Dynamic Modeling of Air Cushion Vehicles
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1 Proceedings of IMECE ASME International Mechanical Engineering Congress Seattle, Washington, November -5, 27 IMECE 27-4 Dynamic Modeling of Air Cushion Vehicles M Pollack / Applied Physical Sciences B Connell / Applied Physical Sciences J Wilson / Applied Physical Sciences W Milewski / Applied Physical Sciences ABSTRACT The motion of an Air Cushion Vehicle (ACV) is a complex process involving the nonlinear dynamics of the ship, free surface waves, air cushion cavity, skirt, and air flow hydraulics (e.g. orifice behavior of the bag feed holes). The overall system is tightly coupled, with the loading of the ship dependent on the pressure field within the cavity, and the dynamics of the cavity dependent on the motion of the ship, free surface, and skirt. Principle excitation to the system is through the free surface motion and the fan flow. The large dimensions of the system introduce low frequency acoustic and mechanical resonances, which lead to complex and coupled system dynamics. The focus of this paper is on analytical modeling of an ACV and its physics to enable verification of a numerical model which is under development. The initial focus is on the dynamics of the air cushion cavity, with emphasis on its resonant frequencies and mode shapes. The mode shapes are important because they define the nature of the dynamic pressure distribution acting on the ship, and associated heave, pitch, and roll excitation. The strong dependence of the cavity resonant characteristics on the impedance of the skirt, which bounds the cavity, is first demonstrated by assessing limiting cases of a high impedance skirt (e.g. massive or rigid) and of a low impedance skirt (e.g. light or soft). The changes in resonant frequency and dynamic pressure distribution associated with the changes in skirt impedance are illustrated. Because the actual skirt impedance will lie between these two idealized cases, we also develop a lumped parameter model of the skirt dynamics. Initial parametric studies with this model, which investigate variations in the skirt mass, further demonstrate the strong dependence of the resonant frequencies and pressure distributions on the skirt impedance. cavity, skirt, and air flow hydraulics (e.g. orifices). The overall system is tightly coupled, with the loading of the ship dependent on the pressure field within the cavity, and the dynamics of the cavity dependent on the motion of the ship, free surface, and skirt. Principle excitation to the system is through the free surface motion and the fan flow. The large dimensions of the system introduce low frequency acoustic and mechanical resonances, which lead to complex and coupled system dynamics. A numerical model is being developed to predict the motion of an ACV, with varying fan and skirt designs, subject to a range of ocean surface excitations. The model encompasses the complex geometry of an ACV and the three-dimensional variation of the air cushion pressure. It is nonlinear, addressing large skirt deformations and large amplitude motions of the ship and surface waves. This represents a significant extension of classical ACV heave models in the literature (, 2, 3), which assume particular characteristics of the subsystems (e.g. rigid skirt boundaries, uniform pressure within the air cavity, and uniform free surface displacement). Although sufficient for particular cases, these past assumptions do not capture the broad range of dynamic situations requiring analysis (e.g. coupled pitch-roll-heave motions). In order to guide the development of the new numerical model, a comprehensive understanding of the relevant physics of the ACV system is required. The focus of this paper is on analytical modeling of an ACV and its physics, to enable verification of the developing numerical model. The focus is on the dynamics of the air cushion cavity, with emphasis on its resonant frequencies and mode shapes. INTRODUCTION The motion of an Air Cushion Vehicle (ACV) involves the nonlinear dynamics of the ship, free surface waves, air cushion Copyright 27 by ASME
2 NOMENCLATURE a, b : in plane dimensions of air cavity h: height of air cavity c: speed of sound in air k xm, k ny : wave numbers in x and y directions p: dynamic pressure in cavity u: particle velocity ρ: air density ω: radian frequency φ: velocity potential t: time x, y: in plane coordinate system Z: impedance DISCUSSION Description of Model: The analytical model is a three dimensional rectangular air cavity bounded above by a rigid deck and below by a rigid water surface. The lateral boundaries represent interfaces with ACV skirts, which have local impedances Z mx and Z my. A sketch of the analytical model, the principal components, and the coordinate system is shown on figure. z Deck Air Cavity x Water Surface Figure : Sketch of Analytical Model Skirt The linearized wave equation is used to analyze the dynamics of the air cavity, subject to boundary conditions at the interfaces with the hull, the water surface, and the skirts. This analysis is limited to the resonant frequencies and dynamic pressure mode shapes of this system (i.e. eigenvalue or free vibration analysis). The mode shapes, which represent dynamic pressure distributions, are of primary interest. They provide insight into the potential for coupling of the air cavity pressure field with rolling and pitching, as well as heaving, motions of the ACV. Governing Equations: A linear model of a rectangular cavity is used to illustrate the relevant physics. The governing equation is the wave equation ( / c ) φ / t 2 φ = () where φ is the velocity potential and c is the speed of sound of the fluid within the cavity. The time dependent pressure, p, and the oscillating particle velocity vector, u, are given in terms of the velocity potential by p = ρ φ / t; (2) u = φ Assume that the ship deck is a rigid surface upper termination to the air cavity, and that the flat ocean surface is a rigid lower termination. The skirt forms the terminations along the four lateral sides. Consider a rectangular cavity of lengths a and b in the in-plane x and y directions, and h in the vertical direction. If the local impedances of the skirt along the boundaries, x=, a and y =, b are assumed to be Z mx and Z my respectively, then the wave equation and the interaction conditions (i.e. equal normal velocities) lead to the following characteristic equations: tan( k xm a) = 2 tan( k yn b) = 2 k yn Z my ω = c k xm [( k xm Z mx / iρω ) ( iρω / k xm Z mx )] [( / iρω ) ( iρω / )] [ ( ) ] / lπ / h k yn k yn Z my (3) In these equations, k xm and k yn are the wave numbers in the two in-plane directions in modes m and n, with m, n, l being integers. The first two equations determine the characteristic wave numbers, and wavelengths, of the system. The third equation determines the resonant frequencies of the system. The pressure distribution within the cavity can be expressed as a superposition of sine and cosine waves in the x and y directions, as follows. The ratios of the cosine to the sine wave amplitudes in the x and y directions are given by B m / A m = ( k xm Z mx / iρω ) βn / αn = ( k yn Z my / iρω ) These equations determine the ratios of the sine and cosine functions in the pressure distributions (i.e. mode shapes) in the in-plane directions. The coefficients along with their trigonometric functions determine the phasing of the pressure distribution applied to the ship by the air cushion cavity. Limiting Cases: (4) 2 Copyright 27 by ASME
3 Consider first the case where the skirt impedance is assumed to be rigid. For that case Z m ~ at all terminations, and k xm = mπ/a; k yn = nπ/b. The resonant frequencies are given by / 2 [( mπ / a) + ( nπ / b) ( lπ / h) ω = c + ] with pressure mode shapes of p (5) iωt = P cos( mπ x / a) cos( nπy / b) cos( lπz / h) e (6) In equation (6), P represents the amplitude of the pressure in mode, and the spatial cosine functions define the mode shapes or pressure distributions. Consider an air cushion cavity with dimensions a=9 ft, b = 4 ft, and h = 6 ft. If the cavity is filled with air, the fundamental resonance frequencies, f = ω /2π, are f = Hz, f = 6 Hz, f = 4 Hz, and f = 94 Hz. The zero frequency mode is the rigid body fluid mode. The first mode with a significant vertical gradient is f at ~ 94 Hz. In the two in-plane directions, there are low frequency fundamental modes (i.e. 6 Hz and 4 Hz modes for an air filled cavity) with antisymmetric pressure distributions about the origin. Given the cosine nature of their pressure distributions in the x and y directions, these modes will effectively excite pitching and rolling motions of the ship. There will also be coupled modes available, which will be able to excite coupled heaving-pitching-rolling motions. Therefore multi-dimensional cavity dynamics is essential to capture the dynamics relevant to coupled pitch, roll and heaving motions assuming there are sources available to excite those modes (e.g. displacement excitations from surface waves). Now consider the case where the skirt impedance is assumed to be very soft at all terminations (i.e. Z m ~ ). For this case the equations for the characteristic wavenumbers and resonant frequencies remain the same. However the modal pressure distributions change from cosine functions to sine functions in the in-plane directions. iωt p = P sin( mπ x / a) sin( nπy / b) cos( lπz / h) e (7) and the integers m, n are,2, For this situation, the lowest modes will have in-plane pressure distributions which are inphase, and more effective in exciting heaving translation than rolling or pitching motions. The lowest nontrivial resonant frequency, for the m= n= l= mode, will be ~ 5 Hz. As frequencies increase, out of phase pressure distributions will become significant. For the length scales assumed above, these out of phase modes can begin ~ 2 Hz for air filled cavities. Skirt Impedance Modeling: The actual local skirt impedances, Z mx and Z my, will lie between these two ideal cases and will be a function of frequency and position. They will depend on the dynamic characteristics of the skirt assembly as well as the flow orifices within the skirt. If the impedances are defined, then the cavity-skirt system resonance frequencies and mode shapes can be determined using the governing equations summarized in this section. For skirt impedances intermediate between rigid and soft, low frequency multi-dimensional resonances will exist within the cavity with complex phasing (i.e. combinations of cosine and sine wave functions). For the cases discussed in this paper, the dynamic pressure mode shapes are illustrated in terms of a normalized pressure. Since only the free vibrations of the air cavity system are being analyzed, in contrast with the forced response, only the distribution of pressure within the cavity, and the relative amplitudes of the cosine and sine waves are of relevance; the absolute pressure amplitude is not of significance. It is the distribution of pressure which lends insight into the potential for exciting rocking and pitching, versus heaving, motion of the ACV. As one example of the effect of skirt impedance, the skirt is assumed to be a mass around the entire perimeter of the air cushion cavity described previously, and the cavity is assumed to be filled with air. The effective spring constant of the skirt for this case is negligible. Three cases are considered: () the effective skirt mass along each lateral boundary is one hundred times the air mass within the cavity, (2) the effective skirt mass along each boundary is equal to the air mass within the cavity, and (3) the effective skirt mass along each lateral boundary of the cavity is one percent of the air mass within the cavity. For cases and 2 the lowest nontrivial resonant frequency is ~ 6 Hz. For case, the dynamic pressure distribution across the cavity is dominated by the out of phase cosine distribution as shown below on Figure 2. Such a pressure distribution would be expected to couple well with rocking and pitching motions of the ACV. 3 Copyright 27 by ASME
4 pressure wave normalized amplitude Pressure Distribution within Air Cavity Case : each lateral boundary skirt mass times air cavity mass sine wave cosine wave -3 Normalized Length Figure 2: Pressure Distribution within Air Cavity for Case For case 2, the dynamic pressure distribution is the superposition of cosine and sine waves which are approximately equal in amplitude. Figure 3 below illustrates this pressure distribution. For this case, the pressure distribution would couple well with both rocking, pitching, and heaving motions of the ACV. pressure wave normalized amplitude Pressure Distribution within air Cavity sine wave cosine wave Case 2: Skirt Mass along each lateral boundary equal Air Cavity Mass - Normalized Length Figure 3: Pressure Distribution within Air Cavity for Case 2 For case 3, the pressure distribution is dominated by the sine wave distribution shown below on Figure 4. For this case the lowest nontrivial resonant frequency is ~ 5 Hz, since the cosine pressure distribution is so small. This pressure distribution is expected to couple well with heaving motions of the ACV. pressure wave normalized amplitude Pressure Distribution through Air Cavity sine wave cosine wave Case 3: Skirt Mass along each lateral boundary equal percent of Air Cavity Mass -.2 Normalized Length Figure 4: Pressure Distribution within Air Cavity for Case 3 The actual skirt impedance is dependent on the effective mass, spring constant, and damping of the fabric and bag forming the skirt. The impedance will also be dependent on the effective spring constant formed by the air within the bag, and on the inertial and damping characteristics of the flow orifices within the skirt. These impedance elements can be modeled as an effective mechanical circuit. To the extent that spatial distributions influence the dynamics, their effects can be captured in a normal modes approach to the impedance modeling. A parametric study was done to investigate the significance of the spring constant of the air within the bag relative to the spring constant of the fabric and bag. The same air cavity geometry as described earlier was used. The effective mass was set at lbm for each lateral skirt, and an effective spring constant of x 6 lb/ft was used for the fabric and bag. A single flow orifice /6 inch in diameter and / inch long was assumed in each lateral skirt, thereby minimizing its contribution to the effective impedance. First an air bag with a width ten times the height of the air cavity was assumed. For this case, the lowest resonant frequency of the cavity-skirt system is 2.8 Hz, slightly less than the 5 Hz frequency associated with a zero impedance skirt. The pressure distribution in both in-plane directions is dominated by the sine wave components, as illustrated in Figure 5 for the x-direction. This pressure distribution will couple effectively with heave motion of the ACV. For this case the effective stiffness of the skirt, which is formed by the mechanical series equivalent of the fabric/bag stiffness and the air within the bag, is dominated by the spring constant associated with the air within the bag. 4 Copyright 27 by ASME
5 pressure wave normalized amplitude in x direction Normalized Distance Figure 5: Pressure Distribution in x direction for Air Bag Width Equal to times the Air Cavity Height When the bag width is reduced to equal the height of the air cavity, the stiffness of the air within the bag increases and the effective stiffness of the skirt becomes rigid. For this case the high skirt impedance yields a cavity resonance frequency of 5.6 Hz, which is close to the first nontrivial frequency of the cavity with infinite termination impedances. The pressure distribution in the x direction for that mode is shown on Figure 6, and is dominated by cosine wave activity. It will couple effectively with rolling and pitching motion of the ACV. pressure wave normalized amplitude in x direction Pressure Distribution in Air Cavity - Normalized Distance Figure 6: Pressure Distribution in x direction for Air Bag Width Equal to the Air Cavity Height The effect of the flow orifice impedance on the cavity dynamics can be seen by changing the flow area of the orifices relative to the surface area of the skirt (i.e. fabric and bag). Consider the case with an effective mass of x 6 lbm and an effective spring constant of lb/ft for each lateral skirt. With a single /6 inch diameter orifice / inch long in each skirt, the skirt effective impedance is massive and provides essentially infinite impedance to the air cavity. For this case, the impedance of the orifices has a minimal effect. The lowest nontrivial resonance has a frequency of 6.25 Hz and a pressure distribution in the long in-plane x direction of a half cosine wave, as shown in Figure 6. For a case where the effective flow area through the orifices is 5 percent of the lateral area formed by the skirt, the surface area of the skirt fabric is reduced by half and the effective impedance is significantly influenced by the flow orifices. In fact, for flow areas of 5 percent or greater the effective skirt impedance approaches that of a pressure release boundary and the pressure distribution approaches a half sine wave, as shown in Figure 7. The resonant frequency is 5.3 Hz, consistent with the low skirt impedance case discussed earlier. pressure wave normalized amplitude in x direction Normalized Distance Figure 7: Pressure Distribution in Air Cavity with Orifice Flow Area equal to 5 Percent of the Lateral Skirt Area CONCLUSIONS The resonant frequencies and mode shapes of the dynamic pressure within an ACV air cavity are strongly influenced by the skirt impedance which bounds that cavity. The pressure mode shapes determine the degree of coupling of the pressure field to translating (e.g. heave) as well as pitching and rolling motions of the ACV. The effective skirt impedance is dependent on the mechanical properties of the skirt and air bag, the geometry of the air bag, and the impedance of the flow orifices. This study focused on the free vibration response (i.e. eigenvalue analysis) of the system. The results provide insight into the need for multidimensional modeling of the air cavity and for sufficient modeling of the impedance of the skirts. Forced vibration analyses of the system would then yield 5 Copyright 27 by ASME
6 insight into the response of the ACV to specific excitations, such as free surface motion and fan flow. ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Office of Naval Research (Dr. Ki-Han Kim, Program Manager) under Contract N4-3-C-69. REFERENCES. Nonlinear Heave Dynamics of an Air Cushion Vehicle Bag and Finger Skirt, J. Chung, P.A. Sullivan, and T. Ma, AIAA CP, Effects of Unsteady Fan Response on Heave Dynamics of an Air Cushion Vehicle Bag and Finger Skirt, J. Chung and P.A. Sullivan, AIAA , Heave Stiffness of an Air Cushion Vehicle Bag and Finger Skirt, P.A. Sullivan, P.A. Charest, and T. Ma, Journal of Ship Research, Vol. 38, No. 4, Dec Copyright 27 by ASME
7 7 Copyright 27 by ASME
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