Abstract
The motion of a spinning rocket inside a smoothbore launcher has drawn some attention in recent years. The inaccuracy of spin-stabilized rockets can be attributed, in part, to the initial motion of the rocket inside the smoothbore launcher.
Recently, tests have indicated that a closer examination of the problem is required. The tests were performed to determine the time history of the spin and forward motion of the rocket by Fastax cameras. The evaluation of the data with respect to engineering-design parameters is difficult since the experimental results must be compared with an analytical model of the system.
In this paper, the results of one test are reported and a nonlinear analytical model is used which includes both the rotary and forward motion of the rocket. The two motions are coupled through a dynamic friction coefficient. The intent of this paper is to show how an analytical model can be made to fit the experimental data, that is, the initial conditions and the coefficient of friction are found which define the solution giving the best least-square fit to the data.
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Abbreviations
- A :
-
nondimensional variable
- F x :
-
friction force in thex-direction, lb
- F ϕ :
-
friction force in the circumferential direction, lb
- \(\bar G(\bar z)\) :
-
vector representation of the nondimensional system
- I :
-
moment of inertia of the round, lb-in.-sec2
- N :
-
reacting force at the contact point, lb
- {d k }:
-
set of constants
- f :
-
nondimensional friction-force vector
- f x :
-
nondimensional axial-friction force
- f ϕ :
-
nondimensional circumferential-friction force
- g :
-
acceleration due to gravity, in./sec2
- \(\{ \bar h^k \} \) :
-
set of vector homogeneous solutions
- l o :
-
length of the launch tube, in.
- l 1 :
-
length of the cylindrical rocket, in.
- m :
-
mass of the rocket, lb-sec2/in.
- \(\bar p\) :
-
vector particular solution
- r 0 :
-
radius of the launch tube, in.
- r 1 :
-
radius of the cylindrical rocket, in.
- s :
-
nondimensional time
- t :
-
time, sec
- w :
-
weight of the round, lb
- x :
-
forward location of the round, in.
- y :
-
redefinition of the friction coefficient
- \(\bar z\) :
-
nondimensional solution vector
- \(\{ \bar z^{(n + 1)} \} \) :
-
set of successive approximations
- β:
-
orientation of the friction-force vector
- Ψ:
-
the mean-square-error fit
- Ω:
-
nondimensional variable
- δ=r 0−r 1 :
-
offset of the rocket in the bore, in.
- ε:
-
nondimensional geometric parameter
- η:
-
nondimensional variable
- θ:
-
angular location of a point on the round
- ν:
-
coefficient of friction
- ζ:
-
nondimensional distance
- ξ1, 2, 3 :
-
nondimensional form of the data
- ρ:
-
nondimensional reacting force
- τ:
-
nondimensional time
- ϕ:
-
angular location of the contact point
- (·):
-
derivative with respect to time
- ()′:
-
derivative with respect to nondimensional time
References
Harrington, W. J. andR. E. Bullock, “The Motion of a Spinner Rocket Inside a Smoothbore Launcher,”Jnl. Franklin Inst.,277 (6)552 (1964).
Roth, R. S., “Data Unscrambling and the Motion of a Rocket Through a Smoothbore Launcher”,Jnl. Math. Anal. and Appl.,27 (2) (Aug.1969).
Roth, R. S., “The Unscrambling of Data: Studies in Segmental Differential Approximation,”Jnl. Math. Anal. and Appl. 14 (1)5 (Augil (1966).
Kalaba, R., “On Nonlinear Differential Equations, The Maximum Operation, and Monotone Convergence,”Jnl. Math. and Mech.,8 (4)519 (1959).
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Roth, R.S. Estimating the dynamic coefficient of friction from experimental data: the motion of a rocket inside a smoothbore launcher. Experimental Mechanics 9, 565–570 (1969). https://doi.org/10.1007/BF02316659
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DOI: https://doi.org/10.1007/BF02316659