Kinematics of Particles: Space Curvilinear Motion ME101 - Division IV Sandip Das Example: The power screw starts from rest and is given a rotational speed which increases uniformly with time t according to , where k is a constant. Determine the expressions for the velocity v and acceleration a of the center of the ball A when the screw has turned through one complete revolution from rest. The lead of the screw (advancement per revolution) is L. Solution: The centre of the ball moves in a helix on the cylindrical surface of radius b. Using Cylindrical coordinate system (r-θ-z). 1 kt
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Kinematics of Particles: Space Curvilinear Motion
ME101 - Division IV Sandip Das
Example: The power screw starts from rest and is given a rotational speed
which increases uniformly with time t according to , where k is a
constant. Determine the expressions for the velocity v and acceleration a
of the center of the ball A when the screw has turned through one
complete revolution from rest. The lead of the screw (advancement per
revolution) is L.
Solution:
The centre of the ball moves in a helix on the
cylindrical surface of radius b.
Using Cylindrical coordinate system (r-θ-z).
1
kt
Kinematics of Particles: Space Curvilinear Motion
ME101 - Division IV Sandip Das
Example:Solution:
Integrating θ = ½ kt2
For one revolution of the screw from rest: θ = 2π
2π = ½ kt2
Therefore,
The Cylindrical coordinates are shown:
γ is the helix angle, which is given by tan γ = L/(2πb)
This is the angle of the path followed by the
centre of the ball.
From the Fig: vθ = vcosγ
Using the available following eqns:
kt
kt /2
kkt 2
2
Along z
222
zr
z
r
vvvv
zv
rv
rv
brv
coscos
bvv
222 4
2cos
bL
b
γ = α
Kinematics of Particles: Space Curvilinear Motion
ME101 - Division IV Sandip Das
Example:Solution:
For one revolution position:
Velocity is given by:
Acceleration:
kkt 2
3
coscos
bvv 222 4
2cos
bL
b
222
2
2
zr
z
r
aaaa
za
rra
rra
Kinematics of Particles
ME101 - Division IV Sandip Das
Relative Motion (Translating Axes)• Till now particle motion described using fixed reference axes
Absolute Displacements, Velocities, and Accelerations
• Relative motion analysis is extremely important for some cases
measurements made wrt a moving reference system
4
Relative Motion Analysis
is critical even if aircrafts
are not rotating
Motion of a moving coordinate system is specified wrt a
fixed coordinate system (whose absolute motion is
negligible for the problem at hand).
Current Discussion:
• Moving reference systems that translate but do not
rotate
• Relative motion analysis for plane motion
Kinematics of Particles
ME101 - Division IV Sandip Das
Relative Motion (Translating Axes)Vector RepresentationTwo particles A and B have separate curvilinear motions
in a given plane or in parallel planes.
• Attaching the origin of translating (non-rotating) axes
x-y to B.
• Observing the motion of A from moving position on B.
• Position vector of A measured relative to the frame x-y
is rA/B = xi + yj. Here x and y are the coordinates of A
measured in the x-y frame. (A/B A relative to B)
• Absolute position of B is defined by vector rB
measured from the origin of the fixed axes X-Y.
• Absolute position of A rA = rB + rA/B
• Differentiating wrt time
Unit vector i and j have constant direction zero derivatives5