Patent Publication Number: US-2003224869-A1

Title: Shaft stiffened by axial pre-stress

Description:
BACKGROUND OF THE INVENTION  
       [0001] The invention concerns a new shaft, in which the bending stiffness of the shaft is enhanced by properly arranged pre-stress in the shaft. Shafts like golf clubs, fishing rod, arms used in precision equipment, etc. may find use of the invention.  
       [0002] In theory of mechanics, it was known that when a cantilever beam is in bending due to a transverse load, its end deflection would be reduced if the beam has simultaneous axial tension force. It is because under the simultaneous load, the neutral axis of the beam will be shifted upward and the axial tension force will produce a counter moment to offset the external bending moment, and consequently the deflection of the end of the beam will be reduced. Axial compression force will have the opposite effect.  
       [0003] A difficulty of application is the involved complicated mechanics, which is not popularly understood; and the net gain in performance regarding the new technology is not quantitatively clear.  
       [0004] The invention suggests a self-equilibrating internal force system, establishes rules governing the stiffness change, and a procedure to estimate the enhanced stiffness. 
     
    
    
     BRIEF DESCRIPTIONS OF THE DRAWING  
     [0005]FIG. 1 shows the cover page of a reference book titled Roark&#39;s Formulas for Stress &amp; Strain, by Warren C. Young, sixth Edition, McGraw-Hill, Inc.  
     [0006]FIG. 2 shows p. 157, Table 9a, Reaction and deflection coefficients for beams under simultaneous loading, taken from the FIG. 1 reference book.  
     [0007]FIG. 3 curve A shows the force and rigidity coefficient plotted against the deflection ratio coefficient based on FIG. 2 tension data. Curve B is a plot related to the data applicable only to the sample shaft of FIG. 4.  
     [0008]FIG. 4 shows a sample shaft constructed according to the findings of the invention.  
    
    
     A PREFERRED EMBODIMENT  
     [0009] We conclude that the new enhanced shaft should follow these rules:  
     [0010] (1), the internal axial force system is self-equilibrating.  
     [0011] (2), the shaft comprises concentric, longitudinal shaft members joined at the ends, which are either in compression or in tension, wherein each member bends according to its own bending rigidity, as affected by the internal stress and constraints.  
     [0012] (3), the sum of the bending rigidities of its tension members should be greater than that of the compression members.  
     [0013] A fourth rule concerning estimation of the stiffness increment will be described later.  
     [0014]FIG. 2 table shows quantitatively that the tension beam becomes stiffer in bending and the compression beam becomes weaker in stiffness and deflects more.  
     [0015] In FIG. 3, the curve A is plotted from the tension table in FIG. 2. In the vertical coordinate of curve A, EI is the beam bending rigidity, wherein E is its Young&#39;s modulus, I the bending moment of inertia of the cross-section of the tensioned portion, P is the axial tension force in kg, and L the beam&#39;s length in cm. The horizontal axis is the deflection ratio d*/d, where d* is the reduced end deflection under the axial tension P, and d is the conventional beam deflection of the tensioned part, without the axial force. Curve B is specific for the sample shaft in FIG. 4, which shall be explained later.  
     [0016] To illustrate the invention, a golf club shaft is shown in FIG. 4:  
     [0017] The shaft  1  has a length L, having a grip end  2  and a small end  3 , which is joined to the golf club head. The shaft consists of an outer member  4  and an inner member  5 ; one inside the other and each bends accordingly. They are joined at the common end  6 . A screw device  7  joins  4  to  5  at the grip end. The axial force P, created by turning the screw, compresses the inner member, which in turn drags the outer member into tension, because the end of  4  restraints the axial movement of  5 . Other forcing arrangement is possible. The bending rigidity EI of the tension member is greater than that of the compression member. The spring device  8  is optional. Through the spring, the axial force may be maintained constant at all times.  
     [0018] To increase the ratio of bending rigidities, one may also choose to reduce the bending rigidity of the compression member. An example is to cut light, parallel transverse surface lines on the surface of a compression member, which does not affect its compressive strength but reduces its bending rigidity.  
     [0019] Other embodiments are possible. For example, there may be three or more co-axial members, or there may be honeycomb-like inner member, or a slender member with discrete spacers for bracing purpose. The outer member is preferred to be in tension.  
     NUMERICAL EXAMPLE OF A PRE-STRESSED SHAFT  
     [0020] A conventional graphite golf shaft is used as an example. All relevant calculations are presented in detail so that we can follow the progressing steps closely.  
     [0021] The shaft length L is 100 cm, weighs 58 g, outer diameter 1.23 cm, inner diameter 0.96 cm, wall thickness 0.135 cm, which has 9 plies of prepreg fiber-reinforced resin tapes. Specific weight of the material is 1.25 g per c.c, and the Young&#39;s modulus E of the tape is 1,339,000 kg/sq.cm. The constant diameter shaft may represent a corresponding, conventional tapered golf club shaft of the same weight and length.  
     [0022] The 9-ply wall is divided into a 6-ply outer member and a 3-ply inner member. They fit snugly, each can bend independently. Having the same end deflection makes compatibility easier between various members&#39; slightly different curvatures.  
     [0023] In this shaft, the total bending rigidity EI of the shaft is 94,700 kg-sq.cm, the EI of the tension member is 70,600, and that of the inner member is a much smaller 24,100.  
     [0024] In general, beam deflection is proportional to its load, but is inversely proportional to its bending rigidity EI. Since the outer and the inner shaft share the same bending load W, and both deflect the same at the end, the outer shaft should take 70,600/94,700=75% of the load W and the inner shaft 25%, in the absence of axial force. Now, having the axial force P, the tension shaft should have less deflection and the compression shaft more. Therefore, the percentages of W on the two component shafts should be adjusted so that their ends can deflect equally. The adjustment depends on P. That&#39;s where the mechanics of the combined load on a beam comes into the picture.  
     [0025] The tension member of the FIG. 4 shaft has EI/L-square=7.06 kg. Using curve A, we can construct a new P-versus d*/d curve, which is shown as curve B in FIG. 3, which is specific only to this tension member of this sample shaft. The dependency of the reduced deflection d* upon the axial force P is very clear.  
     [0026] Given a P value, say 30 kg, curve B yields d*/d=0.33. This means having the tension force P, the deflection d* of the outer member is reduced to 33% of d.  
     [0027] In Ref. 2, the corresponding table for the compression shaft is limited to have the force &amp; rigidity coef. up to 1.0 only. So we have to take d*=1.67d as the effect of axial compression for all ranges of P.  
     [0028] So, with the deflection d* reduced to 0.33d for the tension shaft, and the compression shaft&#39;s deflection d* increased to 1.67d, the share of the W load should be changed, more to the tension member, so that their end deflections can remain the same as before.  
     [0029] Simple calculation leads to the answer: the tension shaft should take 94% of the W load and the weakened compression shaft takes only 6% of the W load.  
     [0030] Studies show that such overwhelming one-sided load distribution is generally true.  
     [0031] Therefore, we now have the fourth rule for the design as follows:  
     [0032] (4), the shaft&#39;s bending rigidity comprises of its tensioned member only, and its deflection is the deflection of the tensioned member under the combined load W and P.  
     [0033] To get the deflection, the EI of the tensioned part is 70,600 and P is 30 kg, curve A yields d*/d=0.33. Convert d of the tension shaft to d of the original shaft, both without the axial force P, we should increase d by a factor of 94700/70600=1.34. Therefore, the final estimated deflection according to rule (4) is d==0.33×1.34=0.44.  
     [0034] In this example, assume the shaft is a golf club shaft of stiffness “s”. The stiff golf shaft of length 100 cm, having “s”, will have an end deflection about 3.5-inch measured on a standard Deflection Measurement Board. Multiplied by a reduction factor of 0.44, the new deflection becomes 1.54 inch. This is a very stiff number. The enhanced shaft now becomes so stiff that it is way beyond the range of the calibration chart of a conventional Measuring Board.  
     [0035] Of course, stiffening is not the only merit of the invention. We also can make a much lighter shaft, keeping the same stiffness. That is what most players are looking for, and that probably has the most appeal. We also can have shaft whose stiffness can be changed by the turn of a screw at the grip end. This on spot adjustment is allowed by USGA. Its rules said that adjustment is allowed if the adjustment is done in a simple way, having the handle end sealed easily, and the process completed before the tournament.