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Catatan Perkuliahan ( Lecture Notes )

MCS220801 - KINEMATIKA DAN DINAMIKA

Topik 10: Dinamika Benda Kaku

GAYA dan PERCEPATAN

Pengelola dan pengajar : Wahyu Nirbito, Ir., MSME, DR.

Departemen Teknik Mesin

Fakultas Teknik

Universitas Indonesia

Depok, 2012

Topik 10: Dinamika Benda Kaku

GAYA dan PERCEPATAN

Chapter Objectives :

To introduce the methods used to determine the mass moment of inertia of a body.

To develop the planar kinetic equations of motion for a symmetric rigid body.

To discuss applications of these equations to bodies undergoing translation,

rotation about a fixed axis, and general plane motion.

Mass Moment of Inertia

The mass moment of inertia is a measure of the resistance of a body to a change in

its angular velocity.

It is defined by and will be different for each axis about which it is

computed.

For a body having axial symmetry, the integration is usually performed using disk

or shell elements.

Many bodies are composed of simple shapes.

If this is the case, then tabular values of I can be used, such as the ones given on

the inside back cover of this book.

To obtain the mass moment of inertia of a composite body about any specified

axis, the mass moment of inertia of each part is determined about the axis and the

results are added together.

Doing this often requires use of the parallel-axis theorem.

Handbooks may also report values of the radius of gyration k for the body.

If the body’s mass is known, then the mass moment of inertia is determined from

PROCEDURE FOR ANALYSIS :

• Consider only symmetric bodies having surfaces which are generated by revolving

a curve about an axis

• An example of such a body which is generated about the z axis

• Two types of differential elements can be chosen

Shell Element

• A shell element with height z, radius r = y, and thickness dy is chosen for

integration, volume, dV = (2πy)(z) dy

• Used to determine moment of inertia Iz of the body about the z axis, since the

entire element, due to its thinness, lies at the same perpendicular distance r = y

from the z axis

Disk Element

• A disk element with radius y, and thickness dz is chosen for integration, volume,

dV = πy2 dz

• Finite in the radial direction and its parts do not lie at the same radial distance r

from the z axis

• Determine moment of inertia of the element about the z axis and then integrate the

result

dmrI 2

2mdII G

2mkI

Planar Equations of Motion

The equations of motion define the translational, and rotational motion of a rigid

body.

In order to account for all of the terms in these equations, a free-body diagram

should always accompany their application, and for some problems, it may also be

convenient to draw the kinetic diagram.

Translation

Here since

Rectilinear translation Curvilinear translation


FBD

• Establish the x, y, a or n, t inertial coordinate system and draw the FBD to account

for all the external forces and couple moments that act on the body

• Direction and sense of the acceleration of the body’s mass center should be

established

• Identify the unknowns

• If the rotational equation of motion is to be used, consider kinetic diagram

Equations of Motion

• Apply the three equations of motion in accordance with the established sign

convention

• To simplify the analysis, the moment equation can be replace by a more general

equation about P where point P is usually located at the intersection of the lines of

action of as many unknown forces as possible

• If the body is in contact with a rough surface and slipping occurs, use the

frictional equation

0GI 0

0 0

GG

tGtyGy

nGnxGx

MM

amFamF

amFamF

Kinematics

• To determine velocity and position of the body

• For rectilinear translation with variable acceleration,

• For rectilinear translation with constant acceleration,

• For curvilinear translation,

Rotation About a Fixed Axis For fixed axis rotation, the kinetic vector m(aG)n produces no moment about the

axis of rotation, and so the rotational equation of motion reduces to a simplified

form about point O.

or


FBD

• Establish the x, y or n, t coordinate system and specify the direction or sense of

the accelerations and the angular acceleration of the body

• Draw the FBD to account for all the external forces and couple moments that act

on the body

• Compute the moments of inertia


• If the rotational equation of motion is used, draw the kinetic diagram for better

visualization

Equations of Motion


convention

• If the moments are summed about the center of mass, G, ∑MG = IGα since (maG)t

and (maG)n create no moment about G

Kinematics

• Use kinematics if a complete solution cannot be obtained strictly from the

equations of motion

• If angular acceleration is variable, use

α = dω/dt αdθ = ωdω ω = dθ/dt

If the angular acceleration is constant, use ω = ωO + αCt

θ = θO + ωO t + ½αCt2

ω2 = ω

2O + 2αC(θ – θO)

OGGG

GtGtyGy

GnGnnGn

IMIM

rmamFamF

rmamFamF

rm

rm

G

2

G

2

tGGGGtGGtGGnG

GOGOGG

OGGGOGGGOGG

GGGGGGGG

advvdsadtdvava

tatvss

ssavvtavv

dtdsvdvvdsadtdva

)()(/)(/)(

2

1)()(

])([2)()(

//

22

2

22

General Plane Motion If the body is constrained by its supports, then additional equations of kinematics

can be obtained by using to relate the accelerations of any two G to

another point A on the body.


FBD

• Establish the x, y coordinate system and draw the FBD for the body

• Specify the direction and sense of the acceleration of the mass center and the

angular acceleration of the body

• Compute the moment of inertia


• If it is decided that the rotational equation of motion is to be used, consider

drawing the kinetic diagram in order to help visualize the moments

Equations of Motion


convention

• When friction is present, there is the possibility for motion with no slipping or

tipping

Kinematics

• Use kinematics if a complete solution cannot be obtained strictly from the

equations of motion

• If the body’s motion is constrained due t its supports, additional equations may be

obtained by using aB = aA + aB/A, which relates the acceleration of any two points

A and B on the body

• When a wheel, disk, cylinder or ball rolls without slipping, then aG = αr

AGAG aaa /

PkPGG

yGyyGy

xGxxGx

MMIM

amFamF

amFamF

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