14. starptautiskÄ konference 2012 - Latvijas JÅ«ras akadÄmija

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Proceedings of 14th International conference „Maritime Transport and Infrastructure - 2012”Excluding the parameter , transfer to the polar coordinates: x r cos;y r sin,calculate:a22 2 2r x y ; (13)2 2k 22k 21 cos r . (14)athe equation of the hodograph in polar coordinates is obtained:ar 1 cos (15)2 k 2The cardioid is obtained using the results of the experiment:r 34,05 1cos . (16)12090606015040303405 ( 1cos(x))1001802000210330240270x300Fig. 2. The hodograph in polar coordinates.ConclusionMathematics apparatus deals with practical tasks. In our case a partial derivative is applied toanalyse the physical process after the experiment with the new composition of fuel.References1. E. Kronbergs, P. Rivža, Dz. Bože. Augstākā matemātika, 2. daļa, Rīga, Zvaigzne, 1988., 528 lpp.2. K. Šteiners. Augstākā matemātika, III. daļa, Rīga, Zvaigzne ABC, 199*., 192 lpp.78
Proceedings of 14th International conference „Maritime Transport and Infrastructure - 2012”DIFFERENTIAL EQUATIONS IN GYROSCOPE CALCULATIONSMiroslavs Mališko, Jevgēnijs Šepeļevs, Jānis Bērziņš*, Voldemārs Barkāns***Latvijas Jūras akadēmija, Flotes iela 5B, LV-1016, Latvija, E-pasts: Janis.Berzins@latja.lv**Latvijas Jūras akadēmija, Flotes iela 5B, LV-1016, Latvija, E-pasts: Voldemars.Barkans@inbox.lvAbstractIn this work are illustrated bachelor programme’s differential equations with constant coefficientswhich are used in gyroscope calculations and their solution.KEY WORDS: differential equations with constant coefficients, constant variation methodIntroductionIn technical programs of bachelor studies the course of Math’s is provided to get studentsacquainted with mathematical apparatus which is necessary for studying technical courses of generaleducation. It is also ment for solving assignments of practical nature in speciality and to waken self –contained practical working skills of students. Finally, the course of Math’s is included in technicalstudies to develop logical thinking and ability of students to reach goals according to a plan.Differential equation of second grade.Gyrocompass which includes gyroscopes is a navigation aid and is used aboard ships tostabilize ship ‘s heading. It is possible to have a look at some examples which show that the workingprinciple of gyroscopes is described using system of differential equations. After solving such systemit is possible to acquire differential equation with constant coefficients. Such type of differentialequations is described in basic course of higher mathematics.After solving differential system of equations of movement of gyro sphere and gyroscopes[1, IV5, VI2], a non-homogeneous differential equation of second grade with constant coefficients isacquired:2 2 y y 2ccost . (1)Coefficients andccinclude parameters of the device. Dots are used for marking fluxionsby time t. Similar differential equation was also solved in Physics course when examing mathematicalmodel of pendulum.The result of such equation is stated by sum:y y y , (2)Where y is a general solution of appropriate homogenous differential equation.2 y y 0(3)y is undetermined solution of given non – homogenous differential equation.Appropriate equation of homogenous differential equation (3) is:79
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