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ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY

Dynamic Force Analysis

Dr. Sadettin KAPUCU

© 2007 Sadettin Kapucu

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IntroductionIntroduction

If the acceleration of moving links in a mechanism is running with considerable amount of linear and/or angular accelerations, inertia forces are generated and these inertia forces also must be overcome by the driving motor as an addition to the forces exerted by the external load or work the mechanism does. So, and are no longer applicable.Governing rules will be:

I and m are inertial (bodily) properties. At this stage we need to know the description of the inertial properties.

0F

0 τ

aFmατI

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CentroidCentroid

Centroid is the point where the resultant of distributed force system is assumed to act and generate the same dynamic results.

Centroid

Resultant Force

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Mass CentreMass Centre

If the distributed force is gravity force acting on each particle of mass, then concentrated force itself is called the “weight” and the centroid is called the “center of gravity” or “mass center”. Mass times distance, mr, is called as the first mass moment. This concept of first mass moment is normally used in deriving the center of mass of a system of particles or a rigid body. In figure a series of masses are located on a line. The center of mass or centroid is located at

x3

x1

x2

x-m1 m2 m3

G 321

332211

mmm

mxmxmxx

n

ii

n

iii

m

mxx

1

1

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Mass CentreMass Centre

The coordinates of the masses located on a plane can be obtained as:

m1

m2

m3

x-

y-

G321

332211

1

1

mmm

mxmxmx

m

mxx

n

ii

n

iii

321

332211

1

1

mmm

mymymy

m

myy

n

ii

n

iii

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Mass CentreMass CentreThis procedure can be extended to masses concentrated in a volume by simply writing an equation for the z axis. A more general form of mass center location for three dimensional body can be obtained by using integration instead of summation. The relations then become

n

ii

n

iii

m

mxx

1

1

n

ii

n

iii

m

myy

1

1

n

ii

n

iii

m

mzz

1

1

m

xdmx

m

ydmy

m

zdmz

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Mass Moment of InertiaMass Moment of Inertia

Mass moment of inertia is the name given to rotational inertia, the rotational inertia analog of mass for linear motion. It appears in the relationships for the dynamics of rotational motion. The Mass Moment of Inertia of a solid measures the solid's ability to resist changes in rotational speed about a specific axis.

r

o

o

m

The moment of inertia for a point mass is just the mass times the square of perpendicular distance to the rotation axis. The mass moment of inertia for a single particle is given as:

mrI 200

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Mass Moment of InertiaMass Moment of Inertia

When calculating the mass moment of inertia for a rigid body, one thinks of the body as a sum of particles, each having a mass of dm. Integration is used to sum the moment of inertia of each dm to get the mass moment of inertia of body. The equation for the mass moment of inertia of the rigid body is

dmrdmrI 2200

r

o

o

dm

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Mass Moment of InertiaMass Moment of InertiaThe integration over mass can be replaced by integration over volume, area, or length. For a fully three dimensional body using the density one can relate the element of mass to the element of volume.

x

y

dm

z

dmzydmrI xxx 222

dmzxdmrI yyy 222

dmyxdmrI zzz 222

rz

ry

rx

These three integrals are called the principle mass moment of inertia of the body. Another three similar integrals are

xydmII yxxy

yzdmII zyyz

xzdmII zxxz

} mass products of inertia of the body

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Radius of gyrationRadius of gyration

GICG, mICG, m

Original Body

Sometime in place of the mass moment of inertia the radius of gyration k is provided. The mass moment of inertia can be calculated from k using the relation

where m is the total mass of the body. One can interpret the radius of gyration as the distance from the axis that one could put a single particle of mass m equal to the mass of the rigid body and have this particle have the same mass moment of inertia as the original body.

2mkI

k

Model

k

Model

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Parallel-axis theoremParallel-axis theorem

The moment of inertia around any axis can be calculated from the moment of inertia around parallel axis which passes through the center of mass. The equation to calculate this is called the parallel axis theorem and is given as

2mdII CG G

ICG, mICG, m

o od

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Example 1Example 1

Solution: Mass moments of inertial of a point mass about an axis passing through itself

What are the mass moments of inertial of a point mass about an axis passing through itself and about an axis r distance away from it ?

o

o

m

002 dmdmrI oo

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Example 1Example 1

Solution: Mass moments of inertial of a point mass about an axis passing through itself

What are the mass moments of inertial of a point mass about an axis passing through itself and about an axis r distance away from it ?

o

o

m 002 dmdmrI oo

o

o

rm

x

x

mrmrmrII xxoo222 0

Mass moments of inertial of a point mass about an axis r distance away from it. Using parallel axis theorem

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Example 2Example 2

Solution:

Find the mass moment of inertia of a slender rod of length L (slender rod means that it has a length, and the remaining dimensions are negligible small) about an axis perpendicular to the rod and passing through its mass center.

L

L/2

o

o

L

L/2

o

o

dxx

2/

0

2*2L

oo dmxI

Let density of the material is in kg/m.

Then, infinitesimal mass dm=dx. Substituting this into above equation,

2/

0

2*2L

oo dxxI 12

2mL2/

3

|3

2 Lo

x3

232

L

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Example 3Example 3An uniform steel bar shown in the figure is used as an oscillating cam follower. Drive the equation of mass moment of inertia of the follower about an axis

through O. Use the density of steel =7800 kg/m3.

k=2kN/m

2 cm 25 cm 50 cm 25 cm

5 cm

y

xO

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Example 3Example 3

222 yxr

Solution:

x

y

z

xyr

dx

dzdy

t

w

l

dmrI zz2

dxdydzdm

dxdydzyxI zz )( 22

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Example 3Example 3Solution:

x

y

z

dx

t

w

l

x

y

z

dy

t

w

l

dyltydxwtxI zz ******** 22

2/

0

22/

0

2 ****2****2wl

dyyltdxxwt

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Example 3Example 3Solution:

*** twlm

1212*

12*

2222 wlm

wm

lmI zz

32

****232

****2

33w

lt

l

wtI zz 12****

12****

22 wwlt

llwtI zz

mass moment of inertia about O can be found by parallel axis theorem

48

47

412*

22

2

2222

0

wlm

lm

wlmdmII zz

mkgI .139.148

05.0*41*78.7

22

0