Dinamika - 1 Kinematika Benda Tegar

Dinamika  TKM 2302 / 3 SKS Dr.. Indraswari Dr Indraswar i Kusumaningtyas Kusumani ngtyas 27-10-2014

1 Kinematika Benda Tegar 

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 particles forming 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.

1 Kinematika Benda Tegar 

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 particles forming 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.

 Types of Rigid Body Motion 1. Translation   Any

straight straight line line inside the body body keeps the same direction during the motion.

  All

the particles forming the body move along parallel parallel paths.

Rectilinear Rectiline ar Translation ranslatio n

Curvilinear Curvilin ear Translation ranslatio n

 Types of Rigid Body Motion 2. Rotation about a fixed axis 

The particles forming the rigid body move in parallel planes along circles centered on the same fixed axis.



If this axis of rotation  intersects the rigid body, the particles located on the axis have zero velocity and zero acceleration. See the difference…

Curvilinear Translation

Rotation

- Parallel Circles

- Concentric circles

 Types of Rigid Body Motion 3. General plane motion  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.

 A

B

 A’ 

B’ 

 Types of Rigid Body Motion 4. Motion about a fixed point 

The three-dimensional motion of a rigid body attached at a fixed point O 

Not a plane motion

 Any motion of a rigid body which does not fall in any of the categories above is referred to as a general motion .

 3D

 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/A  is constant in direction and magnitude. Its derivative with respect to time is zero.

 B r 

 A 

0

 Translation  B Differentiating with respect to time, r 

 r   A  r   B  A





 B  with r 

 A 

0

 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 keep the same direction during the entire motion.



Curvilinear translation: velocity and acceleration change in direction as well as in magnitude at every instant.

Rotation about a Fixed Axis Consider the rotation of a rigid body about a fixed axis AA’. The angle θ is the angular coordinate of the position of P.

The length ∆s of the arc described by P when the body rotates through an angle ∆θ is

Hence, the magnitude of the velocity is

Rotation about a Fixed Axis The vector of the velocity is

Where  Acceleration

is the angular velocity

Rotation about a Fixed Axis The vector of the acceleration is

Where

is the angular acceleration

Rotation of a Representative Slab The rotation of a rigid body about a fixed axis can be defined by the motion of a representative slab in a reference plane perpendicular to the axis of rotation.

Rotation of a Representative Slab Position of line 2 with respect to line 1:

1

2

θ2 = θ1 + β β = constant in a rigid body

β θ2 θ1

Hence, derivative with respect to time: ω1 = ω2  and

α1 = α2

 All lines on a rigid body in its plane of motion have the same angular position, angular velocity and angular acceleration.

Equations for Rotation

Contoh 1

Contoh 1

Contoh 1

Contoh 1

Contoh 1

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

General Plane Motion

General Plane Motion Consider the general motion of a representative slab which displaces particles A 1 and B1 to A 2 and B2. The motion can be divided into two parts: 

Translation from A 1-B1 to A 2-B’ 1



Rotation of B’ 1 to B2 about A 2

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.

Absolute and Relative Velocity  Any plane motion can be replaced by a translation of an arbitrary reference point A and a simultaneous rotation about A.

Absolute and Relative Velocity

  Assuming

the velocity v A  of end A is known, determine the velocity vB of end B and the angular velocity ω of the rod in terms of v A , l, and θ.



The direction of vB and vB/A  are known. Complete the velocity diagram to find the magnitude.

Absolute and Relative Velocity

Absolute and Relative Velocity 

Selecting point B as the reference point and solving for v A  and ω leads to an equivalent velocity triangle



v A/B has the same magnitude but opposite sense of vB/A . The sense of the relative velocity is dependent on the choice of reference point.

Absolute and Relative Velocity   Angular

velocity ω of the rod in its rotation about B is the same as in its rotation about A.

  Angular

velocity is not dependent on the choice of reference point.

Contoh 2

Contoh 2

Contoh 2

Contoh 2

Contoh 3

Contoh 3

Contoh 3