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Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets

Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 483857, 6 pages doi:10.1155/2008/483857 Research Article Dynamics and Stability of Stepped Gun Barrels with Moving Bullets Mohammad Tawfik Mechanical Engineering Department, British University in Egypt, El-Shorouk City, Cairo 11837, Egypt Correspondence should be addressed to Mohammad Tawfik, mohammad.tawfik@bue.edu.eg Received 12 August 2007; Revised 16 November 2007; Accepted 20 January 2008 Recommended by Luc Gaudiller The stability of an Euler-Bernoulli beam under the effect of a moving projectile will be reintroduced using simple eigenvalue analysis of a finite element model. The eigenvalues of the beam change with the mass, speed, and position of the projectile, thus, the eigenvalues are evaluated for the system with differentspeedsand massesatdifferent positions until the lowest eigenvalue reaches zero indicating the instability occurrence. Then a map for the stability region may be obtained for different boundary conditions. Then the dynamics of the beam will be investigated using the Newmark algorithm at different values of speed and mass ratios. Finally, the effect of using stepped barrels on the stability and the dynamics is going to be investigated. It is concluded that the technique used to predict the stability boundaries is simple, accurate, and reliable, the mass of the barrel on the dynamics of the problem cannot be ignored, and that using the stepped barrels, with small increase in the diameter, enhances the stability and the dynamics of the barrel. Copyright © 2008 Mohammad Tawfik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION Recently, the emergence of the need for very light guns that are mounted on combat aircraft reintroduced the sta- The problem of the dynamics and stability of beams carry- bility problem with new conditions. The motion of the bul- ing moving masses drew a lot of attention in the past half lets inside the gun barrels introduces compression force on century due to the applications that require it such as fast the shell walls, in turn, this compression may cause dynamic trains, motion on bridges, and light guns mounted on air- buckling [2] and excessive vibration in the shell wall [8]. This craft. In 1971, Nelson and Conover [1]presented astudy of type of instability, though of major importance, will not be covered in this study. the problem of an infinite thin beam with periodically dis- tributed simplesupports resting on elastic foundation with a The problem of projectiles inside gun barrels, though train of masses moving on it at constant speed. They applied of important application, was not the subject of many Galerkin method to the proposed approximate solution to researches. In [9, 10], the stability problem was studied using get the system equations then applied the Floquet theorem the impulsive parametric excitation theory [11, 12] for a thin to get the stability boundaries for the periodically repeating beam with periodically distributed controllers. The study system. Simultaneously, Benedetti [2, 3] used a similar ap- ignored the effect of the dynamics of the barrel shell. The proach and could present an analytical relation between the problem of the dynamics of the barrel shell under the effect mass parameter and the critical speed parameter using clas- of the moving projectile with shock and expansion waves sical techniques. was studied in [8] but the study did not tackle the stability Since most of the research was directed to civil structures problem. In [13], the author presented one of the very rare and lathe-machined work pieces, the stability regions were studies that handled finite beams. In that study, the beam not of major interest for general structures. Rather, the re- under investigation was modeled as a Timoshenko beam sponse of the structure to moving loads or masses presented a with simple supports and elastic foundation. The results more practical problem. Further, most of the studies were in- presented different cases of multiple masses and foundation terested in problems with periodically supported beams that stiffness but did not present any comparison with published simulate train rails (see [4–7] as examples for such studies). or experimental results. 2 Advances in Acoustics and Vibration In this study, the stability of an Euler-Bernoulli beam un- And the variation of the potential energy may be written as der the effect of a moving projectile will be reintroduced us- 2 2 ing simple eigenvalue analysis of a finite element model. The ∂ δw ∂ w δU = EI + kδw·w − ρAgδw dx 2 2 eigenvalues of the beam change with the mass, speed, and ∂x ∂x position of the projectile, thus the eigenvalues are evaluated − mg (δw) , for the system with different speeds and masses at different x=x (6) positions until the lowest eigenvalue reaches zero indicating the instability occurrence. Then a map for the stability region while the variation of the work done by the bullet may be may be obtained for different boundary conditions. Then the written as dynamics of the beam will be investigated using the New- mark algorithm at different values of speed and mass ratios. ∂δw ∂w δW = mV . (7) Finally, the effect of using stepped barrels on the stability and bullet ∂x ∂x x=x the dynamics is going to be investigated. Applying the Hamilton principle and using the standard beam finite element interpolation function, we may write the 2. MODEL element equation of motion as The model derived in this section will have the following as- sumptions: the gun barrel will be modeled as thin beam that [M]+ M w ¨ + C w ˙ bullet bullet follows the Euler-Bernoulli theorem, the barrel deflections + [K]+ K − K {w}= f + f , are small, and the effect of elastic foundation will be included s bullet w bullet for the purpose of comparison with published data. (8) The Hamilton principle states that where the beam mass matrix is evaluated by t t 2 2 δΠdt = δ T − U + W dt = 0, (1) bullet t t 1 1 [M] = ρA N (x) N (x) dx,(9) where T is the kinetic energy, U is the potential energy, and where N (x) are the beam interpolation polynomials. The ef- W is the work done by the bullet on the beam. The ki- bullet fective bullet mass matrix is evaluated at the bullet position netic energy of the system may be written as as 1 1 ∂w 2 2 2 M = m N (x) N (x) . (10) bullet T = ρAw ˙ dx + m w ˙ +2Vw ˙ + V ,(2) x=x 2 2 ∂x 0 x=x While the beam stiffness matrix is evaluated by where ρ is the beam mass density, A is the beam cross section area, m is the bullet mass, V is the bullet speed, w is the beam [K ] = EI N (x) N (x) dx. (11) xx xx transverse displacement, and x is the position of the bullet. The potential energy of the system may be written as The foundation stiffness matrix is evaluated by 1 ∂ w 2 l U = EI + kw − ρAgw dx − mg (w) , x=x 2 ∂x K = γEI N (x) N (x) dx, (12) 0 s (3) where γ is the foundation stiffness ratio given by γ = k/EI . where E is the beam modulus of elasticity, I is the beam sec- The effective geometric bullet stiffness matrix is evaluated at ond moment of area, k is the foundation stiffness, and g is the bullet position as the gravitational acceleration. And the work done by the bul- let may be written as 2 K = mV N (x) N (x) . (13) bullet xx xx x=xm 1 ∂w The effective bullet Coriolis matrix is evaluated by W = mV . (4) bullet 2 ∂x x=x C = 2mV N (x) N (x) . (14) bullet x x=x The variation of the kinetic energy may be written as The forces due to the barrel and bullet weights are evaluated, respectively, by δT = ρAδw ˙ f =− N (x) ρAg dx, 1 ∂w ∂δw ˙ ˙ ˙ ˙ ˙ ·wdx + m δw·w +2Vδw +2Vw . (15) 2 ∂x ∂x x=x f =− N (x) mg . (5) bullet x=x m Mohammad Tawfik 3 2.1. Stability boundaries To obtain the eigenvalues of the system, including the effect Unstable of all components, we will need to transform the system into a first-order system by the standard transformation, Single bay β 1 z = w. ˙ (16) 0.01 0.110 Stable Using the above transformation, we will obtain the homoge- neous equation of motion in the form w ˙ 0 I w n×n = , (17) 0.1 z MK MC z α −1 where MK denotes − [M]+ M [K ]+ K − K ) bullet s bullet Single bay-γ = 0 Benedetti-γ = 1 −1 CC-γ = 0 Single bay-γ = 2 andMCdenotes − [M]+ M C . bullet bullet Benedetti-γ = 0 CC-γ = 2 For the above system, the eigenvalues should represent Single bay-γ = 1 Benedetti-γ = 2 the natural frequencies of oscillation of the beam with the CC-γ = 1 bullet. The eigenvalues should all be complex with nonzero imaginary parts for all values of the speed that are below the Figure 1: Stability boundaries for different cases. critical speed. As the bullet speed reaches the critical speed, the smallest complex pair will have zero imaginary parts. The search for the critical values of the speed may be done (2) then the velocity and displacement may be evaluated using the following algorithm. using (1) Select the bullet mass. {w ˙} ={w ˙} +(1 − δ)Δt{w ¨} + μΔt{w ¨} , (2) Select the bullet speed. t+Δt t t t+Δt (3) Change the value of the x-location of the bullet and 2 2 evaluate the eigenvalues of the system. {w} ={w} + Δt{w ˙} + − μ Δt {w ¨} + μΔt {w ¨} . t+Δt t t t t (4) If all eigenvalues have nonzero imaginary parts then (20) increase speed and go to step (3), else go to step (6). (5) If all speed values did not reach the critical value, then In the above algorithm, δ and μ are parameters that reset speed value and increase mass up to a given limit have to obey the constraints δ ≥ 0.5and μ ≥ and proceed to step (3). If mass limit is reached, termi- (1/4)(δ +(1/2)) . nate. (6) Store the values of the critical speed and the mass. 3. RESULTS AND DISCUSSION (7) Increase speed up to a given limit and proceed to step (3). If mass limit is reached, terminate. 3.1. Stability boundaries The above procedure may be repeated for all values of mass, A program was written using MATLAB to perform the cal- speed, and boundary conditions and different information culations of the problem using the following data: Beam may be compiled out of the extracted data. modulus of elasticity 71 GPa, Density 2840 kg/m3, inner ra- dius 0.007 m, and outer radius 0.008 m. The projectile mass 2.2. Time response ranged from 0.005 to 0.75 Kg, and its speed ranged from 2.5 to 600 m/s. Two cases were used to demonstrate the valid- The time response of the system presented by (8)may be ob- ity of the procedure, simply supported and clamped at both tained using the Newmark algorithm as presented in [14]. The algorithm may be presented as follows. sides. Each case was run with three values of foundation stiff- ness γ = 0, 1, 2. The program used 10 3-node beam elements, For the system with 5th-order polynomial, (see [15]), and checked for the [M]{w ¨} +[C]{w ˙} +[K ]{w}= f , (18) w eigenvalues at 20 equidistant points in each element. Note that the 2-node element, 3rd-order polynomial, was checked (1) evaluate the acceleration of the system using for accuracy and the results were not of any considerable dif- ference, rather, the program was already written for 3-node [M]+ δΔt[C]+ μΔt [K ] {w ¨} t+Δt elements and thus used. The convergence of the solution was checked and it was found that the 10 elements with 20 inter- = f − [C] w ˙ +(1 − δ)Δt w ¨ t+Δt t t nal points gave adequate accuracy. Figure 1 shows the results of the two cases with the three − [K ] {w} + Δt w ˙ + − μ Δt w ¨ ; t t foundation stiffness values compared to those obtained from the equation given by Benedetti [2]. It can be obviously (19) 4 Advances in Acoustics and Vibration seen in that graph that the results obtained for the clamped- clamped beam arealmost identical for the three values of γ. Meanwhile, all the results obtained using the simply sup- ported beam gave lower values for the critical speed. The re- sults obtained from the equation by Benedetti [2], however, varied from nearest to single bay (simply supported) (γ = 0) β 1 to almost identical to clamped-clamped case (γ = 1) ending 0.01 0.11 1 0 with just being higher than the clamped-clamped case with γ = 2. Note that the fundamental solution of Benedetti used above gives the relation between the critical speed factor, β , cr and the mass factor, α,as 0.1 −0.5 β = 1+ γα , (21) cr Figure 2: Stability boundaries for cantilever beam. where α = m/ρAL and β = VL/(2π ρA/EI ). It has to be noted at this point that the results that were presented by Benedetti [2, 3] and Nelson and Conover [1] were obtained for the case of an infinite beam that is periodically simply supported. Thus such a structure should be expected to be more stable than a single bay of simply supported beam. However, the results showed that the clamped-clamped beam (single bay) showed very little change with the foundation stiffness. Finally, it is clear that the results of Benedettis for- mula gave values that are much like those of a clamped- clamped beam. Now the most important observation that may be taken from Figure 1 is that when the stability boundaries are all 0.1 drawn on log-log scale the results all showed linear trends, 0.01 0.11 1 0 further, all lines are parallel with a slope of −0.5 (note that the formula presented by Benedetti had a relation between RR = 1 RR = 8/10 −0.5 α and β in the form of β = aα , see the relation above). cr RR = 8/9 RR = 8/12 Thus it may be concluded that the procedure presented in Figure 3: Stability boundaries for stepped cantilever beam. this paper can accurately predict the stability boundaries of the problem with different boundary conditions. Further, us- ing regression techniques, we may write down a relation for The above relation reflects the expected increase in the sta- the critical speed factor as bility of the gun barrel when supported at the tip. −0.5 β | ≈ 0.195 4+ γα , cr S-S Now that the stability boundaries for beams with differ- (22) ent boundary conditions are realized to be a simple relation −0.5 β | ≈ 0.77α . cr C-C between the mass and velocity parameters, we need to inves- tigate the effect of creating a stepped gun barrel on the stabil- The main aim of this study is to determine the stability ity boundaries. It may be realized that increasing the radius boundaries for gun barrels. Such structures may be modeled in parts of the gun barrel will automatically increase the sta- by a cantilever beam. Using the cantilever boundary condi- bility range as the stiffness increases. Rather, the study would tions in the above procedure, and setting the foundation stiff- be on how much that increase in radius should be so as not to ness to zero, we get the critical speed boundaries as presented add unnecessary weight. In the following, the cases, the 8 mm in Figure 2. The relation may be approximated by the for- outer radius of the barrel will be the base for the compari- mula son. The barrel will be divided into twelve equal-length parts −0.5 β | ≈ 0.21α . (23) cr Cantilever six of which have 8 mm radius and the other six will have the same radius but with a value more than 8 mm. Figure 3 Another common configuration of the gun barrel is the presents stability boundaries for barrels with different radius clamped-pinned configuration. This configuration is used to ratios (RR). As clearly evident, the barrels with steps of more support the tip of the gun barrel and, hence, increases its sta- than 8 mm have higher stability boundaries. But it is also ev- bility. When that configuration was used in the program, the ident that 8 to 9 ratio achieved the highest change in the sta- results obtained were similar to all the other cases. The re- bility boundaries compared to the changes occurring when lation between the mass parameter and the critical velocity increasing the radius ratio from 8 to 9 to 8 to 10. Thus the parameters may be given by highest achievement, in terms of gain in stability boundaries, −0.5 β | ≈ 0.45α . (24) was by increasing the radius by only 12.5%. cr Clamped-Pinned RR = 1 Mohammad Tawfik 5 0.2 0 00.20.40.60.81 Barrel mass ignored −0.2 00.20.40.60.81 −0.2 −0.4 β = 0.2 −0.4 β = 0.4 −0.6 −0.6 β = 1 RR = 8/10 RR = 8/9 −0.8 Barrel mass considered −0.8 −1 −1 RR = 8/12 RR = 1 −1.2 −1.2 −1.4 −1.4 Length (X) Length (X) Figure 4: Response of the barrel’s tip versus bullet position with Figure 5: Response of the barrel’s tip versus bullet position for and without considering the barrels weight for α = 0.2 and different stepped beams with different radius ratios with α = 0.2and β = 0.2. values of β. 3.2. Time response 00.10.20.30.40.50.60.70.80.91 −0.2 A program was developed for the time response of the barrel −0.4 to moving bullets using the Newmark technique presented −0.6 earlier and the results were compared to response presented RR = 8/10 RR = 8/9 in [16] with the consideration that the reference did not in- −0.8 clude the effect of the external work done by the bullet on the −1 RR = 8/12 barrel. The values used for the algorithm parameters were −1.2 δ = 0.52 and μ = 0.27. In all numerical results for the dynamics, presented in this section, the time step used was −1.4 1/2400 of the total time required for the bullet to transverse −1.6 the barrel. Convergence of the solution was tested using dif- Length (X) ferent values of the time step and it was found that this value was accurate and convenient. Figure 6: Response of the barrel’s tip versus bullet position for The response of the barrel to the motion of the bullet is stepped beams with different radius ratios with α = 0.2and β = 1.0. one of the important aspects that should be considered when designing a gun barrel. As the instability described in the pre- vious section is a static instability, pitchfork bifurcation, it is reflected in higher dynamic response to the external excita- calculated as tion, rather than self-excited vibrations that are associated with Hopf bifurcations. The higher the response becomes, w = . (25) normalized the more the time between the bullets should be to ensure ρALg/(EI/L ) the accuracy of target hits. One main observation from the literature that studied the response of beams to moving masses or loads is that most This normalization of the deflection will ensure the same re- of them ignored the deflection of the beam due to its own sponse curve for the same values of α and β regardless of the weight. That type of deflection was ignored mostly because beam properties and geometry. such studies were directed to cases where the load is much Investigating the effect of the stepping of the gun barrel, more than the beam weight, as in the case of train moving as in the previous section, on the response to the motion of on a railroad. In our problem, the bullet mass, in most guns, the bullet was the following step. Figure 5 presents the re- is much less than the barrel mass. Hence, ignoring the ini- sponse of the barrel motion of the bullet with α = 0.2and tial deflection due to mass may introduce much difference β = 0.2, while Figure 6 presents the response with α = 0.2 in the response. To illustrate the effect of the initial deflec- and β = 1.0. tions, the response of the barrel to the moving mass is plot in The results in Figures 5 and 6 both agree on that the re- the cases where the barrel mass was not considered. Figure 4 sponse of the plain, less stiff, beam is higher than that for shows clearly that the response, while ignoring the mass, is stepped beams in general. It may be also observed that the not a mere shift downwards for the curves, rather, the re- gain, in terms of tip vibration reduction, obtained by in- sponse, especially at high values of the speed parameter, has creasing the step ratio to 8 to 9 is very good especially that acompletelydifferent pattern. The normalized deflection is it presents the least increase in weight. Normalized displacement Normalized displacement Normalized displacement 6 Advances in Acoustics and Vibration 4. CONCLUSIONS [8] M. Ruzzene and A. Baz, “Response of periodically stiffened shells to a moving projectile propelled by an internal pressure In this study, a finite element model was used to predict the wave,” Mechanics of Advanced Materials and Structures, vol. 13, stability boundaries for beams with moving masses subject no. 3, pp. 267–284, 2006. [9] O. J. Aldraihem and A. Baz, “Dynamic stability of stepped to different boundary conditions. Because of the lack of lit- beams under moving loads,” JournalofSound andVibration, erature, the results were compared to classical solutions pre- vol. 250, no. 5, pp. 835–848, 2002. sented for infinite beams simply supported at equal intervals. [10] O. J. Aldraihem and A. Baz, “Moving-loads-induced instabil- As predicted, the solution of SS beams underpredicted the ity in stepped tubes,” Journal of Vibration and Control, vol. 10, stability boundaries compared to classical solution. Mean- no. 1, pp. 3–23, 2004. while, clamped-clamped beams showed, almost, no change [11] C. S. Hsu, “Impulsive parametric excitation: theory,” Journal with foundation stiffness. of Applied Mechanics, vol. 39, pp. 551–558, 1972. An empirical relation between the mass parameter and [12] C. S. Hsu and W. H. Cheng, “Applications of the theory of critical speed parameter could be obtained for simply sup- impulsive parametric excitation and new treatments of general ported, clamped-clamped, cantilever, and clamped-pinned parametric excitation problems,” Journal of Applied Mechanics, beams. Generally, the relation was given by the relation β ≈ vol. 40, no. 1, pp. 78–86, 1973. cr −0.5 [13] S. Mackertich, “Dynamic stability of a beam excited by a se- cα ,where c is a constant that is determined by the bound- quence of moving mass particles,” Journal of the Acoustical So- ary conditions of the beam. ciety of America, vol. 115, no. 4, pp. 1416–1419, 2004. Also the effect of using stepped barrels was studied to in- [14] S. Krenk, “Energy conservation in Newmark based time inte- vestigate the feasibility of such techniques. It was found that gration algorithms,” Computer Methods in Applied Mechanics with, considerably, small increase in the radius of the barrel and Engineering, vol. 195, no. 44–47, pp. 6110–6124, 2006. in some parts, a significant increase in the stability bound- [15] M. M. Alaa El-Din and M. Tawfik, “Vibration attenuation in aries was obtained. rotating beams with periodically distributed piezoelectric con- In this study, for the first time in literature, the stability trollers,” in Proceedings of the 13th International Congress on boundaries were predicted using eigenvalues of the system Sound and Vibration (ICSV ’06), Vienna, Austria, July 2006. rather than using time marching techniques. The results pre- [16] D. R. Parhi and A. K. Behera, “Dynamic deflection of a cracked sented in this paper are, to the extent of the authors knowl- beam with moving mass,” Proceedings of the Institution of Me- chanical Engineers, Part C, vol. 211, no. 1, pp. 77–87, 1997. edge, the first in the literature to present accurately the rela- tion between the mass parameter and critical speed parame- ter for beams with general boundary conditions. Further, the model was used to predict the response of the tip of the gun barrel to the motion of the bullet using New- mark algorithm. The results emphasized that the weight of the barrel should be included in the calculations as constant force distributed on the beam. Also the results showed that the response may also be reduced using the stepped barrels. REFERENCES [1] H. D. Nelson and R. A. Conover, “Dynamic stability of a beam carrying moving masses,” Journal of Applied Mechanics, vol. 38, no. 4, pp. 1003–1006, 1971. [2] G. A. Benedetti, “Transverse vibration and stability of a beam subject to moving mass loads,” PhD dissertation, Civil Engi- neering, Arizona State University, Tempe, Ariz, USA, 1973. [3] G. A. Benedetti, “Dynamic stability of a beam loaded by a se- quence of moving mass particles,” Journal of Applied Mechan- ics, vol. 41, no. 4, pp. 1069–1071, 1974. [4] M. A. Foda and Z. Abduljabbar, “A dynamic green function formulation for the response of a beam structure to a moving mass,” Journal of Sound and Vibration, vol. 210, no. 3, pp. 295– 306, 1998. [5] M. Ichikawa, Y. Miyakawa, and A. Matsuda, “Vibration analy- sis of the continuous beam subjected to a moving mass,” Jour- nal of Sound and Vibration, vol. 230, no. 3, pp. 493–506, 2000. [6] R. Katz, C. W. Lee, A. G. Ulsoy, and R. A. Scott, “Dynamic stability and response of a beam subject to a deflection depen- dent moving load,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 109, no. 4, pp. 361–365, 1987. [7] Y. H. Lin and M. W. Trethewey, “Finite element analysis of elastic beams subjected to moving dynamic loads,” Journal of Sound and Vibration, vol. 136, no. 2, pp. 323–342, 1990. 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Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets

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Hindawi Publishing Corporation
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Copyright © 2008 Mohammad Tawfik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 483857, 6 pages doi:10.1155/2008/483857 Research Article Dynamics and Stability of Stepped Gun Barrels with Moving Bullets Mohammad Tawfik Mechanical Engineering Department, British University in Egypt, El-Shorouk City, Cairo 11837, Egypt Correspondence should be addressed to Mohammad Tawfik, mohammad.tawfik@bue.edu.eg Received 12 August 2007; Revised 16 November 2007; Accepted 20 January 2008 Recommended by Luc Gaudiller The stability of an Euler-Bernoulli beam under the effect of a moving projectile will be reintroduced using simple eigenvalue analysis of a finite element model. The eigenvalues of the beam change with the mass, speed, and position of the projectile, thus, the eigenvalues are evaluated for the system with differentspeedsand massesatdifferent positions until the lowest eigenvalue reaches zero indicating the instability occurrence. Then a map for the stability region may be obtained for different boundary conditions. Then the dynamics of the beam will be investigated using the Newmark algorithm at different values of speed and mass ratios. Finally, the effect of using stepped barrels on the stability and the dynamics is going to be investigated. It is concluded that the technique used to predict the stability boundaries is simple, accurate, and reliable, the mass of the barrel on the dynamics of the problem cannot be ignored, and that using the stepped barrels, with small increase in the diameter, enhances the stability and the dynamics of the barrel. Copyright © 2008 Mohammad Tawfik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION Recently, the emergence of the need for very light guns that are mounted on combat aircraft reintroduced the sta- The problem of the dynamics and stability of beams carry- bility problem with new conditions. The motion of the bul- ing moving masses drew a lot of attention in the past half lets inside the gun barrels introduces compression force on century due to the applications that require it such as fast the shell walls, in turn, this compression may cause dynamic trains, motion on bridges, and light guns mounted on air- buckling [2] and excessive vibration in the shell wall [8]. This craft. In 1971, Nelson and Conover [1]presented astudy of type of instability, though of major importance, will not be covered in this study. the problem of an infinite thin beam with periodically dis- tributed simplesupports resting on elastic foundation with a The problem of projectiles inside gun barrels, though train of masses moving on it at constant speed. They applied of important application, was not the subject of many Galerkin method to the proposed approximate solution to researches. In [9, 10], the stability problem was studied using get the system equations then applied the Floquet theorem the impulsive parametric excitation theory [11, 12] for a thin to get the stability boundaries for the periodically repeating beam with periodically distributed controllers. The study system. Simultaneously, Benedetti [2, 3] used a similar ap- ignored the effect of the dynamics of the barrel shell. The proach and could present an analytical relation between the problem of the dynamics of the barrel shell under the effect mass parameter and the critical speed parameter using clas- of the moving projectile with shock and expansion waves sical techniques. was studied in [8] but the study did not tackle the stability Since most of the research was directed to civil structures problem. In [13], the author presented one of the very rare and lathe-machined work pieces, the stability regions were studies that handled finite beams. In that study, the beam not of major interest for general structures. Rather, the re- under investigation was modeled as a Timoshenko beam sponse of the structure to moving loads or masses presented a with simple supports and elastic foundation. The results more practical problem. Further, most of the studies were in- presented different cases of multiple masses and foundation terested in problems with periodically supported beams that stiffness but did not present any comparison with published simulate train rails (see [4–7] as examples for such studies). or experimental results. 2 Advances in Acoustics and Vibration In this study, the stability of an Euler-Bernoulli beam un- And the variation of the potential energy may be written as der the effect of a moving projectile will be reintroduced us- 2 2 ing simple eigenvalue analysis of a finite element model. The ∂ δw ∂ w δU = EI + kδw·w − ρAgδw dx 2 2 eigenvalues of the beam change with the mass, speed, and ∂x ∂x position of the projectile, thus the eigenvalues are evaluated − mg (δw) , for the system with different speeds and masses at different x=x (6) positions until the lowest eigenvalue reaches zero indicating the instability occurrence. Then a map for the stability region while the variation of the work done by the bullet may be may be obtained for different boundary conditions. Then the written as dynamics of the beam will be investigated using the New- mark algorithm at different values of speed and mass ratios. ∂δw ∂w δW = mV . (7) Finally, the effect of using stepped barrels on the stability and bullet ∂x ∂x x=x the dynamics is going to be investigated. Applying the Hamilton principle and using the standard beam finite element interpolation function, we may write the 2. MODEL element equation of motion as The model derived in this section will have the following as- sumptions: the gun barrel will be modeled as thin beam that [M]+ M w ¨ + C w ˙ bullet bullet follows the Euler-Bernoulli theorem, the barrel deflections + [K]+ K − K {w}= f + f , are small, and the effect of elastic foundation will be included s bullet w bullet for the purpose of comparison with published data. (8) The Hamilton principle states that where the beam mass matrix is evaluated by t t 2 2 δΠdt = δ T − U + W dt = 0, (1) bullet t t 1 1 [M] = ρA N (x) N (x) dx,(9) where T is the kinetic energy, U is the potential energy, and where N (x) are the beam interpolation polynomials. The ef- W is the work done by the bullet on the beam. The ki- bullet fective bullet mass matrix is evaluated at the bullet position netic energy of the system may be written as as 1 1 ∂w 2 2 2 M = m N (x) N (x) . (10) bullet T = ρAw ˙ dx + m w ˙ +2Vw ˙ + V ,(2) x=x 2 2 ∂x 0 x=x While the beam stiffness matrix is evaluated by where ρ is the beam mass density, A is the beam cross section area, m is the bullet mass, V is the bullet speed, w is the beam [K ] = EI N (x) N (x) dx. (11) xx xx transverse displacement, and x is the position of the bullet. The potential energy of the system may be written as The foundation stiffness matrix is evaluated by 1 ∂ w 2 l U = EI + kw − ρAgw dx − mg (w) , x=x 2 ∂x K = γEI N (x) N (x) dx, (12) 0 s (3) where γ is the foundation stiffness ratio given by γ = k/EI . where E is the beam modulus of elasticity, I is the beam sec- The effective geometric bullet stiffness matrix is evaluated at ond moment of area, k is the foundation stiffness, and g is the bullet position as the gravitational acceleration. And the work done by the bul- let may be written as 2 K = mV N (x) N (x) . (13) bullet xx xx x=xm 1 ∂w The effective bullet Coriolis matrix is evaluated by W = mV . (4) bullet 2 ∂x x=x C = 2mV N (x) N (x) . (14) bullet x x=x The variation of the kinetic energy may be written as The forces due to the barrel and bullet weights are evaluated, respectively, by δT = ρAδw ˙ f =− N (x) ρAg dx, 1 ∂w ∂δw ˙ ˙ ˙ ˙ ˙ ·wdx + m δw·w +2Vδw +2Vw . (15) 2 ∂x ∂x x=x f =− N (x) mg . (5) bullet x=x m Mohammad Tawfik 3 2.1. Stability boundaries To obtain the eigenvalues of the system, including the effect Unstable of all components, we will need to transform the system into a first-order system by the standard transformation, Single bay β 1 z = w. ˙ (16) 0.01 0.110 Stable Using the above transformation, we will obtain the homoge- neous equation of motion in the form w ˙ 0 I w n×n = , (17) 0.1 z MK MC z α −1 where MK denotes − [M]+ M [K ]+ K − K ) bullet s bullet Single bay-γ = 0 Benedetti-γ = 1 −1 CC-γ = 0 Single bay-γ = 2 andMCdenotes − [M]+ M C . bullet bullet Benedetti-γ = 0 CC-γ = 2 For the above system, the eigenvalues should represent Single bay-γ = 1 Benedetti-γ = 2 the natural frequencies of oscillation of the beam with the CC-γ = 1 bullet. The eigenvalues should all be complex with nonzero imaginary parts for all values of the speed that are below the Figure 1: Stability boundaries for different cases. critical speed. As the bullet speed reaches the critical speed, the smallest complex pair will have zero imaginary parts. The search for the critical values of the speed may be done (2) then the velocity and displacement may be evaluated using the following algorithm. using (1) Select the bullet mass. {w ˙} ={w ˙} +(1 − δ)Δt{w ¨} + μΔt{w ¨} , (2) Select the bullet speed. t+Δt t t t+Δt (3) Change the value of the x-location of the bullet and 2 2 evaluate the eigenvalues of the system. {w} ={w} + Δt{w ˙} + − μ Δt {w ¨} + μΔt {w ¨} . t+Δt t t t t (4) If all eigenvalues have nonzero imaginary parts then (20) increase speed and go to step (3), else go to step (6). (5) If all speed values did not reach the critical value, then In the above algorithm, δ and μ are parameters that reset speed value and increase mass up to a given limit have to obey the constraints δ ≥ 0.5and μ ≥ and proceed to step (3). If mass limit is reached, termi- (1/4)(δ +(1/2)) . nate. (6) Store the values of the critical speed and the mass. 3. RESULTS AND DISCUSSION (7) Increase speed up to a given limit and proceed to step (3). If mass limit is reached, terminate. 3.1. Stability boundaries The above procedure may be repeated for all values of mass, A program was written using MATLAB to perform the cal- speed, and boundary conditions and different information culations of the problem using the following data: Beam may be compiled out of the extracted data. modulus of elasticity 71 GPa, Density 2840 kg/m3, inner ra- dius 0.007 m, and outer radius 0.008 m. The projectile mass 2.2. Time response ranged from 0.005 to 0.75 Kg, and its speed ranged from 2.5 to 600 m/s. Two cases were used to demonstrate the valid- The time response of the system presented by (8)may be ob- ity of the procedure, simply supported and clamped at both tained using the Newmark algorithm as presented in [14]. The algorithm may be presented as follows. sides. Each case was run with three values of foundation stiff- ness γ = 0, 1, 2. The program used 10 3-node beam elements, For the system with 5th-order polynomial, (see [15]), and checked for the [M]{w ¨} +[C]{w ˙} +[K ]{w}= f , (18) w eigenvalues at 20 equidistant points in each element. Note that the 2-node element, 3rd-order polynomial, was checked (1) evaluate the acceleration of the system using for accuracy and the results were not of any considerable dif- ference, rather, the program was already written for 3-node [M]+ δΔt[C]+ μΔt [K ] {w ¨} t+Δt elements and thus used. The convergence of the solution was checked and it was found that the 10 elements with 20 inter- = f − [C] w ˙ +(1 − δ)Δt w ¨ t+Δt t t nal points gave adequate accuracy. Figure 1 shows the results of the two cases with the three − [K ] {w} + Δt w ˙ + − μ Δt w ¨ ; t t foundation stiffness values compared to those obtained from the equation given by Benedetti [2]. It can be obviously (19) 4 Advances in Acoustics and Vibration seen in that graph that the results obtained for the clamped- clamped beam arealmost identical for the three values of γ. Meanwhile, all the results obtained using the simply sup- ported beam gave lower values for the critical speed. The re- sults obtained from the equation by Benedetti [2], however, varied from nearest to single bay (simply supported) (γ = 0) β 1 to almost identical to clamped-clamped case (γ = 1) ending 0.01 0.11 1 0 with just being higher than the clamped-clamped case with γ = 2. Note that the fundamental solution of Benedetti used above gives the relation between the critical speed factor, β , cr and the mass factor, α,as 0.1 −0.5 β = 1+ γα , (21) cr Figure 2: Stability boundaries for cantilever beam. where α = m/ρAL and β = VL/(2π ρA/EI ). It has to be noted at this point that the results that were presented by Benedetti [2, 3] and Nelson and Conover [1] were obtained for the case of an infinite beam that is periodically simply supported. Thus such a structure should be expected to be more stable than a single bay of simply supported beam. However, the results showed that the clamped-clamped beam (single bay) showed very little change with the foundation stiffness. Finally, it is clear that the results of Benedettis for- mula gave values that are much like those of a clamped- clamped beam. Now the most important observation that may be taken from Figure 1 is that when the stability boundaries are all 0.1 drawn on log-log scale the results all showed linear trends, 0.01 0.11 1 0 further, all lines are parallel with a slope of −0.5 (note that the formula presented by Benedetti had a relation between RR = 1 RR = 8/10 −0.5 α and β in the form of β = aα , see the relation above). cr RR = 8/9 RR = 8/12 Thus it may be concluded that the procedure presented in Figure 3: Stability boundaries for stepped cantilever beam. this paper can accurately predict the stability boundaries of the problem with different boundary conditions. Further, us- ing regression techniques, we may write down a relation for The above relation reflects the expected increase in the sta- the critical speed factor as bility of the gun barrel when supported at the tip. −0.5 β | ≈ 0.195 4+ γα , cr S-S Now that the stability boundaries for beams with differ- (22) ent boundary conditions are realized to be a simple relation −0.5 β | ≈ 0.77α . cr C-C between the mass and velocity parameters, we need to inves- tigate the effect of creating a stepped gun barrel on the stabil- The main aim of this study is to determine the stability ity boundaries. It may be realized that increasing the radius boundaries for gun barrels. Such structures may be modeled in parts of the gun barrel will automatically increase the sta- by a cantilever beam. Using the cantilever boundary condi- bility range as the stiffness increases. Rather, the study would tions in the above procedure, and setting the foundation stiff- be on how much that increase in radius should be so as not to ness to zero, we get the critical speed boundaries as presented add unnecessary weight. In the following, the cases, the 8 mm in Figure 2. The relation may be approximated by the for- outer radius of the barrel will be the base for the compari- mula son. The barrel will be divided into twelve equal-length parts −0.5 β | ≈ 0.21α . (23) cr Cantilever six of which have 8 mm radius and the other six will have the same radius but with a value more than 8 mm. Figure 3 Another common configuration of the gun barrel is the presents stability boundaries for barrels with different radius clamped-pinned configuration. This configuration is used to ratios (RR). As clearly evident, the barrels with steps of more support the tip of the gun barrel and, hence, increases its sta- than 8 mm have higher stability boundaries. But it is also ev- bility. When that configuration was used in the program, the ident that 8 to 9 ratio achieved the highest change in the sta- results obtained were similar to all the other cases. The re- bility boundaries compared to the changes occurring when lation between the mass parameter and the critical velocity increasing the radius ratio from 8 to 9 to 8 to 10. Thus the parameters may be given by highest achievement, in terms of gain in stability boundaries, −0.5 β | ≈ 0.45α . (24) was by increasing the radius by only 12.5%. cr Clamped-Pinned RR = 1 Mohammad Tawfik 5 0.2 0 00.20.40.60.81 Barrel mass ignored −0.2 00.20.40.60.81 −0.2 −0.4 β = 0.2 −0.4 β = 0.4 −0.6 −0.6 β = 1 RR = 8/10 RR = 8/9 −0.8 Barrel mass considered −0.8 −1 −1 RR = 8/12 RR = 1 −1.2 −1.2 −1.4 −1.4 Length (X) Length (X) Figure 4: Response of the barrel’s tip versus bullet position with Figure 5: Response of the barrel’s tip versus bullet position for and without considering the barrels weight for α = 0.2 and different stepped beams with different radius ratios with α = 0.2and β = 0.2. values of β. 3.2. Time response 00.10.20.30.40.50.60.70.80.91 −0.2 A program was developed for the time response of the barrel −0.4 to moving bullets using the Newmark technique presented −0.6 earlier and the results were compared to response presented RR = 8/10 RR = 8/9 in [16] with the consideration that the reference did not in- −0.8 clude the effect of the external work done by the bullet on the −1 RR = 8/12 barrel. The values used for the algorithm parameters were −1.2 δ = 0.52 and μ = 0.27. In all numerical results for the dynamics, presented in this section, the time step used was −1.4 1/2400 of the total time required for the bullet to transverse −1.6 the barrel. Convergence of the solution was tested using dif- Length (X) ferent values of the time step and it was found that this value was accurate and convenient. Figure 6: Response of the barrel’s tip versus bullet position for The response of the barrel to the motion of the bullet is stepped beams with different radius ratios with α = 0.2and β = 1.0. one of the important aspects that should be considered when designing a gun barrel. As the instability described in the pre- vious section is a static instability, pitchfork bifurcation, it is reflected in higher dynamic response to the external excita- calculated as tion, rather than self-excited vibrations that are associated with Hopf bifurcations. The higher the response becomes, w = . (25) normalized the more the time between the bullets should be to ensure ρALg/(EI/L ) the accuracy of target hits. One main observation from the literature that studied the response of beams to moving masses or loads is that most This normalization of the deflection will ensure the same re- of them ignored the deflection of the beam due to its own sponse curve for the same values of α and β regardless of the weight. That type of deflection was ignored mostly because beam properties and geometry. such studies were directed to cases where the load is much Investigating the effect of the stepping of the gun barrel, more than the beam weight, as in the case of train moving as in the previous section, on the response to the motion of on a railroad. In our problem, the bullet mass, in most guns, the bullet was the following step. Figure 5 presents the re- is much less than the barrel mass. Hence, ignoring the ini- sponse of the barrel motion of the bullet with α = 0.2and tial deflection due to mass may introduce much difference β = 0.2, while Figure 6 presents the response with α = 0.2 in the response. To illustrate the effect of the initial deflec- and β = 1.0. tions, the response of the barrel to the moving mass is plot in The results in Figures 5 and 6 both agree on that the re- the cases where the barrel mass was not considered. Figure 4 sponse of the plain, less stiff, beam is higher than that for shows clearly that the response, while ignoring the mass, is stepped beams in general. It may be also observed that the not a mere shift downwards for the curves, rather, the re- gain, in terms of tip vibration reduction, obtained by in- sponse, especially at high values of the speed parameter, has creasing the step ratio to 8 to 9 is very good especially that acompletelydifferent pattern. The normalized deflection is it presents the least increase in weight. Normalized displacement Normalized displacement Normalized displacement 6 Advances in Acoustics and Vibration 4. CONCLUSIONS [8] M. Ruzzene and A. Baz, “Response of periodically stiffened shells to a moving projectile propelled by an internal pressure In this study, a finite element model was used to predict the wave,” Mechanics of Advanced Materials and Structures, vol. 13, stability boundaries for beams with moving masses subject no. 3, pp. 267–284, 2006. [9] O. J. Aldraihem and A. Baz, “Dynamic stability of stepped to different boundary conditions. Because of the lack of lit- beams under moving loads,” JournalofSound andVibration, erature, the results were compared to classical solutions pre- vol. 250, no. 5, pp. 835–848, 2002. sented for infinite beams simply supported at equal intervals. [10] O. J. Aldraihem and A. Baz, “Moving-loads-induced instabil- As predicted, the solution of SS beams underpredicted the ity in stepped tubes,” Journal of Vibration and Control, vol. 10, stability boundaries compared to classical solution. Mean- no. 1, pp. 3–23, 2004. while, clamped-clamped beams showed, almost, no change [11] C. S. Hsu, “Impulsive parametric excitation: theory,” Journal with foundation stiffness. of Applied Mechanics, vol. 39, pp. 551–558, 1972. An empirical relation between the mass parameter and [12] C. S. Hsu and W. H. Cheng, “Applications of the theory of critical speed parameter could be obtained for simply sup- impulsive parametric excitation and new treatments of general ported, clamped-clamped, cantilever, and clamped-pinned parametric excitation problems,” Journal of Applied Mechanics, beams. Generally, the relation was given by the relation β ≈ vol. 40, no. 1, pp. 78–86, 1973. cr −0.5 [13] S. Mackertich, “Dynamic stability of a beam excited by a se- cα ,where c is a constant that is determined by the bound- quence of moving mass particles,” Journal of the Acoustical So- ary conditions of the beam. ciety of America, vol. 115, no. 4, pp. 1416–1419, 2004. Also the effect of using stepped barrels was studied to in- [14] S. Krenk, “Energy conservation in Newmark based time inte- vestigate the feasibility of such techniques. It was found that gration algorithms,” Computer Methods in Applied Mechanics with, considerably, small increase in the radius of the barrel and Engineering, vol. 195, no. 44–47, pp. 6110–6124, 2006. in some parts, a significant increase in the stability bound- [15] M. M. Alaa El-Din and M. Tawfik, “Vibration attenuation in aries was obtained. rotating beams with periodically distributed piezoelectric con- In this study, for the first time in literature, the stability trollers,” in Proceedings of the 13th International Congress on boundaries were predicted using eigenvalues of the system Sound and Vibration (ICSV ’06), Vienna, Austria, July 2006. rather than using time marching techniques. The results pre- [16] D. R. Parhi and A. K. Behera, “Dynamic deflection of a cracked sented in this paper are, to the extent of the authors knowl- beam with moving mass,” Proceedings of the Institution of Me- chanical Engineers, Part C, vol. 211, no. 1, pp. 77–87, 1997. edge, the first in the literature to present accurately the rela- tion between the mass parameter and critical speed parame- ter for beams with general boundary conditions. Further, the model was used to predict the response of the tip of the gun barrel to the motion of the bullet using New- mark algorithm. The results emphasized that the weight of the barrel should be included in the calculations as constant force distributed on the beam. Also the results showed that the response may also be reduced using the stepped barrels. REFERENCES [1] H. D. Nelson and R. A. Conover, “Dynamic stability of a beam carrying moving masses,” Journal of Applied Mechanics, vol. 38, no. 4, pp. 1003–1006, 1971. [2] G. A. Benedetti, “Transverse vibration and stability of a beam subject to moving mass loads,” PhD dissertation, Civil Engi- neering, Arizona State University, Tempe, Ariz, USA, 1973. [3] G. A. Benedetti, “Dynamic stability of a beam loaded by a se- quence of moving mass particles,” Journal of Applied Mechan- ics, vol. 41, no. 4, pp. 1069–1071, 1974. [4] M. A. Foda and Z. Abduljabbar, “A dynamic green function formulation for the response of a beam structure to a moving mass,” Journal of Sound and Vibration, vol. 210, no. 3, pp. 295– 306, 1998. [5] M. Ichikawa, Y. Miyakawa, and A. Matsuda, “Vibration analy- sis of the continuous beam subjected to a moving mass,” Jour- nal of Sound and Vibration, vol. 230, no. 3, pp. 493–506, 2000. [6] R. Katz, C. W. Lee, A. G. Ulsoy, and R. A. Scott, “Dynamic stability and response of a beam subject to a deflection depen- dent moving load,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 109, no. 4, pp. 361–365, 1987. [7] Y. H. Lin and M. W. Trethewey, “Finite element analysis of elastic beams subjected to moving dynamic loads,” Journal of Sound and Vibration, vol. 136, no. 2, pp. 323–342, 1990. 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