Oar Loom OptimizationCEE 201L. Uncertainty, Design, and Optimization

Department of Civil and Environmental EngineeringDuke University

Henri P. GavinSpring, 2015

1 Description

An oar loom is a carbon-fiber beam with a tapered circular tube cross section. Oar loomsare fabricated by wrapping carbon-fiber tape, pre-impregnated with epoxy resin, around asteel mandrel and baking/curing the epoxy in an oven. The oar is finished by attaching ahandle on the wide end, a blade to the narrow end, and a sleeve and button at the oar-lockposition.

Figure 1. An oar and a scull. http://www.vermontc2.com/es/remos-dreissigacker-concept2/

The loom is designed to:

• transfer power generated by the rower and the resistance of the water into the oarlockwithout flexing too much or breaking;

• give minimal inertial resistance when pushed from the extraction of the blade towardsthe stern to take the next stroke;

• be balanced so that the blade rests lightly on the water without upward or downwardforce applied to the handle.

The oar must be very (very) safe with respect to its structural strength and the rower’sstrength; no-one knows how powerful a rower can be. In this analysis, a large factor-of-safetyis used in assessing the safety with regards to structural strength and athlete strength.

In order to protect rowers from back injuries, the loom must not be too stiff. Thestiffness of the loom is quantified in terms of the bending deflection of the blade when loadedwith a mass of 10 kg (98 N).1

1Concept 2, stiffness options, http://www.concept2.com/oars/oar-options/shafts/stiffness

2 CEE 201L. Uncertainty, Design, and Optimization – Duke University – Spring 2015 – H.P. Gavin

The location of the button is determined by the gear-ratio. The gear ratio can bedefined as the ratio of the length from the tip of the handle to the tip of the blade to thelength from the tip of the handle to the button, G = L/Li. Gear-ratio used at the 2006 worldchampionships (Eton, UK) were 3.27-3.32 for the M8+ boat class and 3.23-3.28 for the W8+boat class.2

2 Optimization-Oriented Analysis

An oar loom can be analyzed and optimized for it’s strength, stiffness, and mass (inertia)characteristics. To do so, it is convenient to parameterize the radius of the loom as a functionof the distance from the handle. In the optimization described in these notes, the oar loommay taper non-linearly from the handle to the blade. The inner loom radius, r(x), is expressedas a function of the distance x from the handle and is parameterized in terms of:

• the inner radius at the handle, r0;• the inner radius at the blade, rL; and• a positive-valued exponent that determines how non-linearly the radius tapers from the

handle to the blade, n.

r(x; r0, rL, n) = r0 + (rL − r0)(x/L)n

If n = 1 the taper is linear. In this optimization, the thickness of the tube wall, t, is constantover the length of the loom. Since the loom radius tapers, the cross sectional area A and thebending moment of inertia I also decrease from x = 0 to x = L. The cross section area A(x)and the bending moment of inertia I(x) can be derived from r(x) and t.

A(x) = π [(r(x) + t)2 − (r(x))2]

I(x) = (π/4) [(r(x) + t)4 − (r(x))4]

From these expressions the mass of the oar including a blade mass, Mblade can be derived,

m = Mblade + ρ∫ L

0A(x) dx

The rotational inertia of the shaft and blade about the oar-lock is

J =∫ L

0((x− Li)2ρA(x))dx+Mblade(L− Li)2

This rotational inertia resists the rower in swinging the oar from the “finish” position (whenthe blade comes out of the water) to the “catch” position (when the blade enters the water).

2Valery Kleshnev, Rowing Biomechanics Newsletter, vol. 6, no. 68 (2006). .http://www.biorow.com/rbn en 2006 files/2006rowbiomnews11.pdf

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Oar Loom Optimization 3

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r(x), A(x), I(x)

target flexibility

∆ = 4cm

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r(x) = r(0) + [ r(L)−r(0) ] (x/L)

x r(x)

nθ = 2.5

F=98N

Figure 2. Geometry, loading, deformation, and shape of an oar loom.

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4 CEE 201L. Uncertainty, Design, and Optimization – Duke University – Spring 2015 – H.P. Gavin

The loom is loaded with a point forces at the handle, at the oar-lock, and at the blade.The bending moment diagram therefore has a triangular shape, as shown in Figure 2. Totest the flexibility of the loom, the loom is held fixed at the oar-lock and at the handle, andis loaded with a 10 kg mass at the blade. With this loading, the bending moments withinthe loom, M(x), vary linearly from 0 at the ends to a value of −(L − Li)(98) N.cm at theoar-lock, as shown in Figure 2.

M(x) ={

−Px 0 ≤ x ≤ Li

−Px+R(x− Li) Li ≤ x ≤ L

The bending stress depends on M(x), r(x), and I(x),

σ(x; r0, rL, n, L, t) = M(x) (r(x) + t)I(x)

So, since the loom has a taper, the bending stress, σ does not vary linearly along the loom(as the bending moment does). Note that the maximum bending stress is not necessarily atthe oar-lock (the location where the moment is maximum).

From the principle of real work, the external work Wext of the end load increasing slowlyfrom 0 to F through displacements increasing proportionally from 0 to ∆ equals the internalstrain energy stored in the loom, Uint. The tip displacement can be computed from

∆(r0, rL, n, L, t) = 1F

∫ L

0

(M(x))2

EI(x) dx

3 Optimization Problem Statement

In this exercize the optimization objective is to minimize the rotational inertia of theloom and blade about the oar-lock such that the loom is not too flexible, and such that thestresses are significantly less than the expected stresses that the oar loom would be subjectedto. In other words,

min J = f(r0, rL, n, L, t) =∫ L

0((x− Li)2ρA(x))dx+Mblade(L− Li)2

such that∆(r0, rL, n, L, t) ≤ ∆target

φL ·max |σ(r0, rL, n, L, t)| ≤ Sfail

The set of design parameters is [r0, rL, n, L, t]. Reasonably, these values should lie withinspecific ranges:

p r0 rL n L t

cm cm - cm cmpmin 1.5 0.1 0.5 370 0.1pmax 3.0 2.0 6.0 380 1.0

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Oar Loom Optimization 5

The desired deflection, ∆target, is 4 cm. The safety factor for the constraints on stresses, φL,is set to 5. The mass of the blade, Mblade is set to 1.3 kg. Typical carbon fiber compositeproperties are: mass density ρ = 0.0019 kg/cm3, modulus of elasticity E = 2 × 107 N/cm2,and ultimate tensile strength Sfail = 3.8× 104 N/cm2.

The integrals for J and ∆ can not be solved easily with pencil & paper. But approxi-mating the integrals numerically (for example, with the trapezoidal rule) is easy.

4 Optimized SolutionThis optimization problem was solved using sequential quadratic programming. Nu-

merical results are listed in the tables below and the properties of the optimized oar loomare plotted in Figure 3.

Table 1. Optimal oar loom design parameters.r0 3.0 cm optimal radius at x = 0 (max. limit)rL 0.1487 cm optimal radius at x = L

n 5.5582 - optimal exponentL 370 cm optimal length (min. limit)t 0.1300 cm optimal tube wall thicknessJ 1.05× 105 kg.cm2 minimum rotational inertia

Table 2. Constraint values and Lagrange multipliers at the optimal solution.g1 = 0.000817 . . . deflection active λ1 = 4301.8g2 = -4.1056 . . . stress not active λ2 = 0

• The objective function is sensitive to changes to deflection constraint, the other con-straints do not affect the optimal solution. If the flexibility constraint is met, thebending stresses are just within acceptable limits (1/5-th of the tensile strength).• The radius at the handle optimized to its maximum limit, and the length optimized to

its minimum limit.• The initial design [r0 = 2.5 cm; rL = 1.5 cm; n = 1.0; L = 376 cm; t = 0.5 cm;] is not

feasible and has an objective that is 58% larger than the optimized value of J .• The oar length optimized to the lower limit of 370 cm. This is the logical result of

optimizing for the minimum swing weight. But other considerations, (such as boatspeed, for instance) are used to determine the oar length. Oar lengths measured at the2006 world championships were between 375.1 cm and 377.5 cm for M8+ crews andbetween 371.5 and 375.5 for W8+ crews.3• Oar loom diameters could alternatively be optimized for other critera, such as wind

resistance.3Valery Kleshnev, Rowing Biomechanics Newsletter, vol. 6, no. 68 (2006). .

http://www.biorow.com/rbn en 2006 files/2006rowbiomnews11.pdf

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6 CEE 201L. Uncertainty, Design, and Optimization – Duke University – Spring 2015 – H.P. Gavin

1 function [Inertia , constraints ,mass ,delta ,d_Diag ,s_Diag ,stress ,x] = oar_analysis (param ,c)2

3 rho = c(1); % d e n s i t y o f carbon−f i b e r tube , kg/cmˆ34 G = c(2); % gear−r a t i o o f the oar , L / Li5 E = c(3); % modulus o f e l a s t i c i t y , carbon−f i b e r , N/ sq .cm6 Mb = c(4); % mass o f b lade , kg7 delta_target = c(5); % a l l o w a b l e d e f l e c t i o n at b lade t ip , cm8 S_fail = c(6); % t e n s i l e s t r e n g t h o f carbon f i b e r composite N/ sq .cm9 F = c(7); % weight o f a 10 kg load at end o f b lade , N

10

11

12 r0 = param (1); % radius o f loom near the handle . . . approx 2.5 cm13 rL = param (2); % radius o f loom near the b lade . . . approx 1.5 cm14 n = param (3); % exponent15 L = param (4); % l e n g t h o f oar16 t = param (5); % tube−t h i c k n e s s o f oar loom ( constant )17

18

19 Li = L / G; % ‘ ‘ in−board , ’ ’ cm20

21 R = F * L/Li; % r e a c t i o n f o r c e on the pin22 P = R - F; % f o r c e on the handle ( p u l l i n g f o r c e )23

24 dx = Li /100; % de l ta−x incrementa l l e n g t h25 x = [0: dx:L]; % x−a x i s data p o i n t s26

27 r = r0 + (rL - r0) * (x/L).ˆn; % v a r i a b l e rad ius o f the loom , r ( x )28

29 M = -P*x + R*(x-Li ).*(x>Li ); % bending moment , M( x )30

31 A = pi * ( (r+t).ˆ2 - r.ˆ2 ); % cross−s e c t i o n a l area32

33 I = (pi /4) * ( (r+t).ˆ4 - r.ˆ4 ); % moment o f area34

35 delta = (1/F) * sum ( M.ˆ2 ./ (E*I) )* dx; % d e f l e c t i o n o f b l ade36

37 Inertia = sum ( (x-Li ).ˆ2 .* A ) * rho * dx + Mb * (L-Li )ˆ2 ;38 % po lar moment o f i n e r t i a o f oar+b lade about pin39

40 mass = sum(A) * rho * dx + Mb; % mass o f the loom + b lade41

42 stress = M .* (r+t) ./ I ; % bending s t r e s s43 s_Diag = cumtrapz ( M ./ (E*I) ) * dx ; % r o t a t i o n s44 d_Diag = cumtrapz ( cumtrapz ( M ./ (E*I) ) ) * dx ˆ2 ; % d e f l e c t i o n s45 s_Diag = s_Diag - d_Diag (100) / x (100) ; % zero d e f l e c t i o n at pin46 d_Diag = d_Diag - d_Diag (100)* x / x (100); % zero d e f l e c t i o n at pin47

48 constraints = [ delta - delta_target ; % loom d e f l e c t i o n l i m i t49 5.0 * max(- stress ) - S_fail ]; % s t r e n g t h l i m i t50

51 % −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− o a r a n a l y s i s

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Oar Loom Optimization 7

1 % oar opt .m2 % s c r i p t .m− f i l e to determine the opt imal loom des ign which minimizes the3 % i n e r t i a o f the oar loom such t h a t i t doesn ’ t bend too much and doesn ’ t break4 % and i s ba lanced in the oar−l o c k5

6 % H.P. Gavin , Duke Univers i ty , 27 Jan . 2000 , 4 Feb . 20157

8

9 % dens i ty , modulus , mass , and load ing f o r an oar loom10 rho = 0.0019; % d e n s i t y o f carbon−f i b e r tube , kg/cmˆ311 G = 3.3; % gear−r a t i o o f the oar , L / Li12 E = 2e7; % modulus o f e l a s t i c i t y , carbon−f i b e r , N/ sq .cm13 Mb = 1.3; % mass o f b lade , kg14 delta_target = 4.0; % t a r g e t d e f l e c t i o n at b lade t ip , cm15 S_fail = 3.8 e4; % t e n s i l e s t r e n g t h o f carbon f i b e r composite , N/ sq .cm16 F = 98; % weight o f 10 kg . . . a t end o f b lade , N17

18 c = [ rho , G, E, Mb , delta_target , S_fail , F ]; % a l l cons tant s in a v e c t o r19

20 % −−− i n i t i a l guess f o r loom des ign r0 , rL , n , L, and t21 r0 = 2.5; % radius o f loom at x=022 rL = 1.5; % radius o f loom at x=L23 n = 1.0; % e x p o n e n t i a l v a r i a t i o n o f r24 L = 376; % l e n g t h o f the oar , 366 to 384 cm25 t = 0.5; % t h i c k n e s s o f the loom−tube wal l , cm26

27 param_init = [ r0 rL n L t ]; % v a l u e s o f the des ign parameters28 param_min = [ 1.5 0.1 0.5 370 0.10 ]; % minimum v a l u e s f o r des ign params29 param_max = [ 3.0 2.0 6.0 380 1.00 ]; % maximum v a l u e s f o r des ign params30

31 options (1) = 1; % d i s p l a y convergence r e s u l t s32 options (2) = 1e -3; % t o l on ”X”33 options (3) = 100; % t o l on ”F”34 options (4) = 1e -3; % t o l on ”G”35 options (5) = 500; % maximum i t e r a t i o n s36 options (6) = 1e5; % pena l t y on u n s a t i s f i e d c o n s t r a i n t s37

38 % a l g o r i t h m i c parameters . . .39 % d i s p l a y tolX to lF tolG MaxEvals Penal ty Exponent40 options = [ 1 0.001 0.01 0.001 500 1e5 1.0 ];41

42 [param_opt , I_opt , g_opt , cvg_hst , Hess , lambda ] ...43 = SQPopt ( ’oar_analysis ’, param_init , param_min , param_max , options , c );44

45 % e v a l u a t e the opt imal des ign46

47 r0 = param_opt (1)48 rL = param_opt (2)49 n = param_opt (3)50 L = param_opt (4)51 t = param_opt (5)52

53 lambda54

55 % p opt = [ 2.9827 0.39813 2.3921 370 0.2 ] ;56 [Inertia , constraints ,mass ,delta ,d_Diag ,s_Diag ,stress ,x] = oar_analysis (param_opt ,c);57

58 r = r0 + (rL - r0 )*(x/L).ˆn; % loom inner radius , r , as a func t i on o f x59

60

61 %

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8 CEE 201L. Uncertainty, Design, and Optimization – Duke University – Spring 2015 – H.P. Gavin

62

63 % −−−−−−−−−−−−−−−−− outer rad ius data measured from a commercial oar in 200064 rc = [ 0+53 19.45 /(2* pi) ;65 20+53 19.20 /(2* pi) ;66 40+53 18.61 /(2* pi) ;67 54+53 18.29 /(2* pi) ;68 70+53 17.61 /(2* pi) ;69 90+53 17.39 /(2* pi) ;70 110+53 16.80 /(2* pi) ;71 130+53 16.30 /(2* pi) ;72 150+53 15.71 /(2* pi) ;73 170+53 15.3 /(2* pi) ;74 190+53 14.7 /(2* pi) ;75 210+53 14.2 /(2* pi) ;76 230+53 13.75 /(2* pi) ;77 250+53 13.35 /(2* pi) ;78 258+53 12.35 /(2* pi) ];79

80 % −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−81

82 % −−−−−−−−− Pl o t s83 %e p s P l o t s = 0; i f epsPlots , formatPlot (24 ,15 ,5) ; e l s e formatPlot ( 0 ) ; end84 figure (1);85 c l f86 subplot(4 ,1 ,1) % p l o t the d e f l e c t e d shape o f the oar loom87 plot (x, s_Diag *180/ pi)88 ylabel (’loom rotations ( degrees )’)89

90 subplot(4 ,1 ,2) % p l o t the d e f l e c t e d shape o f the oar loom91 plot (x, d_Diag )92 ylabel (’loom deflection (cm)’)93

94 subplot(4 ,1 ,3) % p l o t the shape o f the opt imal des ign95 plot (x,r,’-b’,x,-r,’-b’,rc (: ,1) , rc (: ,2) , ’or ’,rc(:,1),-rc (: ,2) , ’or ’)96 ylabel (’optimal radius of loom (cm)’)97

98 subplot(4 ,1 ,4) % p l o t the bending s t r e s s99 plot (x,abs( stress )/1000)

100 ylabel (’bending stress (kN/cm ˆ2) ’)101 xlabel (’x-axis (cm)’)102

103 % i f epsPlo ts , p r i n t ( ’ oar opt . eps ’ , ’− co lor ’ , ’− FHelve t ica :8 ’ , ’−S600 , 9 2 3 ’ ) ; end104

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Oar Loom Optimization 9

124 % −−−−−−−−−−−−−−−−−−−−−−−−−−− EXAMPLE OF RESULTS −−−−−−−−−−−−−−−−−−−−−−−−125

126 % ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SQPopt ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗127 % i t e r a t i o n = 25 ∗∗∗ f e a s i b l e ∗∗∗128 % func t ion e v a l u a t i o n s = 211129 % o b j e c t i v e = 1.046 e+05130 % parameters = 3.000 e+00 1.487 e−01 5.556 e+00 3.700 e+02 1.300 e−01131 % max c o n s t r a i n t = 8.174 e−04 (1)132 % Step S i ze = 5.000 e−01133 % BFGS method : r e g u l a r Hessian update .134 % QP method : ok135 % X− convergence : 3.437 e−04136 % F− convergence : 1.260 e−05137 % ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ SQPopt ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗138 % ∗∗∗ Woo−Hoo ! Converged s o l u t i o n found f o r X and F!139 % ∗∗∗ Woo−Hoo ! Converged s o l u t i o n i s f e a s i b l e !140 % ∗∗∗ Optimal Parameters :141 % 2.99989 0.14870 5.55582 370.00130 0.13004142 % ∗∗∗ O b j e c t i v e Function :143 % 1.0457 e+05144 % ∗∗∗ Constra in ts :145 % 8.1736 e−04 −4.1056e+00 −3.3339e−01 −2.6666e+00146 % ∗∗∗ Active Constra in t s :147 % 1148

149

150 % g (1) = 8.1736 e−04 . . . d e f l e c t i o n ∗ a c t i v e ∗ lambda (1) = 4301.3151 % g (2) = −4.1056e−10 . . . s t r e s s ∗not a c t i v e ∗ lambda (2) = 0152 % g (3) = −0.3334e−01 . . . ba lance lambda (3) = 0153 % g (4) = −2.6666e+00 . . . ba lance lambda (4) = 0154 %155 % mass = 3.3639 e+00156 % d e l t a = 4.0000 e+00157 %158 % min( s t r e s s ) = −7.6000e+03

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10 CEE 201L. Uncertainty, Design, and Optimization – Duke University – Spring 2015 – H.P. Gavin

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Figure 3. Result of the oar-loom optimization, top to bottom: v′(x), bending rotation of theloom; v(x), bending displacement of the loom (note v(L) = 4.0 cm.; optimized loom radius(red dots show radii measreud from a comercial oar in 2000; maximum bending stress acrossthe loom (note the maximum is at the blade connection).

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Oar Loom Optimization 11

Figure 4. An early carbon-fiber oar loom strength test. .http://www.concept2.com/oars/oar-options/shafts/

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