Optimization of robotic arms made of composite materials for maximum fundamental frequency

Composite Structwes 31( 1995) 1-8 CI 1995 Elsevier Science Limited

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Optimization of robotic arms made of composite materials for maximum fundamental frequency G. Caprino & A. Langella Department @Materials and Production Engineering,

University ofNaples, P. le Tecchio, 80,80125

Naples, Italy

The paper deals with the design of a composite arm, to be employed in the architecture of industrial, measuring robots. Based on the real working conditions, the arm is reduced to a cantilever beam, shaped as a thin walled tube, supporting a mass at the free end. An optimization study is carried out, aiming to determine the laminate properties capable of providing the maximum fundamental frequency (FF) in bending. The optimization relies on an approximate solution, obtained by modifying the Rayleigh energy method to account for the shear effect. It results in a closed-form formula, explicitly correlating FF with composite elastic properties. Two classes of laminates, namely ( + t3),$and (O/ f 45,,),, are examined in order to assess their ability to fulfil the requirement of a maximum FF value. Moreover, the advantages and drawbacks of composites compared to metals are highlighted for the application under concern. Finally, the validity of the proposed solution is demonstrated by comparison with the results of a finite element analysis.

1 INTRODUCTION

robot architecture plays a fundamental role in determining the machine productivity, and must be carefully evaluated in the design stage. Of course, the use of high modulus and low density materials, such as composites, in a measuring robot would result in higher resonance frequencies. This would allow higher operating speeds without affecting precision, therefore bringing increased productivity. Apart from cost considerations, a technical barrier also exists against a deeper diffusion of highly oriented composites in the field of industrial machines; in fact their anisotropy, together with the variability of the laminate mechanical properties depending on fibre orientations, result in a more difficult approach to design, especially when optimization procedures must be followed for the achievement of the goal. On the other hand, the possibility of designing the material for a specific application is one of the most powerful advantages of composite materials. Only optimization can guarantee the full exploitation of their potentialities. Due to the reasons previously recalled, one of the scopes in optimizing a composite arm is the achievement of higher natural frequences in bending. In general, the main objective is the increase

Although the specific mechanical properties of composite materials have long been appreciated in many engineering fields, their impact in the components of both tooling machines and industrial robots is presently restricted to a few examples.‘-3 Apparently, this situation is mainly due to cost considerations, which render the cheaper conventional materials, such as steels, more attractive for the latter applications. Undoubtedly, composites are not cost-competitive when compared to traditional metals, especially when rigidity requirements demand the use of sophisticated fibres. However, not only the machine cost, but also the economics induced in the overall working cycle by the introduction of these new materials must be taken into account, when their usefulness is evaluated. For instance, of numerically controlled the productivity measuring robots is generally quite low, as a consequence of their limited operating speed. The increase in the latter would imply steeper acceleration and deceleration ramps of the mobile components. However, this could possibly induce resonance phenomena, deleteriously affecting precision. Therefore, thLedynamic response of the 1

2

G. Caprino, A. Langella

of the lowest natural frequency, associated with the fundamental mode of vibration. In spite of its practical importance, this aspect has not been sufficiently covered in the literature. Moreover, although many studies have been published on the dynamic response of composite structures,3-8 the proposed solutions are difficult to manage within an optimization procedure. The scope of this work was the design of a composite arm destined to an industrial robot, to be optimized with reference to its fundamental frequency (FF ). To this end, by the Rayleigh energy method, an approximate closed-form formula accounting for the shear effect, was derived, explicitly correlating FF with laminate elastic properties. The ability to fulfil the requirement of a maximum FF value was then examined with reference to two classes of graphite/epoxy composites, i.e. ( k f3), and (O/ + 45 ,), laminates. The advantages and drawbacks of composites compared to metals for the application under consideration were also assessed, in the light of the proposed solution. Finally, the efficiency of the formula was ascertained by comparison with the results of a finite element (FE) analysis.

2 SCOPE In Fig. 1, a typical architecture of a three-axis measuring robot is schematically depicted. Three carriages travel along three mutually orthogonal directions (x, y and z in the figure), suspended on non-contact, air bearings. A cylindrical arm,

Prohihead

rigidly fastened to the y-carriage, supports the probe measuring head. During motion, the carriages are subjected to rapidly varying velocity profiles; this can result in oscillations of the arm, the carriages themselves, or the machine frame, all of which are detrimental to precision. The most common way of reducing the deleterious effects of vibrations is limiting the acceleration and deceleration maximum values during the working cycle. Of course, this solution does not allow high working speeds, resulting in low productivity. A more reliable solution to the problem would be the extensive use of light as well as stiff materials in the mobile elements: lightness would reduce inertia forces; stiffness would give high natural frequencies; both yielding the possibility of steeper acceleration ramps. Allowing for the previous considerations, composite materials appear as ideal candidates for the mobile components of an industrial robot. However, a quantitative evaluation of their advantages is in order. Therefore, the scope of the present study is twofold: to establish an optimization procedure for the design of a composite arm, with reference to FF; 0 to evaluate the benefits offered by composite materials compared to metals. l

3 ANALYTICAL MODEL Figure 1 shows that if the FF of the arm in bending is under study, the actual system can be easily reduced to a one-dimensional cantilever beam, with a concentrated mass M at the free end (Fig. 2). An exact solution to this problem is available in Ref. 9; although it was obtained for isotropic materials, its usefulness can easily be extended to composites, at least when a specially orthotropic laminate, avoiding all coupling effects, is considered. However, the exact solution is not a closed-form one, so that it can be hardly handled within an optimization procedure; furthermore, it does not account for shear deformation effects:

I

Fig. 2. Fig. 1.

Schematic view of a measuring robot.

-

-

I

One-dimensional cantilever beam, with a concentrated mass at the free end.

3

Optimization of robotic arms

the latter may affect the mechanical behaviour of a composite more markedly than usually happens for metals.‘“* ’ ’ In order to find a simple, yet approximate analytical expression to calculate FFFV, a modified version of the Rayleigh energy methodI is applied. By this meth.od an arbitrary shape, fulfilling boundary conditions, is assumed for the first normal elastic curve of the beam; with this assumption, the maximum potential energy, E,, and maximum kinetic energy, E,, are calculated; finally, FF is evaluated by equating E, and E,. If the shear effect on the beam deformation has to be taken into account, the total deflection y(x, t) of the vibrating beam {Fig. 2) must be considered as the sum of two components, ybx, t) and y,(x, t), due to flexure and shear, respectively: Yk t) = Yfb, 4 + Y,k 4

(1)

where t is time. Looking at yf(x,t), it is supposed that this component of deformation is cosinusoidally shaped in the space dolmain: sin w t

yr(x, t) = y,, 1 - cos 7: i

(2)

i

It can be easily verified that eqn (2) satisfies the boundary conditions; in fact, deflections as well as rotations are zero at the clamped end and the curvature is zero at the free end, irrespective of t. It is worth noting that the zero curvature at the free end is guaranteed by the absence of flexural moments at that point. The latter hypothesis only holds if the rotatory inertia of the mass A4 is negligible, so that this assu.mption is implicitly made throughout the present analysis. In order to resort to a suitable expression for the shear component of deflection, the following relations, given by strength of materials,13 are utilized: M,=

-D$ &!

&EL dx

-

dx3

where the symbols M, T shear force, respectively, stiffness of the beam. Of course, eqns (3) and only for static problelms;

(4 represent moment and and D is the flexural (4) are rigorously valid in using them for a

is implicitly dynamic case, an approximation made, because inertia forces are disregarded. It is well known that y, is correlated to T by the following equation:13

(5)

;dx

y,=

i

where S is the shear rigidity of the beam. Substituting eqns (2), (4) in eqn (5), and solving the integral, it is verified that: 2

y,(x,t)=

-yo;

0 ;[

co+wr+A

(6)

The constant of integration, A, is obtained by the constraint condition at the clamped end: x=o+y,=o

(7)

so that eqn (6) becomes: Y,(X,t)=y~,s(~~(l-cos~)

sinot

(8)

Substituting eqns (2) and (8) in eqn (l), the beam deflection y, accounting for shear effect, is derived: y(x,t)=y,,

1-~0s; i

(l+K)sinot

(9)

i

where: 2

K,E

0 jt

s

(10)

21

From Ref. 13, the potential system is given by:

energy

of the

(11)

Ep=

The maximum value of E, is attained in correspondence of the most deflected position of the beam; of course, this happens when sine t= 1. With this in mind, eqn (11) can be combined with eqns (2) to (4), providing:

n4D(l +K)

Ep=Y:, 64p

(12)

When the beam is undeformed, all the potential energy is converted into kinetic energy; the maximum value of the latter is calculatedt2 from equa-

G. Caprino, A. Langella

4

Dividing term by term eqns ( 15) and ( 18):

tion: y2dx+My:,(l+K)2

1

I

which, taking into account eqn (9), results in: &=~(0.226m+M)(l+K)’

(19)

(13)

(14)

which allows a quantitative evaluation of the error made in disregarding the shear effect. Substituting eqns ( 16) and ( 17) in eqn ( 10):

n2 E R2

Kc-L

Equating eqns ( 12) and ( 14), and considering that o= 2~rh the following expression is finally obtained: I

f2

8

D 21"(0*226m+

M)( 1 + K)

(15)

Equation (15) is the solution to the problem under study, giving FF as a function of the physical, elastic and geometric parameters characterizing the system. If the beam is made of a thin walled cylinder, a more explicit correlation between FF, the laminate elastic moduli and the radius R of crosssection can be obtained by recalling that:r3 D= E,nR 3s

(16)

S= 0.87G,,nRs

(17)

where s is the wall thickness, and E,, Gxs the elastic and shear modulus of the material, respectively. The indexes ‘x’ and ‘0’ affecting the elastic moduli refer to the longitudinal and circumferential direction of the cylinder, respectively. It is worth noting that this solution is conceivably valid in the hypothesis of specially orthotropic laminates, because all the coupling effects have been disregarded in the analysis.

(20)

3.48 G,J2

so that the relative importance of shear deformation is apparently affected by the ratio E,/GXe, depending on the selected laminate, and by the ratio l/R, depending on the arm geometry. In Fig. 3, the non-dimensional FF, f/f*, is reported against the non-dimensional arm length, Z/R, for three E,/G,, values. It can be seen that, the higher E,/G,@, the lower f/f * for a given Z/R; moreover, for a fixed E,/G,@ ratio, a decreasing non-dimensional FF is calculated at decreasing Z/ R ratios. This means that, if the shear effect is not accounted for in the calculations, the actual FF is overestimated the more, the lower the shear modulus, and the shorter the beam. It is interesting to note that in general eqn (18) can be reliably used for isotropic materials (E,/ GxB= 2.7), unless very short beams are concerned: for I/R> 5, an error lower than 10% is made. The same statement no longer holds for highly anisotropic laminates: a 10% error corresponds to l/R = 10 for a unidirectional glass/epoxy composite (E,IG,B = lo), and I/R = 20 for a unidirectional graphite/epoxy composite adopting high strength fibers (E,/GXe = 30). It is worth noting that Z/R

l.zIl---l

4 EFFECT OF SHEAR DEFORMATION

1.0

If an infinite shear rigidity is assumed for the arm, the shear effect is disregarded; in this hypothesis, eqn ( 15) reduces to:

$

c

0.8 -

Ex /GxH

= 2.7

0.6

f*=iJL

(18) _. .

and the solution proposed in Ref. 12 is recovered. In eqn (18), the symbol f * has been used to indicate the FF value accounting for the flexural deformation only.

0

10

20

30

40

I/R

Fig. 3.

Non-dimensional fundamental frequency, against non-dimensional beam length, l/R.

f/f *,

Optimizationof robotic arms

values in the range lo-20 are usually adopted for robot arms. Of course, the approximation in eqn ( 18) becomes the more unacceptable, when the application of very high modulus graphite fibers is considered.

5

adopted in evaluating E, and GXe: E,=230MPa E, = 10 MPa =7MPa

G2

Y,*= 0.23 5 LAMINATE OPTIMIZATION Recalling eqn ( 16), eqn ( 15) can be rearranged the following form: f=Z&

in

The laminate elastic moduli were calculated from lamination theory,14 which provides the following relationships between the laminate strains E,, E,~, yoxe, and the in-plane loads per unit width, N,, N,, N,, for a balanced laminate:

(21)

where

a=-nR"s 21”

(22)

(23)

In eqn (25), [a] is the in-plane compliance matrix, from which E, and GxB are easily obtained as:

‘=M(l+1:226-3

Y=

E,

(27)

(24)

J? ER2 l+--?-3.48 G,J2 It is evident that a is only dependent on the arm geometry; /3 on the masses involved in the phenomenon; finally, y is a function of nondimensional arm length, Z/R, as well as the elastic properties of the selected laminate. If the scope of optimization is the choice of a laminate supplying the highest FF for a given geometric configuration and a given basic lamina, the optimum laminate will be that capable of providing the maximum y value. Of course, an infinity of laminates, exhibiting very different elastic properties, may be obtained from the same basic lamina. However, a simplification during fabrication is achieved, if a limited number of different fiber orientations is chosen, or laminae oriented only at 0” and + 45” are scheduled. The latter arrangement allows us to partially utilize reinforcement in the form of cloth, which is more easy to work than unidirectional laminae. Consequently, the optimization study in the present paper wa.s restricted to ( k t9), and (O/ + 45,), laminates. Tlhe following properties, typical of a high modulus graphite/epoxy lamina, were

In the case of ( + 19)~laminates, the problem of optimization is reduced to finding the 8 value that maximizes J The results of such analysis are reported in Fig. 4, where the non-dimensional f/h, is shown against 8. In the figure, J;, indicates the

1.o 0

s ........+ .. .

0.5 tI 0.0

----O---

I/R-7.5

----W---

I/R-) 0

--+ -

I/R-20

I

0

,/R-5

10

1

20

I

I

40

30

1

50

I

60

J 70

*e Fig. 4. Non-dimensional fundamental frequency, f /fO, against fibre angle, 0, for a ( f &J), laminate. I/R=nondimensional arm length.

6

G. Caprino, A. Langella

FF for 8= O”, calculated by eqn (15); therefore, the advantages of choosing the most efficient fibre angle, as compared to the trivial solution 0= O”, can be immediately appreciated from the figure. As expected, the optimum fibre angle, eopt, is strongly dependent on the non-dimensional arm length. More specifically, decreasing Z/R results in increasing oopt. It is interesting to notice that, for low l/R ratios, the adoption of eopt instead of 0 = 0” can give an improvement as high as 3 5% in the dynamic behaviour of the arm; this conclusion is particularly useful for material handling robots, where short beams are usually employed. Of course, the solution oopt= 0” occurs for long beams. This is easily seen from eqn (24): for high I/R values, the parameter y is essentially affected by the Young’s modulus E,, which attains its maximum value when 0= 0”. However, even for Z/R= 20, 8,,, is different from 0”. Interestingly, the curve is initially quite flat: therefore, a range of 8 values exists, within which FF does not vary significantly. This can help avoid the use of unidirectional laminates exposed to matrix cracking under fatigue conditions, as well as prescribing less stringent tolerances on fibre orientation during manufacturing. Figure 5 reports eopt against the non-dimensional arm length, l/R. As noticed from Fig. 4, when I/R increases, the optimum fibre angle tends to 0”. On the contrary, eopt+ 45” when f/R- 0. The explanation for this behaviour can be found in eqn (24): only the shear modulus G,, which is maximum for 0= 45”, appears in eqn (24) when I/ R=O.

‘n’, that is the ratio between the laminae oriented at + 45” and 0”. In Fig. 6, the variation of f/A, for a (O/ f 45 ,,), laminate is shown as a function of the percentage of f 45” laminae. The trend of the curves is qualitatively in agreement with that observed in Fig. 4: in order to obtain the maximum f/A, value, higher percentages of + 45” laminae, resulting in a higher shear modulus, are required when low I/R ratios are considered. Moreover, by quantitatively comparing the curves in Figs 4 and 6, it is evident that, for a fixed l/R ratio, approximately the same maximum f/f0 value is attained with both a ( k f3), and a (O/ f 45 ,),s laminate. Therefore, the choice of the actual laminate can be based on considerations other than FE In Fig. 7, the percentage of f45” laminae resulting in the maximum FF for a (O/ + 45,),s

: c

1.0

--+-

I/R=20

1

0.5 0

25

50

I

75

%f45”

In considering (O/+45.),V laminates, the only parameter to be defined in view of optimization is

Fig. 6. Non-dimensional fundamental frequency, against percentage of +45” laminae for a (O/&45,,), nate. l/R = non-dimensional arm length.

f /fo, lami-

40 30 F m +I 20 10

I/R

Fig. 5. Optimum angle. for maximum fundamental frequency, /3C,p,r against non-dimensional arm length, l/R.

I/R

Fig. 7. Optimum percentage of f 45” laminae resulting in maximum fundamental frequency, (% f 45),,, against nondimensional arm length, l/R.

7

Optimizationof robotic arms

laminate, (% f 45),,rt, is reported against the nondimensional arm length. Of course, when Z/R= 03 the best laminate is a 0” one, so that (% + 45),,, = 0. For l/R= 0, (Oh+ 45),,, = 100, and a f 45 laminate is obtained. Finally, from eqns (2 1) to (24), it can be noticed that the mass m is of real importance only when the ratio m/M is low compared to unity. Otherwise, FF is only determined by the elastic constants of the constituent material and the arm geometry. From a plhysical point of view, this means that the lighmess of constituent material may not be the best for robot arms intended for manipulation of heavy masses. In the latter case, only composites made of very stiff fibers will be capable of competing with steels.

6 FINITE ELEMENT RESULTS Of course, the conclusions drawn in the previous section are valid, provided the approximations made in obtaining eqn (15) do not bring us to unacceptable estimates of FE To test the validity of eqn ( 15) in predicting FF, the fundamental frequency of selected cylindrical structures was evaluated by FE analysis. In carrying out the FE analysis, the structure geometry was held constant. To emphasize the shear influence on FF, a short beam (Z/R= 7.4) was adopted, with I= ‘700 mm. The cylinder was divided into 200 four-node, thin shell orthotropic elements, 5 mm in thi(ckness. A mass M= 4.8 kg was placed at the free end. The mass m of the beam was 3.4 kg. A schematic view of the mesh used in the FE analysis is shown in Fig. 8. By using the laminae elastic constants previously specified, the elastic constants of ( AI O), laminates, with 8 varying in the range 0” to 35”, were calculated through lamination theory. These constants were used as material input data in subsequent runs of the FE program. The same material constants were utilized in eqn ( 15), to evaluate FE

Fig. 8.

Finite element mesh of the cylindrical structure.

2

200

100 -

"0

-c--C-

Without shear With shear Finite Element

10

20

30

40

50

*Cl Fig. 9.

Fundamental

frequency, f, against fibre angle for a ( f O), laminate.

The numerical results obtained from FE and eqn (15) are represented by black symbols in Fig. 7, where fis reported against 8; the white symbols in the same figure refer to the calculations made by putting K= 0 in eqn (15), i.e. disregarding shear effect. Both the FE results and predictions from eqn (15) follow the same trend. Apparently, the use of eqn (15) produces an overestimate of FF for low 8 values. This is expected, because the Rayleigh method always gives values of the natural frequencies which are higher than the true ones.r2 However, the correlation with FE analysis is within 7% accuracy. On the contrary, disregarding shear deformation results in largely non-conservative errors in evaluating FF, at least when low 8 values are concerned. Beyond 8~ 30”, all the solutions converge to a single curve, so that the shear effect on FF can actually be neglected.

7 CONCLUSIONS

Within the restraints of a study devoted to the design of robot arms made of advanced composite laminates, a modified version of the Rayleigh energy method was used as a closed-form formula, calculating the fundamental frequency (FF ) of a cantilever beam with a concentrated mass at the free end. The proposed formula accounts for the effect of shear deformation on the dynamic behaviour of the structure. Based on it, an optimization procedure was developed, in order to find the fiber orientations capable of providing the maximum FF for a given arm geometry. From the numerical results presented in

8

G. Caprino, A. Langella

paper, the main conclusions are as follows:

this

neglecting the effect of shear results in nonconservative estimates of FF, the error increases with decreasing the ratio of Z/R between the length I and the radius R of the arm; unacceptable errors can be made for l/ R values usually adopted in actual robot arms; l due to the shear effect, in many cases a 0” laminate does not provide the best performances in terms of FF, an optimization procedure, allowing for a proper selection of the most efficient laminate, can give significant improvements in the dynamic response of the arm, without affecting its weight. 0 the proposed formula can be reliably used to evaluate FF, in fact, an excellent correlation between its predictions and numerical results deriving from a finite element analysis was found. l

Although the optimization procedure here proposed was developed for a thin walled cylinder, its extension to robot arms having different cross-sections is straightforward. On the contrary, more work is needed in order to remove the assumption of special orthotropy. The latter limits the applicability of the present solution to laminates in which all the bending/stretching coupling effects have been suppressed. ACKNOWLEDGEMENTS This work was carried out under CNR Contract no. 91.01904.PF67, as a part of the Research

Program ‘Robdtics’. Italian CNR is gratefully acknowledged for its financial support.

REFERENCES 1. Feyerabend, F., Systematic optimization of a robot arm structure. The Industrial Robot, Dec. 1988, 15 (4) 219-22. 2. Belegundu, A. D., Optimizing the shapes of mechanical components. Mech. Engng, (1993) 46-8. 3. Ghazavi, A., Gordaninejad, F. & Chaloub, N. G., Dynamic analysis of a composite-material flexible robot arm. Computers and Struct,, 46 (1993) 315-27. 4. White, J. C., The flexural vibrations of thin laminated cylinders. J. Engngforlndustry (1961) 397-402. 5. Mirsky, I., Vibrations of orthotropic, thick, cylindrical shells. J. AC. Sot. ofAm., 36 (1964) 41-51. 6. Dong, S. B. & Tso, F. K. W., On a laminated orthotropic shell theory including transverse shear deformation. J. Appl. Mech., 39 (1972) 1091-7. 7. Sivadas, K. R. & Ganesan, N., Vibration analysis of orthotropic cantilever cylindrical shells with axial thickness variation. Comp. Struct., 22 (1992) 207-15. 8. Narita, Y. & Ohta, Y., Finite element study for natural frequencies of cross-ply laminated cylindrical shells. Comp. Struct., 26 (1993) 55-62. 9. Belluzzi, O., Scienza delle Costrazioni, IV (1986) 356-70, Zanichelli. 10. Whitney, J. M. & Pagano, N. J., Shear deformation in heterogeneous anisotropic plates. J. Appi. Mech., 37 ( 1970). 11. Murthy, M. V. V., An improved transverse shear deformation theory for laminated anisotropic plates. NASA Technical Paper 1903 ( 198 1). 12. Den Hartog, J. P., Mechanical Vibrations, McGraw-Hill, New York, 1947. 13. Belluzzi, O., Scienza delle Costruzioni, I, p. 94, Zanichelli, (1986). 14. Agarwal, B. D. & Broutman, L. J., Analysis and Performance of Fiber Composites, John Wiley and Sons Publ., New York, 1980.