an attempt to accommodate

these dominant forces while modeling the particle dynamics. Its an initial attempt to pave the

way towards controlled micro-part handling on a flexible surface.

1. Introduction: Motion analysis of macro sized components needs consideration of bodyforces and applied loads but the motion analysis for micro scale particle requires additional

consideration. Surface forces start becoming prominent once the part size starts becoming small

and weight becomes negligible [8, 23]. Modeling the dynamics of small particles requires one to

accommodate this shift of dominance from body to surface forces. Van Der Waals and Casimir

surface forces have a strong dependency on the distance from the surface. The effective distance

from the surface is a function of the surface roughness thus making surface roughness one of the

parameters that affects the magnitude of the surface forces. There have been a number of

attempts to mathematically model the dependence of these forces on the surface roughness. The

Casimir force has been modeled as the distance derivative of difference of total black energy

within the space between two surfaces and the energy in the outer space assuming the only

modes of electromagnetic fluctuations having wavelength smaller than the distance between the

two surfaces can exist [9, 10]. Others looked at it as distance derivative of the difference of

surface energy of in contact surfaces and the surfaces separate from each other as the source of

surface force of attraction and tried to develop contact models for pre-assumed shapes in contact.

Models proposed by Johnson et. al. (JKR) and Derjaguin et. al. (DMT), which are modifications

of Hertz Contact model, are adhesive-contact models to accommodate force of attraction

between a sphere and a flat plate [11] [12]. The JKR model is suitable for softer material havingcompliant contacts while the DMT model is suitable for stiffer materials. The surface roughness

models used to generate a numerical equivalent surface of a given pair of surfaces [13,14], along

with JKR and DMT contact models, are employed to accommodate the surface attraction force.

These solutions were extended to calculate static friction force. The coefficient of friction

calculated from these extensions is dependent strongly on the applied net normal force which

includes the force of attraction as well which is the case at micro scale.


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A mathematical model was proposed by W. R. Cheng, I. Etsion and D. B. Bogy (CEB model) to

calculate real area of contact, force of attraction and friction force between rough surfaces[2, 3,

4]. The CEB model is the extension of the GW surface roughness model by Greenwood and

Williamson [13]. The GW model assumes that one of the surfaces is covered with hemispheres

of known radius, pressed against another flat surface and these peaks are arranged with some

predefined probability density function (PDF). The GW model calculates contact pressure and

area of contact assuming Hertz contacts. However, Hertz contact theory is an elastic contact

theory and it neither assumes any surface force of attraction nor any plastic deformation [28]. To

estimate the force of attraction, CEB model uses the solution of a sphere against a flat surface by

Muller et al [1983 reference] for non-contacting asperities, where for contacting asperities, the

deformation profile of a compressed sphere against a flat is calculated and force of attraction is

the integral of the force contributed by each non contacting point calculated by Lennard-Jones

potential [Muller]. The limit for the initiation of plastic deformation developed by Bush and

Gibson is used to accommodate plasticity of contact points [31]. The CEB model augmented

with this additional pull force and plasticity estimates a different actual area of contact when

compared with GW model. Once the actual area of contact is known, with the assumption thattangential force can be supported only by the areas which are under elastic contact, the friction

force is estimated.

The CEB model assumed that asperities are either at an elastic or fully plastic state. The

transition region between the two states confirmed by Johnson [reference] was not

accommodated in the CEB model. Zhao et al. modified the CEB model to include the

discontinuity in contact load between the elastic and plastic limits and modeled the transition

region between elastic and fully plastic deformation [32]. Kogutand Etsion modified the CEB

model on the basis of FEA results of compression of a sphere with a flat surface and proposed

the KE friction model by curve fitting the FEA data [18, 19, 20]. The CEB model assumption

that tangential force is only supported by the elastically deformed material was modified. In KE

model, the shear load is not only supported by elastically deformed asperities but some of the

asperities in the elastic-plastic transition region also contribute to the net frictional load. Surface

profile of deformed asperities is also modified on the basis of FEA results [19,21,25, 16].

The KE model was extended to calculate dynamic friction between lubricated surfaces [27].

Sliding contact and dynamic friction coefficient between rough surfaces is considered, however

these models neglect the surface force of attraction and cannot be employed for micro scale [33,

34, 36].

The static friction force is often an upper bound of friction force because the strength of junction

of the contacting points increases as the time of stationary contact increases [36]. If the static

friction coefficient is used, the results will be conservative. Also, as noted by Matrinz et al.

Contrary to general opinion, no distinction can be made between static and kinetic coefficient of

friction and experimental observation of the difference between static and kinetic friction

coefficient are not necessarily intrinsic properties of dry contact. Dynamic properties of

experimental apparatus and external perturbations may lead to this difference [33, 36].


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To the best of our knowledge, no mathematical model is available in the open literature to

calculate the dynamic friction coefficient between non-lubricated rough surfaces at micro scale.

Dynamic friction coefficient is a function of roughness and materials of surfaces in contact and

Experimental data in the published literature provides clear evidence that the dynamic friction

coefficient value is close to the static one [17, 21]. Therefore, the proposed analysis considers the

value of static friction coefficient as a safe initial guess of dynamic coefficient of friction.

The KE surface roughness / friction model is can be evaluated according to [22]

=

=

= = 2

.

with

= 23 .

+ 1.03 .

+ 1.4 .

+ 3

= 2

+ 0.98 ..

+ 0.79 ..

+ 1.19 ..

Q =23 HA

K 0.52 I.

+ 0.01I. +0.09I. 0.4I. +0.85I.

where Jnc, J and I are given by [22]

= 43

0.25

=

=

Mean of asperity heights

Flat surface

d

Asperities with

Constant Radius R

Intersecting Peak

z

Mean of surface heights

hs

r

a

z

Asperity in contact with

flat surface

Gaussian distribution of

asperity heights

Figure 1: rough surface in contact with flat surface. Dotted line shows the original asperity profile where as solid red line shows

the profile after compression. The compressed asperity has profile Z = f(r)


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2. Dynamic Model of SystemOur objective in this exercise is to capture the motion of a micro-particle while on a

deforming flexible surface. The equations of KE model are used to estimate force ofattraction and friction force between the surfaces when the distance between them is known. To

capture the particle movements, this set of equations needs to be embedded into the dynamicmodel of the system and a sequence of solution steps needs to be defined to.

The dynamic model to predict the motion of a micro particle is developed considering all the

forces acting on the particle while in contact with or close to a deformable surface. To calculate

inertial force while the particle is on the surface, following set of assumption is used

1. The acceleration of particle perpendicular to the surface is the same as the acceleration ofsurface itself in the same direction.

2. Acceleration of particle in the direction parallel to the surface is determined by therelative velocity of particle with surface and magnitude of friction force; details are

mentioned in friction logic ahead.The schematic of the particle on the flexible surface is shown in the Figure 1. To simulate the

system, two coordinate systems are used simultaneously as shown below. The local system

defined at the center of the particle and is dynamically changing relative to the global system

such that one of its axes is always tangent to the deformable surface at the contact point. Inertia

forces are calculated along local coordinate system.

The system dynamic model is based upon following assumptions

-mytt

ytt

-myttb

-mytta

yt

ytb

yta

(b)(a)

Global CoordinatesX

Y

Local Coordinates

Figure 2: Acceleration and velocity of the particle while on flexible surface. Bold line representsthe instantaneous position of moving surface with any profile (a) Acceleration and resultant

force on the particle (b) velocity decomposed along and perpendicular to the surface


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Surface Compliance: The actuation is achieved by a controlled deformation of a flexible

surface. In the proposed model, the surface is assumed to have infinite localized stiffness through

its thickness, i.e. the deformation profile at the top and bottom of the surface are assumed to be

the same, thus compliance is not considered. The acceleration and velocity generated by the

actuator at the bottom of surface are assumed to be the same on the top of the surface due to no

compliance.

State of the particle to detach from the surface: The forces acting on micro-part while in

contact with a surface are shown in Figure 2.

When the two surfaces are in contact, the surface attraction force pulls the two surfaces together

while the compression of asperities generates a repulsion force between the two surfaces. The net

applied load on the particle is the difference of these two forces. When the distance between two

surfaces is very small it indicates high compression and the force due to the compression of

asperities is larger compared to the force of attraction; this load is balanced by externally applied

load to keep the equilibrium. Increasing this distance will decrease the contact load at a larger

rate than the rate of decrease of force of attraction as shown in Figure 3 (b/c). Therefore by

increasing distance a state reaches when the force of attraction is equal to applied load. This

state represents the compression of asperities due to attraction force between the surfaces.

Applied load is the difference of attraction force and contact load. Further increasing the

distance, the force of attraction is higher in magnitude than force due to the compression of

asperities and it Applied Load gets negative. This shows that one requires a pull force to increase

the distance beyond this point. The applied load becomes negative beyond a certain distance

indicating that the force due to asperity compression is smaller than the surface attraction force.The minimum point in Figure 3(a) represents the maximum pull force required to separate the

two surfaces. In the proposed dynamic model, the part detaches from or flies off the surface once

the acceleration component perpendicular to the surface, generates an inertial force larger than

the maximum pull force shown on Applied Force vs. Distance graph in Figure 3(a).

Base surface

Micro particleParticle

Equivalent

RoughSuface

Fatt = Attraction forces applied by surfaceFcont = Repulsion by asperities compression

Fapl = Net applied loadFapl = Fcont - FattFext = -Fa l

Fapl

Fatt Fcont d

Mean of

Asperities height

Fext

Figure 3: Forces Acting on Micro Particle


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Fig: Applied Distance Vs Force

Particle Motion in Air: NEED TO REFERENCE DUSCUSSION to figure 3 and 4. If the

input acceleration is such that the particle gains enough energy to overcome the attraction and

gravity forces it will detach from the surface and this changes the dynamics of the system. At this

system state, the particle motion is affected only by gravitational force and the surface attraction

force. To, predict the motion of the particle, while detached from the surface, requires initial

conditions which are set to be the states from the previous time step when the particle was in

contact with the surface. The surface attraction force is estimated by continuously (at every

integration time step) evaluating the shortest distance of the particle from the deformed surface

for the current system state using potential field approach as shown in Figure 4. . In addition to

the distance, the direction of the suface attraction force is calculated as well. The resultant forceacting on particle is the vector sum of force of attraction and gravitational force. This net applied

force is resolved into its components along global coordinates to estimate system states for next

time step.

Min force point

No Applied Force

(a) (b)

(c)

Figure 4: Forces acting on Micro Particle. = 20nm

Minimum point marked on the applied force graph represents the maximum pull off force

required to separate the two surfaces.


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Asperity Density: The surface roughness has an apriori defined asperity density. We are aware

that when a surface is deformed it will experience stretching that will cause the characteristics of

the surface roughness to change. In proposed system model, the asperity density is assumed to

be constant.

Particle Velocity Perpendicular to Surface: The relative perpendicular velocity of the part

with respect to the surface is zero while the part stays on or is in contact with the surface due to

the non-compliant surface assumption. The position of micro particle is monitored to identify the

time at which it detaches (flies of the surface) and the time at which it returns or re-touches the

surface. When the particle returns to the surface, its velocity is decomposed to the local

coordinate system into perpendicular and tangential components based on surface deformation.

The particle velocity after contact is the decomposed tangential velocity and this is conserved

and used for the next time step in the analysis while the perpendicular component is eliminated

since no impact dynamics are considered.

Area of contact: Nominal area of contact between the particle and the surface is assumed to be

constant even when the surface is deformed.

Air Damping:The model assumes that the effects of air damping even when the particledetaches from the surface are negligible and therefore not included in the analysis.

Friction Logic: The system dynamics are a function of the nonlinear behavior of the friction

force or friction coefficient which in effect is a function of surface roughness, asperity contact

and deformation, surface attraction force and distance of the particle from the surface. The logic

for defining the friction force/coefficient implemented in the proposed dynamic model as

described by Woods [37] which considers the relative velocity between the two surfaces to

Fatt= Surface Attraction Force

W= Weight

R = Resultant Force

Flexible Surface

Particle with

potential fieldMin. distance

R

W

Fatt

Figure 5: Particle motion in air. Minimum distance of the particle is calculated to estimate themagnitude and direction of force of attraction. Resultant force is the vector sum of both forces


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capture hysteresis and stick-slip behavior. The governing equations of the system are functions

of the state of the particle relative to the surface.

v Relative Velocity Base Velocity Part Velocity

v y u

= =

=

If the relative velocity of the particle with respect to surface is zero then friction is equal to

tangential component of applied force on the particle. If the relative velocity has a non-zerovalue, , the friction force is equal to its maximum value and friction force direction is opposite to

direction of relative velocity. In order to implement this logic in a numerical simulation and to

capture hysteresis and stick-slip behavior, a threshold of relative velocity is defined. The

threshold value of relative velocity depends upon the friction force, integration step size and

particle mass.

The threshold of velocity is defined as the maximum velocity of the mass, when it is applied by

an external force opposing the velocity, it will decelerate it to complete stop (V f = 0) within one

time step. The details can be found in Woods [37] -

( )

0

/

/

/

f

th f

tot

th

V

acceleration V V t

acceleration F m

Combining the above three equations

V FricF t m

=

=

=

=

When the relative velocity is less than threshold velocity (Vth) and the absolute value of

threshold velocity is reducing, the part will come to complete stop with respect to base and the

acceleration and velocity of the part is same as the base. Contrary to this, if the value of relative

velocity is less than Vth , and the absolute value of relative velocity is increasing, the

acceleration of the part is determined by the net applied force and the equation of system will beThe sign of friction force is always opposite to the direction of relative velocity. Numerical

implementation of the logic is

[ ] [ ]th th

th

if abs(v) V OR if abs(v) < V AND abs (total force on mass) Max value of Friction Force

FF = Max value of Friction Force sign(v)

x = velocity of particle

FFx =

mass

if abs(v) < V AND abs (tota

[ ]l force on mass) Max value of Friction Force

x = velocity of base

x = acceleration of base

At micro scale, friction force is a function of normal force, thus the value of threshold velocity

varies and is estimated at each time step.

Inversion of Friction Model: The presented friction model considers the material properties of

contacting surfaces, their surface roughness and the distance between them as input. The friction

model is used to calculate normal force on the part, friction force and true contact area. This

model must be inverted in order to be used in the dynamic model for the part motion. In


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proposed model system dynamics, the applied normal force is input while the friction force and

the surface force of attraction are to be calculated. The friction model is not in the form of

explicit equations which can be easily inverted. In order to accomplish the inversion task, data

for applied load is generated for a range of distance values for defined surface roughness and

material properties. The set of estimated applied load as function of distance is used to identify

an explicit curve fit equation, A plot of generated data is shown in Figure 5(a) where the

identified curve fit equation for two steel surfaces with = 20 nm and = 2.5 is shown in Figure

5(b). The explicit equation is used during simulation to evaluate the friction force at each time

step as function of the mean distance between the surfaces.

A rational curve was fitted on the data with the following values of goodness of fit SSE =

0.02223 and R2 = 0.999 using Curve Fit Toolbox of Matlab.

The resultant equation for distance, d, as a function of applied load, (x), is given by

( )( )

4 3 6 2 11 16

5 4 4 3 8 2 13 18

69.86 0.0494 2.885 10 4.249 10 1.839 10

1.37 7.569 10 4.0314 10 5.7 10 2.436 10

x x x xd

x x x x x

+ + + + =

+ + + + +

This equation is used to calculate the distance which is further used to calculate attraction force

between the surfaces and friction force.

Solution Methodology: The set of derived system dynamic equations, representing a non-linear,

discontinuous system, are solved using a custom written 4th

order Runge-Kutta integration

scheme. The states of micro-particle are monitored at each time step as the particle is moving

along the surface for stick-slip and part detachment and updated accordingly.

Two coordinate systems are employed in parallel; a fixed global coordinate system and a local

coordinate system attached to the particle. The local system not only moves but also orients with

the movement of particle. The ordinate of the local coordinate system is always perpendicular to

(a) (b)

Figure 6: Estimated Applied Load as function of Distance and Curve fitting corresponding to

steel surfaces with with = 20 nm and = 2.5


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the surface at the point of contact with the particle. The two coordinate systems and the particle

at two instances (undeformed and deformed surface) are shown in Figure 6

At each time step, input surface velocity and acceleration at the location of particle are known in

global coordinates and are decomposed in local coordinates. The acceleration along ordinate

(abscissa and ordinate will be used only for local coordinates) is used to calculate the inertia

forces normal to particle which determine instantaneous value of friction factor. The acceleration

along abscissa is used to calculate the force on the particle. The velocities along abscissa and

ordinate are used to estimate updated position in local coordinates. These derivatives of the states

are transformed to global coordinates and all four states in global coordinates are updated for

next time step. The transform is achieved through coordinate transformationcos sin

sin cos

u x y

v x y

=

= +

The friction force, being a function of normal force, varies as the acceleration of surface along

ordinate changes. The threshold velocity is calculated at each time step. The relative velocity of

surface and particle are calculated at the particle location in tangential direction. This value is

compared with threshold velocity to determine the relative motion of particle with respect to

surface during the next time step. If relative velocity is larger than threshold value, the part will

not becapturedor stick to the surface during the next time step. If relative velocity is smaller

than threshold value, the particle could stick to the surface or continue slipping, depending uponthe stick-slip condition. The stick-slip condition is evaluated according to the derivative of the

absolute value of relative velocity; if this derivative is negative, the particle will be captured or

stick to the surface. Once it sticks to the surface, the velocity and acceleration of the particle are

the same as those of the surface during the next time step. If the value of this derivative is

positive, the particle will continue sliding during the next time step.

If the input inertia force due to the motion of the surface causes the particle to detach or fly off

the surface, the forces acting on the particle are gravity and surface attraction force (which is a

Figure 7: Local and Global coordinates in solution. Global coordinates remain fixed while local co-ordinates move and orient with the motion of particle.

X, Y Global Coordinates

, Local Coordinates

Y

X

X

Y

Y X

(a) (b)


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function of distance between them). The states of the particle are monitored and updated in

global coordinate system only. At every time step, the position of particle is estimated relative to

the surface, to calculate force of attraction and to check whether the particle re-attaches to the

surface. If the distance of particle from surface is less than the distance corresponding to the

minimum force point on Applied Force vs. Distance graph in Figure (3) and the velocity

component perpendicular to surface is towards the surface, the particle is considered re-attached

and the system dynamics model switches back to the dynamics of particle while on the surface.

Feasibility of Motion: The objective of this exercise is to establish the feasibility of motion of a

micro particle while on a flexible surface. The feasibility is estimated by checking the initiation

of motion while on a flat surface and by the distance travelled by particle by some possible form

of actuation configuration. Particle on flat surface with acceleration direction along the surface is

the most favorable combination to slide of particle against the surface. If the inertia force

generated by acceleration is more than the friction force, the part will slide along the surface as

shown in Figure 8. Friction force is calculated from KE model and the acceleration required to

initiate acceleration needs to be checked against the capability of available actuation mechanismsto establish possibility of initiation of motion. Data of force of attraction, friction force and

friction coefficient is generated for the following

values of parameters of surface and particle as

shown in Table 1 and the graphs of friction force,

coefficient of friction and Applied Load are shown

in Figure 9.

Table 1: Input Parameters for feasibility studyInput Value

Description Parameter

Variable

Standard deviation of surface roughness 20 nm

Cross-section area (contact area) An 100m x 100m

Thickness of micro particle t 10 m ~100 m

Poissons ratio 0.33

Plasticity index 2.5

Difference of surface energy 1

Hardness of material H 200 HB, Approx 1000 MPA

Assuming the Micro particle mass = 2 x 10 -9 kg

To calculate the friction force from KE model, distance between the two surfaces is needed. To

estimate the distance, see the Applied Load vs. Distance curve for corresponding parameters of

surface roughness and material constants. With no acceleration in normal direction, the only

normal load on the part is its weight. Find the distance from the graph corresponding to this

Acceleration

Direction

Surface

Particle

Friction

Force

Inertia

Force

Figure 8: Part on flat surface with acceleration

along surface


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normal applied load. Once distance is known, the friction force can be calculated easily by using

Distance vs. Friction Force curve or using equation for friction force from the set of KE equation

mentioned above.

Friction Force = 0.8 x 10-7

N

To initiate motion, the inertial force component parallel to the friction force should be large

enough to overcome it. At the moment of initiation of motion these two forces will be equal.

2/ 40 / sec

friction inertia

inertia

friction

friction

F F

F ma

so

F ma

a F m m

=

=

=

= =

With this value of applied acceleration, the particle will start slipping on the base surface. This

value is within the range of current piezoelectric actuators which can be used to actuate the

system.

The second criteria to check the feasibility is a reasonable distance-travel for a specific actuation

configuration. To calculate the distance travelled by the particle, considering the nature of

equations, no analytical solution can be calculated. The numerical simulation scheme elaborated

in Figure 13 is employed in MATLAB to calculate the distance travelled.

Figure 9: variation of friction force and coefficient of friction with the change of distance between the

two surfaces


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The chosen actuation configuration is shown in Figure 10. An actuator placed vertically beneath

a flexible surface deforms it vertically upwards. Micro particle is placed on the surface

experiences an inertia force. The component of inertia force along the surface will move the

particle along the surface. This configuration had been selected because of its viability for real

time application.

Surface deformation profile is assumed to be Gaussian. Simulation is done to estimate the

distance travel. Variation of friction force plotted against the time is in Figure 11 (a). The

horizontal component of velocity of particle (in global coordinate system) with respect to time is

represented in Figure 11 (b). The graph of the distance the particle moved along the surface, with

single stroke of actuator, with the variation of input frequency is shown in Figure 12.

From the Velocity vs. Time curve (Figure 11(b)) it is clear that in the acceleration phase of

forward stroke of actuator the particle gains positive velocity but due to smaller slope of surface,

the velocity gain is small, during deceleration of forward stroke, the component of inertia along

abscissa of local coordinate is high and the velocity gain is larger. The similar situation is

available in the reverse stroke which results into a net unidirectional distance of 1.25 mm

covered by particle.

(a) (b)

Forward

stroke

Reverse

stroke

X

YY

X

(a) (b)

Actuator beneath the surface in

retracted position

Surface deformation by

actuator stroke

Figure 10: schematic of the deformation of a surface and the resultant particle motion

Figure 11: Particle Velocity and Friction Force during Actuator Stroke


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Conclusion: Motion of a micro part on a surface is discontinuous system with non linear

behavior. The dynamics of motion on a surface is modeled on micro scale and simulation scheme

is developed. After estimating the friction force between micro part and base surface, the

initiation of motion has been confirmed using simple mechanics. Viability of process dependsupon the distance travelled by micro particle. With the help of developed simulation scheme, the

estimated value of distance travelled by micro particle is estimated to be 1.25 mm. This lays the

foundation of a new methodology in micro part handling. The process can be tested for variation

of material, surface roughness and actuation parameters.

Figure 12: Distance Travelled by Particle with The Variation in Input

Frequency of Actuation


15/18

Figure 13: simulation scheme to trace the particle motion on the flexible surface


16/18

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with Rough Surfaces

[35] Johnson K. L. Contact Mechanics, Cambridge University Press, Cambridge.

[36] J. T. Oden, J. A. C. Martins Models and Computational Methods for Dynamic Friction

Phenomenoa

[37] (details Dr woods friction paper)


18/18

Nomenclature

c

d

max

e

N = number of asperities in contact

P = total contact load

N = number of asperities per unit area

Fs = force of attraction

Q = total shear forceQ = maximum friction force

A = elastic area of contact

Ap

d

c

= plastic area of contact

R = radius of curvature of asperities

= density of asperities

j = prob.distribution of asperities peak

= molecular distance

= interference at elasticity limit

H = brinnell hardness

= change in surface energy

d = distance between surfaces

E = resultant elastic modulus

= poisson's ratio of material

= plasticity index

= interference of peak with smooth surface

Z = dist

s

th

ance btw flat surf and noncontacting area

= standard deviation of asperity heights = standard deviation of surface heights

v = relative velocity of particle parallel to surface

V = Threshold velovi

t

tb

ty

u = Acceleration of particle parallel to surface

FricF = Friction force on particle

m = mass of particle

y = Velocity of surface at the location of part.

y = Component of velocity perpendicular to t

ta

tt

tta

ttb

he surface.

y = Component of velocity along the surface.

y = Instantaneous acceleration of microparticle.

-my = Component of inertial force along the surface.

-my = Component of inertial force perp. to the surface.

part dyn micro forces mr v4[1]

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