326

International Journal of Mechanical and Materials Engineering (IJMME), Vol.6 (2011), No.3, 326-337

DESIGN, FABRICATION AND ANALYSIS OF DIFFERENT FORM DESIGNS FOR A

CANTILEVER BEAM

S.S. Abuthakeera, V.A.S. Irudayaraj

a, P.V. Mohanram

a and G.M. Kumar

b

aDepartment of Mechanical Engineering, PSG College of Technology, Tamilnadu,India

bPark college of Engineering and Technology, Tamilnadu,India

Email: syathpsgtech@gmail.com

Received 23 February 2011, Accepted 23 August 2011

ABSTRACT

In the past, the design of CNC machine tools focused on

their functional aspects and was hard to acquire any

resonance with customers. Nowadays, despite the needs of

low-price, high-quality, higher cutting speeds and the fact

that machine tools operates at high acceleration and

deceleration with high quality at lower cost machine tools

and more customers request a good-looking machine. To

achieve high performance, the static stiffness of the machine

tool should be increased and the moving mass should be

reduced. The influence and improving methods for possible

structural modifications that could increase its structural

stiffness while enhancing other characteristics including

damping simultaneously. The different form designs were

studied for enhancing the static stiffness. Different form

designs of hollow box with stiffeners in different

orientations were designed to increase the stiffness with

constant mass and constant perimeter employing a

cantilever beam. Modal characteristics and the static

deflection characteristics of those different form designs

were analyzed experimentally. Numerical analysis was done

and results were validated with experiments.

Keywords: Form designs, Stiffness, Damping, Cantilever

Beam.

1. INTRODUCTION

Nowadays design for performance and usability is a longer

empower at a competitive edge. Thus, it is imperative to

design products by engaging customer’s emotions or

attention so as to differentiate among products (Khalid and

Helander, 2004). When designing products, customers’

affective needs must be considered (Jordan, 2003). Affect is

said to be a customer’s psychological response to the

perceptual design details of the product. The main challenge

for affective design is to grasp the customers’ affective

needs accurately and subsequently transform to design

products that match these needs. Presently, for high-price

industrial products such as cars and Computer Numerical

Control (CNC) machine tools, more and more customers are

interesting in both products’ functions and aesthetic

appearances. Studies on improvement in the performance of

machine tool structures are the major area of research. The

two important functional requirements for precision

machine tools are high structural stiffness and high

damping, which cannot be satisfied simultaneously if

conventional metallic materials are used for structures

because conventional high stiffness metals have low

damping and vice versa. Although the stiffness of machine

tool structures can be increased either by employing higher

stiffness materials or by increasing the sectional modulus of

structures (Suh and Lee, 2008). High static stiffness against

bending and torsion, good dynamic characteristics as

reflected by high natural frequency and high damping ratio,

ease in production, good long term dimensional stability,

reasonably low coefficient of expansion, low cost and low

material requirements are the basic properties of machine

tool structures that engineers look for designing and

fabricating. However, from user’s point of view, machine

tool vibration is an important factor because it adversely

affects the quality of a machined surface. To improve both

the static and dynamic performances, the machine tool

structures should have high static stiffness and damping.

Using either higher modulus material or more material in

the structure, the static stiffness of a machine tool may be

increased. But, it is difficult to increases the dynamic

stiffness of a machine tool with these methods and increase

in the static stiffness cannot increase its damping property.

Material distribution is important in the structural strength

and using material in required place can increase static

stiffness with less mass.

Faster cutting speeds can be facilitated only by structures

which have high stiffness and good damping characteristics.

The deformation of machine tool structures under cutting

forces and structural loads are responsible for the poor

quality of products and also aggravate the problem by

introducing noise and vibration. In many a situation, it is the

level of deformation and vibration that determines the upper

limit on the ability of the machine to produce components

with high precision. All these above said deleterious effects

greatly necessitate constant innovations and increasing

amount of research to keep them under check. Increasing

structural stiffness could help in avoiding such problems. To

increase the static stiffness and damping, different form

designs can be used. The high speed machining process

requests completely new demands for the mechanism of

such processing equipment, as due to the process, path

327

speeds exceeding 50m/min can be achieved. In this field,

potential capacities of manufacturing processes require a

dynamic behavior ten times higher than conventional

machine tools and increased accuracy. This can be solved

by the systematical evaluation of suitable machine

kinematics, by the application of linear direct drives as well

as by mass reduction of the axis through light weight

components of sheet metal. The requirements of high speed

machining and ways to improve the performance of

machine tool have been studied (Heisel and Gringel, 1996).

Hollow boxes possess an efficient shape for engineering

components due to their high inherent bending and torsional

rigidities in both directions. For example, box-section steel

girders are a familiar design of beams in bridges and other

civil engineering structures. Currently, industrial interest

exists in the use of tubes for the moving head of a milling

machine. The milling machine heads have the topology of

rectangular tubes with monolithic walls. The overall

compliance of the milling head is partly due to macroscopic

bending of the tube and partly due to the local compliance at

the supports on the guide-rails.

The overall compliance of the hollow, tubular beams are

decomposed additively into a global contribution due to

macroscopic bending and a local contribution associated

with transverse deflection of the walls of the hollow beam

adjacent to the central loading patch for box-section

sandwich beams of various cross sections in three-point

bending. The structural response was analyzed for beams of

square sections with various internal topologies: a solid

section, a foam-filled tube with monolithic walls, a hollow

tube with walls made from sandwich plates, and a hollow

tube with walls reinforced by internal stiffeners. Finite

element analysis was used to validate analytical models for

the overall stiffness of the tubes in three-point bending.

Minimum mass designs were obtained as a function of the

overall stiffness, and the relative merits of the competing

topologies are discussed (Mai et al., 2007). The weights of

optimal compression structures of several types were

studied and estimated. Minimum weights of columns having

solid square or circular cross sections were compared with

those of similar metal foam filled tubes in hollow tubes and

tubes whose walls are foam core sandwiches. Similarly the

minimum weights of wide sandwich compression panels

were studied, solid skin panels and panels in which the skins

and stiffeners are themselves metal foam core sandwiches

(Budiansky, 1999). The minimum deflection and weight

designs of laminated composite plates were studied. The

finite element method using plate theory was used in

conjunction with optimization routines in order to

obtain the optimal designs. Various boundary conditions

were considered and results were given for varying

aspect ratios and for different loading types.

Comparative results were presented for minimum weight

priority design as an alternative to minimum

deflection/minimum weight priority design to investigate

the effect of priority on the deflection and weight (

Walker et al., 1997) analysis for slender beams with a

varying cross-section under large non-linear elastic

deformation was conducted. A thickness variation function

was derived to achieve optimal - constant maximum

bending stress distribution along the beam for inclined end

load of arbitrary direction (Oore and Oore, 2009).

Internal stiffeners support the monolithic walls of the tube

and increase the local bending stiffness adjacent to the

supports. The shape, size, and orientation of stiffeners

decide the stiffness improvement. The compliance of the

machine tool is one of the prime factors for deciding the

static and dynamic characteristics and thus results the

quality and performance. The main objective of current

study is to increase the structural stiffness of beam by

designing and fabricating suitable forms, and experimentally

compare the static and dynamic performance.

2. METHODOLOGY/ANALYSIS/EXPERIMENTAL

SET-UP

2.1 Stiffness design

Stiffness is the capacity of a mechanical system to sustain

loads without excessive changes of its geometry i.e.

deformations. Stiffness is the load per unit deflection. It is

one of the most important design criteria for mechanical

components and systems. There are two major types of

strategies for design. One is design for strength and the

other is design for stiffness. Although strength is considered

as the most important design criterion, there are many cases

in which stresses in components and in connections are

significantly below the allowable levels and dimensions as

well as the performance characteristics of mechanical

systems and their components are determined by stiffness

requirements. Typical examples are aircraft wings, frames,

bed and columns of production machineries and

transmission systems in which stress levels are low

(Koenigsber and Tlusty, 1992). Recently great advances in

improving strength of the mechanical systems were

achieved. The main reason for advancement and

developments are high strength materials, better

understanding of failure mechanism, better method to

compute stress analysis thereby reduction of safety factor.

These advancements often result in reduction of cross

section of structural components. The strength of the

components can be improved by selection of metals and

alloying elements. Stiffness can be modified only by proper

selection of the component geometry-shape and size and its

interaction with other components. The importance of the

stiffness criterion is increasing due to:

• Increasing accuracy requirements due to increasing

speeds and efficiency of machines

• Increasing use of high strength materials resulting in the

reduced cross sections and accordingly in increasing

structural deformations

• Better analytical techniques result in smaller safety

factors which also result in reduced cross sections and

increasing deformations

328

• Increasing importance of dynamic characteristics of

machines since their increase speed and power with light

structures may result in intense resonances and in the

development of self-excited vibrations.

2.2 Influence and effect of different properties

Influence of machining system stiffness and damping

Stiffness effects on performance of mechanical systems are

due to influence of deformation in static and fatigues

strength, wear resistance, efficiency, accuracy, dynamic

stability and manufacturability. Elastic deformations of the

production system, machine tool –fixture-tool-machined

part, under cutting forces are responsible for a significant

fraction of the part inaccuracy. These deformations also

influence productivity of the machining system, either

directly by slowing the process of achieving the desired

geometry or indirectly by causing the self-excited chatter

vibrations. The stiffness enhancement is to reduce these

distortions. When they are repeatable, corrections that

would compensate for these errors can be commanded to

machine by its controller. However the highest accuracy is

still obtained when the error is small. Stiffness of the

production equipment influences not only affects its

accuracy and productivity also the energy efficiency,

dynamic loads and noise generations (Rivin, 1999).

2.3 Choice of Beam Form Designs

A series of beams of square cross-sections, in hollow form

with different stiffening arrangement were selected.

Topology A: a beam of hollow square cross-section, see

Figure 1a. the beam comprises an isotropic elastic solid with

Young’s modulus E, Poisson ratio m, and density ρ.

Topology B: a hollow box with single diagonal stiffener

along length with material parameters (E, m, ρ) see Figure

1b (Koenigsber et al., 1997).

Topology C: a hollow box with two diagonal stiffeners

along length with material constants (E, m, ρ) see Figure 1c

(Koenigsber et al., 1997).

Topology D: a hollow box with two stiffeners along length

with material constants (E, m, ρ) see Figure 1d (Oore and

Oore, 2009; Rivin, 1999).

Topology E: a hollow box with slots on sides with material

constants (E, m, ρ) see Figure 1e (Koenigsber et al., 1997).

Topology F: a hollow box with square honey comb internal

structure with material constants (E, m, ρ) see Figure 1g

(Blodgett, 1997).

Topology G: a hollow box with triangular type honey comb

internal structure with material constants (E, m, ρ) see

Figure 1h (Blodgett, 1997).

2.4 Design of form designs

Dimensions of beams were designed to meet following

constraints:

• The mass of all beams are same

• The material for all forms are same

• The perimeter of all sections are same

• The length of all beams are same

To find the improvement due to form design, the mass and

length were taken as constant and all beams were made up

of same material. The space occupied by a structure is a

constraint and hence the perimeter was taken as constant

and thickness of hollow box and stiffener location, thickness

and orientation were changed to get the form designs

(Makky and Ghalib, 1997). The beam was designed for

length of 240mm and outer height and width are 30mm and

30mm respectively. One end of the beam was fixed for a

length of 80mm which was one third of total length. The

form designs were modeled in ANSYS 11.0 and mesh was

created using 10 node tetrahedron higher order element. The

steel material of E=210GPa, Poisson ratio=0.34,

Density=7850kg/m3 was used. The load was applied at the

free end and the deflection of the beams was found out.

For hollow beam, the thickness was designed as 7mm for

1.2kg mass. For hollow box with single diagonal stiffener,

three configurations were arrived for mass of 1.2 kg

considering that the plates are available in steps of 0.5mm

commercially, Table 1.

a) Topology A b) Topology B c) Topology C

d) Topology D e) Topology E

f)Topology F

g) Topology G

Figure 1 Form Designs

Table1. Design of Hollow box with single diagonal stiffener

Choice

t1

(mm)

t2

(mm)

Deflecti

on (mm)

I 4 7 0.0403

II 5 5 0.0360

III 6 2.5 0.0371

329

Figure 2 Design of Hollow box with single diagonal

stiffener-Deflection of 3 configurations for 1.2 kg mass

Table 2 Dimensions of form designs

Topology

B

(mm)

H

(mm)

t1

(mm)

t2

(mm)

t3

(mm)

Mass

(kg)

30 30 7 - - 1.205

30 30 5 5 - 1.200

30 30 5 2.5 - 1.200

30 30 5 3.5 - 1.199

30 30 8 6 - 1.210

2.5 Fabrication of form design

The form designs were fabricated for the desired dimensions

using standard mild steel plates by conventional welding,

machining, and EDM wire cutting operations. The

fabricated form designs are shown in Figure 3a-g.

a) Hollow Box

b) Hollow box with single diagonal stiffener

c) Hollow Box with two Diagonal Stiffener

d) Hollow Box with plus Stiffeners

e) Hollow Boxes with slots on sides

f) Hollow Box with square honey comb pockets

g) Hollow Box with triangular honey comb pockets

Figure 3 Fabricated form designs

330

2.6 Experimental static stiffness analysis

The static analysis was done by applying the load at the end

(Dado and Al-Sadder, 2005). The strain gauge was fixed

each at the top and bottom of the beam at the free end. The

half bridge strain measurement circuit was developed to

measure strain, Figure4. Strain gauge of 2.1 gauge factor

and 120 ohm resistance was used to measure the strain in

the circuit.

Figure 4 Half bridge circuits

Two strain gauges from specimen and two dummy gauges

were connected as four resistances of half bridge circuit and

to the strain indicating device. Linear Variable Differential

Transducer was placed on the beam at the free end and

connected to meter to measure deflection directly. Strain

gauge was used to measure the strain with a least count of 1

micron and LVDT was used to measure the deflection with

a least count of 10microns. The strain measurement device

and deflection indicating meter are shown in Figure 5 a-b.

a) Ten channel strain indicator b) Digital Displacement

indicator

Figure 5 Measuring devices

Machine devise was used to fix the beam at one end

(Belendez and Wang, 2003). The vise was fixed by

hydraulic jack and loading frame. The loading arrangement

was developed as shown in Figure 7 to apply line type of

loading. The load was given at the end via hydraulic jack

and proving ring. When hydraulic jack was raised, the

weight of loading frame was applied on the beam and

applied load was read in proving ring. The load was applied

in steps of 0.32kN. The strain and deflection were measured

for different loads. Using same setup, the deflection

characteristics of all form designs were measured using

Strain Gauge and LVDT. The experimental setup and its

detailed view are shown in Figure 6-7. The deflection was

calculated from the strain gauge. The static deflection

results for various form designs are shown in Table 3.

Figure 6 Experimental setup

Figure 7 Detailed view of Experimental setup

All beams except beam with triangular pockets show less

deflection values for the applied load than the hollow

section. The stiffness values of all form designs are shown

in table 4. Hollow box with slots on sides shows the best

stiffness value of 10366.12 N/mm which is 18 percent

higher than hollow box section with constant mass. Hollow

box with two diagonal stiffeners shows 8.94 percent and

hollow box with single diagonal stiffener shows 6.9 percent

and Hollow box with plus stiffener shows 6 percent higher

stiffness than hollow box section.

Proving Ring

Fixed End

Fixed End

Strain Gauge

Fixed End

LVDT

Specimen

331

Table 3 Static Deflection results for various Form Designs

SECTION LOAD (N) 320 640 960 1280 1600

Hollow box

D

e

f

l

e

c

t

i

o

n

(mm)

By Strain Gauge 0.039 0.072 0.108 0.144 0.179

By LVDT 0.040 0.070 0.110 0.140 0.180

Hollow box with single

diagonal stiffener

By Strain Gauge 0.035 0.069 0.102 0.135 0.168

By LVDT 0.030 0.070 0.100 0.130 0.170

Hollow box with two

diagonal stiffeners

By Strain Gauge 0.035 0.067 0.101 0.131 0.164

By LVDT 0.030 0.070 0.100 0.130 0.170

Hollow box with plus

stiffeners

By Strain Gauge 0.034 0.070 0.104 0.138 0.170

By LVDT 0.040 0.070 0.110 0.140 0.170

Hollow box with slots

on sides

By Strain Gauge 0.032 0.063 0.092 0.118 0.154

By LVDT 0.030 0.060 0.090 0.130 0.150

Box with square honey

comb pockets

By Strain Gauge 0.038 0.071 0.101 0.135 0.169

By LVDT 0.030 0.070 0.10 0.130 0.160

Table 4 Static Stiffness of various Form designs

Figure 8& 9 Experimental setup

Section

Hollow

box

Hollow

box with

single

diagonal

Stiffener

Hollow

box with

two

diagonal

Stiffener

Hollow

box with

plus

stiffener

Hollow

box with

slots on

sides

Hollow box

with square

honey comb

pockets

Hollow box

with triangular

honey comb

pockets

Stiffness

(N/mm) 8762.07 9367.06 9545.43 9294.50 10366.12

9177.80 8865.51

LabVIEW DAQ

system

Experimental

Setup

Specimen Impact

Hammer

Machine Vise

Accelerometer

332

Table 5 Modal Analysis of Form Designs- First 5 Natural Frequencies in Hz

Section

Hollow

box

Hollow

box

With

single

diagonal

stiffener

Hollow box

With two

diagonal

stiffeners

Hollow

box with

plus

stiffener

Hollow box

with slots on

sides

Hollow box with

square honey

comb pockets

Hollow box with

triangular honey

comb pockets mode

NATURAL FREQUENCY in Hz

1 900 930 925 945 960 1050 980

2 1050 1120 1100 1150 1300 1200 1200

3 2400 2450 2550 2450 2750 2840 2530

4 3000 3150 3100 3250 3250 3450 3350

5 4035 4100 4150 4250 4700 4920 4850

The mode shapes and damping factors of the structures

were calculated using the results of the Fast Fourier

Transform of the vibration signals from the accelerometer.

By using the half-power band width method, the damping

factor of each beam was calculated.

= (f2-f1)/fr

Where (f2-f1) and fr represent the half power band width

and the corresponding natural frequency, respectively

(Jung Do Suh, Ju Ho Kim, Dai Gil Lee, 2008).

The form designs hollow box with square honey comb

pockets, hollow box with triangular honey comb pockets,

and hollow box with slots on sides stiffeners showed

higher improvement in natural frequencies than hollow

section. The other form designs showed little improvement

in natural frequency.

a) Time versus amplitude curve

b) Frequency versus amplitude curve

Figure 10 Sample Response plot taken from Lab VIEW

for Hollow Beam

2.8 Numerical static analysis of form design

The modeling of form designs were done in ANSYS. The

beam was fixed for 0.08m at one end. Loading was given

at the other end of the beam. The material properties of

cantilever beam were listed in the Table 6. Solid 187 was

used as element type for meshing the cantilever with

element edge length of 0.003. Load was applied as line

load at the end. Static analysis was done. The load to

deflection values were tabulated for all form designs. The

finite element model and meshed model were shown in

Figures 11-12.

1

X

Y

Z

MAY 25 2010

12:15:02

VOLUMES

TYPE NUM

U

1

X

Y

Z

MAY 24 2010

00:52:20

ELEMENTS

Figure 11 FEA Model Figure 12 FEA Mesh

The deflection of form designs 320 N applied at the end

for hollow section and hollow section with single diagonal

stiffeners are shown in Figures 13-14.

1

MN

MX

X

Y

Z

-.774E-07

.407E-05.822E-05

.124E-04.165E-04

.207E-04.248E-04

.290E-04.331E-04

.372E-04

MAY 24 2010

10:43:09

NODAL SOLUTION

STEP=1

SUB =1

TIME=1

UY (AVG)

RSYS=0

DMX =.375E-04

SMN =-.774E-07

SMX =.372E-04

Figure 13 Deflection of Hollow Box applying 300N at the

end=0.0375 mm

1

MN

MX

X

Y

Z

-.791E-07

.389E-05.787E-05

.118E-04.158E-04

.198E-04.238E-04

.277E-04.317E-04

.357E-04

MAY 23 2010

18:02:04

NODAL SOLUTION

STEP=1

SUB =1

TIME=1

UY (AVG)

RSYS=0

DMX =.360E-04

SMN =-.791E-07

SMX =.357E-04

Figure 14 Deflection of Hollow Box with single diagonal

Stiffener section applying 300N at the end = 0.0360

mm

333

Table 6 Material properties of beams

Table 7 Numerical Static Analysis of Form Designs

Similarly for different loads, the deflection characteristics

were found and results for different form designs were

shown in Table 7. Stiffness is calculated and compared

Michael R Hatch, 2002). From the numerical static

analysis, hollow box with slots on sides stiffeners show

the higher stiffness values. The same pattern of

improvement in stiffness in the experimental work is

obtained. The difference from the experimental work is

less than ten percent.

2.9 Numerical modal analysis of form designs

The modeling of form designs were done in ANSYS in

meter. Applying the same conditions used in static

analysis, the modal analysis was done using Subspace

method to find first five natural frequencies and mode

shapes (Makky, 1979). The natural frequencies of form

designs were tabulated for first 5 modes. First mode shape

was axial mode. The second mode shape was bending

mode shape, the third and fourth mode shapes were shear

mode. The fifth was combined torsional vibration mode

and bending mode.

Mode I

1

X

Y

Z

APR 16 2010

05:45:41

DISPLACEMENT

STEP=1

SUB =1

FREQ=1018

DMX =2.23

a) Natural Frequency= 1018.4Hz

1

X

Y

Z

APR 16 2010

05:41:35

DISPLACEMENT

STEP=1

SUB =1

FREQ=1026

DMX =2.222

b) Natural Frequency= 1026.5Hz

Figure 15 Mode shape I of hollow box with two axial and

two diagonal stiffeners along length

Material Young’s

Modulus (E)

N/m2

Poisson ratio

()

Density (ρ)

kg/m3

STEEL 2.1x1011

0.34 7850

Section

Hollow

box

Hollow box

With single

diagonal

stiffener

Hollow

box

With two

diagonal

stiffener

Hollow box

with plus

stiffener

Hollow box

with slots on

sides

Hollow box with

square honey

comb pockets

Hollow box

with

triangular

honey comb

pockets

Mass ( n) Deflection in mm

320 0.0375 0.0360 0.0356 0.0358 0.0327 0.0369 0.0377

640 0.0750 0.0724 0.0715 0.0720 0.0657 0.0742 0.0757

9600 0.1125 0.1090 0.1050 0.1080 0.0987 0.1110 0.1140

1280 0.1500 0.1450 0.1430 0.1430 0.1320 0.1490 0.1520

1600 0.1875 0.1820 0.1780 0.1800 0.1650 0.1860 0.1900

BEAM

MASS 1.205 1.190 1.190 1.199 1.210 1.204 1.205

STIFFNESS

(Nm) 8533.3 8831.2 9004.5 8911.3 9729.5 8627.8 8441.1

Improvement

of Stiffness

Ref Hollow

Section (%)

----- 3.49 5.52 4.43 13.42 1.11 -1.08

334

Mode II

1

X

Y

Z

APR 16 2010

05:45:53

DISPLACEMENT

STEP=1

SUB =2

FREQ=1019

DMX =2.23

a) Natural Frequency= 1118.6Hz

1

X

Y

Z

APR 16 2010

05:41:53

DISPLACEMENT

STEP=1

SUB =2

FREQ=1027

DMX =2.223

b) Natural Frequency= 1126.9Hz

Figure 16 Mode shape II of hollow box with two axial and

two diagonal stiffeners along length

Mode III

1

X

Y

Z

APR 16 2010

05:46:50

DISPLACEMENT

STEP=1

SUB =3

FREQ=4445

DMX =2.515

a) Natural Frequency= 2445.2Hz

1

X

Y

Z

APR 16 2010

05:42:18

DISPLACEMENT

STEP=1

SUB =3

FREQ=4409

DMX =2.476

b) Natural Frequency= 2509.1Hz

Figure 17 Mode shape III of hollow box with two axial

and two diagonal stiffeners along length

Mode IV

1

X

Y

Z

APR 16 2010

05:47:03

DISPLACEMENT

STEP=1

SUB =4

FREQ=5320

DMX =2.276

a) Natural Frequency= 3319.6Hz

1

X

Y

Z

APR 16 2010

05:42:38

DISPLACEMENT

STEP=1

SUB =4

FREQ=5284

DMX =2.247

b) Natural Frequency= 3384.1Hz

Figure 18 Mode shape IV of hollow box with two axial

and two diagonal stiffeners along length

Mode V

1

X

Y

Z

APR 16 2010

05:47:20

DISPLACEMENT

STEP=1

SUB =5

FREQ=5321

DMX =2.276

a)Natural Frequency= 4320.6Hz

1

X

Y

Z

APR 16 2010

05:43:17

DISPLACEMENT

STEP=1

SUB =5

FREQ=5286

DMX =2.248

b)Natural Frequency= 4385.9Hz

Figure 19 Mode shape V of hollow box with two axial and

two diagonal stiffeners along length

335

The natural frequencies were tabulated in Table 8 and the

first five mode shapes of hollow box with two stiffeners in

plus arrangement and hollow box with two diagonal

stiffeners were shown in Figures 15-19.

Natural frequencies were higher for the form designs box

with square honey comb pockets, box with triangular

honey comb pockets, and hollow box with slots on side

stiffeners.

Table 8 Numerical Modal Analysis of Form Designs- First 5 Natural Frequencies in Hz

Section

Hollow

box

Hollow

box

With

single

diagonal

stiffener

Hollow box

With two

diagonal

stiffeners

Hollow

box with

plus

stiffener

Hollow box with

slots on sides

Hollow box with

square honey

comb pockets

Hollow box with

triangular honey

comb pockets

Mode Natural frequency in hz

1 923.3 980.7 968.0 995.9 993.2 1051.2 1026.5

2 977.8 1081.9 1071.7 1181.2 1055.5 1152.7 1126.9

3 2860.6 2433.6 2406.3 2350.3 2454.1 2712.2 2509.1

4 3212.6 3099.5 2969.4 3114.3 3063.4 3418.8 3384.1

5 4213.2 4442.6 4384.5 4343.8 4487.1 4921.8 4985.9

3. RESULTS AND DISSCUSSION

Stiffness was calculated by dividing the applied load by

measured deflection. The average of stiffness values was

found and tabulated. Since all the beams were

manufactured for constant mass, length, perimeter, and

same material, the improvement in the stiffness is purely

because of their forms. From the experimental and

numerical static analysis, all form designs except

triangular honey comb form design have higher stiffness

than hollow box section.

From the experimental work, hollow box with slots on

sides showed the best stiffness value and 18 percent higher

than hollow box section with constant mass. So, hollow

box with slots on sides stiffeners are the perfect

alternatives for hollow box section to increase stiffness.

The moment of inertia of this section is higher than the

hollow section results the improvement in the stiffness.

The difference of numerical results compared to

experimental results is less than ten percent. The

percentage improvement and damping ratio comparison is

shown in table 9 and figures 20 and 21.

The comparisons are done without considering the

manufacturability and cost of manufacture. The

comparison is done to find the influence of form on static

stiffness and damping of machine tool structures. The

improvement in stiffness was because of improvement in

section modulus due to its form shape.

The hollow box with slots on side’s stiffeners has the

advantage that both static stiffness and damping factor is

higher than the hollow box section.

Figure 20 Percentage Improvement in Stiffness of Form Designs compared to Hollow Box section

336

Figure 21 Damping ratio of form designs

4. CONCLUSION

Typical form designs of cantilever beam were selected and

dimensions were designed for constant mass, length and

perimeter. The drawing for manufacturing was prepared

and form designs were fabricated. Experimental setup was

designed and developed for cantilever beam arrangement.

The static deflection characteristics and modal

characteristics of beams were found experimentally. The

numerical modeling and analysis of form designs were

done in ANSYS and numerical results were taken and

validated against experimental results. The form designs

having higher static stiffness than the hollow section were

found. Damping and natural frequencies also found for all

form designs. The static stiffness and damping

characteristics can be improved by increasing the section

modulus by choosing proper form design. Hollow box

Section

Hollow

box

Hollow box

with single

diagonal

Stiffener

Hollow box

with two

diagonal

Stiffener

Hollow box

with plus

stiffener

Hollow

box with

slots on

sides

Hollow box with

square honey comb

pockets

Hollow box

with triangular

honey comb

pockets

Mass (kg) 1.205 1.190 1.190 1.199 1.21

1.204 1.205

Stiffness

(experimental)

(N/m)

8762.07 9367.06 9545.43 9294.50 10366.12

9177.80 8865.51

Stiffness

(numerical)

(N/m)

8533.33 8831 9004.5 8911.3 9729.5

8627.8 8441.1

Difference (%) 2.61 5.72 5.67 4.12 6.14

5.99 4.79

Improvement

of Stiffness

than

Hollow

section (%)

-------- 6.905 8.940 6.077 18.307

4.745 -1.181

Damping

Ratio 0.5672 0.5792 0.5869 0.5640 0.5153

0.4922 0.5152

Table 9 Static Stiffness Comparison of Form Designs

337

with slots on the side stiffeners were found to be best for

replacing hollow section to increase the stiffness of

structures

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