学 例如:车轮沿直线轨道作纯滚动,x4一P=0是微分方程 ,但 经过积分可得到xA=C(常数),该约束仍为完整约束 几何约束必定是完整约束,但完整约束未必是几何约東。 非完整约束一定是运动约束,但运动约束未必是非完整约束。 4、单面约束和双面约束 在两个相对的方向上同时OK 对质点或质点系进行运动限制∞ 刚杆 绳 的约束称为双面约束。只能限 y M 制质点或质点系单一方向运动y 的约束称为单面约束。 x2+=12 x2+y2≤P 16
16 在两个相对的方向上同时 对质点或质点系进行运动限制 的约束称为双面约束。只能限 制质点或质点系单一方向运动 的约束称为单面约束。 例如：车轮沿直线轨道作纯滚动， 是微分方程，但 经过积分可得到 （常数），该约束仍为完整约束。 x A −r = 0 xA −r=C 4、单面约束和双面约束 几何约束必定是完整约束，但完整约束未必是几何约束。 非完整约束一定是运动约束，但运动约束未必是非完整约束。 刚杆 x 2+y 2=l 2 绳 x 2+y 2 l 2
Dynamic The equation of constraint of a double face constraint is an equality the equation of constraint of a single face constraint is an inequalit We will discuss on the following only particle or a system of particles which is subjected to steady, double face and holonomic constraints, the general form of their equations is(s is the number of the constraints, n is the number of the particles of the system 1y1 ∠ nyn=n)=0(j=1,2……s) 17
17 The equation of constraint of a double face constraint is an equality, the equation of constraint of a single face constraint is an inequality. We will discuss on the following only particle or a system of particles which is subjected to steady, double face and holonomic constraints, the general form of their equations is (s is the number of the constraints, n is the number of the particles of the system): ( , , ; ; , , ) 0 ( 1,2, , ) 1 1 1 f x y z x y z j s j n n n = =
学 双面约束的约束方程为等式,单面约束的约束方程为不等式。 我们只讨论质点或质点系受定常、双面、完整约束的情况, 其约束方程的一般形式为(s为质点系所受的约束数目,n为质 点系的质点个数) f(x1,y1=1…;xn,yn:二n)=0(=12,…s) 18
18 双面约束的约束方程为等式，单面约束的约束方程为不等式。 我们只讨论质点或质点系受定常、双面、完整约束的情况， 其约束方程的一般形式为（s为质点系所受的约束数目，n为质 点系的质点个数） ( , , ; ; , , ) 0 ( 1,2, , ) 1 1 1 f x y z x y z j s j n n n = =
Dynamic $16-2 Degrees of freedom and generalized coordinates The position of a free particle in space is given by (x,y,z) There are 3 numbers The position of a free system of particles in space is given by(x 另=1)(i=1,2…n). There are3 n numbers. An unfree system of particles, under the action of s holonomic constraints, has only(3n-s )independent The degree of freedo of it is k-3n- dinat The number of the independent coordinates which determine the position of a system under the action of holonomic constraints is called the number of degrees of freedom, or shortly, the degree of freedom For example, in the case of the crankguide, discussed above the position coordinates xA, yA, xB and yB, have to satisfy three equations of constraints, therefore, it has only one degree of 19 freedom
19 §16-2 Degrees of freedom and generalized coordinates The position of a free particle in space is given by（ x, y, z ) . There are 3 numbers. The position of a free system of particles in space is given by ( xi , yi ,zi ) (i=1,2……n). There are 3n numbers. An unfree system of particles, under the action of s holonomic constraints, has only (3n-s ) independent coordinates. The degree of freedom of it is k=3n-s. The number of the independent coordinates which determine the position of a system under the action of holonomic constraints is called the number of degrees of freedom, or shortly, the degree of freedom. For example, in the case of the crankguide, discussed above the position coordinates xA , yA , xB and yB , have to satisfy three equations of constraints, therefore, it has only one degree of freedom
学 §16-2自由度广义坐标 一个自由质点在空间的位置:(x,y,z)3个 个自由质点系在空间的位置:(x12y1,=1)(=1,2…n)3n个 对一个非自由质点系,受s个完整约束,(3n-s)个独立坐标。 其自由度为k3n-s 确定一个受完整约束的质点系的位置所需的独立坐标的数目, 称为该质点系的自由度的数目,简称为自由度。 例如,前述曲柄连杆机构例子中,确定曲柄连杆机构位置的四 个坐标x、yAx2、y2须满足三个约束方程因此有一个自由度。 20
20 §16-2 自由度 广义坐标 一个自由质点在空间的位置：（ x, y, z ) 3个 一个自由质点系在空间的位置：( xi , yi ,zi ) (i=1,2……n) 3n个 对一个非自由质点系，受s个完整约束，（3n-s )个独立坐标。 其自由度为 k=3n-s 。 确定一个受完整约束的质点系的位置所需的独立坐标的数目， 称为该质点系的自由度的数目，简称为自由度。 例如, 前述曲柄连杆机构例子中, 确定曲柄连杆机构位置的四 个坐标xA、yA、xB、yB须满足三个约束方程,因此有一个自由度