Kinematics of Rigid Bodies :: Rotating Axes Notes/ME101-Lecture33-KD.pdf · Kinematics of Rigid Bodies :: Rotating Axes Relative Velocity Relations •The curved slot represents rotating

Kinematics of Rigid Bodies :: Rotating AxesDifferentiating the posn vector eqn to obtain vel & accl eqn:

•The unit vectors are rotating with the x-y axes

time derivatives must be evaluated.

When x-y axes rotate during dt through an angle dθ = ωdt :

•Differential change in i di

di has direction of j

magnitude of di = dθ x magnitude of i = dθ

Therefore, di = dθ j

•Differential change in j dj

dj has negative x-direction

Therefore, dj = - dθ i

Dividing by dt and replacing

Using cross-product: ω x i = ωj and ω x j = - ωi

1ME101 - Division III Kaustubh Dasgupta

Kinematics of Rigid Bodies :: Rotating AxesRelative Velocity Relations

•The curved slot represents rotating x-y frame

•The x-y 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/P

relative to a point P attached to the plate and

coincident with A at the instant under

consideration.

•ω x r has dirn normal to r

= vP/B vel of P rel to origin B of non-rotating

axesComparison betn relative vel eqns for rotating and non-rotating reference axes

• vP/B is measured from a non-rotating 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 non-rotating

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 X-Y system:

Since

ωxV :: difference between time derivative of

the vector measured in fixed and in rotating

reference systemPhysical Significance


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


Kinematics of Rigid Bodies :: Rotating Axes

Relative Acceleration Relations

: tangential component of aP/B of point P wrt B

ω x (ω x r) : normal component of aP/B

This motion would be observed from a set of non-rotating axes moving

with B.

Magnitude of ::

Direction of ::

Magnitude of ω x (ω x r) :: r ω2

Direction of ω x (ω 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:

ρ of the path measured in x-y system


tangent to the circle at P @ B

Kinematics of Rigid Bodies :: Rotating AxesRelative Acceleration Relations

• 2ω x vrel :: Coriolis Acceleration

- Difference between the accln of A relative to

P as measured from non-rotating axes and

rotating axes

- Dirn is always normal to the vrel

- Composed of two separate physical effects



• Coriolis acceleration (2ω 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: xω




:: During an interval dt, x-y axes rotate

with the disk through dθ to x’-y’

- Changes in the two velocity

components

:: Vel increment due to change

in direction of vrel =

:: Vel increment due to change

in magnitude of xω = ωdx

- Both changes along the y-dirn

normal to the slot




:: Dividing each increment by dt

and adding,

- Magnitude of the Coriolis Accln

2ω x vrel

:: xω also changes dirn during dt

- dividing vel increment xωdθ by dt

-

- Acceleration of a point P fixed to

the slot and momentarily coincident

with A




:: Using the relative acceleration eqn for the disk:

Considering the origin B at fixed center O

aB = 0

:: With constant ω:

:: With vrel constant in magnitude and

no curvature to the slot, arel = 0

:: Replacing r by xi, ω by ωk, and vrel by

- Coefficient of the 2nd term is the magnitude of the Coriolis accln




:: 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


Comparison betn relative accln eqns for rotating and non-rotating

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 non-rotating 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 non-rotating system

arel, vrel measured in a rotating system



Relative Velocity and Acceleration Relations

These eqns are also applicable for 3-D motion in space.



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.



Solution

Since motion of A is relative to a rotating path:

Attaching the rotating coordinate x-y axes to the disk at O

Velocity:

since origin is at O vB = 0, Angular vel ω = 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/s2


ME101 - Division III Kaustubh Dasgupta

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Kinematics of Rigid Bodies :: Rotating Axes Notes/ME101-Lecture33-KD.pdf · Kinematics of Rigid Bodies :: Rotating Axes Relative Velocity Relations •The curved slot represents rotating