XV International PhD Workshop OWD 2013, October On the theory of generalized pendulum on a vibrating base

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1 XV International PhD Workshop OWD 03, 9 October 03 On the theory of generalized pendulu on a vibrating base Michail Geraichuk, Yuri Lazarev, Peter Aksonenko, College of Instruent Design and Engineering, National Technical University of Ukraine Kiev Polytechnic Abstract Here we consider the probles of theoretical of non-stationary behavior of the vibrational syste that are described by the following equation α" ζα ' [ B ( τ)]sinα A ( τ)cosα, () at high frequency ( >> ) haronic excitation A ( τ) = Esin( τ) ; B ( τ) = N sin( τ ε),() with significant aplitudes ( E >>, N >> ). Nuerical integration confirs the adequacy/sufficiency of the suggested theoretical odel.. Task forulation The equations of otion of atheatical and physical pendulus at the translational vibration of their suspension point, as well as the equation of otion of two-stage gyro, fixed on a base, aking angular fluctuations around the horizontal axis are reduced to equation in diensionless for of type (). Therefore, the device described by equation () will be called the generalized pendulu. The task is to identify the paraeters of the possible forced oscillations of the device at the angle α provided high frequency ( >> ) and intense indignation in the aplitudes E and N. As shown in [], the in-ax attribute of stability, offered by T.G. Strizhak is an effective eans of theoretical study of the state of the pendulu balance at the intensive high-frequency vibration of the base. However, the deep study leads to the conclusion that this ethod is not universal. It gives good results if applied to the atheatical and physical pendulu, but it does not work when it has to deal with a two-stage gyro-copass (=two-stage gyro/ TSGC). That is the reason why a ore universal ethod was developed [, 3] it is applicable to any of the disturbing haronic vibrational systes and to the one described by equation () in particular, called the balance ethod. Thus, further on the otion described by the following equation is studied α" ζα ' [ N cos( τ ε )]sin α, (3) E cos( τ ) cosα where the prie ark denotes the derivative with respect to the diensionless tie... The balance ethod Coputer odeling, involving nuerical integration of equation (3) has shown that the ain ode of otion of the syste under study is the sustained constrained oscillations of the pendulu relating to a center which is soewhat shifted fro the equilibriu position. Therefore, the su of the two following coponents will be sought as a particular solution of the equation (3) α ( τ) = α α( τ), () where α is assued to be a slow/sluggish variable (the center of sall oscillations of the pendulu), and α(τ) - the high-frequency (fluctuating, centered) coponent of the angle with a ean value equal to zero. Now, equation () can be represented in the following way: α" ζ α ' α " ζα ' A ( τ )cos( α α).(5) [ B ( τ )]sin( α α) Let us suppose that the pendulu derivations/deviations fro the oscillation center are sall in value ( α <<, that is confired by the results of the odeling). Let us expand the trigonoetric functions in Taylor series in the neighborhood of the angle α adjacency and liit the constraint of only the first ter of the expansion. As a result the equation (5) is replaced by the approxiate one α" ζ α ' α " ζα ' A ( ) (cos sin ). (6) τ α [ B ( τ )](sinα α cosα ) The ain idea of the balance ethod is to divide the original equation into two equations, one of which describes the slow otion of the ean position of the pendulu, and the second - fast oscillations relating to this central position, ie, in a separate balancing of the original equation ters α α 30

2 averaged by the fast otion and its centered ebers. In order to get the equation for the slow oveent α (τ), it s necessary to average the resulting equation by the fast otion, taking into account the following α = α' = α" = A ( τ) = B ( τ), as well as considering the average values of the slowly varying quantities equal to these very values α ( τ) = α ; α ( τ) = α ; sin[ α( τ)] = sinα; cos[ α( τ)] = cosα; B ( τ)sin[ α( τ)] = B ( τ) sinα ; A ( τ)cos[ α( τ)] = A ( τ) cosα. On this basis, we coe to the following equation of the slowly varying otion coponent α" ζα ' sin α [ B ( τ)cosα A ( τ)sin α] α, which is ore convenient to use (for periodic processes x (t) and y (t) the equation dx ( t) dy( t) y( t) = x( t) is fair) in the dt dt following way: α" ζα ' sinα [ B( τ)cosα A( τ) sinα ] α ( τ). (7) Let us pass on to the foration of the equation of centered coponent of the otion. To do this, at first we need to subtract the averaged equation (7) fro the original equation (6): α " ζ α ' cosα α ( B α B α )cosα.(8) ( A α A α )sinα = ( A cosα B sinα) Further on we consider the case when the frequency of vibration of the base (and hence the frequency of the variation of the centered coponent of the pendulu otion) is uch higher than the frequency of natural oscillations of the pendulu ( >> ). This iplies that the aplitude of all ebers of the left-hand side of the equation (8) will be uch less than the aplitude of the first suand, and thus can be neglected. The result under rough approxiation will be the following α" [ A ( τ) cos α B ( τ)sin α], or, by eans of integration, α [ A ( τ) cos α B( τ)sin α]. (9) Considering (9), it is possible to specify the equation (7) of slow otion: α " ζα ' sin α [ A ( τ ) ]sin α B ( τ ) B( τ ) A( τ ) cos α After averaging according to the forulae () for a haronic vibration of the base we find (0) α " ζα ' sinα [ N E N cosε cosα E ]sin α. () We cae to the equation, which allows to deterine the slow oveent of the position of possible centers of pendulu s stationary oscillations when defining the paraeters N, E and ε, and oscillations of the point of its suspension. It, in contrast to the original tie-dependent equation is (due to the operation of averaging) a stationary (with constant coefficients) nonlinear differential equation. A distinctive feature of the equation () is that it allows us to investigate not only the equilibriu position, but the transition process of establishing constrained vibrations, as well as to deterine their aplitude. Su of the last three suands of the lefthand side of equation () is a diensionless positional torque P( α ) = sinα [ N E N cosε cosα E ]sin α Its equality to zero defines the position α of equilibriu (ie, the center of the stationary oscillations). Thus, the centers of stationary oscillations can be deterined by the solutions of the trigonoetric equation P( α ) = sinα Within the range [ N E N cosε cosα E π < α π () ]sin α. (3) the equation (3) ay have fro one to four radicals. If to ake a plot () fro the angle α, the solutions of (3) will be the points of intersection of this very plot with the X-axis. Whether the found point corresponds to the stable center is defined by the tangent slope to this plot in the zero-crossing point, ie, the sign of the function P( α ) Φ( α ) = = cosα [ N E ]cosα () α E N cosε sinα at this point. If the tangent to the plot in the selected point of intersection has a positive slope to the X- axis, ie, the condition is hold Φ( α P ( α) ) = > 0, (5) α α=α Then the found center of oscillations α is resistant. If the found eaning is negative in sign, the corresponding center is unsustainable. 3

3 It is easy to see that (5) defines the square of the diensionless frequency of the transient 0 oscillations of the pendulu in establishing the oscillations relating to the center α : = Φ( ). (6) 0 α Going back to the centered part of the solution, let us find fro (9) regarding (3): α = {[ E cosα N sin α cos ε ]sin( τ ),(7) N sin α sin ε cos( τ )} = D sin( τ χ) where D = ( E cosα N sin α cos ε ) ( N sin α sin ε ) = = E cos α N sin α E N sinα cosε = = ( E cosα N sinα ) E N sinα ( cosε) ;(8) N sin α sin ε tg χ =. E cosα N sin α cosε Fro (7) it follows that D α = cos( τ χ). The aplitude of the forced oscillations is defined by the forula D α = = ( E cosα N sinα ). (9) E N sin α ( cosε ).. Methods coparison The task of finding the paraeters of forced oscillations of the syste () has been previously solved with the help of the perturbation ethod and the in-ax ethod of T.G. Strizhak (for a pendulu). The weakness of the perturbation ethod is that the solution of equation () is found out on condition of sall perturbation (aplitudes E and N). Therefore, the centers of oscillations can only be obtained as a sall quantity of the second order regarding to the aplitude of the forced oscillations. However, when high-frequency perturbations are observed, they (perturbations) are ore powerful, the centers of constrained vibrations significantly deviate fro the equilibriu position, nevertheless, the aplitude of constrained vibrations is sall. In other words, the perturbation ethod describes the behavior of a syste under sall perturbations. The in-ax ethod shows good results in deterining the position of the centers of constrained vibrations with significant highfrequency perturbations only when applied to the pendulu systes. Using it for other systes such as two-stage gyro is not resultative, - iniization of the kinetic energy for generalized velocities does not lead to the foration of any kind of siilarity to the "vibration" coponent of the potential energy. In other words, in-ax ethod is not universal and is only valid for pendulu systes. The balance ethod excludes all these drawbacks. It works for the study of any syste described by equation () under the given conditions. In addition to finding the centers of stable constrained vibrations, as in the in-ax ethod, it also allows finding the aplitudes of these constrained vibrations, as well as the oscillation frequency in the transition process of the positioning to the found center..3. Axial vibration of the base Let us consider the otion when the base is vibrating, which we will call the "axial vibration." It corresponds to the oveent of the suspension point of the pendulu in space along the axis line segent that is inclined to the horizontal plane at an angle ϕ. In this case, E = V cos ϕ; N = V sin ϕ; ε, (0) and the laws () of the axial vibration take the following for A ( τ) = V cosϕ sin( τ) ; B ( τ) = V sinϕ sin( τ). As for the pendulu the value V is the aplitude of the diensionless velocity of the base vibration, which is connected to the aplitude v of the vibration velocity of the pendulu suspension point with the following equations: for a copound pendulu l V = v, where the ass of the gj pendulu, J its second oent in relation to the axis of rotation, g the acceleration of gravity, l shift of the ass center in relation to the point of suspension; for a atheatical pendulu v V =. gl In the case of two-stage gyro, axial vibration is a pitching of a base around the axis in the horizontal plane, aking an angle ϕ with the direction of the tangent to the space parallel, and the value V is deterined by the aplitude ϑ of the pitching angle of the base around the axis of vibration: H 0 V = ϑ, Jω3 cosϕ Ã Where H the proper oent of oentu of 0 the gyro; J gyro oent of inertia regarding to its easuring axis; ω the angular velocity of 3 3

4 rotation of the Earth; ϕ the latitude of the gyro à installation place. Hereafter, we will interpret the outcoing results in ters of two-stage gyro otion, because the in-ax stability criterion turns out to be invalid for it. In this case, the angle α characterizes the deviation of gyro spin axis fro the horizontal direction to the north (eridian line), the diensionless frequency is the ratio of the angular fluctuations of the base to the natural frequency of the two-stage H gyro 0ω3 cos ϕ à ω =. If the gyro with the 0 J oent of oentu H 0 = Ns and the 3 oent of inertia J =,607 0 Ns is located at the equator ( ϕ ), then its proper Г frequency will ake ω 0, 506c =, Hz. That is why only actions with frequencies ore than 0.5 Hz. can be rated as highfrequent ones. In this case, the diensionless "speed" is connected with the aplitude of the angular pitching of the base at the ratio V = 337, 7 ϑ, ie, it reaches the unit value when the aplitude of the pitching angle of the base ϑ, = = ', i.e., one angular inute. This eans that in the following plots for two-stage gyro with such paraeter points the value V can be replaced by the aplitude of pitching in angular inutes. It should be noted that the equilibriu position of the northern real axis of the two-stage gyro corresponds to the lower the equilibriu position of the pendulu, Southern equilibriu position of the two-stage gyro to the upper position of the pendulu, pitching of the base of the two-stage gyro around the north line to the vertical vibration of the pendulu base, pitching of the base of the twostage gyro around the tangent to the place parallel to the horizontal vibration of the suspension point of the pendulu. Therefore, all the features of twostage gyro otion can be transferred to the pendulu with the entioned substitutions. In the case of axial vibration (0), equation (3) takes the following for α" ζα ' [ V sin ϕ cos( τ )]sin α, () V cosϕ cos( τ ) cosα Positional torque () in this case is deterined by the expression V P ( α) = sin α sin ( α ϕ). () Fig. shows plots () of the angle α for different angles ϕ of pitching axis inclination to the west-east line and the different values of the aplitude V of the pitching base. Positional torque of the generalized pendulu Positional torque of the generalized pendulu Positional torque of the generalized pendulu Positional torque of the generalized pendulu 33

5 Positional torque of the generalized pendulu Positional torque of the generalized pendulu Fig. Theoretical dependence of the nonlinear positional torque fro angle of the pendulu alteration It follows fro the plots in Fig., at pitching of the base around the west-east line ( ϕ ) and with the aplitude of the vibration of the base V =, the two-stage pendulu has only one stable oscillation center the northward direction. If the aplitude of pitching of the base is large (for exaple, V = 3), then there are two stable centers of oscillation, that are syetrically arranged as related to the eridian of the place. At pitching of o the base around the base eridian line ( ϕ = 90 ) with a sall aplitude (for exaple, V = ), the real axis of the two-stage pendulu has only one stable equilibriu the northward directed one. If the aplitude of vibration of the base is significant (for exaple, V = 3), there is another stable equilibriu the southward directed one. Pitching of the base around the axis that is o positioned at an angle ϕ = 5 to the cardinal directions, the vibration with low intensity is accopanied by the presence of only one stable center of oscillations (see the case when V =, 8), that has a deviation fro the North direction. More intense vibration (for exaple, V = 3) causes the appearance of two stable centers of oscillation. The trigonoetric equation (3) for deterining the position of the center of oscillation will take the following for V sin α sin ( α ϕ). (3) The condition of stability of the center of oscillations becoes V cosα cos ( α ϕ) > 0. () The expression for the frequency of the transition process of the gyro takes the for V 0 = cosα cos ( α ϕ), (5) and the aplitude of forced oscillations of the gyro becoes V α = cos( α ϕ). (6) In relation to the centers of forced oscillations the balance ethod, when applied to the two-stage gyro, allows to get the results siilar to those, resulting fro the application of the in-ax ethod to the pendulu. Thus, northern position of the two-stage gyro s equilibriu of the real axis with a pitching base around the north-south axis is always stable, and the forced oscillations always have the aplitude equal to zero. Southern equilibriu becoes stable if the aplitude V of pitching base around the eridian line exceeds the value of., i.e. subject to the condition that: V >. (7) The aplitude of forced oscillations towards the Southern equilibriu is also equal to zero. Fig.. Dependence of the frequency of the transition process of the Two-stage gyro fro the aplitude of pitching of the base around the eridian line Fig. shows (the dotted lines) the plots (5) of the Northern (blue line) and Southern (red line) natural oscillation frequency towards the equilibriu positions of the two-stage gyro against the aplitude of pitching of the base towards the north-south line. 3

6 In the sae figure the square arkers show the results of easureent of the diensionless frequency of natural oscillations by coputer siulation (nuerical integration of the differential equation of otion ()) at a pitching frequency. It is easy to see that for low vibration aplitudes ( V < ) the balance ethod (leading to 3 the sae results as the ethod of averaging) gives a satisfactory coincidence with the odel experient. However, for high aplitudes ( > V > ), the 3 difference between the experient and the theory is 35%. In case of pitching of the base around the westeast axis the oscillations of the two-stage gyro towards to the North direction are stable, provided the aplitude of pitching of the base is subject to the condition V <, (8) In this case the sustained forced oscillations have the aplitude V α =. (9) and becoe unstable when the aplitude of the velocity is subject to the condition (30). The frequency of the transitional process can be defined by equation 0 V =, (30) i.e., reduced (to zero at V = ) with increasing intensity of the vibration of the base. When pitching of the base around the westeast line, the oscillations of the two-stage gyro towards the southern equilibriu are always unstable. The equilibriu positions defined by the condition = V cosα, (30) have a real significance only in case of the fulfillent of the condition (7). That is why, when the condition (7) is fulfilled, the pendulu oscillations near the relevant equilibriu positions are stable. The aplitude of oscillations of the two-stage gyro concerning these positions, in accordance with (6) and (37), is defined by the equation α =, (3) V and the frequency of the transition process with the ratio V 0 = Φ( α ) =. (3) V Fig.3 shows the plot of the position of the center of oscillation against the aplitude of the pitching of the base around the west-east axis (dots), built in accordance with the forula (30). Also there are siilar plots based on the use of the perturbation ethod (asterisk arkers). Square arkers point at the results of the centers position easureent on condition of the perturbation frequency = 0, obtained by progra siulation (nuerical integration of the equation ()). Fig.3. The dependence of the two-stage gyro s center of oscillation position fro the velocity aplitude of the vibration base Fig.. The dependence of the aplitude of the two-stage gyro s forced oscillations fro the aplitude of pitching of the base around the west-east line Fig. shows the plots (dot arkers) of the aplitude of pendulu s forced oscillations, built in accordance with the forulas (9) and (3) when the case of the perturbation frequency = 0 is fair, as well as siilar plots, based on the results of the perturbation ethod application (asterisk arkers), and the ones obtained by coputer siulation (square arkers).). 35

7 Fig.5. The dependence of the frequency of the two-stage gyro s transition process fro the aplitude of pitching of the base around the west-east line Fig.5 (point arks) shows the plots of the theoretical dependences (30) and (3) of the frequency of the two-stage gyro s transition process towards the northern and side positions of equilibriu against the diensionless aplitude of the pitching of the base. Square arkers reflect odel easureents of the transition process frequency at a frequency of vibration of the base = 0. As it can be seen, the aplitude velocity increase causes the reduction of oscillation frequency towards the lower equilibriu to zero (if V = ). The frequency of natural oscillations towards the lateral centers increases, starting fro zero (at V = ) till infinity. The siulation results coincide with the predictions of the theory in the whole range of existence of forced oscillations. As seen fro the figures, the significant difference can only be observed in definition of the frequency of the transition process, starting with the aplitude V. Starting with 3 the values of the aplitudes V, forced oscillations with a frequency of vibration of the base, are replaced by the paraetric ones with half as uch frequency. The oscillation of base around the axis that is inclined to the cardinal points at an angle of 5 degrees causes forced oscillations with the paraeters that are represented in Fig.6 and 7. The positions of stable centers of forced oscillations found by eans of other ways, are shown in Fig.6. Fig.6. The dependence of the two-stage gyro s equilibriu position fro the aplitude of oscillation of the base at o φ = 5 Fig.7 shows the dependence of the aplitudes of two-stage gyro s forced oscillations, calculated according to the forulas of the balance ethod (dot arkers) and the perturbation ethod (asterisks arkers) for the perturbation frequency = 0. Square arkers show the corresponding aplitude values obtained by the progra siulation of the oveent of the pendulu. Fig.7. The dependence of aplitude of oscillation of the base fro the aplitude of forced oscillations at perturbation frequency = 0 As it can be seen, in this case, the coincidence of the balance ethod with the experient is alost coplete. The perturbation ethod leads to satisfactory values of the aplitudes only at lowintense oscillations of the base ( V )... Conclusions The investigation held allows to conclude the following.. The proposed balance ethod is rather siple and effective in defining the paraeters of forced oscillations of nonlinear vibration syste at an intensive high-frequency vibration perturbation. In 36

8 addition to the position of the centers of stable oscillations, it allows to deterine the aplitude of these oscillations, and the frequency of the transition process leading to it.. The balance ethod is ore universal than the ethod of in-ax criterion of stability proposed by T.G. Strizhak and it leads to results that are ore feasible than the perturbation ethod. 3. Though soe results ay be obtained with the help of other ethods, the balance ethod can yield siilar results easier, uch faster and in a wider range of possible paraeters of oscillation of the base.. The definition of the aplitudes of forced oscillations with the help of the balance ethod also deonstrates good coincidence with the odeling results. 5. The results obtained with the help of the balance ethod agree with the previously studied pendulu s behavior patterns and the two-stage gyro. They allow to deepen and develop both quantitatively and qualitatively our knowledge of these interesting objects. Aong the drawbacks of the balance ethod one can find the next positions: the outlined variant of the balance ethod doesn t allow to predict the occurrence of paraetric oscillations, the appearance of which deonstrates the nuerical integration of the corresponding differential equations, and, oreover, to deterine the paraeters of such oscillations; in deterining the frequency of the transient process, the balance ethod does not allow to establish the influence of frequency of the oscillation of the base on this frequency, which is also deonstrated by the odeling.. Bibliography And Authors [] Geraichuk M., Lazarev Yu., Lasarenko A. On stability of the centers of force pendulu oscillation on base vibration // XIV International PhD Workshop OWD 0, 0-3 October 0, pp [] Лазарев Ю. Ф. К теории обобщенного маятника на вибрирующем основании. // XІI Міжнародна науково-технічна конференція "Приладобудування: стан і перспективи". Збірник тез доповідей. К.: НТУУ "КПІ", с. С [3] Лазарев Ю. Ф., Аксененко П. М. Применение метода баланса для исследования поведения двухстепенного гирокомпаса при вибрации // XІI Міжнародна науково-технічна конференція "Приладобудування: стан і перспективи". Збірник тез доповідей. К.: НТУУ "КПІ", с. Authors: Prof. Michail D. Geraichuk College of Instruent Design and Engineering National Technical University of Ukraine 37 Prospect Pereogy Kiev 03056, Ukraine eail: geraichuk@kpi.ua Ph.D., Associate Professor Yuri Lazarev College of Instruent Design and Engineering National Technical University of Ukraine 37 Prospect Pereogy Kiev 03056, Ukraine eail: laz@pson.ntu-kpi.kiev.ua Graduate student Peter Aksonenko College of Instruent Design and Engineering National Technical University of Ukraine 37 Prospect Pereogy Kiev 03056, Ukraine eail: akc.petia@ail.ru, akc.petia@gail.co 37