The Ultra-light Cellular Structure for The High-end Numerical Control Machine

Tool Optimal Design Applications

C.Y.Ni

25.Jan.2008

Supervised by Prof. T.J Lu

Advised by Prof. C.Q.Chen

Background

1 mHigh-end numerical control machine tool (accuracy ,acceleration

) 5g

On of the 16 grave special projects for the state medium-term and long-term develop plan

Trend of development

High speed

High degress of accracy

Multiplicity

Intelligence

Flexibility and integration

lightweight

stiffness

damp

elimination of heat

Challenges

Background

Proper distribution of the rib reinforcement can increase the stiffness and natural frequency of the beam

(from C.L.Luo et al.)

Box structures are well applied in new style machine tool

As the structure requirement , the beam must be hollowness

Ultra-light cellular structure

The beam is the master part of a machine tool ,the rationality of its structural design can influence the stiffness and the precision of the machine tool directly

Here we just consider the 2D honeycomb structure as the rib reinforcement

Background

Take the beam for example to try to do some optimal design for the structure

Objective

Build simplified theoretical model for the beam

Analyze the static stiffness of the beam with different

cellular structure to find the optimal design in theory

FEA of the natural frequency and modal for the beam

with different cellular structure to find the optimal design

Analysis example

Background

Static stiffness analysis

Simplified model

To the Load

Tool box hanging on the beam

To the Boundary condition

The beam is located on the guide rail ,considering the static stiffness

To the Structure

According to mechanical acknowledge

Simple support

Combination effect of the moment of flexion and torque

a sandwich bar whose outer layer is a closed-cell thin wall bar and the inner layer is a core bar

Static stiffness analysis

Simple mechanical mode

0550 , 209 , 286 , 215 , 30 , 2320H mm h mm L mm l mm t mm X mm

The honeycomb structure can be transferred to the continuous homogeneous structure using equivalent method in order to solve discrete structure by the theory of continuous medium

(from Lorna J.Gibson Michael F.Ashby)

Here we just consider four kinds of classic honeycomb structures

equivalent structure

Relative density

s

t

l

is the cellular structure density

is the material density

For honeycomb cell wall with a thickness to length ratio of

Its relative density can be expressed as

l

tC

s1

It has grave effect on the dynamical character for the cellular structure

(from Lorna J.Gibson Michael F.Ashby)

Static stiffness analysis

Definition:s

r

is the weight and is the density of the material

Considering a core structure composed of one row honeycomb cell

the relative density can be expressed by the macroscopic parameters of the core structure

For the given core structure we can determine the relative density directly

W

Static stiffness analysis

In-plane equivalent stiffness of the honeycomb structure

(from A.-J.Wang and D.L.McDowell)

Static stiffness analysis

The simplified model reveals that the static stiffness of the beam contain the torsional stiffness and the bending stiffness

For outer layer closed-cell thin wall bar

Bending stiffness

1

1 M

K

Solving the geometric parameter:

We conclude that: 91 2.01 sK e E

1K EI92.01zI e

Torsional stiffness24A G

GJS

Solving the geometric parameter

We conclude that: 81 8.87 sD e G

Closed-cell thin bar torsional stiffness formula:

Beam bending formula:

Static stiffness analysis

For the inner layer equivalent continuous bar

Using the same method to conclude that:

Bending stiffness9

2 1.81K e E

82 4.92D e G

Using the principle of superposition to produce the static stiffness of the beam

1 2 1 2P K K D D

Square cross-section bar torsional formula:

32D ab G

Static stiffness analysis

Torsional stiffness

Basing on the conclusion above, we can determine the stiffness using the macroscopic parameters of the beam

For the beam whose inner layer with different core shape we get the conclusion as follow:

Static stiffness analysis

Static stiffness analysis

Conclusion

Analyze the static stiffness of the beam with the theoretical simplified model

Find the hexagon honeycomb core structure is the optimal structure in this four structures.

For the beam we choose several usual rib reinforcement structures to carry on the analysis

The analysis of the natural frequency and modal is a basic and important content in dynamical analysis of the structure

FEA of the natural frequency and modal

Strucure 1 Square honeycomb-like core (original beam structure)

Structure 2 Fold-like core

Structure 3 Dummy plate core

Structure 4 Fold-like structure

Structure 5 Hexagon-like core

Concluding the first 10 grades natural frequency as follow:

FEA of the natural frequency and modal

各结构前三阶固频比较图

200

230

260

290

320

350

380

410

440

结构一 结构二 结构三 结构四 结构五

HZ频率（ ）

237

238

239

240

241

242

243

244

245

246

247HZ频率（ ）

二阶固频三阶固频一阶固频

Conclusion

Hexagon-like structure is the best among these five structures by analyzing the natural frequency and modal of the structures

This conclusion also certificates the rationality of the theory simplification above in certain extent

FEA of the natural frequency and modal

Conclusions

Find that the relative density of the cellular structure is not only relative to the microscopic parameters but also can be determined by the macroscopic parameters

A simplified model has been suggested to analyze the mechanical performance for the beam of the machine tool

The simplified model is applied to analyze the static stiffness of the beam structure and we find that hexagon honeycomb structure is the optimal design of the four structures

FEA of the natural frequency and modal for the beam structure also finds that hexagon honeycomb structure is the best one of the five structures

The conclusion of the modal analysis certificates the rationality of the simplified model in certain extent

Future work

For the static and dynamical performance, we can do the optimal design from theory model building ,computer simulation and experiment research three aspects to expect concluding the general analysis model

The optimal approach should contain topological optimization and general optimization

For the machining accuracy, we may progress the global error analysis ,and reduce the global error by controlling the local accuracy.

Considering the optimal design for the structure undergoing extreme loads

At the same time, we also should make full use of the multifunctional character of the ultra light cellular structure

Thanks