Download Chapter 5 Mutual actions in machinery elements

Chapter 5 Mutual actions in
machinery elements
Bachelor Program in AUTOMATION ENGINEERING
Prof. Rong-yong Zhao
([email protected])
First Semester,2014-2015
Content
• 5.1 Introduction
• 5.2 Solid body contacts
– 5.2.1 Friction in solid body contacts
– 5.2.2 Static friction (adhesion condition)
– 5.2.3 Dynamic friction
– 5.2.4 Rolling contact
– 5.2.5 Rolling friction
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5.1 Introduction
1) Every machine include Relative
motions,
• among the composing elements,
• between the machine and the
surrounding environment.
2) On the dynamic behavior of a
machine are of great relevance
• the contact forces occurring
between solid bodies and the
consequent effects, i.e. wear;
• the forces exchanged between
solids and fluids.
Four-stroke cycle engine
Two meshing gears
transmit rotational
motion.
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5.2 Solid body contacts
• The two main phenomena related to contacts
between solid bodies are friction and wear.
• Friction Definition : Friction consists in a resistance or
opposition to a relative motion between contacting
surfaces;
1) disadvantage when power loss occurs between
components to be kept in relative motion
e.g. friction in bearings, in seals, etc. Hydraulic seals
explosive view of ball bearing
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Friction as advantage
2) advantage or the essential condition for the correct
working of a machine
e.g. rolling without slipping at the wheel – rail or
tire – road ,brakes, clutches, etc.
Wheel-rail
Formula 1 Race Car
brake
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Clutch
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Wear
• Wear happens with a progressive loss of material
from the body surfaces, as a result of the motion
relative to another body.
• Positive factor: surface finishing machining ;
• Negative factor: causes a progressive deterioration
of the mating surfaces.
Grinding for crankshaft
coarse grinding for a shaft
Blade deterioration
due to wear
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Kinematic view of solid contact
• between solid bodies there can be rolling contact,
sliding and impact.
Rolling Contact :pure
rolling contact the relative
speed in the contact point
is zero.
Sliding: a relative velocity
component is present
along the tangent t
Impact: a component
along the normal n of
the colliding profile is
also present.
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Geometric view of body contact
• Rigid body contact: distinguishing among point,
line and surface contacts, depending on the
geometric entity shared between the contacting
bodies;
• Actually, bodies are deformable;
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5.2.1 Friction in solid body contacts
• Friction Definition : Friction is the resistance to motion
occurring when a body slides on another;
• Friction Direction : This resistance action is opposite to the
relative motion;
• Classification : static friction and dynamic friction(Kinetic
friction);
• The static friction force: opposing a motion starting from
static conditions;
• The dynamic friction force: opposing the relative motion of
two bodies already moving with a relative speed
• Comparison : The dynamic force is usually less than the static
force.
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Examples of friction
• Static friction
Dynamic friction
A physical model of the so-called 'Ziegler
column', a two-degree-of-freedom system,
exhibiting flutter instability as induced by
dry friction.
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Microscopic study
• a microscopic model of the contact can be used to
justify the existence of static and dynamic friction
forces;
• actual body surface is rough in microscopic view;
the pressure in those points is
extremely high, thereby causing “micro
welding” phenomena, i.e. the two
surfaces are “glued” in
correspondence of those contact
points.
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Coulomb laws in 5 steps
1) a constant normal load P and where F has to be regarded as a force
(slowly) increasing in value;
2) Initially (i.e. for small value of F) the body stays at rest, since the
resistance offered by the micro-welding points prevents the body
motion;
3) This adhesion effect gives rise to a tangential component T of the
reaction force exerted by the supporting plane on the body;
4) As the force F increases, a situation is
reached where it can shear the microwelding points;
5) Consequently, relative motion starts
and kinetic friction takes the place of
static friction.
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5.2.2 Static friction (adhesion condition)
• normal and tangential components of the
constraint reaction: N and T
• N and T are calculated by the equilibrium equations
and must be verified by the inequality:
If inequality (5.1) is satisfied, the two mating surfaces will not slide.
friction force is independent from the size of contact area
static friction coefficient does not depend on the contact
surface ,nor on the applied load, but only on the nature of the
contacting body materials
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Application example: adhesion on an
inclined plane
• a body of mass M is in contact with an inclined
plane.
• With respect to the direction of the acceleration of
gravity g the inclined plane forms an angle (π/2
−α ) .
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Friction Analysis
• With the sign convention indicated in Fig.5.4, for
the static equilibrium of the body we have:
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5.2.3 Dynamic friction
• an increase of the force F (see Fig.5.3) is not
followed by an increase of the tangent component T,
whose maximum value is Tlim ;
V
the magnitude of tangent
component of the constraint reaction:
Friction force direction is always opposite to the sliding motion.
The dimensionless parameter μk is called coefficient of kinetic
friction ,depending on the characteristics of the contacting
surfaces only
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coefficient of static friction and kinetic
friction
Between two steel objects:
μ s=0.3~0.8 and μk =0.15~0.6
Between steel and Teflon
μs =0.3~0.6 and μk =0.1~0.2
Teflon
two steel objects
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Approximate coefficients of
friction
Static friction,
Materials
Dry and clean
Lubricated
Aluminium
Steel
0.61
Copper
Steel
0.53
Brass
Steel
0.51
Cast iron
Copper
1.05
Cast iron
Zinc
0.85
Concrete
Rubber
1.0
Concrete
Wood
0.62
Copper
Glass
0.68
Glass
Glass
0.94
Metal
Wood
0.2–0.6
0.2 (wet)
Polyethene
Steel
0.2
0.2
Steel
Steel
0.80
0.16
Steel
PTFE (Teflon)
0.04
0.04
PTFE (Teflon)
PTFE (Teflon)
0.04
0.04
Wood
Wood
0.25–0.5
0.2 (wet)
0.30 (wet)
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Application example: sliding with
friction on an inclined plane
• the body slides (down the inclined plane, in Fig. 5.5)
with velocity V and acceleration a.
The corresponding
dynamic equilibrium equations
(D’Alèmbert principle) read:
By adding eq (5.6) to eqs (5.7)
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Analysis
• Acceleration a is less than that of a frictionless
sliding;
• Kinetic friction represents a power loss
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5.2.4 Rolling contact
• The simplest model of a contact of a
wheel rolling on a plane still
involves Coulomb friction laws.
• Both the wheel and the constraining
plane are modeled as rigid, thereby
assuming that they are in contact
on a rectilinear segment, becoming
a point on the directory plane. Pure
rolling contact (no slipping) takes
place if the Coulomb inequality
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Continue
• the disc of radius r, supposed to roll without
slipping and generically loaded as in Fig. 5.6,
the moment equilibrium has been taken
around the contact point
due to the hypothesis of pure rolling ,
contact point is the instantaneous center
of velocity,
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Velocity and acceleration
• The velocity and the acceleration of the center of
gravity G：
analytically represent the pure rolling motion
inequality (5.10) must hold true, otherwise the pure rolling motion
is not possible and sliding at the contact point/line takes place.
The tangent component T is now defined by the equation:
μk is the kinetic coefficient of friction.
T is always directed oppositely to the sliding speed.
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5.2.5 Rolling friction
• Rolling friction definition: Rolling friction is defined
as the resistance encountered by a body rolling
without macroscopically slipping on another body’s
surface.
• Rolling friction occurs when a wheel, ball, or cylinder
rolls freely over a surface, as in ball and roller bearings.
• Necessary to apply a tractive action (force and/or torque)
to keep a wheel in motion at a constant speed.
• Main source of friction in rolling appears to be dissipation
of energy involved in deformation of the objects.
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Microscopic analysis
• The elastic deformation or
compression produced at
the leading section of the
contact area is a hindrance
to motion
• Rolling motion is not fully
compensated as the
substances spring back to
normal shape at the
trailing section.
a disc/wheel pressed
against a plane surface
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Continue
• assume that the deformability
of the disc is much higher than
that of the surface, so that
• The disc is soft；
• The plane is rigid；
a disc/wheel pressed
against a plane surface
the contact zone will change from a
line into a rectangle.
Hertz’s theory, the pressure distribution
in the contact area is parabolic, with its
maximum in correspondence to the
centerline of the contact area (Fig. 5.7).
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Continue
• Consider a point on the disc
on the external surface of the
disc.
• During rolling of the wheel
this point will occupy
successive positions like P1,
P2, ….P5,;
• deforming (shortening and
lengthening) the
corresponding radius;
• compression will increase
from P1 to P3, and decrease
from P3 to P5.
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Continue
• If the wheel material were
perfectly elastic;
• the Hooke’s law : stating
the linear proportionality
between pressure p and
deformation ε, so that the
upload line P1-P3 of
Fig.5.8b) will be covered
exactly by the download
line P3-P5.
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Continue
• If the wheel material is
inelastic, for which the
value of p depends on the
direction followed
describing the p-ε curve,
the load line and download
line are not the same and a
hysteretic cycle will occur.
• The area contained within
the hysteretic cycle
represents the energy, per
unit of volume, dissipated
in one load-download cycle.
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Continue
The energy is the difference between the energy
cumulated in the upload phase and that given back
in the download one.
• The energy not given
back represents an
energy loss which is
transformed into
heat.
A simplified model
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Continue
• The pressure in the front part
is increasing, is always higher
than that in the rear part
(trailing edge), corresponding
to the download phase.
• a non symmetric pressure
distribution, does not pass
through the wheel center, but
is displaced of a quantity u
towards the contact area
leading edge, thus generating
a resistant moment Nu with
respect to the wheel center.
A simplified model
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Continue
• As an example, the wheel in Fig.
5.10 rolls without slipping on a
flat surface.
• Beside the external forces Q and
F and the couple C, the
(constraint) reaction
components T and N are
indicated.
kinetic relationship v=ωr
r is the wheel radius;
v is the speed of the wheel center ;
ω is the wheel angular speed.
T
a
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Continue
• The normal component is
displaced forward of a
quantity u with respect
of the wheel centerline,
• Can be considered
applied to a point P’, the
intersection of N with the
wheel profile.
b
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Continue
• applying the power balance approach of Eq. (4.24)
with the form (4.29)
The power dissipated by rolling friction is
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Rolling Friction Coefficient
• The displacement u towards the leading edge of the
normal reaction component N is assumed to be
proportional to the rolling radius r according to the
expression:
depends on the materials of the contacting bodies.
• This coefficient is determined experimentally with dedicated test
equipment or pulling a wheel on a road.
• To measure the mechanical power necessary to keep a steady motion;
• This power is equal to that dissipated by rolling friction.
Characteristic values of the rolling coefficient Vμ:
Tyre-road contact Vμ = 1~2 10-2
Steel on steel contact Vμ = 1~5 10-3
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Acceleration
Substituting in Eq. (5.12a) we obtain
Solving the scalar products of vectors of Eq. (5.12), we obtain:
rolling without slipping , v=ωr, (and consequently a =ω r)
By eliminating ω, and solving with respect to the acceleration a :
the unknown normal component N, still to be determined.
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Vertical equilibrium
• Beside the power balance, a dynamic equilibrium
equation has to be considered to determine N.
• For instance, by considering the vertical equilibrium
By substituting in Eq. (5.12d) we have
Q
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Tangent component T
• the energy dissipation caused by rolling friction is
not due to a sliding at the contact point;
• the tangent component T must be calculated by
means of a dynamic equilibrium equation
introduced in the inequality (5.10)
This verifies that the wheel is able to roll without slipping.
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Rolling with slipping
• In case of rolling with slipping ,
kinetic friction has to be considered, thus determining
the tangent component T
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