Theoretical Mechanics - Babeș-Bolyai...

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.



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.



(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:



(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



(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.



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.



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)



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.



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.



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.



Further, we deduce the equations of the space and body centrode:

(6.9)

But and using (6.9) we have

(6.10)



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.



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.



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



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)

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Consider the position vector of M:

Velocity:

(7.3)

Lecture 5. Kinematics of the rigid body

(7.4)

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The term

is the relative velocity of point M (the velocity in 𝑂𝑥𝑦𝑧 supposed fixed)

(7.5)


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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Acceleration


(7.8)

=

=

Here

is the absolute acceleration,

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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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Corriolis effect

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Example

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