1. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 Introduction Fundamentals of Analytical Method For analytical kinematic (and dynamic) analysis, the following vector fundamentals are necessary: 1.Convention for describing angle of a vector 2.Projecting a vector onto a Cartesian reference frame 3.Types of vectors 4.Time derivatives of vectors

2. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 There are infinite ways to describe angle of a vector. We adopt the convention to describe the angle of a vector with respect to the positive x-axis in a counter-clockwise (CCW) direction. For vector R (with a magnitude R) the angle  is measured as: Angle of A Vector R x y  1.From the base (tail) of the vector draw a line in the positive x-axis 2. In a counter-clockwise direction, starting from this line, draw an arc until it reaches the vector This is the angle . Regardless of the orientation of a vector, this process provides the angle in such a way that simplifies any further analysis. ► Angle of A Vector ►

3. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 Examples: 1.Vector R 1 2.Vector R 2 3.Vector R 3 4.Vector R 4 Angle of A Vector R1R1 x y  1 = 36.4 o ► Angle of A Vector (cont.) ► R2R2  2 = 160.7 o ► R3R3  3 = 215.6 o R4R4  4 = 310.2 o ►

4. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 For vector R with a magnitude R and an angle , if the angle is described according to the “convention”, it is a simple matter to determine its x-y components. Vector Projection ► R x y  R cos  R sin  The x component is: R (x) = R cos  (a) The y component is: R (y) = R sin  (b) Equations (a) and (b) are always valid, regardless of the angle being in the first quadrant (0-90 degrees) or the second quadrant (90-180 degrees) or so on. ► Projecting A vector onto x-y Axes

5. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 1.Constant magnitude and constant angle 2.Constant magnitude and variable angle 3.Variable magnitude and constant angle 4.Variable magnitude and variable angle Types of Vectors ► ► R4R4 R1R1 R1R1 R1R1 R4R4 R3R3 R2R2 R3R3 R2R2 R2R2 R4R4 ► ► The magnitude and the angle of a vector may be constant and/or variable. Therefore we may consider four types of vectors:

6. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 Constant magnitude and constant angle Position components: R (x) = R cos  R (y) = R sin  First derivative (velocity components): R (x) = d R (x) / dt = 0 R (y) = d R (y) / dt = 0 Second time derivative (acceleration components): R (x) = d 2 R (x) / dt 2 = 0 R (y) = d 2 R (y) / dt 2 = 0 Time Derivatives Time Derivatives of Vectors R....

7. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 Constant magnitude and variable angle Position components: R (x) = R cos  R (y) = R sin  First derivative (velocity components): R (x) = d R (x) / dt =  R (sin  )  R (y) = d R (y) / dt = R (cos  )  V t Where  = d  / dt is the angular velocity of the link. Second time derivative (acceleration components): R (x) = d 2 R (x) / dt 2 =  R (sin  )  R (cos  )    R (y) = d 2 R (y) / dt 2 = R (cos  )  R (sin  )    A t A n Where  = d  / dt is the angular acceleration of the link. Time Derivatives Time Derivatives of Vectors (cont.) R R....

8. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 Variable magnitude and constant angle Position components: R (x) = R cos  R (y) = R sin  First derivative (velocity components): R (x) = d R (x) / dt = R cos  R (y) = d R (y) / dt = R sin  V s Where R = dR / dt = V s is the rate of change in the magnitude of the link. Second time derivative (acceleration components): R (x) = d 2 R (x) / dt 2 =  R cos  R (y) = d 2 R (y) / dt 2 = R sin  A s Where R = d 2 R / dt 2 = A s is the second time derivative of the magnitude of the link. Time Derivatives Time Derivatives of Vectors (cont.) R.......

9. P. Nikravesh, AME, U of A Fundamentals of Analytical Analysis 1 Variable magnitude and variable angle Position components: R (x) = R cos  R (y) = R sin  First derivative (velocity components): R (x) = d R (x) / dt = R cos  R (sin  )  R (y) = d R (y) / dt = R sin  R (cos  )  V s V t Second time derivative (acceleration components): R (x) = d 2 R (x) / dt 2 =  R cos  R (sin  )  R (cos  )    2R (sin  )  R (y) = d 2 R (y) / dt 2 = R sin  R (cos  )  R (sin  )    2R (cos  )  A s A t A n A c Time Derivatives Time Derivatives of Vectors (cont.) R R........