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The motion takes place in a Single Plane (Plane Motion) when all parts of the body
move in parallel planes.
This is equivalent with:
𝜔 has a fixed direction
Lecture 5. Kinematics of the rigid body 1
Theoretical Mechanics
6. The plane parallel motion of the rigid body
(Plane kinematics of rigid bodies)
The rigid body points on a line
parallel with 𝜔 have the same
velocity. Thus, the study is reduced
to the study of a plane section of the
rigid in its plane.
Lecture 5. Kinematics of the rigid body 2
Theoretical Mechanics
Plane MotionTranslation
No rotation of any line in body.
Motion of the body specified by
motion of any point in the body
≈Motion of a single particle.
Rotation about a Fixed Axis
All particles move in circular
paths about the axis of
rotation. All lines perpendicular
to the axis of rotationn rotate
through the same angle.
General Planar Motion
Combination of translation and
rotation
The actual paths of all
particles in the body are
projected on a single plane
of motion.
Lecture 5. Kinematics of the rigid body 3
Theoretical Mechanics
(6.1)
Consider the fixed frame 𝑂1𝑥1𝑦1 and a mobile frame 𝑂𝑥𝑦 linked to a section S of
the rigid body.The plane motion have 3 degree of fredom:
The coordinates of S in the fixed
frame 𝑂1𝑥1𝑦1𝑧1 are:
and the equation of motion
for S are given by:
Lecture 5. Kinematics of the rigid body 4
Theoretical Mechanics
(6.3)
(6.2)
Consider
In 𝑂1𝑥1𝑦1: 𝑀 𝑥1, 𝑦1
In 𝑂𝑥𝑦 : 𝑀 𝑥, 𝑦 Then, Ԧ𝑟1 = Ԧ𝑟0+ Ԧ𝑟
we deduce that the motion of section S is a
instantaneous roto - translation. It means
that the finite motion of the rigid body
between 𝑡𝐴 and 𝑡𝐵 is a succesion of
instantanous roto - translations.
Taking into account the velocity
Lecture 5. Kinematics of the rigid body 5
Theoretical Mechanics
(6.4)
Moreover,
Instantaneous Centre (IC) of Zero Velocity
We will show that if at the moment 𝑡, 𝜔(t) ≠ 0 (i.e. the rigid body performs a
rotation) there is a point 𝐼 ∈ 𝑆 with the property Ԧ𝑣𝐼 = 0.Thus, we have
(6.5)
Using (5.6) it is possible to determine the point 𝐼.This point lies on the Instantaneous Axis of Zero Velocity(or instantaneous axis
of rotation), and intersection of this axis with the plane of motion is known as
Instantaneous Centre of Zero Velocity.
The body may be considered to be in pure rotation in a circular path about the
Instantaneous Axis of Zero Velocity.
Locating the instantaneous centre of zero velocity is important to simplify the
solution of many problems involving rigid body rotations.
Lecture 5. Kinematics of the rigid body 6
Theoretical Mechanics
Locating the Instantaneous Center (IC)Assume that the directions of the absolute velocities of
any points A and B on rigid body are known and are not
parallel.
If C is a point about which A has an absolute circular
motion at the instant considered, C must lie on the
normal to vA through A.
Similarly, point B can also have an absolute circular
motion about C at the instant considered.
Intersection of two perpendiculars will give the absolute
center of rotation at the instant considered.
Point C is the instantaneous center of zero velocity and
may lie on or off the body
Instantaneous center of zero velocity need not be a
fixed point in the body or a fixed point in the plane.
Lecture 5. Kinematics of the rigid body 7
Theoretical Mechanics
Locating the Instantaneous Center (IC)
If the velocities of two points are parallel and the line
joining the points is perpendicular to the direction of the
velocity, the instantaneous center is located by direct
proportion.
As the parallel velocities become equal in magnitude,
the Instantaneous Centre moves farther away from the
body and approaches infinity in the limit that body stops
rotating and translates only.
(Kaustubh Dasgupta http://www.iitg.ac.in/kd/homepage/index.html)
Lecture 5. Kinematics of the rigid body 8
Theoretical Mechanics
If 𝑀 ∈ 𝑆 the velocity is given by
In order to find the position of the IC one multiply eq. (6.5) with 𝜔 :
(6.6)
(6.5)
Thus, we obtain:
(6.7)
(6.8)
If magnitude of velocity at one of the points on the rigid body
under general plane motion is known (vA) the angular velocity
of the body ω and linear velocity of every point in the body can
be easily obtained using: ω= vA/ rA
Velocity of B is vB= rBω= (rB/rA) vA.
Lecture 5. Kinematics of the rigid body 9
Theoretical Mechanics
Once the IC is located, the direction of the instantaneous velocity of every point in
the body is readily found since it must be perpendicular to the radial line joining the
point in question with IC.
Velocity of all such points: v= rω
Note that the spokes are somewhat visible
near IC, whereas at the top of the wheel
they become blurred.
Also note how the direction of velocity
varies at different point on the rim of the
wheel.
Lecture 5. Kinematics of the rigid body 10
Theoretical Mechanics
Locus of the ICs in space (in 𝑂1𝑥1𝑦1) is known as space centrode (baza)
Locus of the ICs on the body (in 𝑂1𝑥1𝑦1) is known as body centrode (rulanta)
At the instant considered, the two curves are tangent at C
The body centrode rolls on the space centrode during the motion of the body.
Lecture 5. Kinematics of the rigid body 11
Theoretical Mechanics
Further, we deduce the equations of the space and body centrode:
(6.9)
But and using (6.9) we have
(6.10)
Lecture 5. Kinematics of the rigid body 12
Theoretical Mechanics
On the other hand,
Thus
(6.11)
Eqs. (6.11) are the equations of the space centrode (bazei) in the fixed frame of
reference 𝑂1𝑥1𝑦1.
Lecture 5. Kinematics of the rigid body 13
Theoretical Mechanics
Using (6.11) and taking into account
we have
(6.12)
Eqs. (6.12) are the equations of the body centrode (rulantei) in the mobile
frame of reference 𝑂 𝑥 𝑦.
Lecture 5. Kinematics of the rigid body 14
Theoretical Mechanics
Lecture 5. Kinematics of the rigid body 15
Theoretical Mechanics
Lecture 5. Kinematics of the rigid body 16
Theoretical Mechanics
Lecture 5. Kinematics of the rigid body 17
Theoretical Mechanics
Lecture 5. Kinematics of the rigid body 18
Theoretical Mechanics
Lecture 5. Kinematics of the rigid body 19
Theoretical Mechanics
Consider a particle M moving in a refernce frame Oxyz. The reference frame Oxyz
is moving relative to a fixed frame O1x1y1z1.
7. Relative motion of the particle
Lecture 5. Kinematics of the rigid body 20
Theoretical Mechanics
Consider a particle M moving in a reference frame Oxyz and the reference frame
Oxyz is moving relative to a fixed frame O1x1y1z1.
The motion in the frame 𝑂1𝑥1𝑦1𝑧1is called absolute motion.
The equations of the absolute
motion are:
(7.1)
The motion in the mobile frame 𝑂𝑥𝑦𝑧 is called relative motion.
The equations of the relative motion are:
(7.2)
21
Theoretical Mechanics
Consider the position vector of M:
Velocity:
(7.3)
Lecture 5. Kinematics of the rigid body
(7.4)
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Theoretical Mechanics
The term
is the relative velocity of point M (the velocity in 𝑂𝑥𝑦𝑧 supposed fixed)
(7.5)
Lecture 5. Kinematics of the rigid body
The term
is the transport velocity of point M (the velocity of the point M considered
fixed in the moving frame of reference 𝑂𝑥𝑦𝑧). The vector 𝜔 is the angular
velocity of 𝑂𝑥𝑦𝑧 about 𝑂.
(7.6)
Using (7.4) -(7.6) we can give the first Coriolis formula:
(7.7)
The absolute velocity is the sum of the relative and transport velocity.
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Theoretical Mechanics
Acceleration
Lecture 5. Kinematics of the rigid body
(7.8)
=
=
Here
is the absolute acceleration,
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Theoretical Mechanics
Lecture 5. Kinematics of the rigid body
The second Coriolis formula is given by:
is the relative acceleration,
is the transport acceleration, and
is the Coriolis acceleration.
(7.9)
(7.10)
(7.11)
(7.12)
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Theoretical Mechanics
Lecture 5. Kinematics of the rigid body
Corriolis effect
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Theoretical Mechanics
Lecture 5. Kinematics of the rigid body
Example
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Theoretical Mechanics
Lecture 5. Kinematics of the rigid body
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Theoretical Mechanics
Lecture 5. Kinematics of the rigid body
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Lecture 5. Kinematics of the rigid body
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Lecture 5. Kinematics of the rigid body
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