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Three River Technologies -- Simulation of the Vibrational Response of a Rifle Barrel During Firing Page 2
To investigate these ideas, a computer program was written to solve the generalized transient beam vibration equation, which is second order in time and fourth order in space. Equation (9.23) from the Shock and Vibration Handbook (Harris and Crede 1976) is reproduced here, and is assumed to be the governing differential equation for this system:
The
coefficients E and I are Young's modulus (ratio of stress
to strain, 
where M is the bending moment on the barrel. As another special case of interest, if the barrel were of zero radial extent, I would equal zero in equation (1), and one is left with the classic damped, forced harmonic oscillator for every point along the barrel. The two terms containing the area moment provide the coupling between points along the barrel. Boundary conditions are assumed to be zero deflection and slope at the breech end of the barrel (where the barrel attaches to the rifle's action), and zero shear and zero bending moment at the muzzle end of the barrel:
where L is the overall length of the barrel. These boundary conditions describe the idealized, "clamped-free" beam problem in mechanics, but do not explicitly model the more complex boundary conditions of mechanical coupling between rifle barrel and stock along the length of the stock, coupling to the shooter (whose shoulder is not rigidly fixed), etc. However, during the 1-2 msec time window of interest, it is believed that these more complex effects are negligible, and the simple, clamped-free set of boundary conditions should suffice. Equation
(1) is fourth order in space and second order in time. To solve this
equation, we first split it into two coupled equations that are each first
order in time. If we set
This can be rearranged as:
Equation
(5) is integrated over time using the "theta-differencing" method,
where the integral
Using the finite difference formula for a fourth-order derivative,
and
integration of equation (5) over time (applying equation
(6) to the This represents a set of five-point-coupled algebraic equations of the form: where the coupling coefficients (which are, in general, a function of position) are defined as follows: These equations are collected into a five-diagonal matrix, referred to here as a "5 x N" matrix, where N is the number of mesh intervals used in the barrel model, which has the following form: The four boundary conditions are incorporated directly into the 5 x N matrix. The last two equations in the matrix are reserved for the imposition of the zero shear and zero bending moment. Using the finite difference formulae for the second- and third-order derivatives results in: The first two equations in the matrix impose the zero displacement and slope boundary conditions. The displacement at x=0, y0, is zero, and so does not have to be calculated. Therefore, the first equation in the matrix is for the first point internal to the barrel, y1. The first equation imposes zero slope at x=0 by requiring reflective symmetry (i.e., by setting y-1 = y1). The second equation in the matrix is a general internal equation, but takes advantage of the fact that y0 = 0. Formally incorporating these boundary conditions into equation (11) imposes a zero time derivative on y and the first three spatial derivatives. However, at t=0, the displacements are all zero, and so are the three spatial derivatives. Requiring a zero time derivative keeps the displacement and the spatial derivatives equal to zero, and so these boundary conditions are equivalent. These equations are solved by direct inversion of the 5 x N matrix for the transverse velocity distribution for a single time step. This in turn is integrated once in time to obtain displacement:
The
speed of sound,
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and
the area moment for the barrel cross section 
where r is measured from the x-axis and dS
is a differential area over the barrel cross section). The coefficients
c1 and c2 are internal and external
damping constants. If an external rib is attached to the vibrating barrel,
k characterizes its linear restoring force. In the acceleration
term, 
and S
are the material density and the cross-sectional area of the barrel. F(x,t)
is the external driving, or excitation, function. The steady-state form
of this equation reduces to the classic beam flexure equation. With transient
terms set to zero (and neglecting the rib constant, k), 
we can rewrite equation (1):
is approximated by a weighted average of future time fj+1
and present time fj (Carnahan et al. 1969):
terms only, while leaving the remaining terms constant over the time step)
results in:
is
approximately equal to 5100 m/sec in steel. Any impulse to the barrel
will result in an acoustical pulse traveling down the barrel, exciting
small transverse waves which lag behind the sound pulse. This acoustic
speed represents a small time constant attribute that makes the vibration
equation a "stiff" system, requiring small time steps to resolve
the response correctly. Small time steps and consideration of internal
damping are required for numerical stability. Given the mesh spacing
used, the sound pulse requires about 10-6 sec to traverse one mesh
interval, and so 10-7 sec time steps were used (which in turn
requires double precision in the simulation). Animations created from
simulated transients display visible disturbances which appear to travel
faster than the primary vibrational response, but lag behind the longitudinal
sound pulse. The initial pulse reaches the end of the barrel in 0.46 msec
(which is about a factor of 3 less than the known speed of sound in steel),
but the primary vibrational response reaches the end of the barrel only
after 1.0 msec.
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