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In the previous paper [1], the formula of the contact pressure distribution in the steady wear state for drum brake system was derived. The contact pressure distributions on the leading and trailing shoes were found to be different and larger on the leading shoe. Using these contact pressure distributions, it is easy way to calculate the braking torque. Assuming the varying position of the shoe pin, the optimization problem can be formulated by requiring the optimal pin position corresponding to maximal braking torque. The steady wear states and optimal designs were specified for both shoes. The elastic displacement field of the drum brake at the optimal contact pressure distribution is calculated by the finite element system ABAQUS. At given rigid body rotation of the shoe, the wear in the leading and trailing shoes in the steady wear state can be easily found.

The analytical expressions of contact pressure distribution in drum brakes were proposed in some papers, starting from the assumptions of Koessler [

A two-dimensional model of the brake shoe was developed by Huang and Shyr [

A complete three-dimensional structure of the drum brake has been treated by Hohmann et al. [

In the paper by Ahmed et al. [

In our analysis, the contact pressure distribution is derived from the variational principle referred to the steady wear state, by minimizing the wear dissipation power, as has been presented in [

Consider now the drum brake system shown in

The following wear rule is assumed in our analysis

w ˙ n = β ( τ n ) b ‖ u ˙ τ ‖ a = β ( μ p n ) b ‖ u ˙ τ ‖ a = β ( μ p n ) b v r a = β ˜ p n b v r a , (1)

where β , a , b are the wear parameters, μ is the coefficient of friction, β ˜ = β μ b , v r = ‖ u ˙ τ ‖ is the relative velocity between two bodies.

In the previous paper [

e R = λ ˙ F + λ ˙ M × Δ r ‖ λ ˙ F + λ ˙ M × Δ r ‖ (2)

the optimality rule is expressed as follows

w ˙ 1 = − w ˙ 1 , R e R , w ˙ 2 = w ˙ 2 , R e R , (3)

where λ ˙ F and λ ˙ M are the rigid body relative translation and rotation velocities induced by wear; Δ r is the position vector.

The total wear rate vector can be decomposed components into normal and tangential directions, thus are

w ˙ n = w ˙ 1 , n + w ˙ 2 , n , w ˙ τ = w ˙ 1 , τ + w ˙ 2 , τ . (4)

The normal and tangential wear rate components now are

w ˙ n = w ˙ R cos χ , w ˙ τ = w ˙ R sin χ = w ˙ n tan χ , (5)

here χ is the angle between n c and e R .

The contact pressure p n ( x ) and the friction induced shear stress τ n = μ p n ( x ) must satisfy the global equilibrium conditions for the body B 1 .

The contact traction of the drum brake now is expressed as follows

t = t 1 c = − t 2 c = − p n ρ c = − p n ( n c ± μ e τ ) = − p n [ ( − sin α e x + cos α e z ) ± μ ( cos α e x + sin α e z ) ] , (6)

where the upper sign (+) corresponds to the anticlockwise drum rotation. The geometrical parameters l x , l z , R 0 , α are shown in

Δ r = ( l x − R 0 sin α ) e x + ( l z + R 0 cos α ) e z , (7)

As the shoe is allowed for free rotation, only the moment equilibrium condition of the shoe should be satisfied.

m = m c − M 0 = 0 , (8)

where the moment of external load F 0 is m 0 = − F 0 L e y = − M 0 e y , the wear rotation angular velocity equals λ ˙ M = − λ ˙ M e y , the moment of contact stress is

m c = ∫ α i α u ( Δ r × t 1 c ) ⋅ e y R 0 t t t d α = ∫ α i α u p n [ ( l z sin α + l x cos α ) ∓ μ ( l z cos α − l x sin α + R 0 ) ] R 0 t t t d α = ∫ α i α u p n m ∓ ( α ) R 0 t t t d α (9)

where m ∓ ( α ) = [ ( l z sin α + l x cos α ) ∓ μ ( l z cos α − l x sin α + R 0 ) ] .

The rigid body wear velocity is coaxial with the unit vector e R , so we have

e R = Δ r × e y | Δ r × e y | = 1 A ( α ) { ( l x − R 0 sin α ) e z − ( l z + R 0 cos α ) e x } , (10)

where

A ( α ) = ( l x − R 0 sin α ) 2 + ( l z + R 0 cos α ) 2 . (11)

Between the vectors n c and e R there is angle χ , its cosine value is

cos χ = n c ⋅ e R = ( l x cos α + l z sin α ) 1 A ( α ) . (12)

and the rigid body wear velocity vector equals

w ˙ R = w ˙ R e R , w ˙ R = w ˙ n cos χ . (13)

Introducing the parameter c = ( b + 1 ) q − 1 and the integral

I D w ∓ ( q ) = ∫ α i α u ( m ∓ ( α ) ( 1 ∓ μ tg χ ) − q ) 1 / c m ∓ ( α ) R 0 t t t d α , (14)

where the upper sign (−) corresponds to the anticlockwise drum rotation, the lower sign (+) corresponds to clockwise drum rotation.

In the previous paper [

D w ( q ) = ∑ i = 1 2 ( ∫ S c ( t i c ⋅ w ˙ i ) q d S ) 1 / q with the control parameter 0 ≤ q ≤ ∞ . The optimal contact pressure and total wear rate for q = 1 (which corresponds to the steady wear state) were expressed by the following formulae

p n ∓ = F 0 L I D w ∓ ( q = 1 ) ( A ( α ) cos χ ) 1 b = F 0 L I D w ∓ ( q = 1 ) ( l x cos α + l z sin α ) 1 / b (15)

w ˙ R ∓ = ∑ i = 1 2 β ˜ i ( F 0 L I D w ∓ ( q = 1 ) ) b ( R 0 ω ) a i A ( α ) (16)

or in the other form, which is dependent only on geometrical data and the friction coefficient

p n , G ∓ = p n ∓ F 0 L = 1 I D w ∓ ( q = 1 ) ( A ( α ) cos χ ) 1 b = 1 I D w ∓ ( q = 1 ) ( l x cos α + l z sin α ) 1 b [ 1 / mm 3 ] (17)

here the dimensions of I D w ∓ ( q = 1 ) and p n ∓ are [mm^{3+1/b}] and [MPa], of w ˙ R ∓ is [mm/s] because β ˜ i has the dimension [mm^{1−a+2b}・s^{a}^{−1}・N^{−b}], which for a = b = 1 is [mm^{2}/N].

The wear rate w ˙ R can be expressed directly in terms of the rigid body wear angular velocity λ ˙ M

w ˙ R ∓ = λ ˙ M ∓ A ( α ) . (18)

An important result can be stated. The rigid body rotation angular velocity λ ˙ M ∓ depends only on the main characteristic of the system and not on local pressure values, thus

λ ˙ M ∓ = ∑ i = 1 2 β ˜ ( F 0 L I D w ∓ ( q = 1 ) ) b ( R 0 ω ) a i . (19)

The brake moment in the steady wear state is expressed in terms of the contact pressure (for the optimal contact pressure distribution) by the following integral

M T = M b r a k e = ∫ α i α u μ ( p n − + p n + ) R 0 R 0 t t t d α

or

M b r a k e 2 F 0 L = 1 2 ∫ α i α u μ ( 1 I D w − ( q = 1 ) + 1 I D w + ( q = 1 ) ) ( l x cos α + l z sin α ) 1 / b R 0 2 t t t d α (20)

since the drum brake has two shoes. The entire moment from the external load is equal to 2 F 0 L , so the relation β f = M b r a k e 2 F 0 L provides the brake factor, indicating the portion of the given load moment used in the braking process. The reaction forces in the pin are

F x ∓ = ∫ α i α u p n ∓ ( sin α ∓ μ cos α ) R 0 t t t d α − F 0 ,

F z ∓ = ∫ α i α u p n ∓ ( cos α ± μ sin α ) R 0 t t t d α , (21)

and their resultant is F r ∓ = ( F x ∓ ) 2 + ( F z ∓ ) 2 . This value is important for the pin design.

In the alternative form, we have

F x ∓ F 0 L = 1 F 0 L ∫ α i α u p n ∓ ( sin α ∓ μ cos α ) R 0 t t t d α − 1 L

F z ∓ F 0 L = 1 F 0 L ∫ α i α u p n ∓ ( cos α ± μ sin α ) R 0 t t t d α . (22)

Let us now discuss the numerical results for the assumed wear parameters a = b = 1 , β 1 = β 2 = 2 × 10 − 8 , μ = 0.25 , angular velocity of drum ω = 2.5 [ 1 / sec ] , geometric parameters R 0 = 100 [ mm ] , l x = 20 [ mm ] , l z = 80 [ mm ] , t t t = 10 [ mm ] and loading force F 0 = 10 [ kN ] , which are typically for small break system. Because D w ( q ) for q = 1 presents the global wear dissipation power but for q → ∞ presents its local value, the control parameter q in our calculation is assumed to vary within the interval (0.0 - 10.0). Instead of q = ∞ we take q = 10 . In

It is seen that the contact pressure and wear rate are localized for the singular value of q = 0.5. The steady wear state regime corresponds to q = 1 with the respective contact pressure and wear rate distributions presented in Figures 2-4.

The brake moment value depends on the pin coordinates l x , l z . At fixed geometric values ( R 0 = 100 [ mm ] , t t t = 10 [ mm ] , α i = 30 ∘ , α u = 150 ∘ ), the brake factor value defined as the brake moment to given moment of the load F 0 : β f = M b r a k e / ( 2 F 0 L ) is demonstrated in

The contact zone can be changed by selecting initial and final angles α i , α u and accounting for the constraint of a positive contact pressure, p n ≥ 0 . Requiring that at the end of contact zone there is l x cos α u + l z sin α u = 0 , (this point corresponds to a shorter distance from pin, and the pressure at this point can vanish). Using this constraint, we find a larger relative value of the brake factor β f = M T / ( 2 F 0 L ) . In

The surface load on the cylindrical surface of radius R 0 is given from the steady wear state contact traction (17) at L = 160 [mm] and at load F 0 that is p s u r f a c e ∓ = p n ∓ = p n , G ∓ F 0 L [MPa], the shear stress is given by Coulomb law. Using the length coordinate s along the circular profile of radius R 0 in the clockwise direction the distribution of normal loads p n ∓ t t t [ N / mm ] are demonstrated in

After solution of the finite element model, the normal displacement is presented in

The normalized distance along the drum surface is specified by the formula s n o r m = s / 2 π R 0 . The deformed body is demonstrated in

For wear calculation, a simplified model is set up. The shoes are assumed as rigid, the elastic deformation is only in the brake drum. We give different rotation values for shoes, λ l for the left shoe, λ r for the right shoe. It is assumed that λ r = λ l u n , max , r i g h t / u n , max , l e f t . In the steady wear state along the whole domain ( α i ≤ α ≤ α u , R 0 ) there is contact, so the gap between the bodies is absent, and we can write geometrical Signorini contact condition

d = u n − u R + w n = 0 , (23)

where u n is the normal displacement of the brake drum in direction of the radius of cylindrical system, u R is the normal displacement from rigid body rotation of the shoe and w n denotes the wear value. We have

u R − = λ M − [ ( R 0 cos α − + l z ) sin α − − ( R 0 sin α − − l x ) cos α − ] ( − ⇒ + ) (24)

where α − = π − ϕ , α + = − π + ϕ , ϕ = s / R 0 , next ? is used to denote the left shoe region, and + denotes the right shoe region. From Equation (23) it is easy to determine the shoe wear. The wear profiles are presented in

We take different y sections for demonstration of the wear distribution (see

Taking the assumed wear parameters, the values a i = b = 1 , q = 1 , friction coefficient, angular velocity of the brake drum, distance L, load F 0 , the calculated integrals (19) are I D w − = 132 × 10 5 [ mm 4 ] , I D w + = 256 × 10 5 [ mm 4 ] and the rigid body wear rotation velocities are λ ˙ M − = 1.51 × 10 − 7 [ rad / s ] , λ ˙ M + = 7.81 × 10 − 8 [ rad / s ] .

We take rotation of the left shoe λ l = 0.028 . Supposing that wear rate is changing approximately in proportion to quadratic law in time and the initial value is equal to 3 λ ˙ M − = 4.53 × 10 − 7 , (this can be calculated from initial contact pressure), then λ ˙ − = λ ˙ M − ( 1 + 2 ( τ − T c ) 2 T c 2 ) . After time integration we find T c = 3 5 λ M − λ ˙ M − = 30.91 h . For the right shoe is λ r = 0.018 = λ M + , λ ˙ M + = 7.81 × 10 − 8 , then T c = 3 5 λ M + λ ˙ M + = 38.41 h . It is clear that the time period for the right shoe is larger than for the left shoe, because the wear rate is smaller. As in the steady wear state λ ˙ M − = c o n s t , the increment of the shoe rotation angle is Δ λ = λ ˙ M − ( T − T c ( λ l = 0.028 ) ) . From this fact we find that T ( λ l = 0.032 ) = 45.77 h , T ( λ l = 0.036 ) = 53.13 h , T ( λ l = 0.040 ) = 60.49 h . Knowing angular velocity ω = 2.5 [1/sec], it is easy to calculate the number of revolutions of the drum n = 3600 T ω / 2 π , n ( λ l = 0.032 ) = 65500 , etc.

For a braking system in the steady wear state, the contact pressure distribution provides a good possibility to find the maximum brake moment in the optimal design problem. This distribution can be determined from the minimization procedure of the wear dissipation power. The wear of the shoes can be easily calculated using the steady contact pressure distribution. In our analysis, the shoe was supposed as rigid body; the drum was assumed as linear elastic; then the wear rate is overestimated. The numerical predictions presented in the paper require experimental verification.

The present research was partially supported by the Hungarian Academy of Sciences, by the grant National Research, Development and Innovation Office―NKFIH: K115701.

The authors declare no conflicts of interest regarding the publication of this paper.

Páczelt, I., Baksa, A. and Mróz, Z. (2020) Analysis of Steady Wear State of the Drum Brake. Open Access Library Journal, 7: e6432. https://doi.org/10.4236/oalib.1106432