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Tham khảo tài liệu 'advanced engineering dynamics 2010 part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 234 Robot arm dynamics HOMOGENEOUS TRANSFORMATION MATRIX FORA SINGLE LINK This matrix is ce -Case SaSe aC0 so CaCe - -SaCe aS6 M 0 Sa Ca d 0 0 0 1 The overall tra T nsformation matrix is 1 1 and the position vector p p.r Py Pz 1 Fig. 8.29 9 Relativity 9.1 Introduction In this chapter we shall reappraise the foundations of mechanics taking into account Einstein s special theory of relativity. Although it does not measurably affect the vast majority of problems encountered in engineering it does define the boundaries of Newtonian dynamics. Confidence in the classical form will be enhanced as we shall be able to quantify the small errors introduced by using Newtonian theory in common engineering situations. The laser velocity transducer employs the Doppler effect which for light requires an understanding of special relativity. The form of the equations derived for cases where the velocities of the transmitter and or the receiver are small compared with that of the signal is the same for both sound and light. This will be discussed later. We shall also consider the definition of force. It is of note that relativistic definitions are such that they encompass the Newtonian. The general theory of relativity raises some interesting questions regarding the nature of force but these do not materially affect the equations of motion already derived. 9.2 The foundations of the special theory of relativity It is not our intention to retrace the steps leading to the theory other than to mention the most significant milestones. In the same way that Isaac Newton crystallized the laws of mechanics which have formed the basis for the previous chapters in this book Albert Einstein provided the genius that solved the riddle of the constancy of the speed of light. James Clerk Maxwell s equations for electrodynamics predicted that all electromagnetic waves travelled at a constant speed in a vacuum. If the value of the speed of light c is evaluated for what we shall assume to be an .
