Dynarnics Geometrical constraint: y=r, Constraint of motion: VA-ro=0, (ro=o) (2) Steady constraints and unsteady constraints A constraint which is time-dependent is called an unsteady constraint. A constraint which does not depend on time is called a steady constraint In the examples above the constraints do not change with the time, so they are all steady constraints For example, the object M is made of a x fixed ring and a rope tied to the ring. At the beginning, the length of the single pendulum is lo, drag the rope with uniform speed v. In the equation of the constraint, x2+y2=(l o vt 2, the time t appears directly. I
11 Geometrical constraint: Constraint of motion: ( 0). 0, , − = − = = x r v r y r A A A A constraint which is time-dependent is called an unsteady constraint. A constraint which does not depend on time is called a steady constraint.In the examples above, the constraints do not change with the time, so they are all steady constraints. (2) Steady constraints and unsteady constraints For example, the object M is made of a fixed ring and a rope tied to the ring. At the beginning, the length of the single pendulum is l0 , drag the rope with uniform speed v. In the equation of the constraint, x 2+y2=( l0 - vt ) 2 , the time t appears directly
几何约束:y4=r 运动约束:v4-rO=0 (x4-r=0) 2、定常约束和非定常约束 当约束条件与时间有关,并随时间变化时称为非定常约束。 约束条件不随时间改变的约束为定常约束。 前面的例子中约束条件皆不随时间变化,它们都是定常约束。 0例如:重物M一条穿过固定圆环的细绳 系住。初始时摆长l,匀速γ拉动绳子。 x2+y2=(1-t)2约束方程中显含时间t 12
12 几何约束： 运动约束： ( 0) 0 − = − = = x r v r y r A A A 当约束条件与时间有关，并随时间变化时称为非定常约束。 约束条件不随时间改变的约束为定常约束。 前面的例子中约束条件皆不随时间变化，它们都是定常约束。 2、定常约束和非定常约束 例如：重物M由一条穿过固定圆环的细绳 系住。初始时摆长 l0 , 匀速v拉动绳子。 x 2+y2=( l0 -vt )2 约束方程中显含时间 t
Dynamic (3)Holonomic and nonholonomic constraints If there appear time-derivatives of coordinates in an equation of a constraint(as in the case of a constraint of motion and if), moreover these derivatives can not be removed by the infinitesimal calculus (hence, the coordinate derivative contained in the equation of the constraint is not a total differential of a certain function and the equation of the constraint can not be changed into a finite form by integration) this constraint is called a nonholonomic constraint Generally, the equations of the nonholonomic constraints can not be expressed in differential form If there are on time-derivatives of coordinates in the equation of a constraint or if such derivatives can be transformed into a finite form by infinitesimal calculus, then this kind of constraint is called a holonomic constraint 13
13 If there appear time-derivatives of coordinates in an equation of a constraint (as in the case of a constraint of motion and if), moreover, these derivatives can not be removed by the infinitesimal calculus, (hence, the coordinate derivative contained in the equation of the constraint is not a total differential of a certain function and the equation of the constraint can not be changed into a finite form by integration) this constraint is called a nonholonomic constraint. Generally, the equations of the nonholonomic constraints can not be expressed in differential form. If there are on time-derivatives of coordinates in the equation of a constraint, or if such derivatives can be transformed into a finite form by infinitesimal calculus, then this kind of constraint is called a holonomic constraint. (3) Holonomic and nonholonomic constraints
学 3、完整约束和非完整约東 如果在约束方程中含有坐标对时间的导数(例如运动约束) 而且方程中的这些导数不能经过积分运算消除,即约束方程中 含有的坐标导数项不是某一函数全微分,从而不能将约束方程 积分为有限形式,这类约束称为非完整约束。一般地,非完整 约束方程只能以微分形式表达。 如果约束方程中不含有坐标对时间的导数,或者约束方程 中虽有坐标对时间的导数,但这些导数可以经过积分运算化为 有限形式,则这类约束称为完整约束。 14
14 如果在约束方程中含有坐标对时间的导数（例如运动约束） 而且方程中的这些导数不能经过积分运算消除，即约束方程中 含有的坐标导数项不是某一函数全微分，从而不能将约束方程 积分为有限形式，这类约束称为非完整约束。一般地，非完整 约束方程只能以微分形式表达。 3、完整约束和非完整约束 如果约束方程中不含有坐标对时间的导数，或者约束方程 中虽有坐标对时间的导数，但这些导数可以经过积分运算化为 有限形式，则这类约束称为完整约束
For example, if a wheel is purely rolling along a linear rai Xro=o is a differential equation, but after integration we get x-r=C. This constraint is a holonomic one A geometrical constraint must be a holonomic one, but a holonomic one is not necessarily a geometrical one. A nonholonomic constraint must be a constraint of motion, but a constraint of motion one is not necessarily a nonholonomic one (4) Single and double face constraints A constraint which limits the O movement of a particle of a system in two opposite 刚杆 directions at the same time is 绳 called a double face constraint A constraint which limits the y M M movement of a particle of a stem in a single direction is x2+y=p x2+y2≤P 15 called a single face constraint
15 A constraint which limits the movement of a particle of a system in two opposite directions at the same time is called a double face constraint. A constraint which limits the movement of a particle of a system in a single direction is called a single face constraint. For example, if a wheel is purely rolling along a linear rail, is a differential equation, but after integration we get . This constraint is a holonomic one. x A −r = 0 xA −r=C (4) Single and double face constraints A geometrical constraint must be a holonomic one, but a holonomic one is not necessarily a geometrical one. A nonholonomic constraint must be a constraint of motion, but a constraint of motion one is not necessarily a nonholonomic one. 刚杆 x 2+y 2=l 2 绳 x 2+y 2 l 2