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EFFECT OF BLADE DESIGN on SHEARING FORCES. i  » -' ;
0F HYDRAULIC SHEAR .ch RODWOOD -  
Thesis for the Degree of M. S. . °
MICHIGAN STATE UNIVERSITY
MANUEL L. LECCA a
1969

 

 

 

   
 
      

LIBRARY ""

Michigan State
University

 

ABSTRACT

EFFECT OF BLADE DESIGN ON SHEARING FORCES
OF HYDRAULIC SHEAR FOR ROD WOOD

By

Manuel L. Lecca R.

Felling of rod trees with machines using hydraulic
powered blade has become a standard practice in forest
mechnization. These machines have proven to be fast,
flexible and safe. However further work must be done to
improve their performance. The magnitude of the shearing
force is dependent on the blade characteristic and the
geometry of the link through which the force is trans-
mitted.

A model two blade felling machine was designed and
constructed. The blades were hydraulically operated and
moved toward each other in the same plane.

The objectives of this study were to investigate:

a. the blade thickness, edge angle and shape

that will be the best for this model.
b. the force required and blade design for a
prototype, which is expected to work with 18

in. tree diameter.

Manuel L. Lecca R.

Three blade thicknesses, each one with three dif-
ferent edge angles were made of steel plate. They were
tested with fresh wood samples 3.125, H.125, and 5.5 in.
in diameter.

The 0.50 in. blade had the poorest performance.
However, after changing its shape by grooving both sides,
a reduction of 30% of the force required was obtained.

The 0.187 in. blade had the best efficiency and it
was selected for model test.

The model tests were done with 0.187 in. blade having
30 degrees edge angle by using wood samples of “.5 in.
diameter. Twenty—two tests were made which gave the data
for computing the Pi terms. From the Pi terms the force
:required for the prototype was calculated to be A2,000 lb.
with. parameters:

t = 0.75 in. grooved blade

18 in tree diameter

d

a:

30 degree blade edge angle

However, during the model tests, the design condi-
tsion equations were not satisfied completely due to the
capacity of the model hydraulic system and mechanical

properties of the samples.

 
  
 

Approve-

I”
Approved M

Depar ment Chairman

 

EFFECT OF BLADE DESIGN ON SHEARING FORCES

OF HYDRAULIC SHEAR FOR ROD WOOD

By

Manuel L. Lecca R.

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Agricultural Engineering

1969

41

f?
fiflid ACKNOWLEDGMENTS

The author wishes to express his sincere thanks
to the following:

Professor Chester J. Mackson, the author's major
professor, who from the beginning fo the study provided
continuing encouragement and guidance.

Professor Rolland T. Hinkle of Mechanical
Engineering, with whom the author consulted about the
construction and testing problems of the blades.

Professor H. F. McColly of Agricultural Engineer-
ing, who provided with great enthusiasm the necessary
equipments.

Mr. Glenn Shiffer and Mr. Larry Foster of Agri-
cultural Engineering Research Laboratory, with whom the
author discuSsed the construction of the model.

Mr. Mounir Marcos, graduate student of Agricultural

Engineering for assisting with the photographs.

11

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . .
LIST OF TABLES . . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . . . .
Chapter
I. INTRODUCTION AND OBJECTIVES . . . . .
II. REVIEW OF LITERATURE . .- . . . . .

III.

Common Shearing Blade Machines for
Felling Rod Trees . .
Research with Hydraulic Blades in
Laboratory . . . . . .
Equipment Commonly Used in Laboratory
Sample Used for Laboratory Testing .
Some Results Obtained in Laboraty
Tests . . . . . . . . . . .
2.5.1 Influence of the Blade on the
Force Required . . . . . .
2.5.2 Influence of Wood Specimen on
Force Required . . . . . .

2.5.3 Influence of Machine Operation

on Force . . . . . .
2.5.A Coefficient of Friction . . .
2.5.5 Damage to Wood Fiber . . . .

THEORETICAL EQUATION FOR SHEARING WOOD . .

3.1

'3.2

3.3

Edge Force . . . . . . . .

Wedging Force . . . . . . .

Friction Force .

3. 3.1 For Blade with Sides Parallel

3. 3. 2 For Blade with Faces Sloping
Backward . . . . . . . .

iii

Page

11

vi

Chapter

IV. SIMILITUDE AND MODELING .

A.l Characteristic of the Prototype

4.2 Modeling Variables . .
4.3 Prediction Equations

V. MODEL CHARACTERISTICS . .
VI. PROCEDURE FOR THE TEST . . .

6.1 Blade Test .
6.1.1 Moisture Content

6.1.2 Maximum Bending Stress

6.1.3 Stem Weight Density

6.2 Model Test . . . . .

6.3 Blade Speed . . . . .

VII. DATA FROM THE TEST . . . .

7.1 Blade . . . . . . .
7.2 Model . . . . .

VIII. RESULTS AND DISCUSSION .

8.1 Blades . . . . . . .
8.2 Model Test . . . . .

IX. 'SUMMARY . . . . . . . .
x. CONCLUSIONS . . . . . . .

XI. SUGGESTIONS FOR FUTURE STUDY .

REFERENCES . . . . . . . . .

iv

Page

LIST OF TABLES

Table Page
Moisture content of samples . . . . . . 43
2. Maximum bending stress . . . . . . . 44
3. Weight density of sample . . . . . . . 45
4. Maximum bending stress . . . . . .. . 46
5. Force, P, required for the .5 in. blade . . I 50
6. Force required (P) for .250 in. blade . . 50
7. Force, P, required for .187 in. blade . . 51
8. P/Qv and t/d values, corresponding to n =

670 and “4 = 30 . . . . . . . 3. . 51

9. P/Qv and od2/Qv values corresponding to flu =
30 and N2 = .042 . . . . _. . . . . 53

10. P/Qv and a values corresponding to n = 670
and v2 = .042 . . . . . . . 3. . . 55

Figure

.2.‘

CDNChU'l

10.
ll.

l2.

l3.
14.
15.
16.

17.
18.
18a.

LIST OF FIGURES

Monohydraulic blade type.
Guillotine blade type.
Sawtooth blade type . . .

Diagram of hydraulic system used in
laboratory testing machine .

Motor, gear case and couplings.
Blades used in laboratory . . . .

Blade shapes used in laboratory . .

Hydraulic machine used in laboratory for '

cutting wood . . . . . . . .

Blade and frame used in determination of
coefficient of friction . . . .

Fiber deflection during cutting process.
Theoretical beam on elastic foundation

Theoretical deflection of a fiber beam

on elastic foundation . . . . .
Theoretical axial pulling stress . .
Wedge force components . . . . .

Friction force components; flat blade .

Friction force components; backward
sloped blade . . . . . . . .

Sketch of top view of the prototype . .
Sketch of blade driver linkage. . . .

Sketch of "latch" driver linkage . .

vi

Page

10

12

14
21

21

22
22
28
28

28
34
35
35

Figure Page

19. Top view of the testing model.' .7 . . 39
20. Hydraulic power generator . . . . . 41
21. Measuring equipments. . . . . . . 41
22. 0.5 in. grooved blade . . . . . . 49
23. Tree diameters vs. Force required for

0.5 in. blade . . . . . . . . 54
24. Tree diameters vs. Force required for

0.250 in. blade. . . . . . . . 56
25. Tree diameters vs. Force required for

0.187 in. blade. . . . . . . . 58
26. Dimensionless value P/Qv vs. t/d. . . 62
27. Dimensionless value P/Qv vs. a . . . 63
28. Dimensionless value P/QV vs. od2/Qv. . 64

vii

CHAPTER I

INTRODUCTION AND OBJECTIVES

In the last ten years, tremendous effort has been
devoted to forest mechanization, especially in harvesting.
The use of the new hydraulic shear blade, when properly
designed, has proven to be a faster, flexible and safer
way of felling rod trees.

However there are still some factors which reduce
the efficiency of this device. The force required to
operate the blades is large and is a function of the
shape, size and cutting direction. Other factors such as
the wood itself and environments also affect the magnitude
of applied force.

There are very few experimental results which relate
the force required with different parameters. Generally
one blade was used against an anvil, with no axial
pressure on the sample. These laboratory processes are
not similar to those actually occurring in the field, but
still these laboratory results help to evaluate different
parameters for the purpose of modeling or designing of
prototypes.

A careful study of these works indicates that the
applied force and damage to the fiber can be reduced with

l

proper blade shape and size and by applying the pushing

force in the right direction.

The

a.

objectives of this study are to:

Study selected variables affecting the
shearing force for design purposes.

Design a prototype shearing mechanism with
two shearing blades traveling toward each
other without overlapping.

Design, construct and test a model built to
one-half scale.

Test three blade thicknesses (.50, .25, and
.187 in.) each one with three different edge
angle (30, 45 and 60 degrees).

Determine the force required and blade design
for a prototype through data obtained from the

model tests.

CHAPTER II

REVIEW OF THE LITERATURE

2.1 Common Shearing Blade Machines for
Felling Rbd Trees

Of all machines used in forest mechanization the
tree felling device has become most important. The
device shears trees with diameters up to 40 inches. The‘
blades, which differ in design are activated by a most
sephisticated hydraulic system; some Of them are combine
operating machines with auxiliary systems which fell and
bunch simultaneously in one operation by using only
blades.

The monoblade type is shown in Figure l; the blade
rotates on a pin. In this way the blade severes the
fibers by compressing the tree stem against an anvil
which does not move.

The guillotine type is shown in Figure 2. This
device usually works in horizontal position and its
blade is longitudinally stroked by one hydraulic cylinder.
In front of the blade there is the "latch" mechanism
which embraces the stem and holds it while the blade is
forced through the wood. The principal feature of this
type is that it allows the severing of the trees without

the transfer of force to the tractor.
3

 

 

 

Latch open Latch closed

Figure 2.--Guillotine blade type.

 

5

The sawtooth type is shown in Figure 3. This
device is similar to the monoblade type except that the
cutting edge is provided of exaggerative teeth.

2.2' Research with Hydraulic Blades
- Tn Laborat 911

So far the reSearch had been concerned with the
study of the influence of factors that affect the force
required in severing "square wood samples" perpendicular
to the fiber and also to study damage resulting from the
cutting process. It has been shown under laboratory con-
ditions that the force required as well as the damage can
be reduced to a minimum by a correct selection of the
blade.

The factors which affect the force required can be
grouped into three types: those inherent to the blade
itself, those inherent to the material to be cut, and
those due to the machine operation. The blade factors
are the most important for design purposes.

The blade factors are:. section shape, thickness,
edge angle, wedge and quality of blade surface.

The principal factors inherent to the material to
be out are: moisture content, wood temperature, density,
sample size and part of the trunk where the sample was
obtained.

The factors important to the machine operation are:

direction of the force applied and blade speed.

Review of one of the laboratory methods nOWr
available in literature will give a better understanding
of principal features and comparative results of equip-
ment, methods etc.

2.3 Equipment CommoniyUsed in
' Laboratony_ ‘”

 

Carl Kample (1965), Swedish mechanical engineer
designed and used successfully a testing machine consisting
of:

a. Power generator

b.. Hydraulic system

c. Shearing blades

d. Force metering

A Volkswagon industrial engine of 30 hp. and Speed
of 1500-300 RPM was selected as a power source. The _
output engine shaft was coupled to a single dry disc
clutch to allow the operator to start or stop the
mechanism. Between the clutch and the hydraulic pump
there was a gear case with a 1.5:1 ratio and an elastic
coupling to reduce wear to the pump due to mis-alignment
of the shaft. A sketch of the coupling mechanism is
given in Figure 5.

The hydraulic system consisted of a pump, two
check valves, two pressure relieve valves, a reservoir,
one hydraulic cylinder and a four way control valve.

The pump was a Vicer's double vane type with a capacity

Figure 4.--Diagram of hydraulic system used in laboratory
testing machine.

   

 

 

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Figure 5.--Motor, gear case and couplings.

8

of approximately 75 1ts./min. at 1990 psi. The four way
control valve when in its rest position guided the flow
back to the tank, and in its extreme position the oil
was forced into the working cylinder. The hydraulic
cylinder had a 3.75 in. internal diameter, 11.8 in.
stroke and 2000 psi. maximum wOrking pressure. A sketch
of the hydraulic system is given in Figure 4.

The blades were made of heat-tempered steel. The
testing was done with six blades, each one having distinct

shape and size. 'The principal feature of them being:

Blade No. l 2 3 4 5 6
Edge angle, '
degree. 45 45 30 45 60 45

Thickness, in. .397 .198 .475 .345 .258 .358/.159
The blades were attached to the machine by two
bolts, one of them to the blade—holder and the other to the
hydraulic cylinder. The form of the blade-holder was such
that it permitted the blade to wOrk only on bending
moment. 5
These blades are shown in Figures 6 and 7. It should
be noted that blades numbered 3 and 4 are a little
eccentric.
Besides, a special blade was used to determine the
coefficient of friction between wood and metal. It was
18 in. long, 6 in. wide and 0.75 in. in thickness. The

edge angle was 45 degrees.

 

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Figure 6.--Blades used in Laboratory.

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Figure 7.-—Blade shapes used in laboratory.

11

The force was measured axially by two strain
gauges; which were glued to the connecting rod between
the piston and blade frame.

The output of the strain gauges was amplified and
recorded by a Visicorder. The output from this recorder
was in the form of a curve giving force per unit time.
Since the knife and the recorder speed is an invariant,
the curve also gives force per unit distance. A sketch
of the testing machine is given in Figure 8.

In order to find the coefficient of friction, the
same testing machine was used with a special blade. The
blade used for friction was mounted directly on the
hydraulic cylinder. The anvil consisted of two steel
bars with a narrow spacing between them to allow the
knife to pass through. In this way the friction force
could be measured when the blade edge passed the anvil.
The axial force was provided by a 25 Ton hydraulic Jack,
and any variation of this force was measured with a
manometer. Knowing the axial force, and the pushing
force exerted by the hydraulic cylinder, the coefficient

of friction is obtained through the following relation-

ship:
P : 2Pf
Pf = uPa
P = 2uPa
and. u = P

2Pa (l)

l2

 

 

 

 

 

 

 

Figure 8.--Hydraulic machine used in laboratory for cutting
wood.

13

Where:
P = knife pushing force
Pa = axial force on the Jack

Pf

friction force

u = coefficient of friction

A sketch of the friction testing machine is given

in Figure 9.

2.4 Sample Used for Laboratory
Testing

The samples used were of rectangular section,

 

because tests with a 10 in. outside diameter circular
rod produced considerable damage due to compression of the
rod smaple against the anvil.

The common rectangular sample section were as

follows:
a. 2.75 x 2.75 in.
b. 2.75 x 3.95 in.
c. 3.95 x 3.95 in.
I d. 3.95 x 5.55 in.

e. 3.94 x 3.17 in.

f. 5.55 x 5.55 in.

These samples were taken from tree sections in such
a way that some of them were sapwood and others heartwood.
The samples b, c, and d were also used in friction tests.
They were 27.5 in. lenth to avoid the risk of buckling and

an uneven distribution of strain due to axial force.

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Figure 9.--Blade and frame used in determination of coeffi-
cient of friction.

15

The damage to the wood fiber produced by compressing
the rod sample against the anvil suggests that shearing
under field practice wOuld be very well performed with a
machine which has two blades acting parallel and against
each other.

2.5 Some Resultsgbtained in
Laborato£y_Tests

 

The results for rectangular sections previously

mentioned are given below:

2.5.1 Influence Of the Blade
on the‘ orce Required

2.5.la Effect of the blade shape.--Blades l, 4

 

and 6 represent three different shapes with the same edge
angle of 45°. If one elects 100 units of force for

blade 1, for comparison the result can be written as

follows:
Blade 1 | 100 units
Blade 4 ‘ 75 units
Blade 6 75 units

As a conclusion one could say that the cut-away
shape (blade 4) and the sloping backward shape (blade
6) tend to reduce the force. In practice, however, these
shapes are not common, perhaps due to manufacturing
difficulties.

2.5.lb Effect of blade thickness.--Blades l and 2

represent two different thicknesses with the same edge

1

16

I

angle of 45°. With blade 1 as the basis (100 units of
force), the results to describe the effect of blade
thickness can be reported as:
Blade 1 100 units
Blade 2 50 units
It is evident that the thinner blade required less
force, however in practice the thickness is limited by
bending stress and deflection.
2.5.1c Effect of blade edge angle.--Blades 3, 4
and 5 represent three different blade edge angles, but
all of them have the same shape. Taking the force
required for blade 4 as 100 units we can write the result ‘

of effect on edge angle as:

Blade 3 104 units
Blade 4 100 units
Blade 5 140 units

The results do not show any significant difference
between blade 3 and 4, but the blade 5 shows much higher
force. This suggests that the edge angle should remain
in the range 30°-45°.

2.5.2 Influence of Wood Specime
Mquired -

2.5.2a Effect of specimen knots.--The cut over a

knot showed about 50% greater force than a cut over a

sample without knots .

1?

2.5.2b Effect of specific gravity.--There was a
significant difference between samples of different
specific gravity. Samples with lower values required
lower force. It was estimated that for each per cent
increase of specific gravity, the force required
increases by about l-2 per cent.'

2.5.20 Effect of moisture content and temperature.-—
Moisture content and temperature are factors related to
each other. To show this, samples of sapwood and heart-
wood were exposed to three different temperatures of
15, -5 and -15 degrees Centigrade.‘ The results showed
that sapwood required more force under frozen than in
warm storage. In any case sapwood required more force_
than heartwood which is contrary to what is expected,
however this can be eXplained by the fact that sapwood
has higher visco-elasticity.

2.5.3 Influence of Machine
Operation oanorce

Only blade speed and pushing force direction are
considered here. As far as blade speed is concerned,
laboratory results indicated that speeds between the
:range of l.58-8.7 in. per second had no significant
effect on force.

The importance of direction of pushing force need
riot be over emphasized. It is obvious that the force

shall be mximum if the direction of the pushing force

18

radial to the stem, i.e. two blades travelling parallel and
against each other radially to the stem may be more

efficient.

2.5.4 Coefficient of Friction
From the results of many experiments under different
moisture content, speed and temperature conditiOns, the
following conclusiOns can be drawn:
a. The coefficient of friction is lower in radial
'shearing than in tangential. Felling trees
involves only radial shearing. A
b. Friction seems to decrease with increases in
speed expecially for the frozen sapwood.
c. Friction seems to increaSe with speed for
heartwood.

d. Knots increase friction considerably.

2.5.5 Damage to Wood Fiber

To determine a qualitative damage to the fiber,
thin samples were taken close to the cut surface. The
samples were studied with a microscope under polarized
light. Cracks and cruShing were observed. Less damage
resulted when using the thinnest blade at 30-45 degree

of edge angle.

CHAPTER III

THEORETICAL EQUATION FOR SHEARING WOOD

Theoretically, the total force required in shearing
can be divided into three components.
Edge force or cutting force
Wedging force or separating force
Friction force
The edge force or cutting force is defined as the
force required to cut the fibers by a knife edge not
ideally sharp.
The wedging force is defined as the force required
for a knife wedge triangle to penetrate into the wood.
The friction force is defined as the force cauSed
by friction between wood and the flat blade surface,
except the wedge portion of the knife.
The theory involved in theSe forces is discussed

below.

3.1 Edge Force

 

This force originates when the blade edge, not
ideally sharp, penetrates perpendicular to the wood

grain. As a result of this the fibers deflect and absorb

l9

20

energy before they are cut off. Figure 10 shows
the fiber deflection during cutting.

Mckenzie (1961) formulated a theory concerned with
edge penetration perpendicular to the grain which allows
the theoretical calculation of the edge force. This
theory is based on the fact that the edge force is due
only to the bending moment under which the fiber deflects
in front of the blade edge and breaks at certain deflection.
However, it was observed that besides that, there is
another component: the pulling force along the axis of
the fiber, which is the result of the shear stress
between the fiber sides. The shear stress between fibers
causes separation from each other when the stresses are
considerable. Mckenzie's theory neglects this force,
because he was concerned with cutting small chips where
axial forces would not really occur.

The cutting force is obtained by assuming a beam on
an elastic foundation with a concentrated load at one end
without axial force. Sketches of the assumed beams are
shown in Figures 11 and 12.

‘ In figure 11:
P1 is the cutting force acting on a layer or
ring of thickness hl; this layer is considered
as a beam and the next layers below as the

elastic foundation.

21

 

Figure 10.--Fiber deflection during cutting process.

 

77

la.

Figure 11.4-Theoretical beam on elastic foundation.

22

 

 

 

 

 

 

'////

  

///7( of ”62% -
6(00/147 ”0.0an

Figure l2.--Theoreticaldeflection of a fiber beam on
elastic foundation.

I
«k

T

W\\\\\\\W\\\ \\ \\\
Tn.

  

1’. «9+

 

 

 

 

Figure l3.--Theoretical axial pulling stress.

23

2 , . . . , Pn are the wedging forces acting
at the end of the cut layers.

t is the thickness of the blade

a is the edge angle

d , . . . , dn are distances from the line of

P

symmetry of the blade to P1 , . . . , n

forces.

According to Hetenyi (1946), the bending moment in

Mckenzie's beam is:

—lx
M=-P'e _.S_1.QTL_X_ (2)
l
where P' = Pl/2
1
I.
A = /__K__ ; foundation modulus (3)
RBI

K is a constant.
E is the beam elastic modulus

I is the moment of inertia of beam
section.
According to Biot (1937), the constant K for a

rectangular beam section is:

/3

. .1
K = .645 9'15“?) (4)

24

where

b is the beam width
h is the beam height

E' is the foundation elastic modulus.
Equations (3) and (4) give:

1.18 E' 1/3

A = h E )

The location of the maximum bending moment is

obtained as follows:

setting g; (M) = 0
sin Ax = cox Ax
tan x = l
_ n
xmax. — HI (5)

By replacing Equation (5) in Equation (2), the

maximum bending moment becomes:

-N/4 ' .
M _ -P' e sin N/4 (6)

Relating Equation (6) to the maximum stress

(a max.) that the beam could stand in compression,

Mmax. h

omax. 3 2I

o = ——— P

max. bh 1 E'

which yields:

. _ ' 1/3
Pl - .455 bh Omax. (E') (7)

And finally, Equation (7) gives the edge force:

5.41/3 (8)

P = .910 bh Omax. (E

1

Mckenzie relates the values E' and E with the
elastic modulus perpendicular and longitudinal to the
grain. E is equal to the longitudinal elastic modulus
and E' is equal to 3.92 times the transverse elastic
modulus. _

When it is necessary to consider the pulling
force along the fiber axis, as shown in Figure 13, Kempe
(1965) gives the following Equation:

a
P; = I 2 S sin ¢ dx (9)
0

where

H
‘6-
H9-

9-
u
-e
[.4
aux

26

Pg is the edge force necessary to overcome the
axial fiber tension in lb. per unit blade length.

Therefore the force in terms of lbs. only is:

sin ——— dx. (10)

where E is the effective blade length.

Adding up the components, the edge force finally

becomes:

P = P + P (11)

It should be noticed that the edge force defined
here is not completely analogous to the force experienced
in actual practice. In realtiy the rings are circles of
finite diameter whereas for theoretical consideration the

rings were assumed to have infinite diameter.

3.2 Wedging Force

 

This force is the separating force which permits
penetration of the blade through the wood. It originated
from the normal pressure against the sIOpe and the
friction as the blade wedge moves. In order to find the
theoretical value of the wedging force (Pw)’ the following
assumptions are made: the pressure is uniform along the

slope and the wedge section is symetric. The forces

27

acting on one side of the symetric wedge are shown in

Figure 14. From this figure the following relation Can

be derived:
Qh = Qv tan a/2
- QV
R — cos a 2
= =——1:1—@-Y-_
F uR cos “/2

From conditions of equilibrium:
Pw/2 - F cos c=/2 - Qh = 0
which yields:
Pw = 2 Qv (u + tan m/2) (12)

where Qv is the vertical force acting on the slope and
in practice represents the weight of the tree supported
by the horizontal projection of the wedge slope, u is the

coefficient of friction and c is the wedge angle.

3:2, Friction Force
These resisting forces act on both sides of the
blade as it moves through the wood. In order to over-
come these (Ff) forces, it is necessary to push the
blade. Acting and reacting forces are shown in Figure

15. Assuming that pressure on both sides are uniform,

the following relations are obtained:

 

28

 

 

Figure l4.--Wedge force components.

   
  
   

    
  
    

 

 

I l
eeGRURIfiIRITCk
\\\‘\

  

Figure 15.--Friction force components; flat blade.

 

 

Figure l6.-—Friction force components; backward sloped blade

29

3.3.1 For Blade with Sides Parallel

From equilibrium condition:

Pf - 2u Qf = 0

which yields

Pr = 2 u Qf (13)

where Qf is the force perpendicular to the blade face,
which in practice represents part of the tree weight
supported by the blade, and u is the coefficient of

friction.

3.3.2 For Blade with Faces
S

-—‘

loping Backward

 

This case is shown in Figure 16. By assuming a
uniform pressure on the blade the following relations

are obtained:

th = va tan 8
R 3 va

cos B
Ff - u R

From equilibrium conditions:

Pf s Ff cos B - th
Which yield:

Pr = 2va (u - tan 8) _- (l4)

30

where B is called the clearance angle. The clearance
angle reduces the friction force as can be seen by
comparing Pf in cases mentioned before.

In summary the theoretical equation for the force

required by the blade is:

Pt = P1 + PS + Pw + Pf (15)

CHAPTER IV

SIMILITUDE AND MODELING

It is desired to determine the behavior of a proto-
type tyrough a model which is designed according to a
predetermined scale.

Modeling is a laboratory technique which gives the
possibility of simulating, as nearly as possible, any
process as it happens in practice. The only inconvenience
here is that scaling the material to be Out is difficult.
Distortion between the model and the prototype is to be
expected when working with biological material. The aim
of this work is to measure the force required by the

model and to predict that required by the prototype.

4.1 Characteristic of the Prototype
The assumed prototype is a double blade hydraulic
shearing machine for felling rod wood to beused in
mechanical harvesting of trees. It could be mounted on
any standard forest tractor.
The prototype is designed to cut trees up to 18 in.
in diameter but using geometrical similitude it is

possible to design devices for trees of any diameter.

31

32

Under actual conditions directional felling is
done by a hydraulic boom which is not a part of this
study.

The machine's Operating parts are two blades_and
two latches. The latches embrace the tree and guide the
blades while cutting. The blades work against each
other without overlapping. This orientation of the
blades theoretically give the following advantages:

a. The force required could be less than that

for a single blade, since with the absence
of the anvil, the value E' from the theoretical
equation is expected to be small.

b. Since the blade frame, tree and tractor con—

stitute a static frame, the influence of PS.

in Equation (10) and P in Equation (8) may

1
be reduced with two moving blades. Ps and P1
are proportional to the blade displacement.

c. The damage to the fiber may be reduced since

there is no anvil compression effect.)

The blades are activated by two hydraulic cylinders
through links with a mechanical advantage of approximately‘
1.2. The hydraulic power normally is supplied by the
same tractor on which the machine is mounted, provided

‘that the pump pressure is in the order of 2000 psi. and

the minimum capacity is 20 GPM.

33

In Figure 18 the linkage that transmits the
hydraulic power from the cylinder to the blade is shown;
any linkage could be designed for this purpose provided
that it occupies the same space and offers a mechanical
advantage greater or equal to one. Figures 17, 18 and 18a

show sketches of the prototype and latch mechanism.

4.2 Modeling Variables

 

The variables to be studies Can be separated into
three groups as follows:
a. Those relating to the material to be cut
d diameter of the rod wood
Qv axial load which represents the tree
weight
a maximum compressive stress that the wood
can stand in bending. .
b. Those relating to the model design
a blade edge angle
t blade thickness
c. Those relating to the machine oepration
P total force required to Operate the blade.
For the experiment, moisture cOntent, coefficient
of friction, weight density, blade speed, temperature and
modulus ratio were considered constant.
The functional equation which involves the dependent

and independent variables may be represented as follows:

34

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lei—ti“

 

 

 

 

 

 

 

 

 

 

 

Figure l7.--Sketch of top view of the prototype.

35

 

73 6/¢ a/e

 

 

 

——‘______ From Ayn/aa—
//.e 6//Ino’cf.

Figure l8.——Sketch of blade driver linkage.

Figure 18a.——Sketch of "latch" driver linkage.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

36
f (P, t, d, Qv, O, a) = 0 , 3 (15)
which yields:
P = F(t, d, Qv, O, a) (16)

By applying the Buckingham Pi theorem from,
Dimensional Analysis the representative dimensionless

equation was obtained:

N1 = F(n2, v3, “4) (17)
where:

N1 = P/Qv

Tr2 = t/d

n3 = Od2/QV

"4 = a
then:

P/Qv = F(t/d, od2/Qv, a) (18)

4.3 Prediction Equations
If we assume that no variable which affects the
force required for the blade has been omitted, and no

distortion is expected, the prediction equation will be:
P v = P v ' 1
/Q p/Q p . .( 9)

(and the design condition equations are:

t/d =

Od2/QV

where the subscript

37
tp/dp
2
op dp/Qvp

OID

p represents the prototype.

(20)

(21)

(22)

CHAPTER V

MODEL CHARACTERISTICS

It is important to notice that if we assume that the
arc displacement (Figure 18), formed by the link between
the hydraulic cylinder and the blade in the prototype
is negligible and the structure of the model is rigid
enough, i.e., with very small deformation during loading,
then the model is exactly geometrically similar to the
prototype and the prediction equation can be applied.

The model was constructed to a 1/2 scale in the
Agricultural Engineering Research Laboratory at Michigan
State University. It mainly consisted of a steel frame
supported by four legs; and two blades which slide toward
each other in a severing operation (Figure 19). The
blades are supported and guided by four square bars
which can be adjusted to guide the blade motion in the
plane of the cylinder.

In order to exert an axial force Qv when Operating
the model, a sliding table was built in the middle and
over the blade frame (Figure 19).

The hydraulic system consisted of a pump, a power
generator, two hydraulic cylinders, a relief valve,

a reservoir and a control valve. The pump was a single

38

39

In

. ,. ‘ $2511. V.?1! .
. trv‘lvw 1er 9:1:

. .94.’ WK
.i 1134*} (‘20

bi

 

'1'
, 1

Figure l9.--Top View of the testing model.

Model blade
2. Sliding table support

1.

4O

vane Vickers, model V 111 E; the pump capacity was 4.3
GPM and a working pressure of 1000 psi. The power
generator was a 5 hp., 1170 RPM, electric motOr. The
hydraulic cylinders had a 3 in. bore and an 8 in. stroke
with a recommended working pressure of 1500 psi.= Figure
20 shows the driver and hydraulic pump.

The force measuring device consisted of two
strain gauges, one Bush amplifier (model BL 520) and a
single Bush oscillograph (Figure 21). The two strain
gauges were fixed axially on the blade rod driver; the
strain gauge output was recorded by the oscillograph
which was calibrated at 10 micro-inch per inch of strain
per chart line.

For the first tests, which were concerned with.
determining the blade to fit the model, three blade
thicknesses were chosen: 0.500 in., 0.250 in. and 0.187
in. These blades were not heat-tempered. Edge angles

of 60°, 45° and 30° were used.

41

Electric motor

b s e
a a
s a
i3

.a‘

.I'-§‘ t‘-

° ts.

lng equipmen

Bush amplifier

1. Bush oscillograph

2

l. Vickers hydraulic pump

2.

{$100.
Vt.le.oo

Figure 20.--Hydraulic power generator.

 

. e.
.\ V t. l O .- e I
t o 0 n. ,
- ntfiicjuvo Kiwi-twitter)“ I ,-

Figure 21.—~Measur

CHAPTER VI

PROCEDURE FOR THE TEST

6.1 Blade Test

 

Initially the log specimens were 28 in. long and
had diameters of 4.0, 6.0 and 8.0 in. After a pre-
liminary test the diameters were changed to 3.125, 4.125,
and 5.5 in. for reasons explained in the next chapter.

The samples were obtained from tree stems and were
very heterogeneous. They showed an irregular distribu-
tion of knots and twisted growth especially in the 3.125
in. and 4.125 in. diameter samples.

The physical prOperties of wood samples were deter-
mined in the Agricultural Engineering Physical Properties

Laboratory. The methods and results are given below.

6.1.1 Moisture Content

 

Small pieces of wood were taken from each sample.
These sample pieces, after being weighed, were placed
into an oven at 100°C. for 72 hours and the dry weight
recorded. The moisture content w.b. was determined by
using the following equation (23) and results are shown

in Table l.

43

= Wet Weight - Dry Weight
M.C. Wet Weight x 100 (23)

 

TABLE l.--Moisture content of samples.

 

 

 

Stem diameter Moisture Content
in. % w.b.
3.125 36.2
4.125 35.1
5.500 ' 34.0

 

6.1.2 Maximum Bending Stress

The maximum bending stress is also called failure
stress, and is the max stress used in equation (7).

To determine the bending stress, small beams 8 in.
long and .375 in. x .250 in. cross section were prepared.
Each beam was tested as a cantilever. The load was
applied gradually until rupture, and the stress cOmputed

by using the bending equation (24).

Maximum bending movement (2”)
Beam Section Modules

 

Max. Stress =

Most of these small beams showed a great plastic
deformation before breaking. The results are given in
Table 2.

For comparison it might be important to remark

that Kempe (1963) used Mckenzie's equation NO. (8) with a

44

TABLE 2.--Maximum bending stress.

 

Cross Section

Stem diameter Stress

 

in. *b Sq“ in. **h lb./sq. in
3.125 .375 .250 7,700
4.125 .275 .250 8,500
5.500 .375 .250 9,700

 

*
b is width of the beam

*
h is the depth of the beam.

maximum stress of 1,100 psi. Also Scofield (1963)
reported that wooden beam failure stress is not constant;
it decreases with beam depth. Therefore the great dif-
ference between values showed in Table 2 and Kempe's
value could be due to beam size, mositure content and

tree variety.

6.1.3 Stem Weight Density
Weight density was determined for each sample by

applying Mohsenin's method. This method consists of
weighing the sample, in air and water, on a platform
scale.

In order to prevent the absorption of moisture by
the wood sample, a very thin layer of grease was applied
to the sample before weighing in water for density deter-
mination. The weight of the grease applied was considered

insignificant compared to the weight of the sample.

 

45

The results are given in Table 3. The densities

were determined by applying the following equation.

Weight = Weight in air x Weight density of water gg (25)
Density Weight of displaced water in.3

TABLE 3.--Weight density of sample.

 

Stem diameter Weight Density

 

in. lb./cu. in.
3.125 .0232
4.125 .0262
5.500 .0259

 

The weight density values in Table 3 show that,
there is not too much difference among samples. Therefore
we may consider that for a tree diameter range of 1.125-
5.500 in., weight density is constant. Moisture content
can also be assumed constant because of the small

variation shown in Table 1.

6.2 Model Test
Stems from the same specimen as those used for
blades were obtained for testing the model. From these
stems, samples with diameters of 3.5, 4.0, and 4.5 in.
in diameter were prepared. Each sample was 26 in.

long and was cut twice at 8 in. intervals.

46

Here, again the samples appeared to be heterogeneous;
. irregular distribution of knots at the level of cut was
observed, but twisted growth was much smaller.

Since the samples were prepared from the same
specimen as those for the blade test, moisture content
and weight density were not determined and were assumed
to be similar to those given in Tables 1 and 2. The
maximum stress was determined from the equation (24) for

the model test. The results are given in Table 4.‘

TABLE 4.--Maximum bending stress.

 

Cross Section

in. x in. Stress

Stem diameter

 

in. lb./sq. in.
*b fiflh

3.250 .375 .250 8,900

4.000 .375 .312 5,700

4.500 .375 .312' 5,300

 

* .
letter b stands for width and

**
h for depth.

It is evident from Tables 2 and 4 that the failure
stress is not constant. This is in agreement with the
results obtained by Scofield. Therefore, selecting a
failure stress value as a model system parameter is a

compromise.

47

6.3 Blade Speed

 

The capacity of the model hydraulic system was a
limiting factor because the no-load speed of the

equipment was limited to 2.0-2.5 i.p.s.

CHAPTER VII

DATA FROM THE TEST

7.1 Blade

 

Twenty-nine experiments were performed,to test the
three blades on the model frame. The force, P, required
for each cut was obtained from the oscillograph chart.

The blades were made of steel plate. The blade of
.500 in. thickness, after working with a flat shape, was
changed to a grooved one. Figure 22 shows its
characteristics.

The force, F, required for 0.5 in. blade with edge
angles of 30, 45, and 60 degrees are tabulated in Table
5.

The force, P, required for the 0.25 in. blade is
tabulated in Table 6 and for the 0.187 in. blade in
Table 7. These forces are recorded for edge angles of
30, 45 and 60 degrees.

In Table 7 each sample was completely cut. .

The values of P in Tables 5, 6 and 7, show that the
performance of the .187 in. blade with a 30 degree edge.
angle was best. Therefore this blade was selected for

the model test. However edge wear was observed especially

48

49

 

Figure 22.--0.5 in. grooved blade.

50

TABLE 5.--Force, P, required for the .5 in. blade.

 

 

 

 

 

 

Edge angle Stem diameter Force
degrees in. lb.
60 3.125 7,800
60 4.125 8,100
60 5.500 9,300
Normal 45 3.125 6,000'
45 4.125 7,200
Blade 45 5.500 7,200
30 3.125 7,500
30 4.125 7,800
30 5.500 7,500
Grooved 30 4.125 6,000*
Blade 30 5.500 7,800
e
The cut was complete.
TABLE 6.--Force required (P) for .250 in. blade.
Edge angle Stem diameter Force
degrees in. lb.
60 3.125 5,100*
60 4.125 5,400*
60 5.500 7,800**
45 3.125 5,100“
45 4,125 7,200*
45 5.500 7,5oofiu
30 3.125 5,100*
30 4.125 6,900?
30 5.500 7,500*

 

*
The cut was complete.

as
The cut was not complete.

51

TABLE 7.--Force, P, required for .187 in. blade.

 

 

 

Edge angle 'Stem diameter ' Force
degrees in. lb.
60 3.125 5,400
60 4.125 6,000
60 5.500 7,800
45 3.125 4,200
45 4.125 5,100
45 5.500 7,800
30 3.125 4,500
30 4.125 5,100
30 5.500 7,500

 

when great knots were present; but this defect could be
avoided by using heat-tempered steel.

As indicated in the tables the hydraulic system had
a limiting working capacity of 7,500-7,800 lbs. Since
this force was not great enough to handle the selected
diameter, smaller diameter samples had to be used.

The model tests were done with .187 in. blade and
a max. stem diameter of 4.500 in. The effect of this
change on the accuracy of the Prediction Equation will be

discussed in the next chapter.

7.2 Model
Twenty-two tests were run on the test model to get

the necessary values to compute the Pi term.

52

The data for dimensionless terms, P/Qv and t/d,

which represent n1 and M2 respectively, were obtained by

varying d and Qv, and keeping the other dimensionless

terms constant, i.e., N which equals od2/Qv and "4 which

3

equals 0°.

The value of P/Qv and t/d are tabulated in Table 8.

TABLE 8.--P/Qv and t/d values, corresponding to n3 = 670
and flu = 30.

 

 

Stem diameter P Qv Dimensionless values
in. lb. lb. t/d P/Qv
3.250 4,800 85 .058 ,56.5
3.250 4,500 85 .058 53.0
4.000 7,500 212 .047 35.4
4.000 5,700 212 .047 27.0
4.500 5,700 172 .042 33.2
4.500 5,100 172 .042 30.0

 

A comparison of values of P in Table 4 for 4 inch diameter
stem.indicate that the mechanical properties of wood were
not constant.

The data for dimensionless terms P/Qv and odz/Qv,
i.e., N1 and n3 were obtained by varying Qv while keeping

t/d and « constant. The results are tabulated in Table 9.

53

TABLE 9.--P/Qv and Od2/QV values corresponding to Nu =

30 and n2 8 .042.

 

Stem diameter Dimensionless values

 

in. 1b. lb. P/Qv od2/Qv
4.500 7,500 110 68.0 ‘1150
4.500 7,800 110 71.0 1150
4.500 6,900 115 60.0 1000
4.500 6,750 115 59.0 1000
4.500 5.750 145 39.0 800
4.500 5,900 145 41.0 800
4.500 7,200 220 32.75 520
4.500 6,300 330 19.10 350
4.500 6,900 330 21.00 350

 

The data for dimensionless terms P/Qv and a was
obtained by changing c and keeping t/d and cd2/Qv
constant. These values were tabulated in Table 10.

Table 10 shows that the 30 degrees edge angle is
the best for the model because the blade with such an
angle requires the lowest P.

The performance of normal and grooved blades were
;p1otted in Figure 23. A look at these curves motivates
(nae to make a comparison between a normal and a grooved
txlade. Taking for instance a 4 in. diameter and assuming

trust the normal blade had made a complete cut, we arrive

54

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6.3.3.620 no;

 

 

 

 

 

 

 

m a. o m w n N
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nooom
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253 use 3.. a 90 n 0 Z0
70000
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race 0.

‘QVPtHnbOI 03003

55

 

TABLE 10.--P/Qv and a values corresponding to n3 = 670
and n2 = .042.
Stem diameter P Qv Dimensionless values

 

in. 1b. lb. P/Qv a
4.500 5,600 172 32.7 30
4.500 4,700 172 27.5 30
4.500 5.700 172 33.2 45
4.500 7,500 172 43.6 45
4.500 6,900 172 40.0 60
4.500 7,500 172 43.6 60

 

at the conclusion that grooves had reduced the force by
30 per cent. These curves also show the maximum force
for 5.5 in. dia. which the normal blade in no case had
cut completely, whereas the grooved blade did.

The .25 in. blade showed better performance than_
the 0.5 in. blade. It cut the samples completely except
in two cases, that was at 45 and 60 degrees edge angles
with a 5.5 in. tree diameter. This was attributed to
sample knots. The values of P versus d for this blade
are plotted in Figure 24. The unusual behavior of the
required force for a 4.125 diameter sample can only be
attributed to the variation of the mechanical properties

of wood.

56

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.:_.:.oEc.v of...

 

 

 

 

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57

The 0.187 in. blade showed the best performance.
It out each sample neatly and completely with all edge
angles used. The values P versus d for this blade were
plotted in Figure 25; these curves show that for a 30'
degree edge angle, less force was required, especially
with the 5.5 in. tree diameter. This blade was selected
for the model due to its performance. 8

Finally the curves in Figures 24 and 25 show that
the capacity of the hydraulic testing system was 7,500-
7,800 lb. which was not enough for working with a 9 in.

tree diameter as was planned.

 

58

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CHAPTER VIII

RESULTS AND DISCUSSION

8.1 Blades

 

Part of the objective of this work, as it was
mentioned before, was to determine the best blade.

The normal .5 in. blade gave a poor performance.
It did not cut the sample completely and in some cases
split the wood longitudinally. In one case this blade
did make a neat cut.

The poor performance of the normal blade was
attributed to the combination of thickness and edge
angle. Therefore it was decided to change the shape of
the blade in such a way that its stiffness was not
affected. The 30 degrees edge angle was used because
with these the acting edge remained sharp.

The blade shape was changed by making grooves
parallel to the cutting direction, on both sides of the
blade. The grooves dimensions were: 0.75 in. width,-

4.0 in. long and 0.125 in. depth.

8.2 (Model Test

 

The aim of these tests was to obtain the required

data to compute the dimensionless terms in the Prediction

59

60

Equation. Given the Prediction Equation, the prototype
behavior will be easily predicted.

According to Larson (1968), both model and proto—
type systems have to have geometric and dynamic simili-
tude; geometric similitude means that the ratio of any
two lengths of one system should be equal to the ratio
in the second system; and dynamic similitude means that
the ratio of any two forces in one system should be
numerically equal to the corresponding force ratio in the
second system.

However, when testing the model, the requirements
mentioned above were not completely satisfied. The
geometric condition was not satisfied because of the low
capacity of the hydraulic system of the model. The
tests were done with 4.5 in. diameter instead of 9.0 in.
This caused a distortion in the Prediction Equation. The
dynamic condition also was not satisfied because the
sample, being a biological material behaved as a heter-
ogeneous and anisotropic system. Mechanical properties
of wood are not constant even in the same diameter
sample. Therefore the measured force (F) was somewhat
greater when knots were encountered. Since Qv is not
affected by knots, the ratio P/Qv for the model was some-
what larger; i.e. distorted.

Distortion is common when working with models,

especially when parts of the system do not have constant

61

properties. However, distortion can be evaluated in
order to correct the Prediction Equation by running
similar models of two or more scales.

The P1 terms for the model were plotted in Figures
26, 27 and 28. Because of the distortion explained above,

and n will be:

the relationship between‘wl 1p

"1p = 5N1

where the subscript p stands for the prototype and 6 is
the factor required to correct for distortion. The value
of 6 could be equal to, less, or greater than one. Since
it is not within the scope of this work to evaluate 6,

it will be assumed to be equal to l and then:

1T =1T

1p 1

From Figure 26, the value of n1 = p-p/C.‘2v = 30, then the

force required for the prototype will be:

60,000 lb.

Ep

when:

Qv = 2000 lbs.

.75 in. blade thickness

tp
dp = 18 in. tree diameter

a = 30 degrees edge angle

62

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_ _

 

 

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65

The calculated value of Pp is for a normal blade;

however during the discussion on blade tests it was shown

that for the grooved blade, a 30% force reduction could

be obtained.

prototype the

when:

Therefore by using grooved blade in the

actual force should be:

42,000 lb.

0.75 in. grooved blade thickness
18 in. tree diameter

30 degrees edge angle

CHAPTER IX

SUMMARY

A model for shearing rod wood, used in forest
mechanization, with two blades traveling parallel and
toward each other without overlapping was built and
tested in the Research Laboratory of the Agricultural
Engineering Department at Michigan State University.

Three blade thicknesses, 0.5, 0.25, and 0.187 in.,
were tested with blade angles of 30, 45 and 60 degrees.
Wood samples of 3.125, 4.125 and 5.5 in. diameters were
used in the test.

Twenty—nine tests were made in order to determine the
best blade for the model. The 0.187 in. blade with 30
degrees edge angle, proved to be the best; therefore it
was selected to be used in the model. Although the 0.5
in. blade had the poorest performance, by changing its
shape it was possible to improve its performance.

Twenty-two tests were made with the model to
obtain the data for computing the Pi terms, which appear
in the Prediction Equation.

In solving the prediction equation for a prototype
shearing tree 18 in. in diameter the required force was

found to be 42,000 lbs. which is considered less than

66

67

values reported in literature. The prototype blade

must be 0.75 in. thick and of the grooved design.

 

CHAPTER X

CONCLUSIONS

1. Hardness as blade material property is very
important when designing shears for felling rod wood.
Hardness is related to the time that a blade can hOld
its sharpness. Therefore this property should have the
highest possible value when designing.

2. Blade thickness, edge angle and shape have a
great influence in the force required for shearing. Of
the edge angles tested (30, 45 and 60 degrees) the 30
degree edge angle gave the best results.

3. The blade thickness effect can be reduced by.
30 per cent by grooving both blade sides. This treatment
did not seem to affect the blade stiffness for the
blades tested.

4. Wood knots increase the force required for
shearing and reduced the blade sharpness.

5. The terms in the design condition equation
could not be precisely satisfied because of the varying
prOperties of the biological materials used. These
properties include such parameters as weight density,

failure stress and so on.

68

CHAPTER XI

SUGGESTIONS FOR FUTURE STUDY

1. Further tests should be made with blade
thicknesses and edge angles. Blade materials with high
stiffness and an improved hardness must be used.

2. Model frames should be designed in such a way
that material deformation can be kept at a minimum. This
is particularly important for devices in which the blades
do not overlap.

3. Further works should be conducted with models
of different scales to determine the distortion factor
and thereby enable correction of the prediction equation.

4. Finally, based on the corrected prediction
equation, a prototype should be built and tested in actual

field conditions.

69

REFERENCES

70

REFERENCES

Bending of an infinite beam

(1937).
Transactions of the

l. Biot, M.A.
on an elastic foundation.
ASME 1959: A 1-7.

2. Hetenyi, M. (1946).
University of Michigan Press,

pp. 19-22.

Beam on elastic foundation.
Ann Arbor.

(1964). The force and damage to the

3. Kempe, Carl.
fiber structure involved in hydraulic shearing of
Thesis for degree of Licentiate in

round wood.

Mechanical Technology of Wood; Royal Swedish

College of Technology, Stockholm. (unpublished)
'Dimensional analysis

4. Langhaar, Henry L. (1957).
and theory of model. John Wiley and Sons Inc.

165 pp-
5. Larson. L. M. (1968). Predicting draft force
using model moldboard plow in agricultural soil.
Transactions ASAE 1968. Vol. 11, No. 5, p. 665.
6. Mckenzie, W. (1960). Fundamental aspect of the
wood cutting process. Forest Product Journal

7. Mohsenin, N. N. (1968).
plant and animal materiaIs.

State University, pp. 73-76.
Similitude in engineering.

Physical properties of
The Pennsylvania

 

 

 

8. Murphy, Glenn. (1959).
The Ronald Press Company, New York, 297 pp.

9. Scofield, W. F. (1963). Modern timber engineering.
Southern Pine Association, New Orleans, pp. 1-17,

Simulation and modeling

10. Young, D. F. (1968).
technics. Transactions ASAE 1968, Vol. 11, No. 4,

pp' 590-59“-

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