Dinamika - 1 Kinematika Benda Tegar

download Dinamika - 1 Kinematika Benda Tegar

of 36

Transcript of Dinamika - 1 Kinematika Benda Tegar

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    1/36

    27-10-2014

    Jurusan Teknik Mesin dan Industri, Fakultas Teknik, Universitas Gadjah Mada

    DinamikaTKM 2302 / 3 SKS

    Dr. Indraswari Kusumaningtyas

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    2/36

    2DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    3/36

    3DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    4/36

    4DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    5/36

    5DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    6/36

    6DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    7/367DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    8/36

    8DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Translation

    Differentiating with respect to time, 0with ABABAB rrrr

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    9/36

    9DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    10/36

    10DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    11/36

    11DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    The vectorof the velocity is

    Where is the angular velocity

    Acceleration

    Rotation about a Fixed Axis

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    12/36

    12DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Rotation about a Fixed Axis

    The vector of the acceleration is

    Where is the angular acceleration

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    13/36

    13DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    14/36

    14DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    12

    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

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    15/36

    15DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Equations for Rotation

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    16/36

    16DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 1

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    17/36

    17DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 1

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    18/36

    18DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 1

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    19/36

    19DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 1

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    20/36

    20DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 1

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    21/36

    21DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    22/36

    22DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    General Plane Motion

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    23/36

    23DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    24/36

    24DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Absolute and Relative Velocity

    Any plane motion can be replaced by a translation of an arbitraryreference point A and a simultaneous rotation about A.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    25/36

    25DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    26/36

    26DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Absolute and Relative Velocity

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    27/36

    27DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    28/36

    28DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    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.

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    29/36

    29DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 2

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    30/36

    30DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 2

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    31/36

    31DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 2

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    32/36

    32DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 2

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    33/36

    33DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 3

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    34/36

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    35/36

    35DinamikaKinematika Benda TegarJurusan Teknik Mesin dan Industri FT UGM

    Contoh 3

  • 8/10/2019 Dinamika - 1 Kinematika Benda Tegar

    36/36

    36Di ik Ki tik B d TJ T k ik M i d I d t i FT UGM

    Contoh 3