Mechanical Vibrations: Theory and Practice

APPENDIX - A

Mathematical Relationships

Vibration analyses frequently require referring to some of the relationships from trigonometry, algebra, and calculus. A few of them are given as follows:

 

sin(α ± β) = sin α cos β ± cos α sin β

cos(α ± β) = cos α cos β ∓ sin α sin β

sin(α + β)sin(α − β) = sin² α − sin² β = cos² β − cos² α

cos(α + β)cos(α − β) = cos² α − sin² β = cos² β − sin² α

where

sin² α + sin² α = 1

 

cos2α = 1 − 2sin² α = 2cos² α − 1 = cos² α − sin² α

Law of cosines for triangles:

 

c² = a² + b² − 2abcosC

 

Π = 3.14159 rad, 1 ...