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Russian Journal of Biomechanics
www.biomech.ac.ru
MATHEMATICAL MODELLING OF THE FOOT PROSTHESIS ELASTIC
ELEMENT UNDER BENDING
M.A. Osipenko, Y.I. Nyashin, R.N. Rudakov, A.V. Ostanin, E.N. Kuleshova, T.N. Zhuravleva
Department of Theoretical Mechanics, Perm State Technical University, 29a, Komsomolsky Prospect, 614600, Perm, Russia, e-mail: oma@theormech.pstu.ac.ru
Abstract: The results obtained on the formulation and solution of the problems of the foot prosthesis elastic element bending and optimum design are reviewed. The elastic element is modelled as a cantilever multiple-leaf spring with initially curved leaves. There is an unbonded contact between the leaves. Hence, the corresponding bending problem is the contact problem in the theory of beams. The new approach is suggested to the formulation and solution of such problems. The uniqueness of a solution in the general case is proved. Analytical solutions are constructed in some particular cases. The problem of stress minimization in the elastic element is formulated. The analytical solution of this problem is constructed for a two-leaf element, and the euristic method of solution of this problem for the multiple-leaf element is suggested.
Key words: foot prosthesis, elastic element, leaf spring, curved leaves, weak bending, contact area, stress minimization
Introduction
The foot prosthesis design forms one of the actual tendencies in prosthetic engineering. About 3 thousand people in Perm region make use of the foot prostheses (including the prostheses of foot and shank or foot, shank and thigh).
The artificial foot should reproduce as many functions of the natural foot as possible. These functions are the following: support function, heel strike absorption function, potential energy storing function, spring function (sprawling under the vertical loading), balance keeping function (posture regulation), adaptation function (the ability for walking along the incline). The artificial foot should also possess sufficient strength.
The aforementioned requirements are fully met by the structures, which include the so-called elastic element representing a cantilever leaf spring. The elastic element can be made of steel. Fig. la shows a serial sample manufactured at the Reutovo experimental prosthetic factory, Moscow region, Russia. Such prostheses are widely distributed at present. But the steel elastic element with the admissible elastic properties turns out to be slim, small-leaf and fragile (according to the Perm prosthetic and orthopaedic enterprise information, there are about 70 breakage incidents a year in Perm region). The elastic elements made of light composite materials are more promising. Such elements can be made multiple-leaf, thicker and more durable. Fig. lb shows the experimental sample made of carbon-filled plastic KMY-4JI at Urals Scientific Research Institute of Composite Materials, Perm, Russia. Fig. lc shows the elastic element of the Carbon Copy prosthetic foot manufactured by Ohio Willow Wood Company, USA [1].
Fig. 2). The lengths and the thicknesses of the leaves are different. The sequence of lengths is non-increasing one. The given loading is applied perpendicular to the lower leaf (Fig. 2; the loading is uniform over the leaf width). The leaves undergo the weak joint bending (with the unbonded contact).
At first consider one-leaf spring (Fig. 3). The natural shape of the leaf is described by the function cp(x), where x is the arc-length of the leaf segment placed between the clamped point and some arbitrary point; (p is the angle formed by the tangent to the leaf profile at a given point (Fig. 3); 0 < x < I; / is the given overall arc-length. It is assumed below that the function cp(x) is continuous, piecewise continuously differentiable, non-decreasing, and
cp(x) < Tt/2 for 0 < x < I. Since 9(0) = 0, cp(x) > 0. The shape of the leaf under the load is described by the analogous function <p(x). It is assumed that the potential energy stored in the leaf under bending is [15]
where
a(x) = —— Ewh (x)
is the given bending flexibility of the leaf, E is Young's modulus, h(x) is the (variable) thickness of the leaf. The function a(x) is assumed to be continuous and positive for 0 < x < /. It is assumed that the bending of the leaves is weak (linear approximation with respect to the load). In this case the shape of the leaf under the load may be defined by the normal displacement y(x) (Fig. 4). Let A be a point of the leaf without bending with the curvilinear coordinate x; let B be the position of the same point of the leaf under bending.
Then y{x) is the projection of the vector AB onto the normal to the leaf at the point A.
It is assumed that the loading is normal to the leaf (Fig. 5); qiow(x), qup (x) are the
load densities on the lower and upper sides of the leaf. If the multiple-leaf spring is considered then this assumption should be made only for the lower side of the lowermost leaf, where the load is given a-priori. For the other surfaces of the other leaves this load form follows from
Fig. 1. The variety of the foot prosthesis elastic elements.
Such complicated structures operate more effectively than the simple ones, but more complicated methods are required for their design; the method of mathematical modelling, in particular. Mainly the experimental investigations of the multiple-leaf elastic elements of the foot prostheses and some simple calculations of the motor car leaf springs have been published elsewhere [2-6]. The present study deals with the mathematical model of the elastic element, which has been developed at the Department of Theoretical Mechanics of Perm State Technical University, Russia [7-14]. Some bending patterns are derived and the problem of the elastic element optimum design is solved on the basis of this model.
Mathematical model of the elastic element
In our mathematical model the elastic element represents the stack of slim curved beams (leaves) with the rectangular cross-sections. The leaves fit each other closely (without loading). There is no friction between the leaves. Each leaf has one end clamped and the other free (Fig. 2). All leaves have the same width w (in the direction perpendicular to the plane of
Fig. 2. The model of the elastic element. Fig. 3. The model of one leaf; the definition of
the function q>(x).
the absence of friction and the weakness of bending. We assume that qiow(x), qup{x) can be represented in the form of
»(*) + 2X8(*-*x)> (2)
a
where u(x) > 0 is the piecewise continuous function, which is continuous on the right at x = 0 and on the left at x > 0; the running integer index a has the finite range of function; xa > 0; Ua > 0. For the lower side of the lowermost leaf, formula (2) a-priori gives the form
of loading. For the other surfaces of the other leaves, formula (2) is used in the following formulation of the problem.
The elastic element consists of N > 2 leaves. Their lengths are lk> 0, and their
bending flexibilities are ak(x); 1 < k < N (Fig. 2). The sequence lk is non-increasing one.
The loading with the given density q(x) (which is of type (2)) is applied to the lower side of
the leaf 1. The shapes of the leaves are described by the functions yk(x) (1 <k< N).
Formulation of the problem of the elastic element bending
The problem of one leaf bending is to find y(x), while qiow{x) and qup{x) are given.
Using Eq.(l), the principle of virtual work and the standard calculus of variations techniques [16], we find the solution of this problem:
i
y(x) - J*G(x, s)(qlow (5) - qup (s))ds , (3)
0
where
G(x, s) = G(ma,x(x,s), min(x,5)) (4)
(Green's function),
m
G{M,m) = ja(s)g(s,M)g(s,m)ds, (5)
0
g(j,H) = J*cos(q>(|i) - cp(0) dt. (6)
t
Integral of type Jf^S^ - s* )ds (which may be contained in Eq.(3) and in the
b
following analogous equations) is considered to be equal to ) in the cases s*=b or c.
For the A-leaf elastic element it is required to find the functions yk (x), while the density q{x) of the loading (which is of type (2)) applied to the lower side of the leaf 1 is given. In order to solve this problem it is convenient to rephrase its formulation so that to regard the densities fk(x) (1<k< N-l) of the forces of interaction between the leaves
k, k +1 as the functions to be sought. The functions yk(x) are expressed in terms of fk(x) as follows (these formulae result from Eq.(3) and the law of equal action and reaction)
k k+1
yk 0) = JGk (*> s)fk-1 (s)ds - j\Gk (x, s)fk (s)ds, (7)
where 1 <k< N and it should be considered that lN+] =0, fN(x) = 0, /0(x) = q(x). The functions Gk(x,s) are obtained by substituting a(x) = ak(x) in Eq.(5). It is assumed below that the range of a variable of index k is 1 <k< N-\ (unless it is given explicitly). We assume that fk (x) are of type (2); it is an a-priori assumption on the leaves interaction. We
introduce the notation rk(x) = yk+i (x) - yk(x); then using (7) we find
k h+i rk (*) = - \Gk (x> s)fk-1 (s)ds + \(Gk (x>s) + Gk+10> s))fk (s)ds -
lk+2
¡Gk+l(x,s)fM(s)ds . (8)
0
The conditions of unilateral (unbonded) leaves contact are expressed by the inequalities rk(x) > 0 ; besides, if fk(x) > 0 then rk(x) = 0. Finally, we come to the following problem.
Problem 1. It is required to find fk (x) ( 0 < x < lk+] ), which are of type (2) and should satisfy the conditions
f=0 (fk(x)>0)
rk (*) 1 » (9)
1*0 (fk (x) = 0)
where rk(x) are expressed by Eq. (8).
Some results obtained on the analysis of the problem of an elastic element bending
The full solution of Problem 1 is far from completion. Only the following theorem has been proved for the general formulation of this problem.
Theorem 1. Problem 1 may have only the unique solution.
The proof in the case N=2, ak(x) = ak = const, k = 1,2 is adduced in [13]. This proof can be easily extended for the general case. Note that if the load q(x) is multiplied by a constant then the solution of Problem 1 is linear with respect to the load (but if two loads are added together then the solution is non-linear with respect to the load).
For N -2, cp(x) = 0, ak(x) = ak = const, k = 1,2 the solution of Problem 1 has been constructed explicitly in [9]. In this case, it follows from Eqs. (4-6) that Gk{x,s) = a/cy(max(x,.s'), min(x,s)), where y(M,m) = m2(3M-m) 6. We introduce the notation
h
M= J(x - l2)q(x)dx>0,
h h
<D(A) = M--1—- f(x - A)(/2 - x)(212 - A - x)q(x)dx,
ill- A) a
where 0 < A < /2. Then the following theorem can be proved.
Theorem 2. The solution /¡ (x) of Problem 1 in the aforementioned special case has the form: (i) if M = 0 then
f\(*) = q(x);
ax +a2
(ii) if 0(0) > 0 then where
fx{x) = F S(x-£2),
F =
a,
(«1 + a2)y(l2,l2)
•i
Jy (l2,x)q(x)dx:
(iii) if O(O) < 0 and M > 0 then
fix) = F 8(jc -12) + P 5(x - X) +
a i
ax + a2 0
q(x) (0<x<X)
(10)
(x>X)
where 0 < X < l2 is the root of the equation O(A) = 0,
F =
ax
•i
J(x - X)q(x)dx
P =
(a}+a2)(l2-k) , v,-rr J(/2 - x)q(x)dx.
x h
x+o
The expression ¿±0 denotes right-hand or left-hand limit. Note that if q(x) in Eq. (10) contains the term of type F0 8(x - A,) then this term is contained in /(x) (according
to nonstrict inequality x < X in Eq. (10)).
The bending pattern (i) may be called close fitting-, the bending pattern (ii) may be called pointwise contact.
For N = 3, cp(x) = 0, ak (x) = ak = const, k = 1,3 only few possible bending patterns have been established; they are analogous to the patterns (ii), (iii) of theorem 2. We denote
A =
a2
a\+ a2 <2j + a2
{ax+a2)A (al+a2)l2 (ax+a2)y{l2,A) (ax +a2)y(l2,l2) ~a2y{l2,l3)
a\
-a2l3
<>2
Jc/(x)i/x
\ I
^xq{x)dx
A h
A
*i
a\ fy(^2' x)q{x)dx
A
A
-a2y(/3,A) -a2y{l2,h) (a2+aMh'h) iy(/3,x>/(x>&
o
O(a) = detA, (0<A</3). Then the following theorem can be proved (it has not been published earlier).
Theorem 3. The solution f (x), f2 (x) of Problem 1 in the aforementioned special case has the form:
(i) if 0(0) < 0 then /j (x) = F 5(x -12), f2 (x) = P 5(x -13), where F and P are determined from the following set of linear algebraic equations
r A A \fr?\ r
^32 ^33
KA42 A43J
1 A=0 0
(ii) if 0(0) > 0 and o(/3) < 0 then
a.
ax + a2 0
q(x) (0 < x < X),
fl(x) = F5(x-l2) + P8(x-X) + -
(x > A,),
/2 (x) = Qd(x-13), where 0 < a, < l3 is the root of equation O(A) = 0 and P, F, Q are determined from the following set of linear algebraic equations
a, a 2 fp> ( à > au
al A22 a23 f - a24
al al as j [qj {a4)
The proof of Theorem 3 is analogous to the proof of Theorem 2 [9] (but it is more cumbersome).
For arbitrary N, <p(x) = 0, ak (x) = ak = const, 1 < k < N only the case q(x) - fq 8(x - /j ) has been considered and in this case only necessary and sufficient conditions for the pointwise contact have been established. The following theorem has been proved [9].
Theorem 4. The solution fk (x) of Problem 1 in the aforementioned special case has the form fk (x) = Fk 5(x - lk+l) if and only if
zkakUkih+i ~lk+2X^+1 + ~h+1)]-
(11)
> (ak +ak+l)lk+l(lk+i ~lk+2)(2lk+l -lk+2), where the values zk are determined recurrently
zk =Kak + ak+\)jk+\,k+\ ~ak+llk+\,k+2;zk+l]/akyk,k+\
and yaj p = y(/a, /p ) (the value of zN * 0 is insufficient because y^ N+x = 0 ); the forces Fk are determined from the set of linear algebraic equations
-aklk,k+iFk-i+(ak +ak+i)yk+i,k+\Fk -ak+iyk+i,k+2Fk+i=Q> (12)
F0 is given, FN - 0.
The set of Eq. (12) may be considered as the second-order finite differences equation (with the variable coefficients). As is well known [17], the solution of the finite differences equation with the constant coefficients can be constructed explicitly. Hence, the question has been investigated [14] whether Eq. (12) is reduced to the equation with the constant coefficients. The following theorem has been proved.
Theorem 5. Let A> 0, b> 1/2, \-b <A<27b/(\ + 4B)3 , 3/(l + 4£)<f3iV <1,
/] > 0, a, > 0 ; then the formulae
p* =3/(1+ 25/(1-2V(pli(3-pfc+i)l k = N-1,...,1; a, =(3-p,)/2B$k -1;
k-1 k-\
h =/iF[Pp ; ak =aiîïa/> (13)
p=1 p=s
determine the non-increasing sequence of lengths lk >0 and the sequence of flexibilities ak>0 for which Eq. (12) has the form -BFk_x +Fk - AFk+l =0. The solution of this equation (under the condition FN = 0 ) is
'B}k/2 sh(N-kfe
Ft =
lF0,
where Ç = ArthVl - 4ab
The inequalities (11) for the lengths and flexibilities determined by Eq. (13) take the
form
sh(/V k-1)| < (B-ljßL,(3-ß,+1)+ 2A
(14)
It could be shown that if A = 275/(1 + 4B)3 and ßiV=3/(l + 45) then ß^ = 3/(l + 45) = ß. In this case the inequalities (14) take the form cp(ß) > N-k, where
cp(ß) = Arth Vl-ß/4/Arth (l-ßX/l-ß/4
These inequalities hold simultaneously (hence,
the pointwise contact takes place) if and only if the first of them ( k = I ) holds.
For the elastic element with the curved leaves ( <p(x) # 0 ) for N =2, ak(x) = ak = const, k = 1,2 two sufficient conditions for the pointwise contact have been obtained [12-13]. The following theorem has been proved.
Theorem 6. If (i) q(x) s 0 for 0 < x < l2 or (ii) c} > 0,c2 > 0, where
h h c\ ~ Fg(0J2)~ jg(0,x)q(x)dx, c2 = F sin cp (l2)~ |sincp(x)^(x)Jx, 0 0
F =
1
W2J2)
Jy (l2,x)q{x)dx,
then the solution /, (x) of Problem 1 in the aforementioned special case has the form
/iO) = '
öj
-Fô(x-/2)
ax +a2
It should be noted that the conditions (i) and (ii) of Theorem 6 are not necessary for the pointwise contact. In fact, if we consider cp(x) = 0,
q(x) = F0S(x - /,) + 2(/5 //2 - l)F0S(x -12/2) then the condition (ii) of Theorem 2 holds
(hence, the pointwise contact takes place), but the condition (i) of Theorem 6 does not hold; hence, the condition (i) is not necessary. It is also easy to construct the example of load for which the condition (i) of Theorem 6 holds, but the condition (ii) of the same theorem does not hold; hence, the latter condition is also not necessary.
For the elastic element with the non-uniform leaves (ak(x) £ const) for N = 2,
(p(x) = 0, q(x) = F08(x-ll) the sufficient condition for the pointwise contact has been obtained. For N = 2, (p(x) = 0, l\=l2 the sufficient condition for the close fitting has been obtained. The following theorems have been proved (they have not been published earlier).
Theorem 7. If N = 2, <p(x) s 0, al (x)/a2 (x) is the non-decreasing function for 0 <x<l2, q(x) = F0 5(x — ) then the solution /,(x) of Problem 1 has the form /j (x) = F 5(x -12), where
W M G1(/2,/2)+G2(/2,/2)-
Theorem 8. If N = 2, (p(x) s0, /] =l2 = /,
F - ■
/M=
ax(x)+ a2(x) J
c )q(s)ds
>0
for 0 < x < / then the solution /, (x) of Problem 1 has the form (x) = /(x).
The problem of the elastic element optimum design
Consider arbitrary N, cp(x) = 0, ak(x) = ak = const (hence, hk(x) = hk = const),
1 < k < N, q{x) = F0 5(x -/¡). The value A = yx (lx) may be called the deflection of the
elastic element. Since the problem of bending is linear, A = A Fq; we call coefficient A the
flexibility of the elastic element. It is known from the theory of beams bending [15] that the maximum modulus of the normal stress in the leaf cross section with the coordinate x is
/ \ Ehk ", v|
yk (4-z I
Consider the value
a= max max ok(x).
\<k<N0<x<ek
Since the problem of bending is linear, a = BFq ; we call coefficient B the stress degree of the elastic element. The reduction of B should be striven for (it leads to the increase of strength and durability of the prosthesis), but lx and A cannot be changed (lx is the prescribed prosthesis size, A is defined by the prosthesis designer or is chosen as the patient wishes). Besides, there are some natural restrictions: the leaves lengths and thicknesses should be positive and the sequence of lengths should be non-increasing one. Finally, we come to the following optimization problem.
Problem 2. Under the aforementioned natural restrictions, it is required to find the leaves lengths l2,...,lN and the leaves thicknesses hx,...,hN, which minimize the stress degree B, the flexibility A being constant.
For N = 2 Problem 2 has been solved analytically according to the following plan [11]. It follows from Theorem 2 that in this case the pointwise contact takes place between the leaves. Thus, the shapes of the leaves under bending and, hence, the flexibility and the stress degree of the elastic element can be easily found as the functions of three variables l2,hx,h2. The condition of the flexibility constancy (i.e. the dependence between the variables) is taken into account by the representation of the stress degree as a function of two new independent variables. Three initial variables are expressed in terms of two new ones, which belong to the bounded (due to the natural restrictions) area. Since the stress degree contains taking of maximum, it represents different analytical expressions in the different sub-areas of the aforementioned bounded area. It turns out that there are three such sub-areas. Thus, Problem 2 is reduced to determining the stress degree minimum point coordinates in the given bounded area. These coordinates are determined by means of a standard method. First, it turns out that there are no internal local minima in three aforementioned sub-areas. Then the minima at the area bound and at the bounds between the sub-areas are determined. These bounds consist of sections (curves) corresponding to different analytical expressions for the minimized function (of one variable). It turns out that there are eight such sections. Finally, two coordinates of the point corresponding to the lowest value of all the aforementioned function values are determined. As it has been mentioned, the optimum values of three initial parameters are expressed in terms of these two coordinates. The result of the described above calculations is as follows. The minimum sought for corresponds to the unique set of variables. The optimum ratio of the leaves lengths is 4/13, and the optimum ratio of their thicknesses is 2/3.
After Problem 2 had been solved for N = 2, it was noticed that the determined optimum structure was equal-stressed, i.e. the values ok(x) did not depend on k and x for
0 < x < lk+x. This fact formes the basis of the euristic method for solving Problem 2 for
arbitrary N. Such method is necessary because the generalization of the plan described above for the case of arbitrary N is rather difficult due to both complicated structure of the sub-areas
of different minimized function representations and the lack of explicit solution of Problem 1 in this case.
The developed euristic method is as follows. It is assumed that there is a pointwise contact between the leaves and that the structure is equal-stressed. On these assumptions the parameters 12,—Jn anc* hx,...,hN can be determined. Further, the condition of Theorem 4 is
verified for the determined parameters. It has been turned out that this condition holds (this fact is not trivial), i.e. the assumption of the point contact is non-contradictory. Finally, the determined parameters are considered to give the (euristic) solution of Problem 2. This solution is as follows. Consider the values ak, which are defined recurrently:
o.N = 0, a;
l + 9(l-a,+1)/2 + 2a
lk+1
2
a/t+l
r
Then
k-l
h
p=l
Ы =7,3
EwA
2-3af+a?), hk = hxf[{l-{l-a p} ft.
p= 2
We give the numerical example. If TV = 3 then
1 - 20164/,
2 39937
1»
I = 80656 3 519181
1'
hx =3
271186700940324
/
h =142 ,
63698076029953 ^Â ' 2 169 1 ' 507 It is theoretically possible that there exist the parameters corresponding to the lower value of the stress degree B. In fact, such parameters are unknown. The parameters determined above are admissible in practice, and they sufficiently reduce the stress degree compared with that of the available elastic elements of the foot prostheses.
l > "3
Ы =
284
1 •
Conclusions
The developed approach to the formulation and solution of the problems of elastic element bending and optimum design allows one to investigate some bending patterns and to determine the optimum parameters of the elastic element. The suggested methods of the foot prosthesis elastic element calculation can be used to provide a patient with the individual prosthesis possessing the prescribed flexibility and a long life. The optimum elastic element contains five leaves made of carbon-filled plastic KMY-4J1 and, according to the calculations, can be operated for three years.
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МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ИЗГИБА УПРУГОГО ЭЛЕМЕНТА ПРОТЕЗА СТОПЫ
М.А. Осипенко, Ю.И, Няшин, Р.Н. Рудаков, A.B. Останин, E.H. Кулешова, Т.Н. Журавлева (Пермь, Россия)
Дан обзор результатов по постановке и решению задач изгиба и задачи оптимизации конструкции упругого элемента протеза стопы. Протез моделируется в виде консольно закрепленной многолистовой рессоры с искривленными в естественном состоянии листами. Между листами имеет место контакт с возможным отставанием; задача изгиба поэтому относится к контактным задачам теории балок. Предложен новый подход к постановке и решению таких задач; доказана достаточно общая теорема единственности; построены аналитически некоторые частные решения. Приведена постановка задачи минимизации напряжений в упругом элементе; построено решение этой задачи для двухлистового элемента и предложен эвристический метод решения для многолистового элемента. Предложенный подход к постановке и решению задач изгиба и оптимизации конструкции упругого элемента протеза стопы позволил изучить ряд картин изгиба и найти оптимальные параметры упругого элемента. Приведенная методика расчета упругого элемента протеза стопы может быть использована для обеспечения пациента индивидуальным протезом, имеющим необходимую податливость и обладающим длительным сроком эксплуатации. Оптимальный упругий элемент, содержащий пять пластин из
углепластика КМУ-4Л, в соответствии с расчетами, может эксплуатироваться в течение трех лет. Библ. 17.
Ключевые слова: протез стопы, упругий элемент, листовая рессора, искривленные листы, слабый изгиб, область контакта, минимизация напряжений
Received 15 April 2001
