Dynarnics Equilibrium position x=0
11 Equilibrium position x=0 position Equilibrium j=0 position Equilibrium j=0
学 O平衡位置 9 pa 衡 平衡 位置 12
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Dynamic a force acting on a vibrating body which is always directed towards its equilibrium position is called a restoring force After the action of an initial disturbance the vibration of a system around its equilibrium position under the action of only a restoring force is called an undamped free vibration For a mass-spring system: mx=-kx x+02x=0(02=k/m For a simple pendulum: ml==mglp, 9+09=0(0=8/1,<5) For a compound pendulum: 10=-mgao, +9=0(0=mga/1,<5) 13
13 A force acting on a vibrating body which is always directed towards its equilibrium position is called a restoring force. After the action of an initial disturbance, the vibration of a system around its equilibrium position under the action of only a restoring force is called an undamped free vibration. , 0 ( / , 5 ) , 0 ( / , 5 ) , 0 ( / ) 2 2 2 2 2 2 2 = − + = = = − + = = = − + = = j j j j j j j j j j I mga mga I m l mgl g l mx k x x x k m n n n n n n For a mass-spring system: For a simple pendulum: For a compound pendulum:
力单 运动过程中,总指向物体平衡位置的力称为恢复力。 物体受到初干扰后,仅在系统的恢复力作用下在其平衡位 置附近的振动称为无阻尼自由振动。 质量一弹簧系统:m=-kx,x+o2x=0(o2=k/m) 单摆:m2=-mg1,i+o2=0(2=g1l2q≤5) 复摆:(=-mga,(+02=0(o2=mga/2q≤5)
14 运动过程中，总指向物体平衡位置的力称为恢复力。 物体受到初干扰后，仅在系统的恢复力作用下在其平衡位 置附近的振动称为无阻尼自由振动。 , 0 ( / ) 2 2 质量—弹簧系统： m x = −k x x +n x = n = k m 单摆： 复摆： , 0 ( / , 5 ) , 0 ( / , 5 ) 2 2 2 2 2 = − + = = = − + = = j j j j j j j j j j I mga mga I ml mgl g l n n n n
Dynamic 2. Differential equation and its solution for an undamped free vibration of a system with one degree of freedom For any system with one degree of freedom, g being the generalized coordinate measured from the equilibrium position, the differential equation of a free vibration for a small displacement is aq+ cq=0 where a and c are constants related to the physical parameters of the system. Writing an =cla the standard form of the differential equation of free vibration is g+a,9=0 Its solution is: g=Asin(@nt +a) 15
15 2. Differential equation and its solution for an undamped free vibration of a system with one degree of freedom For any system with one degree of freedom, q being the generalized coordinate measured from the equilibrium position, the differential equation of a free vibration for a small displacement is aq + cq = 0 where a and c are constants related to the physical parameters of the system. Writing n c / a 2 = the standard form of the differential equation of free vibration is 0. 2 q +n q = Its solution is： q = Asin( t +) n