Page 63 - Dynamics of Mechanical Systems

0593_C02_fm Page 44 Monday, May 6, 2002 1:46 PM

44 Dynamics of Mechanical Systems

, ˆ
ˆ n
then by performing three successive dextral rotations of the about ˆ n n 2 , and ˆ n 3 through
1
i
the angles α, β, and γ. The transformation matrix S between the n and the ˆ n i is, then:
i
1 0 0  c β 0 s   c γ s − γ  0
β

S = ABC = 0 c α s − α  0 1 0   s c γ 0   (2.11.18)


  γ


 0 s α c  − s β 0 c  0 0   1
 

α
β
or,
 − 
 cc c s s β 
β γ
β γ
 cs + c c − s s s ) − s c 
S = ( αγ s s c ) ( α γ α β γ α β (2.11.19)
α β γ
 c s s ) 
s c +
 (  ss − c s c ) ( α γ α β γ cc  
α β
α β γ
αγ
Also, it is readily seen that:
S = C B A T (2.11.20)
T
T
T
and that:
SS = S S = I (2.11.21)
T
T
2.12 Closure
The foregoing discussion is intended primarily as a review of basic concepts of vector and
matrix algebra. Readers who are either unfamiliar with these concepts or who want to
explore the concepts in greater depth may want to consult a mathematics or vector
mechanics text as provided in the References. We will freely employ these concepts
throughout this text.

References

2.1. Hinchey, F. A., Vectors and Tensors for Engineers and Scientists, Wiley, New York, 1976.
2.2. Hsu, H. P., Vector Analysis, Simon & Schuster Technical Outlines, New York, 1969.
2.3. Haskell, R. E., Introduction to Vectors and Cartesian Tensors, Prentice Hall, Englewood Cliffs, NJ,
1972.
2.4. Shields, P. C., Elementary Linear Algebra, Worth Publishers, New York, 1968.
2.5. Ayers, F., Theory and Problems of Matrices, Schawn Outline Series, McGraw-Hill, New York,
1962.
2.6. Pettofrezzo, A. J., Elements of Linear Algebra, Prentice Hall, Englewood Cliffs, NJ, 1970.
2.7. Usamani, R. A., Applied Linear Algebra, Marcel Dekker, New York, 1987.
2.8. Borisenko, A. I., and Tarapov, I. E., Vector and Tensor Analysis with Applications (translated by
R. A. Silverman), Prentice Hall, Englewood Cliffs, NJ, 1968.