（mbbs）chapter 2　vibration.pptx

Medical physics Wu Zhiren (Tel ： 13700667406) Department of Physics, Mathematics and 返 回 computer ,Kunming Medical University 前 页 后...

Medical physics Wu Zhiren (Tel ： 13700667406) Department of Physics, Mathematics and 返 回 computer ,Kunming Medical University 前 页 后 页 Teaching arrangement: theory course: 1-18 week Course Assessment 1.Formative Assessment:40% (three pop quiz) 2.Summative Assessment:60% (examination) 返 回 前 页 后 页  Chapter 2 Vibration electrocardiogram 返 pendulum 回 前 页 后 页  Vibration The reciprocating motion of an object near a certain position. 返 返回 回 前 页 后 页  2.1 Simple Harmonic Motion Spring oscillator: an ideal model of elastic force 返 返 回 回 composed of light spring (k) and oscillator (m) 前 页 F: elastic force 后 F=- 页 2.1.1 Equation of Simple Harmonic Motion Newton’s second law ： F= － kx =ma Acceleration: second derivative of displacement X to time 2 dx 2 2  ω x 0 dt k ω  2 m 返 返 The upper expression is a motion-differential 回 回 前 equation, and its solution is: 页 后 x=A cos （ ωt ＋  ） 页  X=A cos （ ωt ＋  ） — —① Motion: the displacement x of the Simple Harmonic particle varies with time t according to the law of cosine (or sine)function. The velocity and acceleration equation of simple harmonic motion 1. Velocity equation ： V=dx/dt V= － Aωsin(ωt ＋  ) ——② 返 返 回 回 2. acceleration equation ： 前 页 a = dv/dt = － Aω2cos(ωt ＋  ) ——③ 后 页 2.1.2 Characteristic Quantity of Simple Harmonic Motion The three characteristic quantities of simple harmonic motion ： A ω  1. Amplitude (A) : The maximum displacement of a vibrating object leaves away from the equilibrium position. 2. Period. Frequency.Angular frequency (ω) （ 1 ） Period (T) : The time it takes for a particle to 返 回 complete a full vibration is called a period. Unit ： S 前 页 后 页 （ 2 ） Frequency (ʋ) : The number   1 of times a particle completes the full T vibration in a unit time. unit ： Hz （ 3 ） Angular frequency (ω): the number of times the particle 2  2  completes the full vibration in 2 π s. T unit ： rad·S-1 spring oscillator ： T natural period 返 回 前 ʋ natural frequency 页 后 页  3. Phase, initial phase （ 1 ） (ωt ＋ ) ： Phase The physical quantity to determine the motion state of vibration system （ 2 ） Initial Phase ： The phase at the time of zero is called the initial phase. x0= Acos(ωt ＋ )= Acos  返 回 前 V0= － Aωsin(ωt ＋ )= － Aωsin  页 后 页 10  0o 30o 45o 60o 90o 180o 270o 360o(0o) Function π 2π Sin 0 1 0 -1 0 Cos 1 0 -1 0 1 返 回 π = 180o 2π = 360o 前 页 后 页 Example: A spring oscillator makes a simple harmonic motion along the x direction with a amplitude of A and a period of T. when t = 0, the initial displacement x = 0 and the motion in the negative direction of x axis, then the vibration equation is 。 返 回 前 页 后 页  1. A particle vibrates harmoniously in the vertical direction, let the upward direction be positive, and A when t = 0, the particle is at the point 2 and moves downward, then the initial phase is   A B - 4 4   C - D 返 回 3 3 前 页 后 页  2.1.3 Graph method of Rotational Vector of SHM A R w wt j t 0 x 0 x projection :x=Acos(ωt ＋） It is a simple 返 回 harmonic motion equation 前 页 so the projective motion of a uniform rotation vector is a 后 页 simple harmonic motion.  2.1.4 Energy of SHM Let’s discuss the system energy of spring oscillator. ω2=k/m 1 1 kinetic energy ： EK = m V2 = mA2ω2sin 2(ωt +  ) 2 2 1 = 2 kA2sin 2(ωt +  ) Potential energy ： 1 2 EP  kx 返 2 回 前 1 页 = 2 kA2 cos 2(ωt +  ) 后 页  1 2 2 Ek  kA sin (t   ) 2 1 2 EP  kA cos (t   ) 2 2 Conclusion: when the kinetic energy is the maximum (kA2/2), the potential energy is 0. 返 回 前 when the potential energy is the maximum(kA /2), 2 页 后 页 the kinetic energy is 0..  total energy E = EK+ ：EP 1 2  kA[sin2(ωt φ )  cos2(ωt φ )] 2 E = 1 mA2 ω2 = 1 kA2 2 2 The total energy invariance indicates that the total mechanical energy is conserved, and the 返 返 回 回 system is called an isolated system or a closed 前 页 后 system. 页 Example. Spring oscillator is a simple harmonic vibration, the total energy is E. If the vibratory mass of the vibrator is increased three times and the amplitude doubled, the total energy becomes A. 2E B. 8E C. 12E D. 4E E = 1 mA2 ω2 = 1 kA2 返 回 2 2 前 页 后 页 Example: if the mass is m, the velocity is V (t ) V0 sin t the total mechanical energy of the vibration system will be 1 A m 2 B mV02 2 1 1 C mV02 D mV02 sin 2 t 2 2 返 回 E Ek max 前 页 后 页 19 2.2.1 Synthesis of Simple Harmonic Vibration If a particle participates in two or more simple harmonic motions at the same time, its motion is the synthesis of these harmonic motions. The simplest case 1 、 Synthesis of two simple harmonic vibrations in the same direction and frequency: 1 ） x1=A1cos(ωt+ 2 ） x =A2cos(ωt+ 2 返 回 Because of the same direction, the combined 前 页 displacement x is the algebraic sum of x1 and x2: 后 页 x=x1 ＋ x2=A1cos(ωt+1) ＋  Graph Method of Rotational Vector ω x1 x2 rotational vector A1 A2 The length of the sum vector A is invariant and A A2 rotates at the same angular speedω. So the △  projection of A on the x A1 x2 axis is also a simple x 0 x harmonic vibration. 2 x = x1+x2 x1 返 回 x A is the rotational vector. 前 页 后 页 Synthetic vibration equation x = A cos (ωt +  ) A A12  A22  2 A1 A2 cos(2  1) A sinφ 1 A2sinφ 2 1 1 φ tg A1cosφ 1 A2cosφ 2 ω The synthetic vibration is A2 A still a simple harmonic vibration. △  A1 x2 the value of its amplitude A x is determined by the phase 0 返 回 x2 difference 前 页 x1 △ = 2 －  1 后 页 x Discussion: 1. when △ = ±2kπ （ k=0 ， 1 ， 2 ， 3 ，…）， A A12  A22  2 A1 A2 cos(2  1) 2 2  A  A  2 A1 A2  A1  A2 1 2 when two partial vibrations are in the same phase, the synthetic amplitude has the maximum value, which is equal to the sum of the two partial amplitudes. x 返 回 t o 前 页 后 页  = ±(2k ＋ 1)π （ k=0 ， 1 ， 2 ，…， 2. When △  A A12  A22  2A1 A2  A1  A2 when two partial vibrations are inverse, the synthetic amplitude has the minimum value, which is equal to the difference of the two component amplitudes. x t o 返 回 3 、 When the difference is other values, 前 页 the combined amplitude is between the 后 页 minimum │A1 － A2│and the maximum Example. If a particle participates in two  harmonic vibrations such as S1 10 cos(t  )  2 AndS 2 20 cos(t  ) in the same direction, 2 the amplitude of the synthetic vibration is A. 20cm B. 10cm C. 30cm D. 0cm 返 回 △ = ±π A  A1  A2 前 页 后 页  2.2.2 Frequency Analysis A complex periodic vibration is decomposed into a series of simple harmonic motions, which is called frequency analysis. 。 x t Serrated vibration 返 1 1 1 回 x  (  si nt  si n2t  si n3t  ) 前 π 2 3 页 后 页 Frequency analysis is the basis of electroencephalogram (EEG) and electrocardiogram (ECG). It can be used to diagnose diseases. Electrocardiogram average person patients with 返 回 coronary heart 前 页 disease 后 页 Summary: k m x spring oscillator : x=A cos （ ωt ＋  ） Fig.4 1 V= － Aωsin (ωt ＋  ) x0= Acos(ωt ＋ )= Acos  ω2 = K/m V0= － Aωsin(ωt ＋ )= － Aωsin  2 v 02 A x  2 0  E = 1 mA2 ω2 = 1 KA2 2 2 返 A A12  A22  2 A1 A2 cos(2  1) 回 前 页 In-phase strengthening △  = 0 or ±2kπ 后 页 Inverse weakening △ =π or ±(2k ＋ 1)π 28 Exercise. For a spring oscillator with a period of T and a amplitude of A.t = 0, the oscillator moves to the right over the equilibrium position. If the direction to the right is positive, the vibration expression is 2  A S  A cos( t ) T 2 2  B S  A cos( t ) T 3 2  C S  A cos( t ) T 2 返 回 2 前 D S  A cos t 页 T 后 页