tailieunhanh - Computational Physics - M. Jensen Episode 2 Part 6

Tham khảo tài liệu 'computational physics - m. jensen episode 2 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . PHYSICS EXAMPLES 269 L Figure Simple RLC circuit with a voltage source V. Damping of harmonic oscillations and external forces Most oscillatory motion in nature does decrease until the displacement becomes zero. We call such a motion for damped and the system is said to be dissipative rather than conservative. Considering again the simple block sliding on a plane we could try to implement such a dissipative behavior through a drag force which is proportional to the first derivative of . the velocity. We can then expand Eq. to d x dx . . .rfi . . where o is the damping coefficient being a measure of the magnitude of the drag term. We could however counteract the dissipative mechanism by applying . a periodic external force F t Bcos wi and we rewrite Eq. as Px 9 dx . Although we have specialized to a block sliding on a surface the above equations are rather general for quite many physical systems. If we replace by the charge Q V with the resistance -R the velocity with the current I the inductance with the mass m the spring constant with the inverse capacitance Ơ and the force F with the voltage drop V we rewrite Eq. as The circuit is shown in Fig. . How did we get there We have defined an electric circuit which consists of a resistance R with voltage drop R a capacitor with voltage drop QỊC and an inductor with voltage 270 CHAPTER 14. DIFFERENTIAL EQUATIONS Figure A simple pendulum. drop Ldl dt. The circuit is powered by an alternating voltage source and using Kirchhoff s law which is a consequence of energy conservation we have V t IR Ldl dt Q C and using I . . . dt we arrive at Eq. . This section was meant to give you a feeling of the wide range of applicability of the methods we have discussed. However before leaving this topic entirely we ll dwelve into the problems of the pendulum from almost harmonic oscillations to chaotic motion The pendulum a nonlinear .