Thông tin tài liệu
then the real part of the solution ~ U and the imaginary part ~ V are suspected to be solutions of the equations LX U and LX V X m 2.4.1 THE METHOD OF VARIATION OF ARBITRARY PARAMETERS (THE LAGRANGE METHOD) If the general solution of the corresponding homogeneous system of equations (2.21) is known, and one cannot choose a particular solution of the system of equations (2.20), then the method of variation of parameters may be applied. Let X n i1 c i X i be the general solution of the system (2.21). The solution of the nonhomogeneous system (2.20) must be of the form X t n i1 c i tX i Y 2X23 where c i t are the new unknown functions. If we substitute into the nonhomogeneous equation, we obtain n i1 c H i tX i F X This vector equation is equivalent to a system of n equations n i1 c H i tx 1i f 1 t n i1 c H i tx 2i f 2 t ÁÁÁ n i1 c H i tx ni f n tX V b b b b b b b ` b b b b b b b X 2X24 All c H i t are determined from this system, c H i tj i ti 1Y 2Y FFFY n, whence c i t j i tdt " c i i 1Y 2Y FFFY nX The system X 1 x 11 x 21 F F F x n1 V b b b ` b b b X W b b b a b b b Y Y X 2 x 12 x 22 F F F x n2 V b b b ` b b b X W b b b a b b b Y Y FFFY X n x 1n x 2n F F F x nn V b b b ` b b b X W b b b a b b b Y of particular solutions of the homogeneous system of differential equations is said to be fundamental in the interval (aY b) if its Wronskian W tW X 1 Y X 2 Y FFFY X n x 11 t x 12 t ÁÁÁ x 1n t x 21 t x 22 t ÁÁÁ x 2n t ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ x n1 t x n2 t ÁÁÁ x nn t T 0 2. Systems of Differential Equations 829 Differential Equations for all t PaY b. In this case, the matrix M t x 11 t x 12 t ÁÁÁ x 1n t x 21 t x 22 t ÁÁÁ x 2n t ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ x n1 t x n2 t ÁÁÁ x nn t P T T R Q U U S 2X25 is said to be a fundamental matrix. The general solution of the homo- geneous linear system of equations (2.21) is X tM tc c c 1 c 2 ÁÁÁ c n V b b b ` b b b X W b b b a b b b Y Y 2X26 The solution of the homogeneous system dX dt AX that satis®es the initial condition X t 0 X 0 is X tM tM À1 t 0 X 0 X 2X27 The system of equations (2.24) may be written in the form M tc H tF tY and hence ct t t 0 M À1 sF sds ~ cX The general solution of the system of equations (2.16) is X tM t ~ c M t t t 0 M À1 sF sdsY 2X28 and the solution that satis®es X t 0 X 0 is X tM tM À1 t 0 X 0 t t 0 M tM À1 sF sdsX 2X29 THEOREM 2.10 Liouville's Formula Let W t be the Wronskian of solutions X 1 Y X 2 Y FFFY X n of the homogeneous system of equations (2.21). Then W tW t 0 e t t 0 n j1 a jj sds Y 2X30 where t 0 PaY b is arbitrary. The homogeneous linear system of differential equations dx i dt n j1 a ij x j 2X31 830 Appendix: Differential Equations and Systems of Differential Equations Differential Equations for which the functions X 1 Y X 2 Y FFFY X n , X k x 1k x 2k ÁÁÁ x nk V b b ` b b X W b b a b b Y Y are linearly independent solutions, may be written as dx i dt dx i1 dt dx i2 dt ÁÁÁ dx in dt x 1 x 11 x 12 ÁÁÁ x 1n x 2 x 21 x 22 ÁÁÁ x 2n ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ x n x n1 x n2 ÁÁÁ x nn 0 i 1Y 2Y FFFY nX 2X32 EXAMPLE 2.6 Show that the system of vectors X 1 1 t &' Y X 2 Àt e t &' is a fundamental system of solutions for the following system: dx 1 dt t e t t 2 x 1 À 1 e t t 2 x 2 dx 2 dt e t 1 Àt e t t 2 x 1 e t t e t t 2 x 2 X m V b b ` b b X Solution The Wronskian determinant is W t 1 Àt te t e t t 2 T 0Y for all t P RX The vector X 1 1 t &' has the components x 11 t1, x 21 tt and dx 11 dt 0Y t e t t 2 x 11 À 1 e t t 2 x 21 t e t t 2 À t e t t 2 0 dx 11 dt Y dx 21 dt 1 e t 1 Àt e t t 2 x 11 e t t e t t 2 x 21 e t 1 Àt e t t 2 e t tt e t t 2 e t t 2 e t t 2 1 dx 21 dt X Hence, X 1 is a solution for the given system. Analogously, X 2 if a solution. m 2. Systems of Differential Equations 831 Differential Equations Remark 2.4 The given system can be written as dX dt AtX Y At t e t t 2 À1 e t t 2 e t 1 Àt e t t 2 e t t e t t 2 P T T R Q U U S X Replacing X 1 (respectively X 2 ) in the equation yields dX 1 dt 0 1 @A Y AX 1 t e t t 2 À1 e t t 2 e t 1 Àt e t t 2 e t t e t t 2 P T T R Q U U S 1 t @A 0 1 @A Y hence, X 1 is a solution for the given system. m EXAMPLE 2.7 Find the homogeneous linear system of differential equations for which the following vectors are linearly independent solutions: X 1 1 t t 2 V ` X W a Y Y X 2 Àt 1 2 V ` X W a Y Y X 3 0 0 e t V ` X W a Y X m Solution The Wronskian determinant is W t 1 Àt 0 t 10 t 2 2 e t e t 1 t 2 T0 for all t P RX Equations (2.32), in this case, are dx 1 dt 0 À10 x 1 1 Àt 0 x 2 t 10 x 3 t 2 2 e t 0Y or dx 1 dt t 1 t 2 x 1 À 1 1 t 2 x 2 dx 2 dt 100 x 1 1 Àt 0 x 2 t 10 x 3 t 2 2 e t 0Y or dx 2 dt t 1 t 2 x 1 t 1 t 2 x 2 832 Appendix: Differential Equations and Systems of Differential Equations Differential Equations and dx 3 dt 2t 0 e t x 1 1 Àt 0 x 2 t 10 x 3 t 2 2 e t 0Y or dx 3 dt 4t Àt 2 1 t 2 x 1 2t 2 À t 3 À 2 1 t 2 x 2 x 3 X We ®nd the system dX dt AtX Y where At t 1 t 2 À1 1 t 2 0 1 1 t 2 t 1 t 2 0 4t Àt 2 1 t 2 2t 2 À t 3 À 2 1 t 2 1 P T T T T T T T R Q U U U U U U U S Y X x 1 x 2 x 3 V b b ` b b X W b b a b b Y X EXAMPLE 2.8 The following system is considered: dx 1 dt t 1 t 2 x 1 À 1 1 t 2 x 2 dx 2 dt 1 1 t 2 x 1 t 1 t 2 x 2 dx 3 dt 4t Àt 2 1 t 2 x 1 2t 2 À t 3 À 2 1 t 2 x 2 x 3 X V b b b b b b b ` b b b b b b b X (a) Find the general solution. (b) Find the particular solution with the initial condition X 0 1 1 3 V ` X W a Y X m Solution The system of vectors X 1 1 t t 2 V ` X W a Y Y X 2 Àt 1 2 V ` X W a Y Y X 3 0 0 e t V ` X W a Y is a fundamental system of solutions. The general solution is X tc 1 X 1 c 2 X 2 c 3 X 3 c 1 À c 2 t c 1 t c 2 c 1 t 2 2c 2 c 3 e t V ` X W a Y X 2. Systems of Differential Equations 833 Differential Equations The initial condition X 0 1 1 3 V ` X W a Y gives c 1 1, c 2 1, c 3 1, and the solution that satis®es the initial condition is X t 1 Àt t 1 t 2 2 e t V ` X W a Y Y or x 1 t1 Àt x 2 tt 1 x 3 t2 t 2 e t X m V ` X EXAMPLE 2.9 Consider the system dx 1 dt t 1 t 2 x 1 À 1 1 t 2 x 2 t dx 2 dt 1 1 t 2 x 1 t 1 t 2 x 2 t 2 dx 3 dt 4t À t 3 1 t 2 x 1 2t À t 3 À 2 1 t 2 x 2 x 3 e 2t X V b b b b b b b ` b b b b b b b X (a) Find the general solution. (b) Find the particular solution with the initial condition X 0 1 1 3 V ` X W a Y X m Solution The corresponding homogeneous system is that from the previous example, and its general solution is X t c 1 À c 2 t c 1 t c 2 c 1 t 2 2c 2 c 3 e t V ` X W a Y X The general solution of the given system will be found by the method of parameter variation, X t c 1 tÀtc 2 t c 1 tt c 2 t c 1 tt 2 2c 2 tc 3 te t V ` X W a Y X From the system c H 1 tÀtc H 2 tt c H 1 tt c H 2 tt 2 c H 1 tt 2 2c H 2 tc H 3 te t e 2t Y V ` X 834 Appendix: Differential Equations and Systems of Differential Equations Differential Equations we obtain c H 1 tt, c H 2 t0, c H 3 te t À t 3 e Àt . Integrating yields c 1 t t 2 2 ~ c 1 Y c 2 t ~ c 2 Y c 3 te t e Àt t 3 3t 2 6t 6 ~ c 3 X The general solution of the given system is X t ~ c 1 À ~ c 2 t ~ c 1 t ~ c 2 ~ c 1 t 2 2 ~ c 2 ~ c 3 e t V b ` b X W b a b Y 1 Àt 0 t 10 t 2 2 e t P T R Q U S t 2 a2 0 e t e Àt t 3 3t 2 6t 6 V b ` b X W b a b Y X t ~ c 1 À ~ c 2 t ~ c 1 t ~ c 2 ~ c 1 t 2 2 ~ c 2 ~ c 3 e t V b ` b X W b a b Y 1 2 t 2 1 2 t 3 1 2 t 4 e 2t t 3 3t 2 6t 6 V b ` b X W b a b Y Y m or x 1 t ~ c 1 À ~ c 2 t 1 2 t 2 x 2 t ~ c 1 t ~ c 2 1 2 t 3 x 3 t ~ c 1 t 2 2 ~ c 2 ~ c 3 e t 1 2 t 4 e 2t t 3 3t 2 6t 6X V b ` b X (b) The initial condition X 0 1 1 3 V ` X W a Y yields ~ c 1 1, ~ c 2 1, ~ c 3 À6. The solution that satis®es the given initial condition is x 1 t1 Àt 1 2 t 2 x 2 tt 1 1 2 t 3 x 3 te 2t À 6e t 1 2 t 4 t 3 4t 2 6t 8X m V b ` b X 2.5 Systems of Linear Differential Equations with Constant Coef®cients A linear system with constant coef®cients is a system of differential equations of the form dx i dt n j1 a ij x j f i ti 1Y 2Y FFFY n2X33 where the coef®cients a ij are constants. The system (2.33) may be compactly written in the form of one matrix equation dX dt AX F Y 2X34 where matrix A is constant. The linear systems can be integrated by the method of elimination, by ®nding integrable combinations, but it is possible to ®nd directly the 2. Systems of Differential Equations 835 Differential Equations fundamental system of solutions of a homogeneous linear system with constant coef®cients. For the system dx 1 dt a 11 x 1 a 12 x 2 ÁÁÁa 1n x n dx 2 dt a 21 x 1 a 22 x 2 ÁÁÁa 2n x n ÁÁÁ dx n dt a n1 x 1 a n2 x n ÁÁÁa nn x n Y V b b b b b b b b b ` b b b b b b b b b X 2X35 the solution must be of the form x 1 s 1 e lt Y x 2 s 2 e lt Y FFFY x n s n e lt Y 2X36 with s i i 1Y 2Y FFFY n and l constants. Substituting Eqs. (2.36) in Eqs. (2.35) and canceling e lt yields a 11 À ls 1 a 12 s 2 ÁÁÁa 1n s n 0 a 21 s 1 a 22 À ls 2 ÁÁÁa 2n s n 0 ÁÁÁ a n1 s 1 a n2 s 2 ÁÁÁa nn À ls n 0X V b b ` b b X 2X37 The system of equations (2.37) has a nonzero solution when its determinant is zero, D a 11 À l a 12 ÁÁÁ a 1n a 21 a 22 À l ÁÁÁ a 2n ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ a n1 a n2 ÁÁÁ a nn À l 0X 2X38 Equation (2.38) is called the characteristic equation. Let us consider a few cases. 2.5.1 CASE I: THE ROOTS OF THE CHARACTERISTIC EQUATION ARE REAL AND DISTINCT Denote by l 1 Y l 2 Y FFFY l n the roots of the characteristic equation. For each root l j , write the system of equations (2.37) and ®nd the coef®cients s 1j Y s 2j Y FFFY s nj X The coef®cients s ij i 1Y 2Y FFFY n are ambiguously determined from the system of equations (2.37) for l l i , since the determinant of the system is zero; some of them may be considered equal to unity. Thus, j For the root l 1 , the solution of the system of equations (2.35) is x 11 s 11 e l 1 t Y x 21 s 21 e l 1 t Y FFFY x n1 s n1 e l 1 t X j For the root l 2 , the solution of the system (2.35) is x 12 s 12 e l 2 t Y x 22 s 22 e l 2 t Y FFFY x n2 s n2 e l 2 t X 836 Appendix: Differential Equations and Systems of Differential Equations Differential Equations FFF j For the root l n , the solution of the system (2.35) is x 1n s 1n e l n t Y x 2n s 2n e l n t Y FFFY x nn s nn e l n t X By direct substitution into equations, the system of functions x 1 c 1 s 11 e l 1 t c 2 s 12 e l 2 t ÁÁÁc n s 1n e l n t x 2 c 1 s 21 e l 1 t c 2 s 22 e l 2 t ÁÁÁc n s 2n e l n t ÁÁÁ x n c 1 s n1 e l 1 t c 2 s n2 e l 2 t ÁÁÁc n s nn e l n t Y V b b ` b b X 2X39 where c 1 Y c 2 Y FFFY c n are arbitrary constants, is the general solution for the system of equations (2.35). Using vector notation, we obtain the same result, but more compactly: dX dt AX X 2X40 The solution must have the form X ~ Se lt ~ S s 1 s 2 ÁÁÁ s n V b b b ` b b b X W b b b a b b b Y X The system of equations (2.37) has the form A À lI ~ S 0Y 2X41 where I is the unit matrix. For each root l j of the characteristic equation jA À lI j0 is determined, from Eq. (2.41), the nonzero matrix S j and, if all roots l j of the characteristic equation are distinct, we obtain n solutions X 1 S 1 e l 1 t Y X 1 S 2 e l 2 t Y FFFY X n S n e l n t Y where S j s 1j s 2j ÁÁÁ s nj V b b ` b b X W b b a b b Y X The general solution of the system (2.35) or (2.40) is of the form X n j1 S j c j e l j t Y 2X42 where c j are arbitrary constants. 2. Systems of Differential Equations 837 Differential Equations 2.5.2 CASE II: THE ROOTS OF THE CHARACTERISTIC EQUATION ARE DISTINCT, BUT INCLUDE COMPLEX ROOTS Among the roots of the characteristic equation, let the complex conjugate roots be l 1 a ibY l 2 a ÀibX To these roots correspond the solutions x i1 s i1 e aibt i 1Y 2Y FFFY n x i2 s i2 e aÀibt lY i 1Y 2Y FFFY nX & 2X43 The coef®cients s i1 and s i2 are determined from the system of equations (2.37). It may be shown that the real and imaginary parts of the complex solution are also solutions. Thus, we obtain two particular solutions, ~ x i1 e at ~ s H i1 cos bt ~ s H i2 sin bt ~ x i2 e at ~ s HH i1 cos bt ~ s HH i2 sin bt Y @ 2X44 where ~ s H i1 , ~ s H i2 , ~ s HH i1 , ~ s HH i2 are real numbers determined in terms of s i1 and s i2 . 2.5.3 CASE III: THE CHARACTERISTIC EQUATION HAS A MULTIPLE ROOT l k OF MULTIPLICITY r The solution of the system of equations (2.35) is of the form X tS 0 S 1 t ÁÁÁS rÀ1 t rÀ1 e l s t Y 2X45 where S j s 1j s 2j ÁÁÁ s nj V b b ` b b X W b b a b b Y Y s ij are constants. Substituting Eq. (2.45) into Eq. (2.40) and requiring an identity to be found, we de®ne the matrices S j ; some of them, including S rÀ1 as well, may turn out to be equal to zero. EXAMPLE 2.10 Solve the system dx 1 dt Àx 1 À 2x 2 dx 2 dt 3x 1 4x 2 X m V b b ` b b X Solution The characteristic equation À1 Àl À2 34À l 0Y or l 2 À 3l 2 0Y 838 Appendix: Differential Equations and Systems of Differential Equations Differential Equations [...]... phenomena, 723, 730 Decomposition method, 15 17 De¯ection analysis beams and, 131±132, 152 153 , 163, 726 Castigliano's theorem, 163±164, 286 central loading, 165±169 columns and, 165±171 compression and, 165 compression members, 171 deformation and, 3, 160±163, 389 eccentric loading, 170 expression for, 157 impact analysis, 157 159 maximum values, 158 springs and, 150 151 stiffness, 149±172 strain energy,... 296±297 compression of, 150 152 distortion-energy theory, 283 elastic constant for, 388 elastic force of, 374±375 ends, 288 extension, 284 helical, 284±290 linear, 83, 150 linear characteristic, 374±375 mass-damper system, 617±618 materials for, 283 mechanical work and, 374±375 multileaf, 292±296 nonlinear, 150 potential energy, 85±86, 374±375 rates, 150 151 , 287±288 spring constant, 150 , 287, 388 spring... rates, 150 151 , 287±288 spring constant, 150 , 287, 388 spring index, 286 stiffness, 150 , 342, 386 tension, 150 152 torsion, 150 152 , 290±293 Spur gears, 253, 425 Square threads, 247±248, 251 Stability analysis of, 414± 415 criteria for, 415 of linear feedback systems, 639±649 of nonlinear systems, 685±688 vibration and, 414± 415 Standard controllers, 650 Standard hydraulic symbols, 605 State variable models,... stresses, 178 impact analysis, 157 159 mechanics of, 119±187 Mohr's circle, 121±124 randomly varying loads, 181±182 shear moment, 131 shear stresses, 140±142 short compression members, 171 singularity functions, 132, 153 spring rates, 150 151 strain energy, 160±162 stress, 120±147 torsion, 143±146 See also speci®c concepts, methods Maximum principle, 801 Mayer's law, 451 Mechanical impedance, 369±370... axes and, 18, 25 cartesian coordinates of, 15 decomposition, method of, 15 ®rst moment, 13, 17±20 Guldinus-Pappus theorems, 21±23 loading curve, 20 mass center and, 16 parallel-axis theorems, 25±26 points and, 13 15 polar moment, 27 position vector, 12 principal axes, 28±29 product of area, 24 second moments, 24 solid, 15 statics and, 12±28 surface properties, 13 15 symmetry and, 18 transfer theorems,... 580 Cramer criteria, 415 Crank, de®ned, 146, 198 Crank slider mechanism, 227±228 Critical damping, 631 Critical load, 165, 167 Cross product, 9±10, 223 Cryogenic systems, 446 Curl, 87, 502 Curvature, 15 17 correction factor, 286 de®nition of, 136 differential equation for, 720±721 envelope of, 758, 772 force and, 564 instantaneous radius of, 61 of plane curve, 152 of surface, 15 17 Curvilinear motion,... diagram, 712±714 Signals, control theory and, 613± 615 Signi®cant digits, 52 Silicone oils, 322 Similarity analysis, 512±519 Similarity variable, 542 Simple couple, 34 Singular integral curve, 772 Singular point, in phase plane, 682±683 Singular solutions, 772±774 Singularities, of system, 621 Singularity functions, 132±135, 153 157 Sinusoidal input, 615 Sinusoidal stress, 179 Size factor, 176 Skeleton... linear shafts, 429 machine tool, 416±444 magni®cation factor, 365 mechanical impedance, 369 mechanical models, 386±391 natural frequencies, 407± 415 nonharmonic exciting force, 358 with one degree of freedom, 374±379 phase angle, 364 polar diagram, 416 rotating shafts and, 374±379 self-excited, 441 simple harmonic, 359±364, 375 stability, 414± 415 steps for solving problems, 385 superposition of, 385 864... I-controller See Integration controller Ideal elements, 705 Ideal gas, 451 Ideal radiator, 454 Identi®cation method, 360 Idler, 261 Immersed bodies, 548, 549 Impact analysis, 157 ±160 conservation in, 90 de¯ection stiffness, 157 159 direct impact, 90±93 oblique impact, 93±94 perfectly plastic, 90 Impedance, 369±370 Impermeable conditions, 503 Impulse, 87±88 angular, 94 Impulse function, 614 Inclined... Nonlinear controllers, 691±695 Nonlinear equations of motion, 346 Nonlinear springs, 150 Nonlinear systems, 678±691 Normal component, of force, 59±73, 77±78 Normal stress, 120, 135±139, 494, 495, 726 Normal vector, 68 Null entropy interaction, 449 Numerical methods, 171 Nusselt number, 507, 515, 540 Nuts, 244 Nyquist criterion, 415, 641±647, 711 O Oberbeck±Boussinesq approximation, 539 Octahedral shear stress, . 94 derivative of, 398 principle of, 113± 115 Angular units, 53±54 Angular velocity diagrams of, 269±270 of rigid body, 98±99 RRR dyad and, 3 Annular Bearing Engineers& apos; Committee (ABEC) grades,. Systems of Differential Equations Differential Equations Index A ABEC grade. See Annular Bearing Engineers& apos; Committee Absolute temperature scales, 449 Absorption, of heat, 454, 466 Acceleration analysis. e Àt 1 4 e 2t À 1X m V b ` b X 2. Systems of Differential Equations 843 Differential Equations EXAMPLE 2 .15 Solve the system of the second-order differential equations d 2 x 1 dt 2 a 11 x 1 a 12 x 2 d 2 x 2 dt 2
Ngày đăng: 13/08/2014, 16:21
Xem thêm: Mechanical Engineers Handbook Episode 15 pot, Mechanical Engineers Handbook Episode 15 pot
