# Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets

Authors: Mohammad Tawfik
Journal: Advances in Acoustics and Vibration
 (2008)

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## Abstract

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 different speeds and masses at different 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.


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## Body

## 1. Introduction and Literature Survey

The problem of the dynamics and stability of beams carrying moving
masses drew a lot of attention in the past half century due to the applications
that require it such as fast trains, motion on bridges, and light guns mounted
on aircraft. In 1971, Nelson and Conover [1]
presented
a
study of the problem of an infinite thin beam with periodically distributed
simple supports resting on elastic foundation with a train of masses moving on
it at constant speed. They applied Galerkin method to the proposed approximate
solution to get the system equations then applied the Floquet theorem to get
the stability boundaries for the periodically repeating system. Simultaneously,
Benedetti [2, 3]
used a similar approach and could present an analytical relation between the
mass parameter and the critical speed parameter using classical techniques.Since most of the research was directed to civil structures and
lathe-machined work pieces, the stability regions were not of major interest
for general structures. Rather, the response of the structure to moving loads or
masses presented a more practical problem. Further, most of the studies were
interested in problems with periodically supported beams that simulate train
rails (see [4–7] as examples
for such studies).Recently, the emergence of the need for very light guns that are mounted
on combat aircraft reintroduced the stability problem with new conditions. The
motion of the bullets inside the gun barrels introduces compression force on
the shell walls, in turn, this compression may cause dynamic buckling [2]
and excessive vibration in the shell wall [8].
This type of instability, though of major importance, will not be covered in
this study.The problem of projectiles inside gun barrels, though of important
application, was not the subject of many researches. In [9, 10],
the stability problem was studied using the impulsive parametric excitation
theory [11, 12] for a thin beam
with periodically distributed controllers. The study ignored the effect of the
dynamics of the barrel shell. The problem of the dynamics of the barrel shell
under the effect of the moving projectile with shock and expansion waves was
studied in [8]
but the study did not tackle the stability problem. In [13], the author
presented one of the very rare studies that handled finite beams. In that
study, the beam under investigation was modeled as a Timoshenko beam with
simple supports and elastic foundation. The results presented different cases
of multiple masses and foundation stiffness but did not present any comparison
with published or experimental results.In this study, 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 different speeds and masses at different 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.


## 2. Model

The model derived in this section will have the following assumptions:
the gun barrel will be modeled as thin beam that follows the Euler-Bernoulli
theorem, the barrel deflections are small, and the effect of elastic foundation
will be included for the purpose of comparison with published data.The Hamilton principle states that(1)∫t1t2δΠdt=∫t1t2δ(T-U+Wbullet)dt=0,  where  T is the kinetic energy,  U is the potential energy,
and Wbullet is the work done by the bullet on the beam. The
kinetic energy of the system may be written as (2)T=12∫0lρAw˙2dx+12m(w˙2+2Vw˙∂w∂x+V2)x=xm,  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 transverse displacement, and xm is the position of the bullet. The potential
energy of the system may be written as
(3)U=12∫0l(EI(∂2w∂x2)2+kw2-ρAgw)dx-mg(w)x=xm,  where E is the beam modulus of elasticity, I is the beam second moment of area, k is the foundation stiffness, and  g is
the gravitational acceleration. And the work done by the bullet may be written
as
(4)Wbullet=12(mV2(∂w∂x)2)|x=xm.  The variation of the kinetic energy may be written as
(5)δT=∫0lρAδw˙⋅w˙dx+12m(δw˙⋅w˙+2Vδw˙∂w∂x+2Vw˙∂δw∂x)x=xm.  And the variation of the potential energy may be written as
(6)δU=∫0l(EI(∂2δw∂x2)(∂2w∂x2)+kδw⋅w-ρAgδw)dx-mg(δw)x=xm,  while the variation of the work done by the bullet may be written as
(7)δWbullet=(mV2∂δw∂x∂w∂x)|x=xm.  Applying the Hamilton principle and using the standard beam finite
element interpolation function, we may write the element equation of motion as
(8)([M]+[Mbullet]){w¨}+[Cbullet]{w˙}+([K]+[Ks]-[Kbullet]){w}={fw}+{fbullet}, where the beam mass matrix is evaluated by
(9)[M]=∫0lρA{N(x)}T{N(x)}dx,  where Nx are the beam interpolation polynomials. The effective
bullet mass matrix is evaluated at the bullet position as
(10)[Mbullet]=m{N(x)}T{N(x)}|x=xm.  While the beam stiffness matrix is evaluated by
(11)[K]=∫0lEI{Nxx(x)}T{Nxx(x)}dx.  The foundation stiffness matrix is evaluated by
(12)[Ks]=∫0lγEI{N(x)}T{N(x)}dx,  where γ is the foundation stiffness ratio given by γ=k/EI. The
effective geometric bullet stiffness matrix is evaluated at the bullet position
as
(13)[Kbullet]=mV2{Nxx(x)}T{Nxx(x)}|x=xm.  The effective bullet Coriolis matrix is evaluated by
(14)[Cbullet]=2mV{N(x)}T{Nx(x)}|x=xm.  The forces due to the barrel and bullet weights are evaluated, respectively,
by
(15){fw}=-∫0l{N(x)}TρAgdx,{fbullet}=-{N(x)}Tmg|x=xm.


## 2.1. Stability Boundaries

To obtain the eigenvalues of the system, including the effect of all
components, we will need to transform the system into a first-order system by
the standard transformation,(16)z=w˙.
Using the above transformation, we will obtain the homogeneous equation
of motion in the form
(17){w˙z˙}=[0In×nMKMC]{wz},
where MK denotes -([M]+[Mbullet])-1([K]+[Ks]-[Kbullet]) and MC denotes -([M]+[Mbullet])-1[Cbullet].For the above system, the eigenvalues should represent the natural
frequencies of oscillation of the beam with the bullet. The eigenvalues should
all be complex with nonzero imaginary parts for all values of the speed that
are below the 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 using the
following algorithm.(1)
Select the bullet mass.(2)
Select the bullet speed.(3)
Change the value of  thex-location of the bullet and evaluate the eigenvalues
of the system.(4)
If all eigenvalues have nonzero imaginary parts then increase speed and
go to step (3), else go to step (6).(5)
If all speed values did not reach the critical value, then reset speed
value and increase mass up to a given limit and proceed to step (3).
If mass limit is reached, terminate.(6)
Store the values of the critical speed and the mass.(7)
Increase speed up to a given limit and proceed to step (3). If mass limit
is reached, terminate. The above procedure may be repeated for all values of mass, speed, and
boundary conditions and different information may be compiled out of the
extracted data.


## 2.2. Time Response

The time response of the system presented by (8) may be obtained using
the Newmark algorithm as presented in [14]. The
algorithm may be presented
as follows.For the system(18)[M]{w¨}+[C]{w˙}+[K]{w}={fw},(1)
evaluate the acceleration of the system using(19)([M]+δΔt[C]+μΔt2[K]){w¨}t+Δt={fw}t+Δt-[C]({w˙}t+(1-δ)Δt{w¨}t)-[K]({w}t+Δt{w˙}t+(12-μ)Δt2{w¨}t);(2)
then the velocity and displacement may be evaluated using(20){w˙}t+Δt={w˙}t+(1-δ)Δt{w¨}t+μΔt{w¨}t+Δt,{w}t+Δt={w}t+Δt{w˙}t+(12-μ)Δt2{w¨}t+μΔt2{w¨}t.
In the above algorithm,  δ and  μ are parameters that have
to obey the constraints δ≥0.5 and μ≥(1/4)(δ+(1/2))2.


## 3. Results and Discussion

## 3.1. Stability Boundaries

A program was written using MATLAB to perform the calculations of the
problem using the following data: Beam modulus of elasticity 71 GPa, Density
2840 kg/m3, inner radius 0.007 m, and outer radius 0.008 m. The projectile mass
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 validity of the procedure, simply supported
and clamped at both sides. Each case was run with three values of foundation
stiffnessγ=0,1,2. The program used 10 3-node beam elements, with 5th-order
polynomial, (see [15]), and checked
for the eigenvalues at 20 equidistant points in each element. Note that the
2-node element, 3rd-order polynomial, was checked for accuracy and the results
were not of any considerable difference, rather, the program was already
written for 3-node elements and thus used. The convergence of the solution was
checked and it was found that the 10 elements with 20 internal points gave
adequate accuracy.Figure1 shows the results of the two cases with the three
foundation stiffness values compared to those obtained from the equation given
by Benedetti [2]. It can be
obviously seen in that graph that the results obtained   for the clamped-clamped
beam are  almost identical for the three values of
γ. Meanwhile, all the results
obtained using the simply supported beam gave lower values for the critical
speed. The results obtained from the equation by Benedetti [2], however,
varied from nearest to single bay (simply supported) (γ=0) to almost identical
to clamped-clamped case (γ=1) ending with just being higher than the
clamped-clamped case with γ=2.Figure 1
Stability
boundaries for different cases.Note that the
fundamental solution of Benedetti used above gives the relation between the
critical speed factor,βcr, and the mass factor, α, as
(21)βcr=121+γα-0.5,where α=m/ρAL and β=VL/(2πρA/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 formula gave values that
are much like those of a clamped-clamped beam.Now the most
important observation that may be taken from Figure1 is that when the stability boundaries are all drawn
on log-log scale the results all showed linear trends, further, all lines are
parallel with a slope of −0.5 (note that the formula presented by Benedetti had
a relation between α and β in the form of βcr=aα-0.5, see
the relation above). Thus it may be concluded that the procedure presented in
this paper can accurately predict the stability boundaries of the problem with
different boundary conditions. Further, using regression techniques, we may
write down a relation for the critical speed factor as
(22)βcr|S-S≈0.1954+γα-0.5,βcr|C-C≈0.77α-0.5.
The main aim of
this study is to determine the stability boundaries for gun barrels. Such
structures may be modeled by a cantilever beam. Using the cantilever boundary
conditions in the above procedure, and setting the foundation stiffness to
zero, we get the critical speed boundaries as presented in Figure 2. The relation may be approximated by the formula
(23)βcr|Cantilever≈0.21α-0.5.
Another common
configuration of the gun barrel is the clamped-pinned configuration. This
configuration is used to support the tip of the gun barrel and, hence,
increases its stability. When that configuration was used in the program, the
results obtained were similar to all the other cases. The relation between the
mass parameter and the critical velocity parameters may be given by
(24)βcr|Clamped-Pinned≈0.45α-0.5.
The above relation
reflects the expected increase in the stability of the gun barrel when
supported at the tip.Figure 2
Stability
boundaries for cantilever beam.Now that the
stability boundaries for beams with different boundary conditions are realized
to be a simple relation between the mass and velocity parameters, we need to
investigate the effect of creating a stepped gun barrel on the stability
boundaries. It may be realized that increasing the radius in parts of the gun barrel
will automatically increase the stability range as the stiffness increases.
Rather, the study would be on how much that increase in radius should be so as
not to add unnecessary weight. In the following, the cases, the 8 mm outer radius
of the barrel will be the base for the comparison. The barrel will be divided
into twelve equal-length parts six of which have 8 mm radius and the other six
will have the same radius but with a value more than 8 mm. Figure3 presents stability boundaries for barrels with
different radius ratios (RR). As clearly evident, the barrels with steps of
more than 8 mm have higher stability boundaries. But it is also evident that 8
to 9 ratio achieved the highest change in the stability boundaries compared to
the changes occurring when increasing the radius ratio from 8 to 9 to 8 to 10.
Thus the highest achievement, in terms of gain in stability boundaries, was by
increasing the radius by only 12.5%.Figure 3
Stability
boundaries for stepped cantilever beam.


## 3.2. Time Response

A program was
developed for the time response of the barrel to moving bullets using the
Newmark technique presented earlier and the results were compared to response
presented in [16] with the
consideration that the reference did not include the effect of the external
work done by the bullet on the barrel. The values used for the algorithm
parameters were δ=0.52 and μ=0.27. In all numerical results for the dynamics,
presented in this section, the time step used was 1/2400 of the total time
required for the bullet to transverse the barrel. Convergence of the solution
was tested using different values of the time step and it was found that this
value was accurate and convenient.The response of
the barrel to the motion of the bullet is one of the important aspects that
should be considered when designing a gun barrel. As the instability described
in the previous section is a static instability, pitchfork bifurcation, it is reflected in higher dynamic
response to the external excitation, rather than self-excited vibrations that
are associated with Hopf bifurcations. The higher the response becomes,
the more the time between the bullets should be to ensure 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 of them ignored the deflection of the beam due to
its own weight. That type of deflection was ignored mostly because such studies
were directed to cases where the load is much more than the beam weight, as in
the case of train moving on a railroad. In our problem, the bullet mass, in
most guns, is much less than the barrel mass. Hence, ignoring the initial
deflection due to mass may introduce much difference in the response. To
illustrate the effect of the initial deflections, the response of the barrel to
the moving mass is plot in the cases where the barrel mass was not considered. Figure4 shows clearly that the response, while ignoring the
mass, is not a mere shift downwards for the curves, rather, the response, especially
at high values of the speed parameter, has a completely different pattern. The
normalized deflection is calculated as
(25)wnormalized=wρALg/(EI/L3).
This normalization
of the deflection will ensure the same response curve for the same values of α
and β regardless of the beam properties and geometry.Figure 4
Response of the
barrel’s tip versus bullet position with and without considering the barrels weight
forα=0.2 and different values of β.Investigating the
effect of the stepping of the gun barrel, as in the previous section, on the
response to the motion of the bullet was the following step. Figure5 presents the response of the barrel motion of the
bullet with α=0.2 and β=0.2, while Figure
6 presents the response with α=0.2 and β=1.0.Figure 5
Response of the
barrel’s tip versus bullet position for stepped beams with different radius
ratios withα=0.2 and β=0.2.Figure 6
Response
of the barrel’s tip versus
bullet position for stepped beams with different radius ratios withα=0.2 and β=1.0.The results in Figures5 and 6  both agree on that the response of the plain, less
stiff, beam is higher than that for stepped beams in general. It may be also
observed that the gain, in terms of tip vibration reduction, obtained by
increasing the step ratio to 8 to 9 is very good especially that it presents
the least increase in weight.


## 4. Conclusions

In this study, a finite element model was used to predict the stability
boundaries for beams with moving masses subject to different boundary
conditions. Because of the lack of literature, the results were compared to
classical solutions presented for infinite beams simply supported at equal
intervals. As predicted, the solution of SS beams underpredicted the stability
boundaries compared to classical solution. Meanwhile, clamped-clamped beams
showed, almost, no change with foundation stiffness.An empirical relation between the mass parameter and critical speed
parameter could be obtained for simply supported, clamped-clamped, cantilever,
and clamped-pinned beams. Generally, the relation was given by the relationβcr≈cα-0.5, where c is a constant that is determined by the
boundary conditions of the beam.Also the effect of
using stepped barrels was studied to investigate the feasibility of such
techniques. It was found that with, considerably, small increase in the radius
of the barrel in some parts, a significant increase in the stability boundaries
was obtained.In this study, for the first time in literature, the stability
boundaries were predicted using eigenvalues of the system rather than using
time marching techniques. The results presented in this paper are, to the extent
of the authors knowledge, the first in the literature to present accurately the
relation between the mass parameter and critical speed parameter 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 Newmark 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.



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*Source: 483857-2008-03-06.xml*