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Tham khảo tài liệu 'advanced engineering dynamics 2010 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 54 Hamilton s principle f h q2 tj u J T TIT ỗudxdí J t 0 ox2 3.20 The first term is zero provided that the ends are passive that is no energy is being fed into the string after motion has been initiated. This means that either Sm 0 or ôu õx 0 at each end. The specification of the problem indicated that 5u 0 but any condition that makes energy transfer zero at the extremes excludes the first term. Combining equations 3.19 and 3.20 and substituting into equation 3.18 yields and because ỖU is arbitrary the integrand must sum to zero so that finally Õ2U _ Ô2 u pa ôt2 õx 2 3.21 This is the well-known wave equation for strings. It is readily obtained from free-body diagram methods but this approach is much easier to modify if other effects such as that of bending stiffness of the wire are to be considered. Extra energy terms can be added to the above treatment without the need to rework the whole problem. This fact will be exploited in Chapter 6 which discusses wave motion in more detail. 4 Rigid Body Motion in Three Dimensions 4.1 Introduction A rigid body is an idealization of a solid object for which no change in volume or shape is permissible. This means that the separation between any two particles of the body remains constant. If we know the positions of three non-colinear points i j and k then the position of the body in space is defined. However there are three equations of constraint of the form r - I constant so the number of degrees of freedom is 3 X 3 - 3 6. 4.2 Rotation If the line joining any two points changes its orientation in space then the body has suffered a rotation. If no rotation is taking place then all particles will be moving along parallel paths. If the paths are straight then the motion is described as rectilinear translation and if not the motion is curvilinear translation. From the definitions it is clear that a body can move along a circular path but there need be no rotation of the body. It follows that for any pure translational motion .