Mass Efficiency in Mechanical Design

Transcription

1 Proceedings of the World Congress on Engineering 008 Vol II WCE 008, July - 4, 008, London, U.K. Mass Efficiency in Mechanical Design Subbiah Raalinga Abstract Using the axiu strain energy density in a linear echanical eleent stressed in tension as the ass efficiency reference, a direct ethod for deterining the ass efficiencies in transversely loaded and in torsionally stressed echanical eleents is developed and presented. It is shown that this ethod also allows the deterination of relative contribution of each geoetric feature in a cross section to ass efficiency of the eleent. Coon loading conditions and structural geoetries are analyzed and it is shown that the ass efficiencies realized in practice offer great any opportunities for iproveent. It is suggested that ore efficient use of aterials and energy is feasible by paying closer attention to ass efficiency considerations during echanical design. Index Ters Mass and voluetric efficiency, Sustainable and green design exaples. Two- and three-diensional trusses and fraes, and stiffened panels and shells used for airfraes and in spacecrafts are exaples where axial copression is efficiently handled. This need is still largely unfulfilled in both transverselyand torsionally loaded eleents and structures. Mass or voluetric efficiency easures are used in this paper to evaluate the solutions now in use to withstand transverse and torsional loads. Maxiu strain energy storable in a given aount of aterial is taken as the reference easure. With this reference easure, coon transversely loaded and torsionally loaded designs and coponents are evaluated to assess relative efficiency in the use of aterials and energy. It is shown that ost coon designs offer significant opportunities for iproveents. I. INTRODUCTION Global consuption of engineering aterials in the early years of the 1st Century is estiated at kg per year [1]. Since nearly all engineered products and systes are ipleented with aterials drawn fro non-renewable resources it will be difficult to sustain this level of aterial consuption and its likely growth in the coing years. Ebedded energy in the aterial, energy requireents to process the aterials into useful fors and the use energy of engineered systes also ake ajor deands on available energy resources. Efficient use of aterials is therefore a fundaental requireent to sustain the global counity and its rapidly iproving living standards. By replacing gaseous electronics with solid state electronics, ajor econoies have been achieved both in aterial and energy consuption in coputer, control and counication industries. Coparable progress has also been achieved in a few segents of transportation industries and in soe parts of civil construction industry. Siilar progress is now necessary in industries related to consuer durables and individual/group obility. Many of the coponents and structures used in engineered products and systes are priarily linear, i.e., their linear diensions are uch greater than their cross sectional diensions. In such structures, four distinct loadings can be identified - axial tension, axial copression, torsion and transverse loading. Major progress has been ade in design when linear structures and eleents are subjected to axial loads either in tension or in copression. Many efficient designs can be cited to withstand tensile loading. Suspension bridges and various fors of cable-stayed bridges are aong II. REFERENCE MEASURE FOR MASS EFFICIENCY Structures and eleents subject to tensile loads offer no difficulty in developing efficient designs since it is only necessary to distribute the aterial and the applied load uniforly across the load-carrying section so that yielding or plastic collapse can occur everywhere at the sae tie when a liiting stress is reached. Mass efficiency can then be defined in ters of strain energy density, i.e., strain energy per unit volue U o in J, perissible in a structure/eleent subject to pure tensile loading. If the axiu perissible design stress is σ, σ < σ y where σ y is the yield stress, the strain energy density in uniaxial tensile loading U o is a reasonable reference easure for ass efficiency. With E representing the elastic odulus, the reference easure for ass efficiency can now be taken as σ Uo = (1) E III. MASS EFFICIENCY OF TRANSVERSELY LOADED STRUCTURES Unlike uniaxial tensile loading where the displaceent under load is proportional to length, displaceent is proportional to the third power of length in transverse loading. Larger elastic displaceents and larger stresses, and a non-unifor stress distribution across the cross section occur. One way to iprove ass efficiency is to replace transverse loading with axial loading when feasible, as in trusses. Although this and other epirical principles have been developed to handle transverse loading, quantitative easures are rarely used to evaluate efficient use of aterials and energy for the solutions advanced. In this work, it is proposed that strain energy density Manuscript received January 17, 008. Subbiah Raalinga is with the Institute of Technology, University of Minnesota, MN USA (phone: ; fax: ; e-ail: ra@ e.un.edu). ISBN: WCE 008

2 Proceedings of the World Congress on Engineering 008 Vol II WCE 008, July - 4, 008, London, U.K. Figure 1. A cylindrical shaft or bea subject to an end oent M. ratio η = U/U o, with the strain energy density U o in uniaxial tensile loading as the ass efficiency reference, is an appropriate easure for the objective evaluation for the ass efficiency η of a design solution. Green design is facilitated by requiring high ass efficiency both fro the aterial and the energy conservation perspective. A. Cylindrical Shaft Subject To Pure Bending Consider a cylindrical shaft/bea of radius R and length l subjected to an end oent M, shown in Figure 1. According to Euler-Bernoulli bea theory, the state of stress in the bea is planar. And the bending stress σ x generated by the oent M applied is solely a function of y, i.e., the distance fro the neutral axis. For the strongest bea with axiu perissible design stress ± σ and oent of inertia I, the bending stress at any point between the two ends, with the axis syste shown, is given by: y R σx = M and σ = M () y so that, σx = σ (3) R It is enough to consider the first quadrant of one-half of the cylindrical volue of length l to calculate strain energy density due to proble syetry with respect to y = 0, z = 0 and x = 1. For a volue V, the strain energy density U due to the oent M is: R R 1 σ x 1 σ y dy = dy (4) V V E 0 E 0 R Using polar coordinates, a volue eleent r dr dθ dx, and a volue V = (4/π R l) for 1/8th of the cylinder, strain energy density is: l R / 4 σ π r sin θ r dr dθdx (5) R l E π R Integrating and applying liits, strain energy density U is: 1 σ (6) 4 E Mass efficiency η of the strongest cylindrical bea subjected to the end oent M is calculated by dividing strain energy density given by (6) with the strain energy density reference easure given by (1). Mass efficiency of the strongest cylindrical bea stressed by an end oent is deterined to be η = U/U o = 1/4 or 5.0 % Yielding failure will ensue due to the non-unifor stress distribution generated by the oent applied, if the strain energy density exceeds 5.0 % of the axiu possible within the shaft or bea volue. Non-unifor stress distribution iposed leads to inefficient use of aterial. B. A Square Shaft/Bea Subject To Pure Bending Consider a square bea, h x h in cross section, and of Figure. A Square bea of length l subject to a oent M. length l subjected to an end oent M shown in Figure. Bending stress σ x generated is a function of y, as in the case of the cylindrical bea. In the strongest bea with axiu design stress ± σ and oent of inertia I, the bending stress σ x at a point between the two ends is: y h σx = M and σ = M (7) y so that, σx = σ (8) h It is enough to consider 1/4 of the square bea volue to calculate strain energy density due to proble syetry with respect to y = 0 and x = l. Considering a unit length and a unit width of the bea and calculating as before, strain energy density U is: 1 σ h y 1 σ dy = (9) h E 0 h 3 E Mass efficiency η of the strongest square bea subjected to the end oent M is η = U/U o = 1/3 or %. The higher ass efficiency in the square bea, when copared with the cylindrical bea, is due to better distribution of load carrying aterial along the y direction. Still, the non-unifor stress distribution iposed, leads to inefficient use of aterial. C. A Rectangular Bea Subject To Pure Bending Consider a rectangular bea of height h and unit width along z and of length l along x, subjected to an end oent M, as in the case of the square bea of the previous section. Bending stress and strain energy density equations for this case are identical to those shown in (7) to (9). Mass efficiency η of the strongest rectangular bea subjected to the end oent M is therefore the sae as that for a square bea, i.e., η = U/U o = 1/3 or %. It is well known that rectangular beas are preferred over square beas to withstand transverse loads and oents. The notion of a shape factor has been introduced to ephasize the preference for rectangular beas over square beas to carry such loads [], [3]. However, it is seen that in ters of ass efficiency, there is little to choose between a rectangular bea and a square one. D. Mass Efficiency Of A Mid-Span Loaded Rectangular Bea. Consider now a siply-supported rectangular bea of height h, unit width and length l along x with a id span load noral to the longitudinal axis. The bending stress σ x generated is a function of both x and y, so that in the strongest bea, y x y σx = M(x) = σ (10) l h As before, just ¼ of the bea volue is considered to calculate the strain energy density U. ISBN: WCE 008

3 Proceedings of the World Congress on Engineering 008 Vol II WCE 008, July - 4, 008, London, U.K. σ 1 1 l h x y dx dy (11) h l E 00 l h 1 1 σ 1 σ x = (1) 3 3 E 9 E Mass efficiency of the rectangular bea with a id span load is η = U/U o = 1/9 or %. Since the axiu bending oent occurs at id span, ost of the aterial used in the bea is inefficiently deployed for the load carrying function. Since the choice of height and width of the rectangular bea is of no consequence in obtaining (10) and (11), ass efficiencies of square beas and of rectangular flats, i.e., in beas where the bea width >> depth, reain at the sae value. That is, η = U/U o = 1/9 or %. E. Mass efficiencies of cylindrical beas with a id span load With a siilar analysis as in the last section, it is easily shown that the ass efficiency of a id span loaded and siply supported strongest bea of solid circular cross section is η =1/1 or 8.33 %. For an identically loaded tubular bea with a tube wall thickness t << tube diaeter D, ass efficiency can be shown to be twice as large as that for a solid circular bea, i.e., η =1/6 or %. Beas of circular cross section, coonly used in gear boxes, power transission systes and echanical shafting, are less efficient than square beas since ore of the load carrying aterial is placed closer to the neutral axis. In contrast, positioning the aterial away fro the neutral axis, as in tubular shafts, is a ore effective approach to raise ass efficiency. F. Mass Efficiency Of A Rectangular Bea With An Interediate Load Between Supports. Using an identical analysis, it can be shown that the ass efficiency of a rectangular bea subjected to a load noral to the bea span, applied at any location between the two siple supports, is the sae as for the bea with id span load. The sae result, naely η = U/Uo = 1/9 or % also holds for ass efficiencies of square beas and for rectangular flats carrying a noral load applied anywhere between the two supports. For beas/shafts of solid circular cross section, ass efficiency is η =1/1 or 8.33 % when the transverse load is applied any where between its two siple supports. Mass efficiency is twice as large as that for an identically loaded solid circular bea, i.e., η =1/6 or % when the eleent is a (circular) tubular bea. G. Mass efficiency of a rectangular bea with a distributed transverse load. When the transverse load applied to a rectangular bea is a distributed load of constant agnitude per unit length, bending oent rises ore rapidly with position x along bea length. The bending stress generated is a function of both x and y so that for the strongest bea of rectangular cross section, y x y σx = M(x) = σ (13) l h Mass efficiency for this case is deterined by replacing σ x of (10) with σ x of (13) and calculating strain energy density as with (11). Mass efficiency is deterined to be η = 1/15 or 6.66 %. With no change in bea geoetry, application of distributed load is found to greatly reduce ass efficiency when copared with the application of a single id span load. H. Mass efficiency of a cylindrical beas with a distributed load. When the transverse load applied to a circular shaft is a distributed load of constant value per unit length, bending stress σ x and strain energy density U are given by (Fig. 1): x y σx = σ (14) R l 4 σ l x 4 R π/ r sin θ dx r dr dθ πr l E 0 l R (15) 1 σ = 0 E Mass efficiency is η = U/U o = 1/0 or 5.0 %. With a change in cross section fro a rectangle/square/flat to circle, the application of distributed load is found to reduce the bea ass efficiency η fro 1/15 to 1/0. It can be shown that when the solid cylindrical bea is replaced with a tubular bea with a tube wall thickness t << tube diaeter D, ass efficiency η rises fro 1/0 to 1/10. Poor ass efficiency obtained in bending of solid circular shafts leads to reduced critical speeds in siply supported rotating shafts (self-aligning bearings or universal joints), where the self-weight of the rotating shaft represents the uniforly distributed load. This proble is encountered in the design of autootive propeller shafts. Raising the ass efficiency by resorting to tubular propeller shafts is a reedy when higher critical speeds are needed. IV. MASS EFFICIENCY OF TORSIONALLY LOADED ELEMENTS In torsionally loaded eleents, axiu displaceent due to the torsion applied is a function both of length and radius or the largest distance fro the axis of torsion. Since the stresses induced are a function of position with respect to rotational axis, ass efficiency of the torsionally loaded structure is also low. A. A circular rod subjected to a torque T Consider a circular rod of length l and radius R fixed at one end and subjected to a torque T at the other end. Shear stress τ is a function of radius with axiu shear stress τ at r = R. Shear strain energy du in an eleent r dθ dr dz at a distance r fro the torsion axis is τ τ r d r dr dθdz = r dr dθdz (16) G G R so that the strain energy stored per unit length U s is ISBN: WCE 008

4 Proceedings of the World Congress on Engineering 008 Vol II WCE 008, July - 4, 008, London, U.K. 1 R 3 τ π r 1 τ 1 Us = dr dθ = ( π R ) V G 0 0 (17) R π R G 4 Hence, the calculated strain energy density in torsion is 1 τ σ s 0.0 (18) G E The last ter on the right hand side of (18) is obtained by replacing the axiu shear stress τ with the yield stress σ, replacing the shear odulus G with an expression using Young s odulus E and Poisson s ratio ν, and taking ν as 0.3. Mass efficiency of a circular cylinder subject to torsion is deterined to be η = U s / U o 1/5 = 0.0 %. Thus, torsionally stressed achine eleents such as coil springs, echanical and power transission shafting, torsion bars, etc., are found to have ass efficiencies not uch different fro those for transversely loaded echanical eleents. When torsionally stressed eleents are also subjected to bending, ass efficiency attainable becoes even lower. Raising ass efficiency in torsionally stressed structures now requires use of tubular eleents where the torsional ass efficiency η is U s / U o /5 = 40.0 %. It is shown in a following section that this value is uch lower than that possible by geoetric shaping when rectangular sections are subjected to bending! V. GEOMETRY CHANGES TO IMPROVE MASS EFFICIENCY One way to iprove ass efficiency in transversely loaded eleents or structures is to place the load carrying aterial farther away fro the neutral axis in the plane of bending. Higher ass efficiencies can also obtained by placing the available aterial such that the aount of load carrying aterial increases with the increase in bending oent applied. In this instance, aterial is deployed preferentially in the plane of the bea so that a constant stress bea geoetry results - cross section reains rectangular but the bending stress no longer varies with distance along the longitudinal axis. A. Modification of bea width to iprove ass efficiency In a transversely loaded bea with a id span load, if the bea width w is allowed to vary with bending oent, it is possible to ake the axiu bending stress at the upper and lower surfaces of the bea to reain constant. Bending stress σ x is now solely functions of distance y fro neutral axis. Plan view of a rhoboidal bea with linearly varying width is shown in Figure 3 below. Bending stress σ x generated by the id span load applied in this variable-width bea is: y y σx = M = σ (19) h 0 Z Figure 3. Plan view of a 'constant stress' bea of variable width Figure 4. Plan view of a constant stress bea of variable width odified for ease of ounting Bending stress σ x given by (19) is the sae as that in a bea of constant width subject to pure bending. Mass efficiency for the constant stress bea shown in Figure 3 is therefore the sae, i.e., η = U/U o = 1/3 or %. The geoetry shown is representative of leaf springs constructed by slicing the rhobus parallel to x axis into a set of fixed width eleents of variable length, stacking and claping the eleents to produce the failiar wagon and autootive springs [4]. Mounting requireents ake it necessary to odify the variable width bea as shown in Figure 4 (in plan view). Leaf spring structure is now an assebly of one rectangular eleent and a set of sliced, fixed width, eleents of variable length, all claped together. If f 1 and f are the weight fractions of central ounting eleent and that of all the sliced eleents, ass efficiency η of the assebled leaf spring is the weighted su of two ass efficiencies η 1 and η and is given by: η = ( η1f 1 + ηf ) = 100( 0.111f f ) % (0) Mass efficiency of this transversely loaded bea of variable width is a function both of the loading and of the bea cross section (geoetry). It is lower than the theoretical axiu of 33.33%. Autootive suspensions with ass efficiency less than 33% have been in use for over a hundread years even though it is known fro vehicles for ass transport (passanger buses) that pneuatic suspensions provide very uch greater ass efficiencies. B. Changing bea geoetry to iprove ass efficiency It is known that variable depth beas of constant width can be ade to yield constant stress beas uch like the variable width beas [4]. But, constant stress beas, designed whether by varying width or by varying depth, do not sufficiently displace the load carrying aterial away fro the bending axis to raise ass efficiency beyond %. A better geoetry can now be visualized in which all of the bea aterial is disposed within a pair of uniaxially loaded, thin rectangular regions syetrically disposed in the plane of bending but uch farther reoved fro the neutral axis. Such a structure, with a pair of uniaxially stressed flanges A and B, each of thickness t f, shown in Figure 5, is an idealized I-Bea [5] or an open web I-Bea. x ISBN: WCE 008

5 Proceedings of the World Congress on Engineering 008 Vol II WCE 008, July - 4, 008, London, U.K. In an ideal I-Bea subjected to pure bending of the for shown in Figure 1 or, the flange A above the neutral axis at a ean distance h is in unifor copression and flange B, below the neutral axis at the sae ean distance h, is in unifor tension. For the strongest bea with design stress ± σ, as each flange is under a siple uniaxial stress state in copression or (no shear web present). With distributed loading of constant agnitude or of triangular loading with the peak load at id span, attainable best ass efficiency in an ideal I-Bea, again with a flange of constant width and thickness, will be uch lower than %. In all these instances, both the ideal ass efficiencies and the real ass efficiencies in the presence of a shear web are calculable by following the sae steps as those shown for real Figure 5. An Ideal I-Bea subjected to a oent M and its section (left) tension with a stress σ, the strain energy density U is σ (1) E Mass efficiency of an ideal I-bea is therefore seen to be its axiu possible value, i.e., η = 1 = 100 %. An ideal I-Bea with 100% ass efficiency is, of course, unattainable, since a shear web necessary to generate the two different uni-axial flange stress states is not present. C. Mass efficiency of a real I-Bea in pure bending. A real I-Bea of total depth h can now be constructed with a pair of flanges, each of thickness t f, and a singe web of thickness t w. When the I-Bea eets all the requireents of an Euler-Bernoulli bea, the flanges are ainly in uniaxial tension or copression. A variable bending stress state, sae as that in a transversely loaded rectangular bea, prevails in the shear web. Mass efficiency of the bea is then given by the weighted su of ass efficiencies of the flanges and of the rectangular shear web, which is 11.11%. To get an accurate estiate of flange efficiency, define half-depth h f as h f = (h - t f ) and nondiensionalise it with h as h f * = (h f /h). Then, by following the sae steps as for derivation of (7) and (8), it can be shown that without the shear web, the flange efficiency η f of the I-bea (i.e., for a flange width less shear web thickness) is 1 (h * 3 (h * ) 3 f ) 1 η f f = = x 100 % () 3(1 h f ) 3(1 h f ) To derive this expression, bending stress in the flange is allowed to vary as a function of distance fro neutral axis (ideal flange assuption is not used). A siilar expression can also be derived for tubular beas of rectangular cross section. Shear web efficiency η s is already known to be 11.11%. Hence the weighted su of ass efficiencies of the flanges and of the shear web is readily deterined. High ass efficiencies are obtained in pure bending with bea shaping. D. Mass efficiency of an I-Bea under other loading conditions. If the I-Bea were id-span loaded with a single load, bending stress in the flange is a function of position along the span - hence the best ass efficiency attainable in an ideal I-Bea of constant flange width and thickness is 33.33% Figure 6. An I-Bea with variable flange thickness and high ass efficiency I-beas in pure bending. E. Other ethods for iproving ass efficiency As noted earlier, for the id-span loaded I-bea, bending stress is a function of distance x along the span. Varying the flange width offers one eans of raising ass efficiency. It is also possible to vary the flange thickness as a function of position x along the span when rolled I-Beas are used. This is accoplished by plate welding or riveting on to the flanges so that the flange thickness varies fro zero at the bea extreities to its axiu value at id span as shown in Figure 6. Mass efficiency η attained with such flange odifications can be substantially above the ass efficiency liit value of Calculation of the ass efficiency follows the sae steps as in previous exaples. Tapered beas, i.e., beas of variable depth, offer an another eans for raising ass efficiency. Tapered bea approach is accessible for solid and tubular rectangular beas as well as I-beas and castellated I-beas. Reliable welding technology is available to coercially ipleent variable depth or tapered beas. As long as the taper is not too great, Euler-Bernoulli bea odeling is adequate for the analysis of variable depth beas [6]. Nuerical calculations are now necessary to deterine ass efficiency, especially for end-ounted echanical eleents of coplex geoetry such as turbine blades (which are basically cantilevers). Fabricated, variable depth beas in wide coercial use in the for of fabricated etal building structures also require nuerical calculations. Methods outlined here are sufficient for this purpose. VI. SUMMARY AND CONCLUSIONS A direct ethod to deterine the ass efficiencies possible in transversely loaded and in torsionally stressed echanical eleents is presented here. Maxiu strain energy per unit volue stored in a linear echanical eleent stressed in uniaxial tension, is used as the ass efficiency reference. This ethod of deterining the ass efficiency also allows the evaluation of contribution of each geoetric feature in a cross section to ass efficiency of the entire echanical eleent. Coon loading conditions and several ISBN: WCE 008

6 Proceedings of the World Congress on Engineering 008 Vol II WCE 008, July - 4, 008, London, U.K. bea geoetries have been analyzed. Loading odes are frequently not taken into account in odeling for iniu weight design in transversely and torsionally loaded eleents. Taking the loading ode into consideration and evaluating the ass or voluetric efficiency, it is found that the ass efficiencies realized in practice are rather low. In nearly all cases, local axiu stresses generated invariably lead to grossly inefficient use of engineering aterials. Efficient use of available aterials and energy requires significant iproveents in ass efficiency. This requires uch greater attention to ass efficiency considerations during echanical design. If life cycle energy requireent [1] is taken to be coprised of ebodied energy in the aterial H, processing energy H p, the use-energy of energy using products H use (including the energy associated with aintenance and service over the useful life of the product) and the energy of disposal H disp, it is seen that ost echanical designs ake very inefficient use of available energy resources. Superior shaping, preferred geoetric designs and aterials are necessary to ake better use of available aterial and energy resources. Mass efficiency can be significantly raised in transversely loaded echanical eleents by selective placeent of load carrying aterial. Shaped and tapered eleents offer eans of iproving efficient use of aterials. Doing so is feasible using several different cost effective and energy efficient processing ethods. Increasing use of etal buildings, castellated beas, and other structures suggest that this is a viable approach. Significantly raising ass efficiency is not straightforward when a design is doinated by torsion. Large reduction in ass efficiency accopanies torsion when it is accopanied by transverse loading. This condition, cobined torsion and bending, is encountered in any echanical eleents such as shafts used in gear boxes, power transission systes and drive trains. Use of such echanical eleents and structures now calls for closer exaination and rigorous engineering justification. Where efficiency iproveents by shape or geoetric odifications do not appear feasible, sustainable or green design deands adaptation of alternate designs or even alternate product ipleentation technologies. Direct drive, as in robotic systes and hard disk drives (both for read-head actuation using a voice coil driver and for platen-drive with axial field otors) [7] is one option. In general, direct drive systes dispense with gears/drive trains to raise overall ass efficiency especially in instances where torsion predoinates in the presence of bending. Other exaples where direct drive systes have served to raise total ass efficiency include refrigeration systes (directly driven scroll copressors as opposed to reciprocating copressors) [8] and wind energy converters (gearless, directly-coupled ulti pole synchronous generators)[9]. Inability to iprove ass efficiency in torsionally loaded eleents will also force acceleration of the on-going trend towards hybrid systes and other direct drive systes (hydraulically-driven earth oving and far achinery). Efforts underway to develop electric propulsion for surface and suberged vessels [10]as well as aircrafts [11] tends to reinforce this view. Mass efficiency is considered here priarily fro the static loading perspective. It is noted in closing that poor ass efficiencies will invariably lead to lowered natural frequencies (in bending) and reduced critical speeds (in torsion) in virtually all dynaically excited echanical systes. Iproved ass efficiency offers a ore rational path to better aterial and energy conservation in all such instances when dynaic loading is a cause for concern. REFERENCES [1] M. F. Ashby, H. Shercliff, and D. Cebon,, "Materials : Engineering, Science, Processing and Design", Elsevier Butterworth-Heinann, UK, 007, ch. 0. [] M. F. Ashby, "Materials Selection in Mechanical Design", Elsevier Butterworth-Heinann, UK 005, pp [3] F. R. Shanley, "Weight-strength analysis of aircraft structures", McGraw-Hill NY 195, pp [4] S. Tioshenko, S., Strength of Materials, 3rd Ed., Van Nostrand Co Princeton NJ 1955, pp [5] M. J. French, "Conceptual design for engineers", Berlin, Springer, 1985, pp [6] B. A. Boley, On the accuracy of Bernoulli-Euler theory of beas of variable section, Journal of Applied Mechanics, Vol. 30, No. 3, 1963, pp [7] J. S. Heath, "Design of a Swinging Ar Actuator for a Disk File", IBM J. Res. Develop., Vol. 0, No.7, 1976, pp [8] E. Morishita, M. Sugihara and T. Nakaure, "Scroll Copressor Dynaics : 1st Report, The Model for the Fixed Radius Crank", Bulletin of JSME, Vol. 9, No. 48, 1986, pp [9] E. Hau and H. von Renouard, "Wind Turbines", nd ed., Springer, Heidelberg, 006, Ch. 9. [10] D. S. Parker and C. G. Hodge, "The electric warship", Power Engineering Journal, Vol.1, No. 1, 1998, pp [11] P J Masson, G V Brown, D S Soban and C A Luongo, " HTS achines as enabling technology for all-electric airborne vehicles", Super Conductor Science and Technology, Vol 0, 007, pp ISBN: WCE 008