学 二、单自由度系统无阻尼自由振动微分方程及其解 对于任何一个单自由度系统,以q为广义坐标(从平衡位 置开始量取),则自由振动的运动微分方程必将是: aq+cq=0 a,c是与系统的物理参数有关的常数。令On=c/a 则自由振动的微分方程的标准形式: 9+ang=o 解为: g=As(on t +a) 16
16 二、单自由度系统无阻尼自由振动微分方程及其解 对于任何一个单自由度系统，以q 为广义坐标（从平衡位 置开始量取 ），则自由振动的运动微分方程必将是： aq + cq = 0 a, c是与系统的物理参数有关的常数。令 n c / a 2 = 则自由振动的微分方程的标准形式： 0 2 q +n q = 解为： q = Asin( t +) n
Dynamic assuming that at t=0, q=go and q =qo we get 90 +2,a=arct 0 or: q=C, coS@n t+Casin@nt with Ci=go Therefore, q=go cos@, t+sin @,t 7
17 Assuming that at t = 0 ， 0 0 we get q = q and q = q , arctg , 0 0 2 2 2 0 0 q q q A q n n = + = or： q C t C t n n = 1 cos + 2 sin with , / . 1 0 2 q0 n C = q C = Therefore, cos sin . 0 0 t q q q t n n n = +
力单 设t=0时,q=40,q=4则可求得: A=19+ q a= arc ctg 0 或 q=C, coS@n t+Casin@nt C1,C2由初始条件决定为C1=qo,C2=qon g=go cos@nttsin nt 18
18 0 0 2 2 2 0 0 , arctg q q q A q n n = + = 设 t = 0 时， q = q0 , q = q 0 则可求得： 或： q C t C t n n = 1 cos + 2 sin C1，C2由初始条件决定为 q n C1 =q0 , C2 = 0 / t q q q t n n n cos sin 0 0 = +
Dynarnics 3. Properties of a free vibration without damping: A-The quantity, which is the maximum distance of a vibrating body from the equilibrium position, is called the amplitude On, t+ a-The quantity, which defines the position of the vibrating body at any given time, is called the phase of the vibration C The quantity, which defines the initial phase, at which the motion starts T=The time T' during which the vibrating body makes one complete vibration is called the period of vibration, T=<7 f-The quantity, which specifies the number of oscillations per second and is the inverse of the period is called the frequency of the vibration, f=1/T On-The quantity which specifies the number of oscillations in 2/seconds, is called natural frequency of the vibration. It characterizes the dynamics of a given vibrating system and depends on the inherent parameters describing the system! 9
19 n 3.Properties of a free vibration without damping: A——The quantity ,which is the maximum distance of a vibrating body from the equilibrium position, is called the amplitude. n t + ——The quantity, which defines the position of the vibrating body at any given time, is called the phase of the vibration. ——The quantity, which defines the initial phase, at which the motion starts. T ——The time T during which the vibrating body makes one complete vibration is called the period of vibration, f ——The quantity, which specifies the number of oscillations per second and is the inverse of the period, is called the frequency of the vibration, f = 1 / T. —— The quantity, which specifies the number of oscillations in 2 seconds, is called natural frequency of the vibration. It characterizes the dynamics of a given vibrating system and depends on the inherent parameters describing the system. . 2 n T =
学 三、自由振动的特点: A——物块离开平衡位置的最大位移,称为振幅 ω,t+α相位,决定振体在某瞬时t的位置 α—初相位,决定振体运动的起始位置、 7—周期,每振动一次所经历的时间。T=2n ∫——频率,每秒钟振动的次数,f=1/T 固有频率,振体在2π秒内振动的次数。 反映振动系统的动力学特性,只与系统本身的固有 参数有关。 20
20 三、自由振动的特点： A——物块离开平衡位置的最大位移，称为振幅。 n t + ——相位，决定振体在某瞬时 t 的位置 ——初相位，决定振体运动的起始位置。 T ——周期，每振动一次所经历的时间。 f —— 频率，每秒钟振动的次数， f = 1 / T 。 —— 固有频率，振体在2秒内振动的次数。 反映振动系统的动力学特性，只与系统本身的固有 参数有关。 n T 2 = n