第5章-振动概要.ppt

The walking speed of Tyrannosaurus rex can be estimated from its leg length L = 3.1 m and its stride length S = 4.0 m. leg length L stride length S * Example: An astronaut on a body mass measuring device (BMMD), designed for use on orbiting space vehicles, its purpose is to allow astronauts to measure their mass in the ‘weight-less’ condition in earth orbit. The BMMD is a spring mounted chair, if M is mass of astronaut and m effective mass of the BMMD, which also oscillate, show that Solution: * §5.2 简谐振动系统的能量 1．以弹簧振子为例 平衡位置时势能 = 0， 总机械能 Epmax Ekmax * t E t x O O 讨论 (1) Ek、Ep 随 t 周期性交替变化 平衡位置 Ep 0 Ek max |x| 最大 max 0 (2) = 恒定 ∴ 机械能守恒 简谐运动是无阻尼自由振动 (3) Ek、Ep 的变化频率 = 2w * 2．一周期 T 内平均值 3．w —— 固有属性， A —— 能量， j —— 初条。 振动 波动 适用于 系统 经典 量子 * [例] 作简谐运动的小球，速度最大值为 vm = 3 厘米/秒，振幅 A = 2 厘米，若从速度为正的最大值的某时刻开始计算时间，求：(1) 振动的周期；(2) 加速度的最大值；(3) 振动表达式；(4) t = 何值时，动能 = 势能。 解：(1) (2) (3) (4) 具体化 * Example: A wheel is free to rotate about its fixed axle, a spring is attached to one of its spokes a distance r from axle. Assuming that the wheel is a hoop of mass m and radius R, spring constant k. a) obtain the angular frequency of small oscillations of this system; b) find angular frequency and how about r = R and r = 0. Solution: (a) (b) If r = R the result of part (a) reduces to And if r = 0 then w = 0 (the spring exerts no restoring torque on the wheel so that it is not brought back towards its equilibrium position ) * 小结 1. 方程 (解析法) 2. 相量图法 1. 方程 (解析法) 2. 判据 孤立系统 3. A、w、j 的确定 * §5.3 简谐运动的合成 5.3.1 同方向简谐运动的合成? 数学分析法 ① ② ①2 + ②2 ②/① —— 同频率的简谐运动 * 旋转矢量图法 x j1 x2 x1 x j w w w N M M1 M2 x2 P Q a b R O j2 * 讨论 a) 同相： k = 0, ?1, ?2, ?… b) 反相： k = 0, ?1, ?2, ?… 振动减弱 c) 2kp Dj (2k + 1)p, k = 0, ?1, ?2, ?… = 2kp, Dj = j2 - j1 cos(j2 - j1) = 1 振动加强 = (2k + 1)p, Dj = j2 - j1 cos(j2 - j1) = -1 若 A1 = A2, 则 A = 2A1, j = j1 若 A1 = A2, 则 A = 0, 质点静止 相位差对合振动起着重要作用! * [例] 某一质点同时参与两个同方向、同频率的谐振动，其振动规律为： x1 = 0.4cos(3t + p/3)， x2 = 0.3cos(3t - p/6) (SI)。 求： (1) 合振动的表达式; (2) 若