It. J. Adv. Appl. Math. ad Mech. 5() (207) 6 (ISSN: 2347-2529) IJAAMM Joural homepage: www.ijaamm.com Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics Stoppig oscillatios of a simple harmoic oscillator usig a impulse force Research Article C. Mabega, R. Tshelametse Departmet of Mathematics, Uiversity of Botswaa, Gaboroe, Botswaa Received 8 November 206; accepted (i revised versio) 6 Jue 207 Abstract: MSC: The harmoic oscillator is famously used amog may systems to model the motio of a mass-sprig system. A either damped or drive harmoic oscillator is kow as a simple harmoic oscillator. Its solutio commoly kow as simple harmoic motio comprises of cotiuous siusoidal oscillatios with a costat amplitude ad period. I this paper, we determie the magitude of a impulse force required to stop the oscillatios of a simple harmoic oscillator of a mass sprig system. The magitude of the impulse force is obtaied as the product of the mass, frequecy ad amplitude of the oscillatios. Depedig o the directio ad time istat this impulse force is applied, the resultig amplitude of the oscillatios ca attai a miimum value of zero or a maximum value of double the iitial amplitude. We propose the optimal time istats ad directio to apply the impulse force i order to stop the oscillatios. 34K 40D0 Keywords: Simple harmoic oscillator Mass-sprig system Impulse force 207 The Author(s). This is a ope access article uder the CC BY-NC-ND licese (https://creativecommos.org/liceses/by-c-d/3.0/).. Itroductio The harmoic oscillator is widely used to model systems from various fields of sciece ad egieerig such as the mass sprig system i mechaics, the RLC series circuits i the study of electrical circuits, ad the blood glucose level regulatory system i the huma body [ 3. The harmoic oscillator equatio for a mass sprig system is give by the secod-order, liear, costat coefficiets ordiary differetial equatio mẍ(t) + cẋ(t) + kx(t) = F (t), x(0) = x 0, ẋ(0) = v 0. () Here ẍ(t) is the acceleratio, ẋ(t) the velocity, x(t) the displacemet of the mass m from the equilibrium positio at time t, k > 0 the sprig costat, c 0 the viscous dampig costat, F (t) the drivig force applied to the system, x 0 ad v 0 the iitial displacemet ad velocity, respectively [ 4. The mass is take to start from a poit below the equilibrium positio if x 0 > 0 ad vice versa [. For better aalysis of the solutios, equatio () is ofte expressed as ẍ(t) + 2ξ ẋ(t) + ω 2 x(t) = m F (t), x(0) = x 0, ẋ(0) = v 0, (2) c k after dividig by m ad lettig ξ = 2 km, ad =, be the dampig ratio ad the atural frequecy of the system, respectively [5. The behaviour of the solutio of (2) is solely determied by the value of the dampig ratio, ξ [5. m I the absece of both dampig (ξ = 0), ad the drivig force (F (t) = 0) we have what is kow as a simple harmoic Correspodig author. E-mail addresses: mabegac@mopipi.ub.bw (C. Mabega), TSHELAME@mopipi.ub.bw (R. Tshelametse).
2 Stoppig oscillatios of a simple harmoic oscillator usig a impulse force oscillator or a udamped harmoic oscillator. Its solutio referred to as simple harmoic motio is characterised by cotiuous siusoidal oscillatios with a costat amplitude ad period [ 3. A impulse force is a force that is defied to be zero elsewhere except at the time istat at which it is applied. It is ofte deoted by F (t) = F 0 δ(t t 0 ), (3) where F 0 is the magitude of the force ad t 0 the time istat at which it is applied [2, 6. For a mass-sprig system a impulse force ca be thought as a sharp blow or a udge. I this paper, we determie the magitude of a impulse force eeded to stop the oscillatios of a simple harmoic oscillator. The correct time istats ad directio of the force are also established. 2. Simple harmoic oscillator drive by a impulse force I the absece of dampig the harmoic oscillator equatio (2) drive by a impulse force is give by ẍ(t) + ω 2 x(t) = m F 0 δ(t t 0 ), with x(0) = x 0, ẋ(0) = v 0. (4) We solve (4) usig the Laplace trasform method as it is suitable for discotiuous forcig fuctios such as impulse fuctios [2, 6. Takig the Laplace trasform both sides of (4) we have L { ẍ(t) + ω 2 x(t)} = L { m F 0 δ(t t 0 ) } (5) which simplifies to X (s) = f ree Respose x 0 s + v 0 s 2 + ω 2 + F 0 e t 0 s m(s 2 + ω 2 ) (6) f ree Respose x 0 s + v 0 L {X (s)} = L F 0 m e t 0 s F (s) (7) = s 2 + ω 2 + where X (s) is the Laplace trasform of x(t), ad F (s) = s 2 + ω 2 [2, 6, 7. Takig the iverse Laplace trasform both sides of (7) we have f ree Respose x 0 s + v 0 which gives f ree respose s 2 + ω 2 + F 0 m e t 0 s F (s) F 0 m u(t t 0 )si[ (t t 0 ) (9) x(t) = x 0 cos( t) + v 0 si( t) + where u(t t 0 ) is a uit step fuctio [2, 3, 6, 7. More explicitly the geeral solutio is give by x 0 cos( t) + v 0 si( t), t < t 0, x(t) = x 0 cos( t) + v 0 si( t) + F 0 si[ (t t 0 ), t t 0. m Expadig the term si[ (t t 0 ), the solutio (0) simplifies to x 0 cos( t) + v 0 si( t), t < t 0, x(t) = [ x 0 F [ 0 v0 si( t 0 ) cos( t) + + F 0 cos( t 0 ) si( t), t t 0. m m For both the cases t < t 0, ad t t 0 the solutio () ca be expressed i amplitude-phase form x(t) = A cos( t φ), (2) where A is the amplitude, is the frequecy ad φ is the phase agle of the oscillatios [, 2, 4. The solutio havig the form (2) is called simple harmoic motio. Its period is give by T = 2π ad is idepedet of the iitial coditios x 0 ad v 0. The sie fuctio ca also be used istead of cosie [, 2. The mass passes through the equilibrium positio whe the displacemet is zero. From (2), x(t) is zero whe cos( t φ) = 0 givig the time values as t = [ π 2 + 2π + φ, t = [ 3π 2 + 2π + φ, Z. (3) (8) (0) ()
C. Mabega, R. Tshelametse / It. J. Adv. Appl. Math. ad Mech. 5() (207) 6 3 2.. Amplitude ad Phase of the geeral solutio 2... Before applyig a impulse force Expadig (2) ad equatig like terms to solutio () for the case before a impulse force is applied (t < t 0 ), we obtai ad A cos(φ) = x 0 (4) A si(φ) = v 0. (5) From (4) ad (5), the iitial amplitude of the solutio A = A ii t is obtaied as A ii t = x 2 0 + ( v0 ) 2. (6) Similarly from (4) ad (5), the iitial phase agle of the solutio φ = φ ii t is obtaied as ( ) φ ii t = ta v0. (7) x 0 2..2. Impulse force applied Similarly from () for the case whe a impulse force is applied (t t 0 ), the resultig amplitude of the solutio A ew is give by A ew = which simplifies to A ew = x 2 0 + ( v0 A 2 ii t + F 2 0 m 2 ω 2 ) 2 ( ) 2 F0 + + 2F [ 0 v0 cos( t 0 ) x 0 si( t 0 ) m m 2F 0 A ii t m si( t 0 φ ii t ) (9) where A ii t ad φ ii t are give by (6) ad (7), respectively. The resultig phase agle φ ew is give by ( ) m φ ew = ta v0 + F 0 cos( t 0 ). (20) m x 0 F 0 si( t 0 ) (8) The aim of this paper is to determie the value of F 0 eeded to stop the oscillatios. That is, the value of F 0 for which A ew is zero. So equatig (9) to zero ad simplifyig gives F0 2 2m A ii t si( t 0 φ ii t )F 0 + m 2 ω 2 A2 ii t = 0. (2) Solvig (2) for F 0 gives F 0 = m A ii t [si( t 0 φ ii t ) ± si 2 ( t 0 φ ii t ). (22) Equatio (22) has real roots whe the discrimiat D = si 2 ( t 0 φ ii t ) is zero. That is, whe si( t 0 φ ii t ) = ±, (23) givig the time istats as t 0 = [ π 2 + 2π + φ ii t, t 0 = [ 3π 2 + 2π + φ ii t, Z (24) ad the magitude of the impulse force as F 0 = ±m A ii t. (25) The plus ad mius idicates that the force acts i the positive ad egative directio, respectively.
4 Stoppig oscillatios of a simple harmoic oscillator usig a impulse force 2..3. Miimum ad maximum values of the resultig amplitude, A ew Next we determie the value of F 0 for which A ew attais its miimum ad maximum values. Mathematically, if A ew attais a miimum at some value of F 0, the the derivative A ew (F 0) is zero [8, 9. Differetiatig (9) with respect to F 0 gives A ew (F 0) = Equatig (26) to zero we obtai A 2 ii t + F 0 2 m 2 ω 2 2F 0 A ii t si( t 0 φ ii t ) m [ F0 m 2 ω 2 A ii t si( t 0 φ ii t ). (26) m F 0 = m A ii t si( t 0 φ ii t ). (27) Similarly, if A ew has a maximum at some value of F 0, the A ew (F 0) = 0 [8, 9. Thus the impulse force givig miimum ad maximum values of the resultig amplitude has a maximum value of F 0 = m A ii t which is equivalet to the value give by (25). Notice that a impulse force of magitude F 0 = m A ii t eeded to stoppig the oscillatio ca also result i oscillatios of various amplitudes. Next we determie the maximum value of the resultig amplitude that ca be produced by this impulse force. Substitutig for F 0 = m A ii t ito (9) ad simplifyig gives A ew = A ii t 2 [ si( t 0 φ ii t ) (28) givig the maximum value of the resultig amplitude as A ew [max = 2A ii t. It occurs whe si( t 0 φ ii t ) = givig the time istats as t 0 = [ 3π 2 + 2π + φ ii t, Z. (29) Also from (28), the resultig amplitude is zero whe si( t 0 φ ii t ) = givig the time istats t 0 = [ π 2 + 2π + φ ii t, Z. (30) These time istats coicide with those obtaied i (24) for a positive impulse force of magitude F 0 = m A ii t. I coclusio, a impulse force eeded to stop the oscillatios of the simple harmoic motio has magitude F 0 = m A ii t. However, this impulse force ca either uchage, reduce or icrease the amplitude by a factor less tha oe whe applied at time istats differet from those give i (24). For example, usig (28) o chage i the amplitude occurs (A ew = A ii t ) whe the force is applied at the time istats t 0 = [ π 6 + 2π + φ ii t, t 0 = [ 5π 6 + 2π + φ ii t The amplitude is halved (A ew = 2 A ii t ) whe the force is applied at the time istats, Z. (3) t 0 = [ si ( ) 7 ω 8 + 2π + φii t, t0 = [ π si ( ) 7 ω 8 + 2π + φii t, Z. (32) The amplitude is icreased by half (A ew = 3 2 A ii t ) whe the force is applied at the time istats t 0 = [ π + si ( ) ω 8 + 2π + φii t, t0 = [ 2π si ( ) ω 8 + 2π + φii t, Z. (33) 2.2. Directio of impulse force The directio i which the impulse force is applied is a crucial. Practically, a miimum value of the amplitude ca be achieved oly if the impulse force acts i the opposite directio of the motio. Similarly, a maximum value of the amplitude ca oly be obtaied whe the impulse force acts i the same directio of the motio. Thus if at a certai time istat the resultig amplitude attais a miimum value whe the force is applied i oe directio, a maximum value is obtaied whe the same force of the same magitude is applied i the opposite directio ad vice versa. 3. Results ad aalysis We cosider a mass-sprig system havig a mass of 2 kg with sprig costat 8 N/m. The mass is iitial displaced a distace of 0.25 m below the equilibrium positio ad the released with a iitial velocity of m/s. The parameter values are: m = 2 kg, k = 8 N/m, x 0 = 0.25 m, v 0 = m/s, = 3 rads/sec, A ii t = 0.467 m, F 0 = m A ii t = 2.5000 N, ad φ ii t = 0.9273 radias. The s for a impulse force havig a magitude of F 0 = 2.5000 N applied i the positive ad egative directio for = 3 are give i Fig. ad Fig. 2. From Fig., a positive impulse force at the give time istat termiates the iitial amplitude while a egative impulse force doubles the iitial amplitude. From Fig. 2, a positive impulse force at the give time istat doubles the iitial amplitude while a egative impulse force termiates the amplitude.
C. Mabega, R. Tshelametse / It. J. Adv. Appl. Math. ad Mech. 5() (207) 6 5 (a)f 0 = +2.5000 N (b)f 0 = 2.5000 N Fig.. Free ad s at t 0 = [ π2 + 2π + φ ii t (a)f 0 = +2.5000 N (b)f 0 = 2.5000 N Fig. 2. Free ad s at t 0 = [ 3π2 + 2π + φ ii t 4. Coclusio The impulse force eeded to stop the oscillatios of a mass sprig system udergoig simple harmoic motio is equal to the product of the mass, frequecy ad amplitude of the oscillatios. This impulse force must be applied at the time istats correspodig to the equilibrium positios of the displacemet ad i the opposite directio to the displacemet. Oscillatios with maximum amplitude of double the iitial amplitude are obtaied whe the force is applied i the same directio as the displacemet. Applyig this impulse force at time istats other tha the equilibrium positios either brigs o chage, a decrease or icrease i the iitial amplitude by a factor less tha oe. A decrease occurs whe the force acts i the opposite directio to the displacemet ad vice versa. Refereces [ P. Blachard, R. L. Devaey ad G. R. Hall. Differetial Equatios, Fourth Editio, Iteratioal Editio, Brooks/Cole, Cegage Learig, USA, 20. [2 M. Brau. Differetial Equatios ad Their Applicatios: A Itroductio to Applied Mathematics, Fourth Editio, Spriger-Verlag, New York, USA, 993. [3 D. Schott, Modelig of Oscillatios: geeral framework ad simu;latio projects, Global Joural of Egieerig Educatio 2 () (200) 7-23. [4 M. D. Mosia, J. Akade, D.K.K. Adjai, L.H. Koudahou ad Y.J.F. Kpomahoa, A theory of eaxact itegrable quadratic Lie ÌAard type equatio, It. J. Adv. Appl. Math. ad Mech. 4 () (206) 2-25. [5 H. Gavi ad J. Dolbow. Buildig Vibratios: Aalysis, Visualizatio ad Experimetatio, Departmet of Civil ad Evirometal Egieerig, Duke Uiversity, Durham, http://weave.duke.edu/hpgavi@duke.edu. [6 K. Asher, A Itroductio to Laplace Trasform, Iteratioal Jourals of Sciece & Research, 2 () (203) 60-606.
6 Stoppig oscillatios of a simple harmoic oscillator usig a impulse force [7 H. K. Jassim, Homotopy perturbatio algorithm usig Laplace trasform for Newell-Whitehead-Segel equatio, It. J. Adv. Appl. Math. ad Mech. 2 (4) (205) 8-2. [8 M. G. Voskoglo, Solvig problems with the help of computer: A fuzzy logic approach, It. J. Adv. Appl. Math. ad Mech. 2 (2) (204) 62-7. [9 A. Rivera-Figueroa, J. C. Poce-Campuzao, Derivative, maxima ad miima i a graphical cotext, It. J. of Mathematical Educatio i Sciece & Techology 44 (2) (203) 284-299. Submit your mauscript to IJAAMM ad beefit from: Rigorous peer review Immediate publicatio o acceptace Ope access: Articles freely available olie High visibility withi the field Retaiig the copyright to your article Submit your ext mauscript at editor.ijaamm@gmail.com