King Fahd University of Petroleum

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King Fahd University of Petroleum Minerals

• Mechanical Engineering
• Dynamics ME 201
• BY
• Dr. Meyassar N. Al-Haddad
• Lecture 6

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Objective

• To investigate particle motion along a curved
  path Curvilinear Motion using three coordinate
  systems
• Rectangular Components
• Position vector r x i y j z k
• Velocity v vx i vy j vz k
  (tangent to path)
• Acceleration a ax i ay j az k
  (tangent to hodograph)
• Normal and Tangential Components
• Position (particle itself)
• Velocity v u ut
  (tangent to path)
• Acceleration
  (normal tangent)
• Polar Cylindrical Components

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Curvilinear Motion Cylindrical Components

• Section 12.8
• Observed and/or guided from origin or from the
  center
• Cylindrical component
• Polar component plane motion

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Application Circular motion but observed and/or
controlled from the center
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Polar Coordinates

• Radial coordinate r
• Transverse coordinate
• q and r are perpendicular
• Theta q in radians
• 1 rad 180o/p
• Direction ur and uq

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Position

• Position vector
• r r ur

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Velocity

• Instantaneous velocity time derivative of r
• Where

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Velocity (con.)

• Magnitude of velocity
• Angular velocity
• Tangent to the path
• Angle q d

d
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Acceleration

• Instantaneous acceleration time derivative of v

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Acceleration (con.)

• Angular acceleration
• Magnitude
• Direction Not tangent
• Angle q f

f
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Cylindrical Coordinates

• For spiral motion cylindrical coordinates is used
  r, q, and z.
• Position
• Velocity
• Acceleration

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Time Derivative to evaluate

• If r r(t) and q q(t)

• If r f(q) use chain rule

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Problem

• The slotted fork is rotating about O at a
  constant rate of 3 rad/s. Determine the radial
  and transverse components of velocity and
  acceleration of the pin A at the instant q
  360o. The path is defined by the spiral groove r
  (5q/p) in., where q is in radians.

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Example 12-20
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Problem
A collar slides along the smooth vertical spiral
rod, r (2q) m, where q is in radians. If its
angular rate of rotation is constant and equal 4
rad/s, at the instant q 90o. Determine - The
collar radial and transverse component of
velocity - The collar radial and transverse
component of acceleration. - The magnitude of
velocity and acceleration
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Thank You