International Journal of Advance Engineering and Research Development OSCILLATION AND STABILITY IN A MASS SPRING SYSTEM

Transcription

1 Scientific Journal of Ipact Factor (SJIF): 5.71 International Journal of Advance Engineering and Researc Developent Volue 5, Issue 06, June -018 e-issn (O): p-issn (P): OSCILLATION AND STABILITY IN A MASS SPRING SYSTEM A. George Maria Selva 1, R. Janagaraj 1 1 Departent of Mateatics, Sacred Heart College (Autonoous), Tirupattur , Vellore Dist., Tail Nadu, S.India. Abstract Tis paper exaines te dynaical beavior of Daped and Undaped otions of ass spring syste represented by Hoogeneous Differential Equations as well as Discrete Fractional order Equations. For eac syste te equilibriu position is establised and te nature of te syste is analyzed. Also tie line plots and pase diagras for appropriate nuerical values are produced. Keywords- Mass Spring, Hoogeneous, Oscillation, Stability, Fractional differential equations. []1 ESTABLISHMENT OF MASS SPRING SYSTEM Mass spring syste is one of te ost iportant applications of second order differential equations [, 5, 8]. Mass spring syste is described as d x x F() t Tis derived fro Newton s second law of otion ([, 5]). Here is ass; d x Ft () is acceleration; x is restoring force ( is spring constant); is daping force ( is daping constant); and denotes external forces acting on te ass, wic depends on bot ass and tie. Tus d x x F() t Te equation is oogeneous or nonoogeneous depending on weter forces oter tan te spring and daping forces act on te ass. In tis paper we are considering only two forces acting on te ass, ignoring te external forces. [] MASS SPRING SYSTEM WITH DAMPING VIBRATIONS All vibrations are subject to daping of soe sort, wen no oter forces act on te ass, besides te spring, and gravity for vertical oscillations. Differential equation (1) becoes oogeneous of te for, d x 0 x 4 Solving (), we get te roots r1,. We sall sow tat tree types of otion can occur [, 5]; 1. Over daped if 4 0 (te roots are real and distinct).. Critically daped if 4 0 (te roots are real and equal). 3. Under daped if 4 0(te roots are iaginary)..1. Syste of Differential Equations Second order oogeneous differential equation is converted into a syste of first order differential equations by taing yt () [4]. Equation () taes te for yt () (3) dy y( t) x( t) Te syste (3) as only one fixed point naely (0,0) and te Jacobian Matrix J for te syste (3) is [4, 7] (1) All rigts Reserved 37

2 Volue 5, Issue 06, June-018, e-issn: , print-issn: Te eigen values of (3) are , 4. Next we discuss te types of otion wit illustration. (4).1.1. Over daped Motion. Te syste (3) is over daped if 4 0, so tat 4. Let 1; 4;and wit te initial condition x(0) 0.5; y(0) 0.5. Fro te Jacobian atrix (4), we obtain te roots and 3.414, wic are real and distinct. In figure - 1 is overdaped otion; daping is so large tat oscillations are copletely eliinated. Te ass siply returns to te equilibriu position witout passing troug it..1.. Critically Daped Motion. Te syste (3) is critically daped if Figure 1. Over daped Motion of (3) wit te 4 0, so tat 4. Taing 1; 4;and 4 initial condition x(0) 0.5; y(0) 0.5. Using te Jacobian atrix (4), te roots are 1,, wic are real and equal. In figure - sows critically daped otion; Once again no oscillations occur. Tis situation fors te division between over daped and under daped otion Under daped Motion. Te syste (3) is under daped if Figure. Critically daped for te syste (3) 4 0, so tat 4. Considering 1; 0.;and wit te initial condition x(0) 0.5; y(0) 0.5, te roots are 1, 0.1 i1.4101, wic is iaginary. In figure - 3, we get underdaped otion; we ave daped oscillation and te otion goes to 0 wen te tie All rigts Reserved 373

3 Volue 5, Issue 06, June-018, e-issn: , print-issn: Figure 3. Underdaped for te syste (3).. Discrete Fractional Order Syste. Let us consider te syste of fractional differential equations as D x( t) y( t) D y( t) y( t) x( t) were is te fractional order. [1, 3, 6] Wen we apply te process of discretization, we get its discrete version in te for, x( t 1) x( t) y( t) (1 ) (5) y( t 1) y( t) y( t) x( t) (1 ) Te fixed point of te syste (5) is (0,0) and te Jacobian Matrix J for te syste (5) is 1 s s s 1 s s were s, te eigen values of te syste (5) are 1, 1 4. Now we investigate te types of (1 ) otion and plot te nuerical siulation wit different paraeters...1. Over daped Motion. Te syste (5) is over daped if 4 0, so tat (6) 4. Let 0.8; 0.04; 1; 4;and wit te initial condition x(0) 0.5; y(0) 0.5. Using te Jacobian atrix (6), te roots are and 0.711, wic is real and distinct. In figure - 4 is Overdaped otion. Figure 4. Over daped Motion for All rigts Reserved 374

4 Volue 5, Issue 06, June-018, e-issn: , print-issn: Critically Daped Motion. Te syste (5) is critically daped if 4 0, so tat 4. Taing te values 0.8; 0.04; 1; 4;and 4 wit te initial condition x(0) 0.5; y(0) 0.5. Using te Jacobian atrix (6), te roots are 1, , wic is real and equal. In figure - 5 is critically daped otion...3. Under daped Motion. Te syste (5) is under daped if Figure 5. Critically daped Motion for (5) 4 0, so tat 4. Considering 0.8; 0.04; 1; 0.;and wit te initial x(0) 0.5; y(0) 0.5. Te roots are 1, i0.1153, wic is iaginary. In figure 6 is underdaped otion. FIGURE 6. Underdaped for te syste (5) In Figures 7, we execute te grap of te dynaical syste for different orders of te fractional derivative; naely, (0.4 All rigts Reserved 375

5 Volue 5, Issue 06, June-018, e-issn: , print-issn: Figure 7. Overdaped and Crically daped otions for te syste (5) wit various fractional order s In figure 8, exibits te underdaped otions for te syste (5) wit various fractional order [0.7,0.8]. Fractional order 'sfro , presents te unbounded oscillations followed by te unifor oscillation wen finally te syste attains stability for [0.78,0.8]. 0.7, and for FIGURE 8. Underdaped otions for te syste (5) wit various fractional order s III. MASS SPRING SYSTEM WITH UNDAMPING VIBRATIONS In tis section, we consider only spring force acting on te ass and ignore te daping forces, wic of te d x 0 x Solving te equation (7), we get te roots r1, i. Now te equation (7) is cange into te syste of differential equations given by yt () (8) dy xt () Te fixed point of te syste (8) is (0,0) and te Jacobian atrix of te syste is Te eigen valus are 1, i. Taing te suitable paraetric values 1; 36 wit te initial conditions x(0) 0.5; y(0) 0.5, we get te eigen values 6. i Te ass oscillates about its equilibriu position forever. Tis is a (7) All rigts Reserved 376

6 Volue 5, Issue 06, June-018, e-issn: , print-issn: direct result of te fact tat daping as been ignored. Te ass oscillates up and down after a long tie period and it is reduced to a stable position wen te springs weaen, see figure - 9. Figure 9. Mass Spring syste (8) for Free Daped Motions Let us consider te syste fractional order differential equations of te for D x( t) y( t) D y( t) x( t) is te fractional order. Now using te discretization process, te above fractional order differential equations is odified into te for of syste of discrete fractional order equations, x( t 1) x( t) y( t) (1 ) Te fixed point (10) is were (0,0) y( t 1) y( t) x( t) (1 ) and te Jacobian Matrix J for (10) is 1 s s 1 s and te eigen values (10) are 1, 1 is. Now coosing te paraeter values (1 ) 0.8; 0.01; 1; 0. wit te initial conditions x(0) 0.5; y(0) 0.5, we get te eigen values 1, 1 i Te syste is undaped, te solution as led to oscillations tat becoe unbounded, refer figure (10) (11) FIGURE 10. Discrete Fractional Order Mass Spring syste (10) of Free Daped All rigts Reserved 377

7 Volue 5, Issue 06, June-018, e-issn: , print-issn: REFERENCES [1] A.George Maria Selva, Janagaraj, R.Dinesbabu, Dynaical Analysis of a Discrete Fractional Order Prey Predator 3 D Syste, International Journal of Researc & Developent Organization, Vol., No. 1, 4-31 (016). [] Dennis G. Zill, A first Course in Differential Equations wit Modelling Applications, Nint Edition, Broos/Cole, Cengage Learning (009). [3] El-Sayed, AMA, Salan, SM: On a discretization process of fractional-order Logistic differential equation, J. Egypt.Mat. Soc. (accepted) [4] Lawrence Pero, Differential Equations and Dynaical Systes, Tird Edition, Springer International Edition, First Indian Reprint, (009). [5] Paul Blancard, Robert L. Devaney, Glen R. Hill, Differential Equations, Fourt Edition, Broos/Cole, Cengage Learning (01). [6] Ravi P Agarwal, Aed MA El-Sayed and Sanaa M Salan, Fractional-order Cua s syste discretization, bifurcation and caos, Agarwal et al. Advances in Difference Equations, 30 (013). [7] Saber Elaydi, An Introduction to Difference Equations, Tird Edition, Springer International Edition, First Indian Reprint, (008). [8] Willia E. Boyce and Ricard C. Di Pria, Eleentary Differential Equations and Boundary Value Probles, Jon Wiley and Sons, Inc., All rigts Reserved 378