Friction Brakes and Clutches

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```					ZEIT3700                                                             2010
Mechanical Design 1                                                  Semester 1

Friction Brakes and Clutches
While brakes and clutches are most often used as essential components of rotating
machines there are a number of important examples of linear motion brakes – for
example, those used to provide emergency stopping of lifts – and a small number of
linear motion clutches.

We will be concerned with the analysis and design of rotary motion devices using
friction to transmit torque between the input and output members of the devices.
There are also positive-engagement clutches, such as dog clutches, which are used to
connect rotating elements, but these cannot be used where there is substantial torque
and a large speed differential between input and output.

Function
The function of a friction clutch is to permit two co-axial rotating members to be
brought to the same angular velocity while a torque is transmitted between them, and
also to permit them to be uncoupled and re-coupled. If the clutch slips, a torque is
transmitted between the input and output members but these have different angular
velocities; the magnitude of the torque depends on the design of the clutch and its
operating force. When there is no slipping the torque transmitted depends on the
external machinery and is only known to be less than the clutch slipping torque.

The function of a brake is to permit a rotating member to be brought to rest, or at least
decelerated. It can be viewed as a clutch in which the rotational speed of the “output”
member is always zero. All brakes slip, and some also have a non-slipping mode.

For both brakes and clutches the torque on the input member is equal in magnitude
but opposite in direction to that on the output member.

Non-friction clutches and brakes
In addition to the devices we will be considering, rotating machines may incorporate a
range of devices which transmit torque between input and output but do not use
friction to do so. Examples of such devices are:

Non-friction clutches:        Dog clutches
Sleeve clutches
Overrunning clutches
Hydrokinetic couplings
Torque converters
Overspeed (throwout) clutches
Non-friction brakes:          Electric brakes
Compressor brakes
Fan brakes

Characteristics of friction brakes and clutches
The primary parameter is [1] The relationship between the operating force and the
slipping torque. The torque may also depend on rotational speed and/or time, but the
performance is more predicable if it does not. An intermediate case is the centrifugal
clutch, where the operating force is speed-dependent.

Additional parameters are [2] The maximum value of the slipping torque, which
depends on the geometry, the coefficient of friction and the maximum allowable
pressure on the friction surface materials, and [3] The maximum energy which the
brake or clutch can dissipate as heat.

Dynamics of brake and clutch systems
In analysing applications of brakes and clutches we need to consider machines
rotating at both constant and changing angular velocities. In many brake applications
the essential quantity is the energy which has to be dissipated, and this energy may be
kinetic (for example, for vehicle braking) or potential (for example, for lift braking).
The brake system must not only dissipate the required energy but must also be
configured to “convert” it into a shaft torque of suitable magnitude.

Through appropriate gearing both clutches and brakes can control the operating
speeds of machines: centrifugal clutches provide engagement when the shaft speed
exceeds a prescribed (sometimes controllable) cut-in value, and disconnect when the
speed falls below a cut-out value ( which may be the same as or different from the
cut-in value). Similarly, centrifugal brakes apply a retarding torque when shaft speed
exceeds the cut-in value, and this torque may increase with increasing overspeed.

The principles of the linked block or the wedge block discussed in earlier notes on
friction may be used to make brakes and clutches self-energising, so that the local
normal pressure is the sum of that due to the operating force and due to friction on
other parts of the contact surfaces.

A particular problem in the braking of wheeled vehicles arises from the limitations of
rolling friction between wheel and surface; it is generally undesirable to lock wheels
during braking, so that anti-skid systems are desirable when maximum braking
retardation is required. There is an additional problem in mixing vehicles with
substantially different wheel coefficients of friction – for example, motor cars and
trams – which has led to the development of special emergency braking systems
(obviously for the vehicles with lower coefficients of friction).

Clutches used for accelerating a load
One of the important uses of friction clutches is to bring a machine up to the same
speed as its prime mover. This operation is particularly important when the prime
mover will not operate, or operates very inefficiently, at low rotational speeds.
Reciprocating internal combustion engines are typical of the former, and electric
induction motors operating at mains frequency are typical of the latter.

If the load is initially at rest and the prime mover is running at some desirable
operating speed, on engagement of the clutch the latter will transmit its slipping
torque (set by the clutch operating force) and will continue to do so until the torque
transmitted falls below this value because the rotational speeds of load and prime
mover are the same.

During the slipping phase the clutch dissipates energy, whose magnitude is the
integral of the product of the slipping torque times the difference in rotational speed

Also, unless the torque output of the prime mover exceeds the slipping torque the
rotational speed of the prime mover will decrease so that part of the “driving” torque
on the load comes from the angular deceleration of the prime mover (and is equal to
the product of mass moment of inertia of the prime mover and the angular
deceleration).

When slipping ceases, and the torque is less than the slipping torque, the torque output
of the prime mover will have to accelerate both the load inertia and the inertia of the
prime mover itself, so that the clutch transmits only a part of the generated torque.

We can summarise the two phases as follows:

Slipping phase:

Clutch

1        QS            QS  2    Load
Prime Mover
IL

.
2 < 1                    2 = QS
IL
Accelerating (non-slipping) phase:

Clutch

IPM        Q PM                                            IL

.
Q PM < Q S                       2 = Q PM
IL + IPM

If the (mass) moment of inertia of the prime mover is Ipm and of the load Il and the
prime mover generates a torque Qpm then the equations of motion are, for the slipping
phase:

I pm1  Q pm  Qs and I l  2  Qs
                      

and for the non-slipping phase:

I   pm    I l   Q pm


Sign of angular velocity and torque
Angular velocity and torque are vectors represented by vector arrows directed along
the axis of rotation with a positive right-hand screw convention. For machines with
rotating shaft inputs and outputs a consistent convention is required, and the suggested
procedure (as used in the above diagrams) is:

♦Introduce breaks in the shafts between machines.

♦For any such break, the shafts on both sides have the same angular velocity, but the
torques applied to the shaft stubs are equal and opposite.

♦If the stub is an input shaft the torque on the stub has the same sense as the
angular velocity.

♦If it is an output shaft, the torque on the stub has the opposite sense to the angular
velocity.
Friction linings and wear
Most friction brakes and clutches use a rubbing pair consisting of one metal face and
one non-metallic face. This prevents the welding of the surfaces and largely confines
the abrasive wear to the non-metallic face or lining.

It is a characteristic of practical lining materials that they have a maximum useable
normal pressure: if this is exceeded the material is crushed and the rate of wear will
be greatly increased due to fracture in the material.

The actual rate of wear – provided the maximum allowable pressure is not exceeded –
is fairly accurately proportional to the product pV of local pressure and sliding
velocity. For a rotating clutch, velocity V is proportional to radius r, so that a uniform
rate of wear requires that pr = constant, that is p is proportional to 1/r.

In service, the pressure distribution on the faces in a flexible clutch will be determined
by the loading device, since the flexibility will accommodate different rates of wear at
different radii, but for a rigid clutch where the dimensions are controlled we must
have a uniform rate of wear, so that the lining will rapidly take up such shape that the
pressure will be proportional to 1/r.
Axial plate clutches
Axial plate clutches usually have two friction faces per plate, and linings extending
from an inner radius a to an outer radius b. The axial pressure p is in general a
function of radius r, and if we consider any small annulus of width dr at radius r the
elementary contribution to torque is

dQ  pr 2rrdr  2pr r 2 dr
So that, for each friction face,
r b
Q  2     pr r
2
dr
r a

For constant pressure p, this gives:
2
Q  p b 3  a 3
3
             
While for uniform wear p = pmax(a/r) and


Q  p max a b 2  a 2         

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