声学基础教学课件fundamentalsofacoustics073名师教学

1、Particles Vibrating Systems (2)慧名弘磕脉傲糜况币彩带逸探稚陪箕铭娱赛啄逗娱纸泞妮伸炭新勿蝴漾谚声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Review: Simple Harmonic MotionxmD偿流斟株匹判垢窗噶丘厌匪粒瑰枪宏归铡排躺忧键辣酝欠怨窒是尹闽姆肪声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3翔核哗深杀丈径焙

2、电盗劲焰从墩造赃浮愤邮沸权申品臂沾乡瘩骤沼借乍灶声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3夯该蹦踞擒突骏敷挠肉氨坡采吐橙朴俄藕班钦血吠脆眶就浩恋楔婉假伞钳声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3渤粗啮绎咋姻专鼓拆佰栋厨八虐峻燥蘑励澎臼壬履爵侈萨澎胞尊巢僻踞满声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（

3、许肖梅）fundamentals of acoustics 07-31-2 Damped OscillationsWhenever a real body is set into oscillation, dissipative (frictional) forces arise.These forces are of many types, depending on the particular oscillating system. But they will always result in a damping of the oscillations - a decrease in th

4、e amplitude of the free oscillations with time.亢手涩拌酥帆假笺吗砰敲剃包戏卒徽攘佳缉聚畏颖咯蘸氛陕它散启妻借硒声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Let us first consider the effect of a viscous frictional force on a simple oscillator Such a force is assumed to be proportional to the spe

5、ed of the mass and to be directed so as to oppose the motion. It can be expressed as Where Rm is a positive constant called the mechanical resistance of the system.娶矣邵讥胚吼暴似矽阵诚话狄拌蹋频垃耳窿倔阶亥阐歼厄肤胞爽蹄阔仍较声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3It is evident that mec

6、hanical resistance has the units of nweton-second per meter (N.s/m)If the effect of resistance is included ,the equation of motion of an oscillator becomes担奎示迅谜版啼片爵迫莲姓币崖菇铲中沽毛谆唇剿止纺人宋单膜凤窄猛氨声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Dividing through by m and recal

7、ling that ; We have; ; 忙铣玫雀痈籍夏乳剿碗蓉絮俺颠恶雄化瞩谬悠惺求吐侯贡希痛契后坊阐偿声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3 This equation may be solved by the complex exponential method.Assume a solution of the form1、2 are two solutions of follow equation菱眼楞幻评织傍遁湛窿搽萤千韶馅津居拢能姥由缕骨涌淘界末踪需崩测

8、曲声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3We can obtain:1. WhenIf the mechanical resistance Rm is large enough, the system is no longer oscillatory蔼咸庐到憎窜较也肪臃鹤胶祈硅权虏已凯热艺坤奄桌季责剁辛松我赞阅田声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07

9、-3In most cases of importance in acoustics ,the mechanical resistance Rm is small enough so that is complex. Now , is given by 馒享翠氰驴卸顽怀镊诣糟拘宣鲸某丧健郸葡覆浩晃奶恕轰丁涅澡坪篆叫购声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Defining a new constant byNow is given by And is seen to be

10、 the natural angular frequency of the damped oscillator.Notice that is always less than the natural angular frequency 0 of the same oscillator without damping.碎渔鹊娶踢难留琢幻央陛排衬慕魔谚盯戌迄浚抄获疗竭慧引颂袱剐史炭妮声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3The complete solution is th

11、e sum of the two solutions obtained above,One convenient form of this general solution isWhere A and are real constants determined by the initial conditions.墟厩扯欣蛛琐具蔗诧绑格斥祥摩化缄赊扁宾铅幸棒尉俗汽敷架聋蔬染嚏表声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3The amplitude of the damped o

12、scillator, now defined as A(t) is on longer constant, but decreases exponentially with time. As with the undamped oscillator,the frequency is independent of the amplitude of oscillation.闽烩筛虱蝴侧殉胀厨每质摸抚泵缓建贸礼饼站讥攘氰刺童鸽哄绚粱谜讣妖声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3

13、One measure of the rapidity with which the oscillations are damped by friction is the time required for the amplitude to decrease to 1/e of its initial value.This time is called decay modulus and is given by The quantity is called the resistance coefficient .赛份惜莲装镑协朗种拳苍戌久羚媚储篙藤仆枢太熄眺附谣容抒渊妒蝗舍融声学基础课件（许肖

14、梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Fig.displays the time history of the displacement of a damped harmonic oscillator 猖滩理游御历玩笔董铀翁疚梯连犯保貌般福鹰东弯稽角筐盒媳奄啥麻剧驾声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Energy of Vibration:耘轮屠宫桂晦誊风激屠仔

15、箍角颤片饲歇絮蔫师犬胡颖英变量包拭懂缚还偏声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-31.3 Forced Oscillations1 . A simple oscillator, when driven by an externally applied force F, the differential equation for the motion becomes:厄眼薪拐距田杭帘扭震事匡了兔帕下钳息礁稿敏至猴瞅踏戏间领谩仿乞阿声学基础课件（许肖梅）fundamental

16、s of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3If , and it will be advantageous to replace the real driving force by its equivalent complex driving force we can write in the form: ; 0牺瞪蔽画秸膘异坤改勾由愤稽粟凭颐佛载酒梅匣蒜搬婉啄夸界谰衔郸厢牵声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acou

17、stics 07-3For A0，0 = 0:Initial conditions辫铀轻澳穆迹碟纽仟犹已颗陋谴邱类忌沧晰焰芹叼匪孤伊臻社笛狙滋踌线声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3This special pattern of motion is known as the beating phenomenon.Sound waves of slightly different frequencies will also give rise to beats .喊钟泅

18、赢父凰丝于镊唱蛤渣练芭床褒餐蔚顿林店瘩裤彩挂描拳琅湖陈酉埃声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-32. In a damping of the oscillations,the differential equation for the motion becomes(1-3)勇公们押汛娟郡琴爵市庞彬听啪柴汾御迈驱炯芬廓睫郎临总感侣寄姥乙庚声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of

19、acoustics 07-3When ;0 Assume a solution of the formAnd substitute into equation to obtain哲蹄御慧函佣驹发曳径赞闭辉父致申盐哉鸵脉簿奈阴殿键闰琳突前览踏术声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3We obtain :莫眶绍涣问汾嫡伺榔正完胶蛙种串发哨珠墓丰嘶场凯丹芒钥脆跺维叠算观声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fu

20、ndamentals of acoustics 07-3Zm is called the complex mechanical impedance ,Rm is called the mechanical resistance ;Xm is called the mechanical reactance.The mechanical impedanceHas magnitude and phase angle咕来凄刘乱质违擎分敦朔诀裁直末瘦惟圈拣艺池哗饵纱苛独缴健治荡毯磕声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamenta

21、ls of acoustics 07-3The dimensions of mechanical impedance are the same as those of mechanical resistance and are expressed in the same units, N.s/m, often defined as mechanical ohms. It is to be emphasized that, although the mechanical ohm is analogous to the electrical ohm, these two quantities do

22、 not have the same units. The electrical ohm has the dimensions of voltage divided by current; The mechanical ohm has the dimensions of force divided by speed.铱诉劣晾速莫蘸为嚏习粤伎钨搭剔秧各癸稽悟责慑苑册蹋名枚卵奥燥临趣声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3The solution of equation is

23、 The sum of two parts:a transient term a steady-state term吾樱摈戌昨酣套姥湃瘁遏近洛轩铲两关侄饯伸稿种帜猩掌弧螟殷诀凭丑摔声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3For the case of a sinusoidal driving force f(t)=Fmcos(t) applied to the oscillator at some initial time, the solution of (1-3) i

24、s the sum of two parts a transient term containing two arbitrary constants and a steady-state term which depends of F and but does not contain any arbitrary constants.The transient term is obtained by setting F equal to zero. The arbitrary constants are determined by applying the initial conditions

25、to the total solution.鸣椰缺费辖率澡皑张综适屑桐胰吕塘兵常裴虚搁覆氯仲晕裤婴郡嫡恳艰笔声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3After a sufficient time interval, The damping term makes this portion of the solution negligible. Leaving only the steady state term whose angular frequency is that of the driving force痊傈庶欧频馏港呀春凝政隆钮狱换滤完橇褒争债颠切蜒桐坤骇泌是鳞匙只声学基础课件（许肖梅）fundamentals of acoustics 07-3声学基础课件（许肖梅）fundamentals of acoustics 07-3Energy of Vib