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Research Papers

# Probability Density Function of Underwater Bomb Trajectory Deviation Due to Stochastic Ocean Surface Slope

[+] Author and Article Information
Peter C. Chu1

Naval Ocean Analysis and Prediction Laboratory, Naval Postgraduate School, Monterey, CA 93943pcchu@nps.edu

Chenwu Fan

Naval Ocean Analysis and Prediction Laboratory, Naval Postgraduate School, Monterey, CA 93943

1

Corresponding author.

J. Dyn. Sys., Meas., Control 133(3), 031002 (Mar 23, 2011) (13 pages) doi:10.1115/1.4003378 History: Received April 05, 2010; Revised November 23, 2010; Published March 23, 2011; Online March 23, 2011

## Abstract

Ocean wave propagation causes random change in an ocean surface slope and in turn affects the underwater bomb trajectory deviation $(r)$ through a water column. This trajectory deviation is crucial for the clearance of obstacles such as sea mines or a maritime improvised explosive device in coastal oceans using bombs. A nonlinear six degrees of freedom (6DOF) model has been recently developed and verified at the Naval Postgraduate School with various surface impact speeds and surface slopes as model inputs. The surface slope $(s)$ randomly changes between 0 and $π/2$ with a probability density function (PDF) $p(s)$, called the $s$-PDF. After $s$ is discretized into $I$ intervals by $s1,s2,…,si,…,sI+1$, the 6DOF model is integrated with a given surface impact speed $(v0)$ and each slope $si$ to get bomb trajectory deviation $r̂i$ at depth $(h)$ as a model output. The calculated series of ${r̂i}$ is re-arranged into monotonically increasing order $({rj})$. The bomb trajectory deviation $r$ within $(rj, rj+1)$ may correspond to one interval or several intervals of $s$. The probability of $r$ falling into $(rj, rj+1)$ can be obtained from the probability of $s$ and in turn the PDF of $r$, called the $r$-PDF. Change in the $r$-PDF versus features of the $s$-PDF, water depth, and surface impact speed is also investigated.

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## Figures

Figure 1

The concept of airborne sea mine/maritime IED clearance

Figure 9

Attack angle (α), center of volume (ov), center of mass (om), and drag and lift forces (exerted on ov). Note that δ is distance between ov and om with positive (negative) value when the direction from ov to om is the same (opposite) as the unit vector e. The unit vector eu is in the direction of the bomb velocity (from Chu 2010 (4)).

Figure 2

MK-84 warhead with (a) tail section and four fins, (b) tail section and two fins, (c) tail section and no fins, and (d) no tail section

Figure 3

Dependence of underwater bomb trajectory, orientation, and horizontal deviation (r) on the ocean surface slope or on different locations of the waves

Figure 4

Ocean surface inclination angle (μ) and bomb impact angle (ϕ) relative to the normal direction of the surface

Figure 5

Air cavity (a) with β<γ (tail section not hitting the cavity wall) and (b) with β=γ (tail section hitting the cavity wall)

Figure 6

Wave effect on the air-cavity orientation, which may cause β>γ (tail section hitting the cavity wall)

Figure 7

The s-PDFs for various surface characteristics: (a) n=2, (b) n=4, (c) n=10, and (d) n=100

Figure 8

Position vectors rh and rt and the unit vector e (from Chu 2010 (4))

Figure 10

Calculation of the probability for the bomb’s horizontal drift r taking values between rj and rj+1 from m intervals of surface slope s. Here, m=1 and m=2.

Figure 11

(a) Positively and (b) negatively skewed PDFs

Figure 12

Probability distribution of the bomb’s horizontal drift (scaled by the depth) r/H with n=2, σ=0.2, and V=300 m/s for various depths: (a) 12.2 m (i.e., 40 ft), (b) 50 m, (c) 91.4 m (i.e., 300 ft), (d) 150 m, (e) 200 m, and (f) 250 m

Figure 13

Probability distribution of the bomb’s horizontal drift (scaled by the depth) r/H with n=2, σ=0.02, and V=300 m/s for various depths: (a) 12.2 m (i.e., 40 ft), (b) 50 m, (c) 91.4 m (i.e., 300 ft), (d) 150 m, (e) 200 m, and (f) 250 m

Figure 14

Probability distribution of the bomb’s horizontal drift (scaled by the depth) r/H with n=100, σ=0.2, and V=300 m/s for various depths: (a) 12.2 m (i.e., 40 ft), (b) 50 m, (c) 91.4 m (i.e., 300 ft), (d) 150 m, (e) 200 m, and (f) 250 m

Figure 15

Probability distribution of the bomb’s horizontal drift (scaled by the depth) r/H with n=100, σ=0.02, and V=300 m/s for various depths: (a) 12.2 m (i.e., 40 ft), (b) 50 m, (c) 91.4 m (i.e., 300 ft), (d) 150 m, (e) 200 m, and (f) 250 m

Figure 16

Probability distribution of the bomb’s horizontal drift (scaled by the depth) r/H with n=2, σ=1.0, and V=300 m/s for various depths: (a) 12.2 m (i.e., 40 ft), (b) 50 m, (c) 91.4 m (i.e., 300 ft), (d) 150 m, (e) 200 m, and (f) 250 m

Figure 17

Probability distribution of the bomb’s horizontal drift (scaled by the depth) r/H with n=2, σ=0.2, and V=200 m/s for various depths: (a) 12.2 m (i.e., 40 ft), (b) 50 m, (c) 91.4 m (i.e., 300 ft), (d) 150 m, (e) 200 m, and (f) 250 m

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