- Email: [email protected]
Stability of a floating cylinder
International Journal of Engineering Science 42 (2004) 691–698 www.elsevier.com/locate/ijengsci
Stability of a floating cylinder D.S. Dugdale Industrial Research Centre, Northumbria University, Newcastle NE1 8ST, UK Received 12 March 2003; accepted 30 April 2003 (Communicated by B.A. BILBY)
Abstract A cylinder having a length less than 0.707 of its diameter will float with its axis vertical while one of length exceeding one diameter floats with its axis horizontal. This transition range was investigated and under certain circumstances, two stable positions of the cylinder axis were found in each quadrant of rotation. Details of behaviour in the transition range were attributed to the edges of the square-ended cylinder. Ó 2003 Published by Elsevier Ltd.
1. Conventional approach A practical method for examining the stability of a floating body has been covered in many text books, for example, that of Douglas et al. [1]. First, the argument of this method will be followed through in summary to obtain some standard results and later an alternative method will be used. When a solid body floats, its weight W acts downwards through its centre of gravity. Forces are also exerted on elements of the immersed surface by fluid pressure, this being proportional to the depth of the element below the surface of the fluid. If the solid body is removed it will leave a cavity of volume V . If this displaced volume is then filled with fluid, this volume of fluid will obviously be in equilibrium with the surrounding fluid. Therefore, instead of considering forces exerted by the fluid on the solid body, an alternative is to consider the force exerted on the solid body by the displaced volume. This buoyancy force is F ¼ qgV , where q is the density of the fluid and g is the gravitational acceleration. This force acts vertically upwards through the centre of buoyancy. The centre of buoyancy is at the centre of gravity of the displaced volume. When the body floats freely, F ¼ W , and the centre of buoyancy lies on the vertical line drawn though the
E-mail address: [email protected] (D.S. Dugdale). 0020-7225/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/S0020-7225(03)00237-4
692
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
centre of gravity of the solid body. The moment about the centre of buoyancy of pressure forces acting externally on the displaced volume is always zero. This situation is represented in Fig. 1. The position of the centre of buoyancy B is found by calculating the first moment VG of elements of the displaced volume about the horizontal plane passing through the centre of gravity G of the floating body, VG ¼
Z
ð1Þ
y dV
and the depth a of the centre of buoyancy follows from a ¼ VG =V . The displaced volume is now tilted though a small angle b. So that the displaced volume will have a surface conforming with the surface of the fluid, a small wedge of fluid must be added on the left-hand side thus adding to the buoyancy force on this side, while a similar wedge must be removed from the right-hand side. This has the effect of applying a torque T to the displaced volume, such that T ¼ qgI0 b where I0 is the second moment of the area A of the fluid surface about an axis O drawn through the centroid of this surface, I0 ¼
Z
x2 dA:
ð2Þ
This torque means that the buoyancy force must now act through a new centre of buoyancy B which has moved through distance b to the left, given by T ¼ Fb. Meanwhile, the centre of gravity G has also moved to the left by distance c ¼ ab. If the couple acting on the floating body is to be such as to restore its original position, it is necessary that length b should be greater than c. By using the above equations the stability criterion becomes ð3Þ
I0 > VG :
In the literature dealing with stability of ships, this argument is taken a step further to discuss stability in terms of metacentric height.
Fig. 1. Stability criterion.
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
693
2. Basic results In the present context, a cylinder is understood to be in static balance. It may consist of uniform material or it may be a square-ended shell containing weights, provided that these weights exert no moment about the geometrical centre. The cylinder then has a mean density equal to its total weight divided by its volume. A cylinder of radius r and length 2l can be described in terms of two parameters, its length/diameter ratio l=r and its density r relative to the fluid in which it floats. First, a cylinder is considered which floats with its axis horizontal, as shown in Fig. 2. The immersed segment is defined by angle h0 . From Eqs. (1) and (2), VG ¼ 4=3r3 l sin3 h0 and I0 ¼ 4=3rl3 sin h0 . The stability criterion (3) shows that stability is lost when l=r is reduced to a value ð4Þ
l=r ¼ sin h0 :
When this is combined with the expression for relative density r ¼ ðh0 1=2 sin 2h0 Þ=p, a line can be plotted in Fig. 3, this being the line passing through point D.
Fig. 2. Cylinder with its axis horizontal and vertical.
Fig. 3. Cylinder characterized by its length/diameter ratio l=r and mean density r relative to the fluid.
694
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
Next, a cylinder is considered which floats with its axis vertical, also shown in Fig. 2. It is immersed to depth h, so relative density is r ¼ h=2l. From Eqs. (1) and (2), VG ¼ pr2 hðl 1=2hÞ and I0 ¼ pr4 =4. The stability criterion (3) now shows that stability is lost when l=r has been increased so as to give r2 ¼ hðl 1=2hÞ, or 2
ðl=rÞ ¼ 1=8rð1 rÞ:
ð5Þ
This line is plotted in Fig. 3, this being the line passing through point B. It should be mentioned that results (4) and (5) have been known for a century or more and equivalent relationships were given by Lamb [2]. Some further remarks may be made about Fig. 3. Within the enclave marked I, the cylinder is stable with its axis vertical but not in any other position. Within the enclave marked II, the cylinder is stable with its axis horizontal but not in any other position. In enclave III the cylinder is not stable with its axis either vertical or horizontal but stability is reached with the axis inclined at some intermediate position. In enclaves marked IV, the cylinder is stable with its axis horizontal and also with its axis vertical. 3. Direct integration of hydrostatic moment When the body floats in equilibrium, the method previously described can be used for establishing whether the equilibrium is stable or unstable, but will not reveal orientations of equilibrium when the axis is tilted to positions between the horizontal and the vertical. A cylinder has only one effective degree of freedom as it is rotated. Stability can be more fully investigated by drawing a graph of the hydrostatic moment exerted on the cylinder while it is rotated through a full quadrant of rotation. Fig. 4 shows three phases of rotation. The axis, initially horizontal, rotates through angle a until edge P reaches the fluid surface OO. Rotation then continues until edge Q reaches the fluid surface. The third phase of rotation brings the axis to the vertical position, when the depth of immersion is h. Hydrostatic moment must be calculated for each phase of rotation. The sign convention takes anticlockwise rotation as positive with moments also considered positive when exerted in this direction. The method of integration is explained in Fig. 5. With its axis initially horizontal, the cylinder is tilted relative to the fluid surface OO by angle a. On the true view of the end of the cylinder, angles h1 and h2 are related by rðcos h1 cos h2 Þ ¼ 2l tan a:
Fig. 4. Three phases of rotation of a floating cylinder.
ð6Þ
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
695
Fig. 5. Integration process for finding displaced volume and hydrostatic moment.
Displaced volume V can be found in terms of these angles by integrating the volume of slices of the cylinder at right angles to its axis as indicated at E, to obtain an expression for this volume and this volume pr2 h ¼
r3 sin h2 sin h1 1=3ðsin3 h2 sin3 h1 Þ ðh2 cos h2 h1 cos h1 Þ : tan a
ð7Þ
With selected values of a, l=r and h=r, Eqs. (6) and (7) were solved iteratively to obtain angles h1 and h2 . These angles were then inserted into integrals of the moment exerted by fluid pressure on the curved surface and ends of the cylinder, in the knowledge that displaced volume will remain constant. In this way, graphs of moment M versus angle a were drawn. For the particular case when angle a is small, it was found that moment could be expressed as M ¼ 4=3qgrl sin h0 ðl2 r2 sin h0 Þ sin a:
ð8Þ
It can be seen that for large values of l=r, stability persists (with negative moment) until this ratio is reduced to l=r ¼ sin h0 , which is the criterion of stability previously given in (4). For finite values of a, the method of ensuring constant displaced volume must be followed. For an inclination b to the vertical (where b ¼ a 90°), a closed expression for moment was found, ð9Þ M ¼ p=4qgr2 r2 ð1 þ 1=2 tan2 bÞ 2hð2l hÞ sin b: This expression is valid for 0 < tan b < h=r. Stability at b ¼ 0 is lost when the term in square brackets becomes equal to zero, which gives a stability criterion as already found in (5).
4. Moments for finite angles of rotation Graphs are now presented of hydrostatic moment exerted on the cylinder as it is rotated through a quadrant. This moment was taken about the centre of gravity of the cylinder. Stable
696
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
equilibrium can be recognized from such a graph when the curve crosses the base line with a negative gradient, so that moments on either side of this point represent restoring moments. Fig. 6 shows the moment for various points marked in Fig. 3, selected on the line r ¼ 0:5. Starting from point A, stability is obtained with the axis vertical, that is, at inclination a ¼ 90°. When point B is reached at l=r ¼ 0:707, the moment curve is tangential to the base line at inclination a ¼ 90° so there is no positive stable equilibrium. Stability then develops at an intermediate position of the cylinder axis, as at point C (l=r ¼ 0:875). A neutral state is again reached at point D, while at point E, stable equilibrium has become established with the axis horizontal, that is, at a ¼ 0. The moment curves shown in Fig. 7 refer to points marked in Fig. 3 on the vertical line l=r ¼ 0:875. Curve F (r ¼ 0:255) has its stability point near to angle a ¼ 30° and is similar to curve C except that it has become rather distorted. On reducing relative density to 0.230 at point
Fig. 6. Calculated curves of hydrostatic moment M against inclination a of the axis to the horizontal for points marked in Fig. 3.
Fig. 7. Further curves of hydrostatic moment for points marked in Fig. 3.
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
697
G, it can be seen that a loop in the curve has crossed over the base line, so that the curve now indicates two positions of stable equilibrium at inclination angles around 22° and 68°. At position H (r ¼ 0:205), which is shown in Fig. 3 as the crossing point of the two limit curves, we see from Fig. 7 that there is no angular position of the cylinder where a state of positive stability exists, but merely a state of neutral stability at a ¼ 0° and 90°. Finally, on passing to point I (r ¼ 0:180), two positions of stability appear, with the cylinder axis vertical and also with the axis horizontal, as expected from the enclave marked IV in Fig. 3. The occurrence of stability at two inclined positions of the axis appears to be confined to the small area shown in Fig. 3 by a broken line. However, restoring moments for points in the vicinity of nodal point H are seen to be small value compared with those at points some distance away. To obtain physical values of moments, the numerical values plotted should be multiplied by the factor qgr4 .
5. Cylinders with rounded edges or rounded ends First, a disc is considered having outside radius r and axial length 2l with its edge rounded to a semicircular profile of radius l, as shown on the left of Fig. 8. By applying Eq. (3) it was found that this cylinder floats in stable equilibrium with its axis vertical but is unstable when its axis is horizontal. When the ratio l=r is increased to a value of unity, the disc becomes a sphere, which is in a neutral state of stability and experiences zero restoring moment, whatever the relative density r may be. Secondly, a cylinder is considered having over-all length 2l with hemispherical ends, as shown on the right of Fig. 8. By applying the stability criterion (3), it was found that this cylinder floats in stable equilibrium with its axis horizontal but is unstable when its axis is vertical. When the ratio l=r is reduced to a value of unity, the cylinder becomes a sphere. Therefore, a continuous variation in the shape of these cylinders reveals a transition at l=r ¼ 1:0 when the position of the axis for stable floatation rotates abruptly through 90°.
Fig. 8. Cylinder with fully rounded edges or hemispherical ends.
698
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
A further remark may be made about the boundary lines in Figs. 3 and 8. Attention is focused on the point marked B in Fig. 3, which is the point where a cylinder with its axis vertical is expected to lose its stability. If a small chamfer is now formed on the sharp edges at both ends of the cylinder, the first moment of immersed volume VG will be reduced while the second moment I0 of the surface of the displaced volume will remain unchanged. Eq. (3) then indicates that stability is retained and that the boundary line moves a little to the right. With a gross amount of rounding of the edges the boundary line will finally reach the position shown in Fig. 8. 6. Conclusions Fig. 3 shows that for l=r < 0:707, a cylinder is always stable when its axis is vertical and for l=r > 1:0, it is always stable when its axis is horizontal. Over the intermediate range of values, the cylinder usually attains stable floatation with its axis at one particular inclination between the horizontal and the vertical. Under restricted circumstances, there may be two positions of the axis for stable equilibrium as the cylinder moves through a quadrant of rotation. When a small change is made to length/diameter ratio or to relative density, the curve of hydrostatic moment versus inclination angle changes its shape progressively, and no discontinuous changes of moment have been found. Depending on the intercepts of the moment curve with the base line, a state of stable equilibrium either appears or disappears in a predictable way. When the edges or ends of the cylinder are fully rounded, the transition from stability with axis vertical to stability with axis horizontal occurs at the single value l=r ¼ 1:0. This suggests that for a simple cylinder, the complexities of the transition range arise on account of the sharp rightangled edges of the cylinder ends. Conclusions reported here have been confirmed experimentally, but this work was of such a simple kind that details need not be given. The cylinders used should be of some uniform material so that irregularities in density distribution do not obscure the expected behaviour. Models made of wood were sometimes found to be unsatisfactory in this respect.
Acknowledgement The author is indebted to Professor B.A. Bilby for his advice which has led to improvements in the presentation of this paper.
References [1] J.F. Douglas, J.M. Gasiorek, J.A. Swaffield, Fluid Mechanics, Pitman Books, London, 1979, p. 79. [2] H. Lamb, Statics, Cambridge University Press, London, 1928 (Reprinted 1955), p. 241.
Stability of a floating cylinder D.S. Dugdale Industrial Research Centre, Northumbria University, Newcastle NE1 8ST, UK Received 12 March 2003; accepted 30 April 2003 (Communicated by B.A. BILBY)
Abstract A cylinder having a length less than 0.707 of its diameter will float with its axis vertical while one of length exceeding one diameter floats with its axis horizontal. This transition range was investigated and under certain circumstances, two stable positions of the cylinder axis were found in each quadrant of rotation. Details of behaviour in the transition range were attributed to the edges of the square-ended cylinder. Ó 2003 Published by Elsevier Ltd.
1. Conventional approach A practical method for examining the stability of a floating body has been covered in many text books, for example, that of Douglas et al. [1]. First, the argument of this method will be followed through in summary to obtain some standard results and later an alternative method will be used. When a solid body floats, its weight W acts downwards through its centre of gravity. Forces are also exerted on elements of the immersed surface by fluid pressure, this being proportional to the depth of the element below the surface of the fluid. If the solid body is removed it will leave a cavity of volume V . If this displaced volume is then filled with fluid, this volume of fluid will obviously be in equilibrium with the surrounding fluid. Therefore, instead of considering forces exerted by the fluid on the solid body, an alternative is to consider the force exerted on the solid body by the displaced volume. This buoyancy force is F ¼ qgV , where q is the density of the fluid and g is the gravitational acceleration. This force acts vertically upwards through the centre of buoyancy. The centre of buoyancy is at the centre of gravity of the displaced volume. When the body floats freely, F ¼ W , and the centre of buoyancy lies on the vertical line drawn though the
E-mail address: [email protected] (D.S. Dugdale). 0020-7225/$ - see front matter Ó 2003 Published by Elsevier Ltd. doi:10.1016/S0020-7225(03)00237-4
692
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
centre of gravity of the solid body. The moment about the centre of buoyancy of pressure forces acting externally on the displaced volume is always zero. This situation is represented in Fig. 1. The position of the centre of buoyancy B is found by calculating the first moment VG of elements of the displaced volume about the horizontal plane passing through the centre of gravity G of the floating body, VG ¼
Z
ð1Þ
y dV
and the depth a of the centre of buoyancy follows from a ¼ VG =V . The displaced volume is now tilted though a small angle b. So that the displaced volume will have a surface conforming with the surface of the fluid, a small wedge of fluid must be added on the left-hand side thus adding to the buoyancy force on this side, while a similar wedge must be removed from the right-hand side. This has the effect of applying a torque T to the displaced volume, such that T ¼ qgI0 b where I0 is the second moment of the area A of the fluid surface about an axis O drawn through the centroid of this surface, I0 ¼
Z
x2 dA:
ð2Þ
This torque means that the buoyancy force must now act through a new centre of buoyancy B which has moved through distance b to the left, given by T ¼ Fb. Meanwhile, the centre of gravity G has also moved to the left by distance c ¼ ab. If the couple acting on the floating body is to be such as to restore its original position, it is necessary that length b should be greater than c. By using the above equations the stability criterion becomes ð3Þ
I0 > VG :
In the literature dealing with stability of ships, this argument is taken a step further to discuss stability in terms of metacentric height.
Fig. 1. Stability criterion.
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
693
2. Basic results In the present context, a cylinder is understood to be in static balance. It may consist of uniform material or it may be a square-ended shell containing weights, provided that these weights exert no moment about the geometrical centre. The cylinder then has a mean density equal to its total weight divided by its volume. A cylinder of radius r and length 2l can be described in terms of two parameters, its length/diameter ratio l=r and its density r relative to the fluid in which it floats. First, a cylinder is considered which floats with its axis horizontal, as shown in Fig. 2. The immersed segment is defined by angle h0 . From Eqs. (1) and (2), VG ¼ 4=3r3 l sin3 h0 and I0 ¼ 4=3rl3 sin h0 . The stability criterion (3) shows that stability is lost when l=r is reduced to a value ð4Þ
l=r ¼ sin h0 :
When this is combined with the expression for relative density r ¼ ðh0 1=2 sin 2h0 Þ=p, a line can be plotted in Fig. 3, this being the line passing through point D.
Fig. 2. Cylinder with its axis horizontal and vertical.
Fig. 3. Cylinder characterized by its length/diameter ratio l=r and mean density r relative to the fluid.
694
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
Next, a cylinder is considered which floats with its axis vertical, also shown in Fig. 2. It is immersed to depth h, so relative density is r ¼ h=2l. From Eqs. (1) and (2), VG ¼ pr2 hðl 1=2hÞ and I0 ¼ pr4 =4. The stability criterion (3) now shows that stability is lost when l=r has been increased so as to give r2 ¼ hðl 1=2hÞ, or 2
ðl=rÞ ¼ 1=8rð1 rÞ:
ð5Þ
This line is plotted in Fig. 3, this being the line passing through point B. It should be mentioned that results (4) and (5) have been known for a century or more and equivalent relationships were given by Lamb [2]. Some further remarks may be made about Fig. 3. Within the enclave marked I, the cylinder is stable with its axis vertical but not in any other position. Within the enclave marked II, the cylinder is stable with its axis horizontal but not in any other position. In enclave III the cylinder is not stable with its axis either vertical or horizontal but stability is reached with the axis inclined at some intermediate position. In enclaves marked IV, the cylinder is stable with its axis horizontal and also with its axis vertical. 3. Direct integration of hydrostatic moment When the body floats in equilibrium, the method previously described can be used for establishing whether the equilibrium is stable or unstable, but will not reveal orientations of equilibrium when the axis is tilted to positions between the horizontal and the vertical. A cylinder has only one effective degree of freedom as it is rotated. Stability can be more fully investigated by drawing a graph of the hydrostatic moment exerted on the cylinder while it is rotated through a full quadrant of rotation. Fig. 4 shows three phases of rotation. The axis, initially horizontal, rotates through angle a until edge P reaches the fluid surface OO. Rotation then continues until edge Q reaches the fluid surface. The third phase of rotation brings the axis to the vertical position, when the depth of immersion is h. Hydrostatic moment must be calculated for each phase of rotation. The sign convention takes anticlockwise rotation as positive with moments also considered positive when exerted in this direction. The method of integration is explained in Fig. 5. With its axis initially horizontal, the cylinder is tilted relative to the fluid surface OO by angle a. On the true view of the end of the cylinder, angles h1 and h2 are related by rðcos h1 cos h2 Þ ¼ 2l tan a:
Fig. 4. Three phases of rotation of a floating cylinder.
ð6Þ
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
695
Fig. 5. Integration process for finding displaced volume and hydrostatic moment.
Displaced volume V can be found in terms of these angles by integrating the volume of slices of the cylinder at right angles to its axis as indicated at E, to obtain an expression for this volume and this volume pr2 h ¼
r3 sin h2 sin h1 1=3ðsin3 h2 sin3 h1 Þ ðh2 cos h2 h1 cos h1 Þ : tan a
ð7Þ
With selected values of a, l=r and h=r, Eqs. (6) and (7) were solved iteratively to obtain angles h1 and h2 . These angles were then inserted into integrals of the moment exerted by fluid pressure on the curved surface and ends of the cylinder, in the knowledge that displaced volume will remain constant. In this way, graphs of moment M versus angle a were drawn. For the particular case when angle a is small, it was found that moment could be expressed as M ¼ 4=3qgrl sin h0 ðl2 r2 sin h0 Þ sin a:
ð8Þ
It can be seen that for large values of l=r, stability persists (with negative moment) until this ratio is reduced to l=r ¼ sin h0 , which is the criterion of stability previously given in (4). For finite values of a, the method of ensuring constant displaced volume must be followed. For an inclination b to the vertical (where b ¼ a 90°), a closed expression for moment was found, ð9Þ M ¼ p=4qgr2 r2 ð1 þ 1=2 tan2 bÞ 2hð2l hÞ sin b: This expression is valid for 0 < tan b < h=r. Stability at b ¼ 0 is lost when the term in square brackets becomes equal to zero, which gives a stability criterion as already found in (5).
4. Moments for finite angles of rotation Graphs are now presented of hydrostatic moment exerted on the cylinder as it is rotated through a quadrant. This moment was taken about the centre of gravity of the cylinder. Stable
696
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
equilibrium can be recognized from such a graph when the curve crosses the base line with a negative gradient, so that moments on either side of this point represent restoring moments. Fig. 6 shows the moment for various points marked in Fig. 3, selected on the line r ¼ 0:5. Starting from point A, stability is obtained with the axis vertical, that is, at inclination a ¼ 90°. When point B is reached at l=r ¼ 0:707, the moment curve is tangential to the base line at inclination a ¼ 90° so there is no positive stable equilibrium. Stability then develops at an intermediate position of the cylinder axis, as at point C (l=r ¼ 0:875). A neutral state is again reached at point D, while at point E, stable equilibrium has become established with the axis horizontal, that is, at a ¼ 0. The moment curves shown in Fig. 7 refer to points marked in Fig. 3 on the vertical line l=r ¼ 0:875. Curve F (r ¼ 0:255) has its stability point near to angle a ¼ 30° and is similar to curve C except that it has become rather distorted. On reducing relative density to 0.230 at point
Fig. 6. Calculated curves of hydrostatic moment M against inclination a of the axis to the horizontal for points marked in Fig. 3.
Fig. 7. Further curves of hydrostatic moment for points marked in Fig. 3.
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
697
G, it can be seen that a loop in the curve has crossed over the base line, so that the curve now indicates two positions of stable equilibrium at inclination angles around 22° and 68°. At position H (r ¼ 0:205), which is shown in Fig. 3 as the crossing point of the two limit curves, we see from Fig. 7 that there is no angular position of the cylinder where a state of positive stability exists, but merely a state of neutral stability at a ¼ 0° and 90°. Finally, on passing to point I (r ¼ 0:180), two positions of stability appear, with the cylinder axis vertical and also with the axis horizontal, as expected from the enclave marked IV in Fig. 3. The occurrence of stability at two inclined positions of the axis appears to be confined to the small area shown in Fig. 3 by a broken line. However, restoring moments for points in the vicinity of nodal point H are seen to be small value compared with those at points some distance away. To obtain physical values of moments, the numerical values plotted should be multiplied by the factor qgr4 .
5. Cylinders with rounded edges or rounded ends First, a disc is considered having outside radius r and axial length 2l with its edge rounded to a semicircular profile of radius l, as shown on the left of Fig. 8. By applying Eq. (3) it was found that this cylinder floats in stable equilibrium with its axis vertical but is unstable when its axis is horizontal. When the ratio l=r is increased to a value of unity, the disc becomes a sphere, which is in a neutral state of stability and experiences zero restoring moment, whatever the relative density r may be. Secondly, a cylinder is considered having over-all length 2l with hemispherical ends, as shown on the right of Fig. 8. By applying the stability criterion (3), it was found that this cylinder floats in stable equilibrium with its axis horizontal but is unstable when its axis is vertical. When the ratio l=r is reduced to a value of unity, the cylinder becomes a sphere. Therefore, a continuous variation in the shape of these cylinders reveals a transition at l=r ¼ 1:0 when the position of the axis for stable floatation rotates abruptly through 90°.
Fig. 8. Cylinder with fully rounded edges or hemispherical ends.
698
D.S. Dugdale / International Journal of Engineering Science 42 (2004) 691–698
A further remark may be made about the boundary lines in Figs. 3 and 8. Attention is focused on the point marked B in Fig. 3, which is the point where a cylinder with its axis vertical is expected to lose its stability. If a small chamfer is now formed on the sharp edges at both ends of the cylinder, the first moment of immersed volume VG will be reduced while the second moment I0 of the surface of the displaced volume will remain unchanged. Eq. (3) then indicates that stability is retained and that the boundary line moves a little to the right. With a gross amount of rounding of the edges the boundary line will finally reach the position shown in Fig. 8. 6. Conclusions Fig. 3 shows that for l=r < 0:707, a cylinder is always stable when its axis is vertical and for l=r > 1:0, it is always stable when its axis is horizontal. Over the intermediate range of values, the cylinder usually attains stable floatation with its axis at one particular inclination between the horizontal and the vertical. Under restricted circumstances, there may be two positions of the axis for stable equilibrium as the cylinder moves through a quadrant of rotation. When a small change is made to length/diameter ratio or to relative density, the curve of hydrostatic moment versus inclination angle changes its shape progressively, and no discontinuous changes of moment have been found. Depending on the intercepts of the moment curve with the base line, a state of stable equilibrium either appears or disappears in a predictable way. When the edges or ends of the cylinder are fully rounded, the transition from stability with axis vertical to stability with axis horizontal occurs at the single value l=r ¼ 1:0. This suggests that for a simple cylinder, the complexities of the transition range arise on account of the sharp rightangled edges of the cylinder ends. Conclusions reported here have been confirmed experimentally, but this work was of such a simple kind that details need not be given. The cylinders used should be of some uniform material so that irregularities in density distribution do not obscure the expected behaviour. Models made of wood were sometimes found to be unsatisfactory in this respect.
Acknowledgement The author is indebted to Professor B.A. Bilby for his advice which has led to improvements in the presentation of this paper.
References [1] J.F. Douglas, J.M. Gasiorek, J.A. Swaffield, Fluid Mechanics, Pitman Books, London, 1979, p. 79. [2] H. Lamb, Statics, Cambridge University Press, London, 1928 (Reprinted 1955), p. 241.