[1]
are presented. The discussed system was brought down to the form of a structural model with a discrete constitution, therefore it is described with differential equations with ordinary derivatives. The study is, in its nature, a generalized deliberation of the posed problem. The presented mathematical model makes it possible to examine the systems with parameters different from those initially assumed. Thus the analysis conducted for a hypothetical system with standard values of parameters constitutes a base for a thorough examination of similar objects. The formulated model allows for describing physical phenomena resulting from the launcher control of the anti-aircraft assembly with a similar structure. The mathematical model presented below consists of eight motion equations describing the performance of a hypothetical anti-aircraft assembly in conditions determined by the assumed guidelines, three equations of equilibrium and two equations representing the launcher control algorithm. On the basis of the obtained equations of the assembly motion it is possible to classify the mathematical model. The formulated mathematical model of the launcher based onboard a ship is: · Geometrically nonlinear, · determined, · variable in time, · dissipative, · bounded. Motion equations: (1) (2) (3) (4) (5) (6) (7) (8) where: Equilibrium equations: (9) (10) (11) Equations representing launcher control algorithm: (12) (13) , , (14) , , (15) (16) where: – controlling forces moments applied to the platform and guide accordingly; – corrective forces moments of the launcher; – launcher corrective forces of the launcher; – angles determining the assigned launcher position in space – amplification coefficients of the controller; – coefficients of the controller damping. Obtained results and final remarks Examples of results of the conducted numerical simulation of the performance of the controlled launcher exposed to instantaneous impulses will be presented. They were obtained for the parameters of the physical model of a hypothetical warship launcher assumed in the introduction(cf. figures 2-4). The obtained exemplary examination results are presented in figures 5-12. In the case of low damping values in the launcher axis bearings the system is unstable. It can be seen in figures 5-7. If the above mentioned damping represents higher values the transitory process fades in time, which is shown in figures 8-10. Satisfactory dynamics of the controlled warship launcher (fast fading transitory process) may be obtained only after the application of the corrective controls described with equations (12)-(16). It can be seen clearly in figures 11 and 12. Fig. 5. Linear relocation variation of the launcher in the time function (small damping) Fig. 6. Angular relocation variation of vibrating motion of the launcher in the time function (small damping)
Fig. 7 Angular relocation variation of the launcher primary motion in the time function (small damping)
Fig. 8 Angular relocation variation of vibrating motion of the launcher in the time function (large damping)
Fig. 9. Angular relocation variation of vibrating motion of the launcher in the time function (large damping)
Fig.10. Angular relocation variation of the primary motion of the launcher in the time function (large damping) Fig. 11. Angular relocation variation of vibrating motion of the launcher in the time function with the application of the corrective controls
Fig. 12 Angular relocation variation of the primary motion of the launcher in the time function with the application of corrective controls Summing up it must be stated that the mathematical model derived in this paper of the dynamics of the warship launcher of the short-range missiles allows for conducting an in-depth analysis of the searching, tracing and homing process of a missile onto a low-flying air target. The andcontrols described with equations (12) and (13) set the launcher into a desired programme motion allowing for detection of the target by the missile head. Whereas the moments and the corrective forces (equations 14-16) damp the dynamic effects of the launcher occurring the moment the disturbances and kinematic ship deck interaction begin to act and the moment the very controlling moments are turned on. These deliberations are merely an introduction to further investigations. As it was mentioned in the introduction there is a need to identify the phenomena accompanying the control process of the launcher at the pre-launch stage. They may have an essential influence on the performance of the missile while launching, and thus determine the realized trajectory and as a result the act of destroying the target. Acknowledgment The authors acknowledge support from the Ministry of Science and Higher Education through Project ON501312638 conducted in the years 2010-2013. Bibliography
Google Scholar
[1]
Cochran J.E., Foster W.A.: Dynamics of flexible missile/launcher systems. Technical Papers, American Institute of Aeronautics and Astronautics, pp.358-366, Washington (1992)
Google Scholar
[2]
Dziopa Z.: Modelling the Dynamics of Missile Launcher Based Onboard the Warship. Automation and Operation of Control and Communication Systems, Volume 1, Naval Academy- Gdynia 2003, ISBN 83-87280-60-7, pp.69-76
Google Scholar
[3]
Dziopa Z., Koruba Z., Krzysztofik I.: Analysis of the Dynamics and Control of the Scanning and Tracking System Placed on the Warship Launcher of the Target-Homing Missiles. Scientific Brochures of the Naval Academy no. 177B, YEAR XLX, Gdynia 2009, ISSN 0860-889X, pp.59-68
Google Scholar
[4]
Dziopa Z., Koruba Z., Krzysztofik I.: Dynamics of a Controlled Anti-Aircraft Missile Launcher Mounted on a Moveable Base. Journal of Theoretical and Applied Mechanics. No.2, Vol. 48, Warsaw 2010, ISSN 1429-2955, pp.279-295
DOI: 10.1063/2.1204308
Google Scholar
[5]
Dziopa Z. Koruba Z., Krzysztofik I.: Elements of the Formulation Method of the Model of the Missile Launcher Placed Onboard the Warship. Fire Control of the Air (Anti-Aircraft) Defence Systems. Naval Academy, Gdynia 2010, ISBN 978-83-60278-50-5, pp.85-94
DOI: 10.4028/www.scientific.net/ssp.180.269
Google Scholar
[6]
Dziopa Z., Małecki J.: The Influence of the Hybrid Vibroisolation System of the Launcher Turret on the Missile Motion along the Guides of the Mobile System of the Waterfront Defence. Scientific Brochures of the Naval Academy 3(182), Gdynia 2010, pp.65-73
Google Scholar
[7]
Inman D.J.: Vibration with Control. John Wiley & Sons, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England (2006)
DOI: 10.1017/s0001924000086929
Google Scholar
[8]
Kaczorek T: Control Theory, Vol. 1; Continuous and Discrete Linear Systems, Vol.2; Non-Linear Systems, Stochastic Processes and Static and Dynamic Optimization. PWN (Polish Scientific Publishers), Warszawa 1977, (1981)
Google Scholar
[9]
Lock S.: Guidance. D. Van Nostrand Company, Inc. Princeton, New Jersey, Toronto, New York, London, (1955)
Google Scholar
[10]
Osiecki J.: Machines Dynamics. WAT(Military University of Technology), Warszawa (1994)
Google Scholar
[11]
De Silva C.W.: Vibration Fundamentals and Practice. Taylor & Francis Group, Boca Raton, London, New York, (2007)
Google Scholar
[12]
Babickij V.I.: Vibrations in Strongly Non-linear Systems- Nonlinearity of the Threshold Type. Science, Moscow (1985)
Google Scholar
[13]
Biessonov A.P.: Basis of the Dynamics of Mechanisms with a Variable Links Mass. Science, Moscow (1967)
Google Scholar
[14]
Svietlickij V.A.: Dynamics of the Flying Machines Take-Off. Science, Moscow1963
Google Scholar
