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Chapter Four

Chapter Four. Oscillation, wave motion and sound. New words and expressions. simple harmonic motion ( 简谐运动 ); spring ( 弹簧 ) ， elastic ( 弹性的 ) ， Phase ( 位相 ) reciprocal ( 倒数 ) ， unavoidable ( 不可避免的 ) amplitude ( 振幅 ); damp ( vt. 阻尼 ,n. 湿气 ) ; deduce ( vt. 推论 ), deduction

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Chapter Four

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  1. Chapter Four Oscillation, wave motion and sound

  2. New words and expressions • simple harmonic motion (简谐运动); • spring (弹簧)，elastic (弹性的)， Phase (位相) • reciprocal (倒数)，unavoidable (不可避免的) • amplitude (振幅);damp (vt.阻尼,n.湿气) ; • deduce (vt. 推论), deduction • vibration (振动)；oscillation (振动) • Resonance (共振）

  3. x §4.1 Simple harmonic motion (SHM) • 4.1.1 Equation of SHM • Definition of SHM • Simple harmonic force (简谐力): The force on a body is proportional to its displacement from the origin and always directed towards the origin. If we choose the direction of displacement as the x-axis, the equation is given by • F = - k x (4.1) • k is the elastic constant. o

  4. SHMIf a body moves in a straight line under the simple harmonic force, the motion of the body is called simple harmonic motion. • 2. Equation of SHM • Generally a Hooke’s law spring satisfies the equation (4.1). If a body’s mass is m and is exerted by a simple harmonic force, its equation of motion can be obtained by using Newton’s second law of motion.

  5. On the other hand, considering eq. (4.1), we have Or Define 2 = k/m and we have (4.2)

  6. This is the differential equation of the SHM. Its solution can be expressed as (4.3) The motion described by a cosine or sine function of time is called Simple Harmonic Motion. It is necessary to point out that the two definitions for SHM are equivalent. One is from the force type and the other is from the equation of motion. Differentiating the equation (4.3) with respect to t, the velocity and acceleration of the SHM can be obtained

  7.  = t +  d(cos x)/dx = -sin x d(sin x)/dx = cos x (4.4)

  8. (4.5)

  9. It is known that the equation (4.5) is equivalent to the equations (4.2) and (4.3). Therefore the equation (4.3) is indeed the solution of (4.2). 4.1.2 The characteristic quantities of SHM In the equation of SHM, A, and  are constants and any individual SHM can be determined by them.

  10. A is called Amplitude (振幅). It is the maximum displacement of a vibrating body from equilibrium position. • Period (周期) and frequency (频率) • The period, denoted by T, is the time taken for a complete vibration which is independent of the position chosen for the starting point of the complete vibration.

  11. Find its amplitude and period. x t

  12. The frequency, denoted by f,is the number of complete vibrations per second, it is the reciprocal (倒数) of the period (4.6) The angular frequency or angular velocity is defined as ∴ (4.7)

  13. 3. Phase and initial phase （初位相） In the equation of SHM, t   is called the phase of SHM, where  is the phase at t = 0, called initial phase (its unit is radian). At t = 0, equations (4,3) and (4.4) becomes respectively (4.8) This is called initial condition.

  14. Squaring both sides of the above equations, the amplitude of the SHM can be found

  15. ∴ (4.9) On the other hand, the initial phase can also be worked out from equation (4.8), so we have (4.10) ∴

  16. Example 1. A particle with mass m = 2.00× 10-2kg is in SHM at the end of a spring with spring constant k = 50.0 N/m. The initial displacement and velocity of the particle is 3.00 × 10-2 m and –1.32 m/s respectively. Calculate: (1) the angular frequency; (2) the initial phase; (3) the amplitude of the vibration; (4) the period; (5) the frequency.

  17. Solution: the known quantities are: m = 2.00 × 10-2 kg k = 50.0 N/m x0 = 3.00 × 10-2 m v0 = –1.32 m/s Now using the formulae we have learned, the problem can be solved easily. (1). the angular frequency, we have (rad/s)

  18. (2). The initial phase of the vibration can be found using the initial displacement and initial velocity. At t = 0, we know x0 = A cos = 3.00 × 10-2 m v0 = - A  sin = -1.32 m/s The  can be obtained by solving above equations. On the other hand, it can be calculate directly by

  19. (3). The amplitude can be calculate by the formula (4). The period can be found through the relation between the angular frequency and the period; (5)

  20. M A M0  t+  P P0 Fig. 4.2 the circle of reference of SHM. 4.1.3 The reference circle of SHM Consider a vector A torotate (旋转) around point O with a constant angular velocity . Suppose that its initial position is OM0 and the angle with x-axis is . At time t, the angle between A and x-axis is  t+. When M moves in a circle motion, the projecting point P on x-axis moves in SHM. It is easy to find that the equation of motion of point P at any t moment is x = A cos ( t+) It is the equation of SHM. O

  21. 4.1.4 The energy of SHM When particle moves in a SHM, the kinetic energy and potential energy are transformed each other based on the position of the particle. For a spring vibrator, F = -kx, the expressions of kinetic energy and potential energy of the system are given by (4.11) ∴ It is conservative!

  22. E Ek Ep x x Fig. 4.3 Total energy of the vibrating system the total mechanic energy of the vibrating system is conservative in the process of harmonic motion. This result is valid to all the SHM systems. The systematic diagram of the energy conservation of the vibrating system is expressed in Fig. 4.3.

  23. §4.2 Damped (阻尼的) harmonic motion, forced vibration and resonance • 4.2.1 The damped harmonic motion • Simple harmonic motion is only an ideal model, in such a ideal case The friction and other damped forces are ignored. • Real vibrating systems have damped force and friction.

  24. In a real vibrating system, the friction and other damped forces are unavoidable (inevitable, 不可避免的). Therefore, the mechanical energy and amplitude of the vibration will decrease gradually. This kind of vibration is called damped vibration or damped Harmonic motion. • How to determine the damped force?

  25. Experiments show that when object moves in medium at a lower speed, the damped force is proportional to the speed of the object but in an opposite direction to its velocity.  is called damping coefficient. Its value depends on the factors of its shape, magnitude, surface condition and its material property.

  26. Now we can write the vibrational equation in the damped condition based on F = ma. Set:

  27. This is the dynamic equation for damped vibration and it is a linear homogeneous differential equation.

  28. When damped force is small (  0), it has a solution: With A0 and  are determined by the initial conditions.

  29. x C b a o t Damping oscillations

  30. For example:  = 1, 0 = 3, A0=2 and  = /4, the equation of motion becomes

  31. The magnitude of the damped force is described by a damped factor, denoted by . When the damped force is small, the period of the damped vibration is given by (4.12)

  32. Where 0 is the proper (固有的) angular frequency of the vibrating system. It can be deduced (推论) that because of the damped factor, the period is getting longer and the frequency (f = 1/T) of the system will become smaller. That is, the vibration goes slower. The properties at this case can be summarized as follows:

  33. Due to the damped force, (1) The mechanical energy becomes smaller; (2) The amplitude decreases; (3) The period increases but not with time (due to ，bigger amplitude faster, smaller amplitude corresponding to slower motion); (4) The frequency decreases but (5) Vibration goes slower and slower. does not change with time ;

  34. 4.2.2 The forced vibration (受迫振动) The forced vibration is defined as the vibration exerted by an external periodic force. Assume that the exerted periodic force is f = F0 cos(t) (4.13) where F0 is the amplitude of the external force,  is the angular frequency of the force.

  35. Solution = General solution + particular solution When vibration is steady, the periodic external force does work and puts in energy that is just equal to the lost energy due to the resistance. The characteristic quantities of the vibration can be derived, given as follows:

  36. (4.14) So we conclude that the vibrating system is also vibratory but with the frequency of the external force. The amplitude of the forced vibration depends on F0, angular frequency () of the force, proper frequency (0) of the system and damped coefficient ().

  37. 4.2.3 Resonance (共振） When the frequency of the external force approaches to the proper frequency of the vibrating system, the amplitude of the forced vibration will increase rapidly. （dA/d=0, extremum condition) This phenomenon is known as resonance. Mathematically we have (4.15)

  38. A small Big   0 Note that when  →0, the resonance frequency is equal to the proper frequency of the system and the amplitude of the forced vibration approaches infinity. Fig. 4.4 Property of resonance

  39. The danger of resonance 1940 年11月7日美国 Tocama 悬索桥因共振而坍塌

  40. 共振故事18世纪中叶,法国昂热市一座102米长的大桥上有一队(a covey of)士兵经过.当他们在指挥官的口令下迈着整齐的步伐过桥时,桥梁突然断裂,造成226名官兵和行人丧生. 类似的事件还发生在俄国.1906年俄国首都彼德格勒有一支全副武装的沙皇军队,步伐整齐的通过爱纪毕特大桥.可是正在指挥官洋洋得意的时候,突然间哗啦一声巨响,大桥崩塌了.顿时间,官兵,辎重,马匹纷纷落水,马嘶人号,狼狈不堪......经过长期追查研究,发现并不是有人故意破坏,肇事的就是受害者.伤亡事故的根本原因就是"共振".此后世界各地都先后规定:凡大队人马过桥时必须碎步走,极力避免这种破坏性的共振现象重演.

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