Mathematical Modeling Vibration Protection. System for the Motor of the Boat

Transcription

1 Applied Mathematical Sciences, Vol. 9, 05, no. 9, HIKARI Ltd, Mathematical Modeling Vibration Protection Sstem for the Motor of the Boat Sergei Evgenievich Ivanov ITMO National Research Universit (ITMO Universit) 970, Saint Petersburg, 49 Kronverksk Pr., Russian Federation Vital Gennadievich Melnikov ITMO National Research Universit (ITMO Universit) Head of the Department of Theoretical and Applied Mechanics 970, Saint Petersburg, 49 Kronverksk Pr., Russian Federation Copright 05 Sergei Evgenievich Ivanov and Vital Gennadievich Melnikov. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. Abstract The nonlinear mathematical model of the sstem to mount the motor in a niche of the outboard motor boat, which protects against external influences b means nonlinear shock absorbers and dampers, has been developed. The mathematical model of the sstem is represented b three nonlinear differential equations of second order. The nonlinear shock absorbers and dampers, as a function of the generalized coordinates are represented b polnomials up to the fifth degree inclusive. For approximate analtical solution of the nonlinear mathematical model is applied the modified method of polnomial transformations and the numerical method of Runge-Kutta. Kewords: Vibration protection sstems, Method of polnomial transformations, Nonlinear mathematical model, Outboard motor boat Introduction When driving on the water at the hull of the boat affect periodic forces, which are transmitted to the outboard motor. To protect the motor from the periodic influ-

2 595 S. E. Ivanov and V. G. Melnikov ence the motor is mounted on the non-linear dampers and the shock absorbers [-6]. The nonlinear mathematical model for mount the engine on niche of the boat has been developed. External forces, acting on the hull of the boat, is transmitted to the motor and represented as periodic functions. It is assumed that the sstem has no resonance. As result the mount of engine in a niche b the nonlinear shock absorbers is a significant reduction in engine vibration amplitudes on the all three axes. Nonlinear rubber shock absorbers [7-] and dampers in the suspension of the motor are widel used on boats "Yamaha", "Tridin", "Erslot". Mathematical model of the vibration protection sstem for motor of the boat Consider mounting the motor of the boat in the niche b means of non-linear rubber shock absorbers and dampers that can be represented in the form of polnomials up to the fifth degree inclusive. Assumed, the impact on the boat an external periodic force in three directions: f A sin( t) B cos( t), f A sin( t) B cos( t), x x x f A sin( t) B cos( t), z z z where Ax, Bx, A, B, Az, B z amplitude,,, the frequenc of external periodic forces. For the equations of motion the dnamical sstem is applied the sstem of Lagrange equations: d L L d L L d L L Qx; Q; Qz, dt x x dt dt z z where L - Lagrange function, Q x - the force corresponding coordinate x. Here x,, z the coordinates of the center of mass of the motor. The generalized forces are equal: 4 5 Q b ( x f ) b ( x f ) b ( x f ) b ( x f ) b ( x f ), x x x x 4 x 5 x Q l ( f ) l ( f ) l ( f ) l ( f ) l ( f ) Here m - the weight of the motor, ci, k i the spring constant, bi, l i coefficients of friction the damping devices. Substituting the expressions for the Lagrange equations, we obtain a sstem of three nonlinear differential equations of the second order: 4 5 mx b ( x f x) b ( x f x) b ( x f x) b4 ( x f x) b5 ( x f x) 4 5 c( x f x) c( x f x) c( x f x) c4( x f x) c5( x f x) m l( f ) l( f ) l( f ) l4( f ) l5( f ) p( f ) mz k( z f z) k( z f z) k( z f z) k4( z f z) k5( z f z) 0

3 Mathematical modeling vibration protection sstem 595 Let us consider the relative coordinates of the center mass the motor: x x f ; f ; z z f ; x z To simplif further entries we omit the sign ~ for coordinates: x x; ; z zz. In the relative coordinates the mathematical model can be written. 4 5 mx b x cx d b x bx b4 x b5x 4 5 cx cx c4x c5x A sin t B t 4 5 m l l l l4 l5 d A t B t 4 5 mz kz kz kz k4z k5z d A t cos p sin cos sin B cost Divide each equation on the weight of the motor. For simplicit, we write the sstem without renaming all the coefficients of the sstem: c c / m; k k / m; b b / m; l l / m. i i i i i i i i We write the nonlinear sstem of equations with the coefficients divided to unit weight of the motor. 4 5 x b x cx d b x bx b4x b5x 4 5 cx cx c4x c5x A sin t B cost 4 5 l l l l4 l5 p d A sin t B cost 4 5 z kz kz kz k4z k5z d A sin t B cost The method of calculation the vibration protection sstem For the calculation of nonlinear dnamic sstems appl different methods [4-9]. Is applied known methods: the method of Van der Pol, harmonic balance method, averaging method, the method of small parameter, the method of Krlov-Bogolubov, method of polnomial transformations, the Poincare perturbation method [0-5]. For the method of Van der Pol and for the method of averaging the truncated equation is considered. In the method of harmonic balance approximate solution takes into account onl the basic frequenc components. In the method of small parameter and perturbations the approximate solution is found, provided that the series converges. For solved the nonlinear dnamic sstem was applied the method polnomial transformation [6]. The sstem of equations is written in matrix form: X X * * RX * P, where [ x,, z, x,, z, exp( i t),exp( i t),exp( i t),exp( i t)exp( i t),exp( i t)] T

4 5954 S. E. Ivanov and V. G. Melnikov P - nonlinear vector the sstem. * * Y AX, we obtain the linear sstem As a result, the linear change of variables: with a diagonal matrix: Y * * ~ Y P. B replacing the variables in accordance with the polnomial transformation method [6]: z a * * S S 5 S, up to terms of the fourth order we obtain an autonomous differential sstem: * * * * * * * * * * * z ( q ) z, z z, z ( q ) z, z z, z ( q ) z, z z. * The solution of the autonomous sstem of equations can be written as: * * * z exp( qt ), z exp( qt), z exp( qt) The solution of the nonlinear dnamical sstem in the original variables x,,z can be written as: F F F F 5 4 t Q Ab 5 F Ab F A F 5 5F 6 8c; F A B ; R 5c b c 5 4 ( ) sin 5 5 F 6 F / 5 4 cost 5Al 5 F 6Al F 8Al 8B p 8 B / R F 7l4 F 8l 8 d / (8 p); F A B, R 8l p ( ) / sin t 8AQ Q 8 / R cost 8BQ Q 8 / R; ; ; R 5 k x( t) 7 b c 8b 8c 8d Q 8 64 b / R sin t Q 5b B 6b B AQ 8A 8 b B / R 8cos b B Q 8 B / R ; Q c c t t B l B l A p A B l R z t F k F k d Q R Q F k F k k F A B k We obtained the stead mode of motion with the following parameters of the dnamic sstem: A 0.4; A 0.; A 0.; B 0.; B 0.; B 0.; ;.; 0.8; m ; b 0.; b 0.0; b 0.0; b 0.0; b 0. 0; c 0.5; c 0.0; c 0.0; 4 5 c 0.0; c 0.0; d 0.0; d 0.0; d 0.0; p 0.; k 0.04; k 0.0; k ; k4 0.0 k5 0.0; l 0.0; l 0.0; l 0.0 l4 l5 ; ; 0.0; 0.0;

5 Mathematical modeling vibration protection sstem 5955 The stationar mode of motion corresponding to the functions: x sin( t) cos( t) sin(. t) cos(. t) z 0.47sin(0.8 t) 0.558cos(0.8 t) We obtained the graphics of motion the center mass the motor (Figure ). Fig.. The relative and absolute movement of the center mass the motor. Legends: x,, z The analtical solution is compared with the numerical solution b the Runge-Kutta method. Error of solving for the polnomial transformation method does not exceed %. Figure shows the dependence of the error ( in percent) on the parameter c and on the time, when c 0.5 the error does not exceed 0.05%. Fig.. The dependence the error on the c and on the time when c 0.5 Figure shows the dependence the error ( in percent) on the parameter b and on the time, when b 0. the error does not exceed 0.5%.

6 5956 S. E. Ivanov and V. G. Melnikov Fig.. The dependence the error on the b and on the time when b 0. We obtained graph the maximum amplitude of the dnamic oscillation sstem depending on the frequenc (Figure 4) Max x,mm ,rad Fig.4. The dependence maximum amplitude of the oscillation on the frequenc We performed the analsis the effect of nonlinear parameters of shock absorbers and dampers to the maximum vibration amplitude (Figure 5). Max x,mm c, Max x,mm b, Fig.5. The dependence the maximum amplitude of the vibrations on the parameters of the shock absorber c and b.

7 Mathematical modeling vibration protection sstem 5957 c, c, c, c, c Legends: 4 5 b, b, b, b, b, 4 5 From the figure 5, it follows that for reduce the amplitude of oscillations is expedient to increase the non-linear parameters of shock absorbers c, c 5, to reduce the non-linear parameters c, c 4, the linear parameters c 0.. For reduce the amplitude of the vibrations is necessar to increase parameter dampers b and to reduce the nonlinear parameters b, b, b 4. 4 Conclusion Thus, the approach in the proposed work, combining analtical and numerical methods, allows to complete analsis of the nonlinear dnamic sstem and to gets the basic characteristics the motion of dnamical sstem. Evaluation the accurac of analtical solution of the nonlinear dnamic sstem shows an error of less than two percent. The effectiveness of nonlinear dnamical sstem essentiall depends on the tpe of external influence and the selected parameters. When considering the parameters dnamic sstem for shock absorbers mounted in the motor niche is necessar to increase the nonlinear parameters c, c 5, to reduce parameters c, c 4 and linear parameters c 0.. For parameters of dampers should be to increase the parameters b and to reduce the nonlinear parameters: b, b, b 4. References [] R. L. Mishkov, V. S. Petrov, Nonlinear adaptive observer with asmptoticall stable parameter estimator, Journal of Advanced Research in Dnamical and Control Sstems, 7 (05), [] N. L. Bochkareva, E. P. Kolpak, On stabilit of arch damper, Vestnik Sankt-Peterburgskogo Universiteta, Ser. Matematika, Mekhanika, Astronomia, 4 (99), [] E. P. Kolpak, S. E. Ivanov, Mathematical modeling of the sstem of drilling rig, Contemporar Engineering Sciences, 8 (05), no. 6, [4] O. Sedova, Y. Pronina, Generalization of the Lamé problem for three-stage decelerated corrosion process of an elastic hollow sphere, Mechanics Research Communications, 65 (05), [5] E. P. Kolpak, L. S. Maltseva, S. E. Ivanov, On the stabilit of compressed plate, Contemporar Engineering Sciences, 8 (05), no. 0,

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10 5960 S. E. Ivanov and V. G. Melnikov Received: September, 05; Published: September 8, 05