Kinematics of Rigid Bodies :: Rotating Notes/ME101-Lecture33-KD.pdf · PDF fileKinematics...

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Transcript of Kinematics of Rigid Bodies :: Rotating Notes/ME101-Lecture33-KD.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 x-y axes

    time derivatives must be evaluated.

    When x-y 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 x-direction

    Therefore, dj = - d i

    Dividing by dt and replacing

    Using cross-product: 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 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/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 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

    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 non-rotating 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 x-y 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 non-rotating 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, x-y axes rotate

    with the disk through d 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 x = dx

    - Both changes along the y-dirn

    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 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

    Kinematics of Rigid Bodies :: Rotating Axes

    12ME101 - Division III Kaustubh Dasgupta

  • Relative Velocity and Acceleration Relations

    These eqns are also applicable for 3-D 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 x-y 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