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27-10-2014
Jurusan Teknik Mesin dan Industri, Fakultas Teknik, Universitas Gadjah Mada
DinamikaTKM 2302 / 3 SKS
Dr. Indraswari Kusumaningtyas
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Kinematika Benda Tegar1 Rigid body is a system of particles which distances between
them do not change.
Kinematics of a rigid body discusses the relations between time
and the positions, velocities and accelerations of the particlesforming a rigid body.
A rigid body undergoes a plane motion if all parts of the body
moves in parallel planes. Then the motion plane is the plane that
has the centre of mass, and we can assume the body to be 2D.
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1. Translation
Types of Rigid Body Motion
Any straight line inside the body keeps the same direction during
the motion.
All the particles forming the body move along parallel paths.
Rectilinear Translation Curvilinear Translation
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The particles forming the rigid body move in parallel planes along
circles centered on the same fixed axis.
If this axis of rotationintersects the rigid body, the particles located
on the axis have zero velocity and zero acceleration.
2. Rotation about a fixed axis
Types of Rigid Body Motion
See the difference
Curvilinear Translation
- Parallel CirclesRotation
- Concentric circles
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All the particles of the body move in parallel planes.
Any plane motion which is neither a rotation nor a translation is
referred to as a general plane motion.
3. General plane motion
Types of Rigid Body Motion
A
B
A
B
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The three-dimensional motion of a rigid body attached at a fixed
point O
4. Motion about a fixed point
Types of Rigid Body Motion
Not a plane motion 3D
Any motion of a rigid body which does not fall in any of the
categories above is referred to as a general motion.
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Translation
Consider a rigid body in translation:
The direction of any straight line inside the body is constant, all
particles forming the body move in parallel lines.
For any two particles in the body:
Because A and B belong to the same rigid
body, then rB/Ais constant in direction and
magnitude. Its derivative with respect totime is zero.
0ABr
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Translation
Differentiating with respect to time, 0with ABABAB rrrr
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Translation
When a rigid body is in translation, all the points of the body have
the same velocity and the same acceleration at any given instant.
Rectilinear translation: all particles of the body move along
parallel straight lines, and their velocity and acceleration keepthe same direction during the entire motion.
Curvilinear translation: velocity and acceleration change in
direction as well as in magnitude at every instant.
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The length s of the arc described by P when
the body rotates through an angle is
Hence, the magnitudeof the velocity is
Rotation about a Fixed Axis
Consider the rotation of a rigid body about a fixed axis AA. Theangle is the angular coordinate of the position of P.
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The vectorof the velocity is
Where is the angular velocity
Acceleration
Rotation about a Fixed Axis
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Rotation about a Fixed Axis
The vector of the acceleration is
Where is the angular acceleration
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Rotation of a Representative Slab
The rotation of a rigid body about a fixedaxis can be defined by the motion of a
representative slab in a reference plane
perpendicular to the axis of rotation.
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1
2
Position of line 2 with respect to
line 1:
2= 1+
= constant in a rigid body
Hence, derivative with respect
to time:
1= 2 and 1= 2
All lines on a rigid body in its plane of motion have the sameangular position, angular velocity and angular acceleration.
Rotation of a Representative Slab
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Equations for Rotation
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Contoh 1
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General Plane Motion
General plane motion is neither a translation nor a rotation
General plane motion can be considered as the sum of a
translation and a rotation
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General Plane Motion
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General Plane Motion
Consider the general motion of a
representative slab which displaces
particles A1and B1to A2and B2.
The motion can be divided into twoparts:
Translation from A1-B1to A2-B1
Rotation of B1to B2about A2
Relative motion of a particle with respect to a moving frame:
To an observer moving with A but not rotating, particle B will
appear to describe an arc of circle centered at A.
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Absolute and Relative Velocity
Any plane motion can be replaced by a translation of an arbitraryreference point A and a simultaneous rotation about A.
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Absolute and Relative Velocity
Assuming the velocity vAof end A is known, determine the velocity vBof end B and the angular velocity of the rod in terms of vA, l, and .
The directionof vBand vB/Aare known. Complete the velocity diagram
to find the magnitude.
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Absolute and Relative Velocity
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Absolute and Relative Velocity
Selecting point B as the reference point and solving for vAand leads to an equivalent velocity triangle
vA/Bhas the same magnitude but opposite sense of vB/A. The sense
of the relative velocity is dependent on the choice of reference point.
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Absolute and Relative Velocity
Angular velocity of the rod in its rotation about B is the same asin its rotation about A.
Angular velocity is not dependent on the choice of reference point.
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Contoh 3