Brake shoe analysis

The illustration below explains this contact in more detail. An element   dA of the lining is shown with the braked body moving past at velocity   v. Since sliding must occur the force resultant on dA comprises the elemental component   dN = p.dA   orthogonal to dA together with the elemental friction component   dF_f = μp.dA   coplanar with dA and in the sense of v.

Shoes are classed as being either short or long. A   short shoe is one whose lining dimension in the direction of motion is so small that contact pressure variation is negligible, ie. the pressure is everywhere uniform, at   p_m say. The contact resultant therefore consists of the normal reaction   N =   Σ dN =   Σ p dA   -->   ∫ p dA =   p_m ∫dA =   p_m A   together with the friction force   Σ μdN = μ.p_mA.   Short shoes are statically determinate.

In a   long shoe the variation of contact pressure in the direction of motion is not negligible and integrals of the above form   ( ∫ pdA ) cannot be evaluated unless the p-variation is known. A long shoe is thus statically indeterminate and must be appraised by the three -pronged attack used for cylinders, welds, thick curved beams and so on - a plausible spatial distribution of lining deformation ( geometric compatibility ) is first assumed, then the lining constitutive law ( presumed elastic ) and finally equilibrium are applied to establish completely the pressure variation for a given actuation. Long shoes are more complex than short.

The pivot G of a hinged shoe introduces a second degree of freedom permitting the lining to adopt a position which tends to spread wear more uniformly over the lining and hence to increase lining life.

The braked body may translate or it may rotate about a fixed shaft axis O. External rotational shoes only are sketched above - the treatment of internal shoes is identical to that of external.

All these configurations behave qualitatively in identical fashion, and they are all analysed in a similar manner - it's only their differing geometries which introduce complications. We shall lead up to the analysis of practical rotational long shoes by first considering the following simpler cases

-	translational shoes - although these are uncommon since mechanical power is usually transmitted rotationally, they are geometrically simpler than rotational shoes, and
-	short shoes - although these are generally impractical because of their small lining area   ( recall the 'Linings' section above ), they are simpler than long shoes as contact pressure is uniform.

A shoe behaves very differently depending upon whether it   trails or   leads. If the braked body drags the lining away from the hinge then the lining trails behind the hinge and the shoe as a whole is said   'to trail.' The shoes in the lower row of the above sketch all trail while the shoes in the upper row all lead for the motion senses drawn.

In each of the following shoe analyses, the shoe geometry is defined and the coefficient of friction μ between lining and braked body is known. In the first instance it is required to find the resultant of the lining/braked-body contact for a given actuation. Determination of the braking effect, shoe performance parameters, and hinge & shaft reactions is thereafter straightforward.
It is useful to realise that effect is always proportional to cause for constant geometry and friction coefficient - thus if a shoe's actuation is doubled then the braking effect, lining pressure, hinge reaction etc. all double.

Short translational shoe

The geometry of a short translational shoe is sketched. The uniform pressure over the contact area   A is   p_m . The braked body may move in either direction at a fixed distance,   h, from the hinge axis.

Free bodies of the shoe are shown for the two possible directions - evidently the shoe in the left configuration trails while that in the right leads. The three contacts which each free body makes with its surrounds are

• the actuation force   P
• the braked body characterised by the normal reaction   N = p_mA   (see previous section), together with the friction force   μN   in the sense of the braked body's motion, and
• the hinge support reaction   R_H .

The normal reaction depends upon the moment   Pe rather than on   P itself. The product   Pe is termed the   actuation moment,   M, so the two equilibrium equations may be written

( iii)       N   =   p_m A   =   M / ( a ± μh )     - the sign depending upon whether the shoe trails or leads

All terms on the RHS of this equation are geometric - apart from the actuation and the (presumably constant) friction coefficient - so the normal reaction and the frictional braking force are   directly proportional to the actuation. The difference between the two behaviours is as follows

• In the LH sketch the moment of the friction force   opposes the actuation moment; friction tends to throw the shoe off the braked body, to reduce the actuation - the shoe is said to be   counter- actuating.
• In the RH sketch the moment of the friction force   assists the actuating moment; friction tends to drag the shoe onto the braked body, to augment the actuation - the shoe is said to be   self- actuating.

Once the normal and friction forces have been found via ( iii) for a given actuation, the hinge reaction,   R_H, may be ascertained easily from force equilibrium of the shoe free body. All forces are then known, allowing safe design to be carried out.

Long translational shoe

A long shoe is analysed by dividing it up conceptually into infinitessimal elements - each of whose contact with the translating body is identical to that of the short shoe above - then summing (integrating) the equilibrating effect of all the elements.
The free body of the long shoe is similar to that of the short in having three contacts, however the contact with the braked body is distributed. Consider such a trailing shoe whose lining, of constant width   w normal to the sketch plane, extends from   x_min to   x_max as sketched { nomenclature explanation}. The lining pressure is   p on the typical element of length   dx located at   x from the hinge, so the normal force on the element is   dN = p.dA = p w dx, while the corresponding friction force will be   μ p w dx.

Moment equilibrium of the assembly of elemental short shoes about the hinge (to eliminate the as yet unwanted hinge reaction) requires
            M   - Σ dN.x   - Σ μdN.h   =   0     or in the limit on substituting for dN :     M   =   w ∫ p ( x   + μh ) dx

Integration is possible only if the pressure distribution   p(x) is known, so the above- noted proportionality between pressure   p and distance   x is formalised as   p   =   p_max ( x / x_max ) - the severity of contact loading being characterised by the peak pressure   p_max which depends upon the actuation.   Inserting this into the equilibrium equation and integrating between the lining limits   x_min and   x_max leads to

( iv)       p_max   =   ( M x_max / w ) / [ ( x_max³ - x_min³ )/3   + μh ( x_max² - x_min² )/2 ]

The relation ( iv) for a long shoe is similar to ( iii) for a short shoe as it expresses the essential proportionality between actuation and the resulting contact effects - if the actuation is doubled then the peak pressure   p_max and the pressures at all points of the lining are also doubled. The total braking effect, hinge reaction etc. produced by a certain actuation follow from ( iv) - for example
              total braking effect   =   Σ μpw dx   =   ∫ μ [ p_max( x/x_max )] w dx
                                            =   μ w p_max ( x_max² - x_min² ) / ( 2 x_max )       or substituting for p_max from ( iv)
                                            =   M / [ h  + 2 ( x_max³ - x_min³ ) / ( 3 μ ( x_max² - x_min² ) ) ]

The details of this final result notwithstanding, it is clear that the braking effect is again proportional to actuation since only constant geometric and frictional terms occur.

Short rotational shoe

The shoe free body is shown at ( b) with the actuation moment   M = Pe and the effect of the braked body (drum) contact on the lining expressed by the normal reaction   N and the friction force   μN in the sense of braked body motion - just like the short translational shoe.
For rotational equilibrium about the hinge to eliminate the as yet unwanted hinge reaction

( v)       N   =   p_m A   =   M / ( a.sin θ_m   - μ ( r   - a.cos θ_m ) )

This is analogous to ( iii) in expressing the essential proportionality between the normal reaction and the actuation - other forces follow from force equilibrium.

This is really all there is to short shoe brake analysis - however it is useful to introduce two artifices to simplify later work

• ( v) applies only to an external trailing shoe. Rather than derive other relations for internal and/or leading shoes, we shall employ a   "double delta" notation and re-derive a general form of ( v) which will describe all possible configurations simply by inserting appropriate ±1 values for the deltas.
• When short shoe results are extended to long shoes it will be seen that the normal and friction components vary in both magnitude and direction around the lining contact. It is much easier to handle these if they are moved to the shaft centre,   O. It will be recalled from basic Statics that a force may be moved transverse to its line of action without altering the equilibrium of a body on which it acts provided a corresponding moment is introduced equal to the force multiplied by the transverse distance moved.

The double delta notation utilises

• upper case   Δ   to designate shoe   position - if a shoe is external then Δ = +1;   if the shoe is internal then Δ = -1.
• lower case   δ   to designate shoe   sense - if a shoe is trailing then δ = +1;   if the shoe is leading then δ = -1.

• if the shoe leads rather than trails then the friction force on the shoe μN will act in the SE direction, ie. negatively in the NW sense sketched since δ = -1 for the leading shoe;
• if the shoe is internal to the drum rather than external, then the normal reaction N will act radially inwards on the lining while, to rotate the shoe clockwise against the surrounding drum, the actuation M must be clockwise ie. negative in the senses sketched since Δ = -1 for the internal shoe.

p_mis the uniform pressure over the short lining whose contact area is   A. The reaction to the torque   T is the braking effect exerted by the shoe on the drum.
Taking moments about the hinge for the free body ( d) - [ ( c) serves equally well ] -

  ( vi)       ΔM   =   a F_y   - δT   =   N [ a ( δμ cosθ_m + Δ sinθ_m )   - δμ r ]     - and solving for   N

  ( 6a)         N   =   M / [ a sin θ_m   - δΔ μ ( r   - a cos θ_m ) ]     - a generalisation of ( v) applicable to all layouts
                          =   M / [ μr ( m   - δΔ n ) ]     - in which   m ≡ ¹/_μ ^a/_r sin θ_m     and     n ≡ 1 - ^a/_r cos θ_m
                                                      are constant dimensionless characteristics of the shoe.

Summarising the analysis steps for a short shoe of known geometry and friction coefficient :

• Evaluate the shoe constants,   m & n from ( 6a);
• for a given actuation   M ascertain the corresponding normal reaction   N from ( 6a) with δ, Δ appropriate to the shoe's configuration;
• determine contact resultants including braking torque from ( 5a), then other forces if desired from force equilibrium.

Shoe figures of merit

The   mechanical advantage (or   "brake factor"), η  
( vii)           η   =   ^output/_input   =   ^{braking effect}/_{corresponding actuation necessary}

The   sensitivity, S, reflects the proportional variation of braking torque with fixed actuation as the friction coefficient varies, thus
( viii)         S   =   ^μ/_T . [ ^∂T/_∂μ ]_M   =   ^μ/_η . ^dη/_dμ

For a single shoe, use of ( 5a), ( 6a) leads to

( 8)             η   =   ^T/_M   =   ¹/_{( m   - δΔ n )}         and         S   =   m η

These are plotted below for representative short shoes of typical proportions. The curves demonstrate that the same trends apply for both figures of merit, namely

δΔ = -1

External leading and internal trailing shoes of the given proportions are counter- actuating. Both mechanical advantage and sensitivity are low and the shoes are inherently stable.

δΔ = +1

External trailing and internal leading shoes of the given proportions are self- actuating. Both mechanical advantage and sensitivity can become high - giving rise to instability, to vibration and squeal, to judder if the assembly is flexible, or even more dangerous, to grabbing of the brake. Physically this is due to friction not merely assisting actuation but swamping it.
Self- actuation occurs if   m and   n   in ( 6a) & ( 8) are approximately equal due to design, or to the shoe becoming out of adjustment because of hinge wear, or if the coefficient of friction increases after the ingress of dirt, drum scoring, or other environmental changes. While significant self- actuation may be advantageous for an emergency brake, generally it should be avoided.

This example analyses a brake with two separately- actuated short shoes, and considers braking torque, sensitivity and bearing loads.

It is now appropriate to examine long shoes - which are more practical than short - before proceeding to look at complete twin shoe brakes.