Kinematics of Rigid Bodies :: Rotating Notes/ME101Lecture33KD.pdf · PDF fileKinematics...
date post
02Sep2018Category
Documents
view
272download
1
Embed Size (px)
Transcript of Kinematics of Rigid Bodies :: Rotating Notes/ME101Lecture33KD.pdf · PDF fileKinematics...
Kinematics of Rigid Bodies :: Rotating AxesDifferentiating the posn vector eqn to obtain vel & accl eqn:
The unit vectors are rotating with the xy axes
time derivatives must be evaluated.
When xy axes rotate during dt through an angle d = dt :
Differential change in i di
di has direction of j
magnitude of di = d x magnitude of i = d
Therefore, di = d j
Differential change in j dj
dj has negative xdirection
Therefore, dj =  d i
Dividing by dt and replacing
Using crossproduct: x i = j and x j =  i
1ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating AxesRelative Velocity Relations
The curved slot represents rotating xy frame
The xy axes are not rotating themselves.
Vel of A measured relative to the plate = vrel.
Magnitude of vrel will be ds/dt
vrel may also be viewed as the vel vA/Prelative to a point P attached to the plate and
coincident with A at the instant under
consideration.
x r has dirn normal to r
= vP/B vel of P rel to origin B of nonrotating
axesComparison betn relative vel eqns for rotating and nonrotating reference axes
vP/B is measured from a nonrotating posn
vP = absolute velocity of P and represent the effect of the
moving coordinate system (both translational @ rotational)
Last eqn is the same as that developed for nonrotating
axes
2
Kinematics of Rigid Bodies :: Rotating AxesRelative Velocity Relations
Transformation of time derivative of the
position vector between rotating and non
rotating axes
Generalized for any vector: V = Vxi + Vyj
The total time derivative wrt XY system:
Since
xV :: difference between time derivative of
the vector measured in fixed and in rotating
reference systemPhysical Significance
3ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating AxesRelative Acceleration Relations
From relative velocity eqn:
Also,
Therefore, the third term in the accln eqn:
The last term in relative accln eqn:
Substituting
4ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating Axes
Relative Acceleration Relations
: tangential component of aP/B of point P wrt B
x ( x r) : normal component of aP/B
This motion would be observed from a set of nonrotating axes moving
with B.
Magnitude of ::
Direction of ::
Magnitude of x ( x r) :: r 2
Direction of x ( x r) :: from P to B
Accln. of A relative to the plate along the path arel
Tangential comp has magnitude:
Normal comp has magnitude:
of the path measured in xy system
5ME101  Division III Kaustubh Dasgupta
tangent to the circle at P @ B
Kinematics of Rigid Bodies :: Rotating AxesRelative Acceleration Relations
2 x vrel :: Coriolis Acceleration
 Difference between the accln of A relative to
P as measured from nonrotating axes and
rotating axes
 Dirn is always normal to the vrel
 Composed of two separate physical effects
6ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating Axes
Coriolis acceleration (2 x vrel)
Consider,
:: A rotating disk with a radial slot
in which a particle is sliding
:: Constant angular velocity of disk
:: Constant speed of particle relative
to the slot,
:: Two components of velocity of A
(a) Due to motion along the slot:
(b) Due to motion along the slot: x
7ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating Axes
Coriolis acceleration (2 x vrel)
:: During an interval dt, xy axes rotate
with the disk through d to xy
 Changes in the two velocity
components
:: Vel increment due to change
in direction of vrel =
:: Vel increment due to change
in magnitude of x = dx
 Both changes along the ydirn
normal to the slot
8ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating Axes
Coriolis acceleration (2 x vrel)
:: Dividing each increment by dt
and adding,
 Magnitude of the Coriolis Accln
2 x vrel
:: x also changes dirn during dt
 dividing vel increment xd by dt

 Acceleration of a point P fixed to
the slot and momentarily coincident
with A
9ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating Axes
Coriolis acceleration (2 x vrel)
:: Using the relative acceleration eqn for the disk:
Considering the origin B at fixed center O
aB = 0
:: With constant :
:: With vrel constant in magnitude and
no curvature to the slot, arel = 0
:: Replacing r by xi, by k, and vrel by
 Coefficient of the 2nd term is the magnitude of the Coriolis accln
10ME101  Division III Kaustubh Dasgupta
Kinematics of Rigid Bodies :: Rotating Axes
Coriolis acceleration (2 x vrel)
:: Same results if the equation developed
for plane curvilinear motion in polar
coordinates is used by replacing r by x
and
:: If the slot in the disk is curved, normal component of accln. relative to
slot will not be zero arel 0
11ME101  Division III Kaustubh Dasgupta
Comparison betn relative accln eqns for rotating and nonrotating
reference axes
From the third eqn: relative accln term aA/P arel
Coriolis accln is the difference betn the accln aA/P of A relative to P as
measured in a nonrotating system and the accln arel of A relative to P as
measured in a rotating system
aP/B, aA/P, aA/B measured in a nonrotating system
arel, vrel measured in a rotating system
Kinematics of Rigid Bodies :: Rotating Axes
12ME101  Division III Kaustubh Dasgupta
Relative Velocity and Acceleration Relations
These eqns are also applicable for 3D motion in space.
Kinematics of Rigid Bodies :: Rotating Axes
13ME101  Division III Kaustubh Dasgupta
Example: At the instant represented, the disk with the radial slot is rotating
about O with a counterclockwise angular velocity if 4 rad/s, which is
decreasing at a rate of 10 rad/s2. Motion of slider A in the slot is separately
controlled, and at this instant, r = 150 mm, r = 125 mm/s, r = 2025 mm/s2. Determine the absolute velocity and acceleration of A for this position.
Kinematics of Rigid Bodies :: Rotating Axes
14ME101  Division III Kaustubh Dasgupta
Solution
Since motion of A is relative to a rotating path:
Attaching the rotating coordinate xy axes to the disk at O
Velocity:
since origin is at O vB = 0, Angular vel = 4k rad/s
vA = 4k x 0.150i + 0.125i = 0.600j + 0.125i in the dirn shown
Magnitude: vA = (0.6002 + 0.1252) = 0.613 m/s
Acceleration:
since origin is at O aB = 0
4k x (4k x 0.150i) = 4k x 0.6j = 2.4i m/s2
10k x 0.150i = 1.5j m/s2
2(4k) x 0.125i = 1.0j m/s2
2.025i m/s2
aA = 0.375i 0.5j m/s2 and magnitude: aA = 0.625 m/s
2
Kinematics of Rigid Bodies :: Rotating Axes
ME101  Division III Kaustubh Dasgupta