Billet expansion as a mechanism for noise production in impact forming machines

Journal of Sound and Vibration (1975)

42(3), 325-335

BILLET EXPANSION PRODUCTION

AS A MECHANISM

IN IMPACT

FORMING

FOR NOISE MACHINES

D. C. HODGSON Department of Mechanical Engineering, University of Birmingham, Birmingham B 15 2 TT, England

AND J. E. BOWCOCK Department of Mathematical Physics, University of Birmingham, Birmingham B 15 2TT, England (Received 7 January 1975, and in revised form 11 March 1975)

In this paper a new mechanism for the production of impulsive noise in impact forming machines is proposed; namely the sudden radial expansion of the billet at the instant of impact. First of all a mathematical model of a forging machine is described which, though simplified in order to render it amenable to calculation, contains the essential features necessary for a realistic calculation of the sound pulse. A theory then is developed which enables the sound pulse to be computed given the rate of expansion of the billet surface. It is shown that the main part of the pulse from this source is produced during the first few microseconds of impact, the remaining few milliseconds of impact time producing a relatively small pulse. It being assumed, then, that the acceleration of the billet surface is essentially a delta function, the sound pulse produced by such an acceleration is computed and shown to constitute a significant part of the peak pressure measured. Finally the variation of pulse height with billet dimensions and impact velocity are given.

I. INTRODUCTION As a step towards reducing the noise produced by impact forming machines, it is desirable have been to identify the mechanisms which produce it. So far three main mechanisms established: (a) the sudden deceleration and acceleration of the dies and platen on impact [I, 23; (b) the air squeezed out from between the approaching dies [3]; (c) the vibration of the machine structure in its various modes [4], In this paper a fourth source of noise is proposed: namely, the sound pulse produced by the sudden radial expansion of the billet at the instant of impact. The theory of this pulse is developed in detail and it is shown that, in most cases, it will contribute significantly towards the peak impulsive sound pressure.

2. THEORY

OF THE SOUND

PULSE

During impact most of the surfaces of a forging machine are in a state of acceleration and therefore emit noise. In order to isolate the sound produced by the expanding surface of a billet, it is convenient to consider a simple system in which only the billet surface is allowed to move, all other surfaces being held fixed. For the purposes of this calculation, therefore, it is sufficient to consider the simplified mathematical model of a forging machine shown in Figure 1. (See Appendix C for notation.) A cylindrical billet, of radius R, is sandwiched between two semi-infinite cylindrical dies, of radius L, which remain fixed for all time. At t = 0 the sides of the billet expand radially outwards through a distance small compared with R, with an acceleration a(t). In this way 325

326

D. C. HODGSON AND J. E. BOWCOCK

Figure 1. Simplified model of forging machine used in calculations.

the pulse produced is caused only by the billet expansion and is not complicated by the additional pulses which in a realistic situation would be produced by the sudden acceleration and deceleration of the dies, air ejection and the “ringing” of the structure. Cylindrical co-ordinates (r, 4,~) can be used to describe the system but owing to the complete cylindrical symmetry about the z axis, the sound pulse produced does not depend on the azimuthal angle 4. For convenience, the regions of air between and outside the dies will be labelled I and II, respectively, as shown in Figure 1. Mathematically, the problem is to solve the wave equation V2P=7s

1 azp

(1)

in cylindrical co-ordinates subject to the boundary condition ap -=-

-au,

an

(2)

p at

be satisfied on the different surfaces of the machine. Equation (2) merely corresponds to equating the normal velocities of the air and machine surface, u,, at their common interface [5, see p. 243 J. The details of this calculation are given in Appendix A. The final result is

to

s m

p(r, z, f) = f

‘@)

H:“(kL) + G(k) H:“(kL) H;“(j&)

+ G(k) H\2’(&3)

I(” ” k, e-iwr do’

(3)

-02

where Hb”(kL) - ‘; I(L, z, k) H:“(U) G(k) =

(4) -HF’(kL)

+f

I(L, z, k)Hi2’(kL)

and I(r,

z, k) =

O” sin k, H cos k, z H~‘(~k~~r) o

k,m

H:“(yI’iiL)

dk . ’

Thus, from knowledge of how the billet surface moves (i.e., of a(t) and hence A(o)) the pressure pulse can be calculated by doing the Fourier integral (3) numerically, by means of the Fast Fourier Transform algorithm of Cooley and Tukey [6].

NOISE IN IMPACT FORMING MACHINES

3. CALCULATION

321

OF a(t)

The nature of a(t) depends on the way in which the billet deforms under load. In a recent set of experiments it was found that the billet maintains both a constant volume and cylindrical shape during impact [7]. In this case the volume of the billet always is given by the expression V = 2nxZ h. Solving this for x gives X = (V/27$” h- l/2

=

RHl/2

h-1’2.

(6)

By differentiating equation (6) with respect to time it is easy to see that xc---

RH’12

ii

2

/23’2

(7)

and RH’12 z=2h-

3/2

-h

+--

[

3h2 2h

I

’

(8)

Hence the initial radial velocity of the billet surface is i(t = 0) = RU/2H.

(9)

This follows since at t = 0 both the top surface of the billet and the upper die are in contact and moving together with a velocity U. In other words, at t = 0, 1 jumps discontinuously from 0 to RU/2H, to a good approximation. The acceleration in the neighbourhood of t = 0 therefore can be approximated reasonably by a delta function, u(t) = (RU/2H) s(t),

(10)

for which the Fourier transform is A(o) = RU/2H. What happens during the remainder of the impact time depends on the properties of the billet. But it is reasonable to expect that after the billet surface has accelerated impulsively to its maximum velocity RU/2H, it spends the remaining time decelerating steadily to rest. This

Time (ms)

Figure 2. Analytical approximations to experimental radial velocity and acceleration. (a) J?= (RU/ZH)cos (462t), 0 < t < 3.4 ms; (b) a = _?= (RC//ZH)[6(t) - 462sin(462r)], 0 < t 6 3.4 ms.

328

D. C. HODGSON AND J. E. BOWCOCK

Time (ms) Figure 3. (a) Pressure pulse due to delta function acceleration of cylindrical billet (full line) and spherical billet (dotted line); (b) pressure pulse due to relatively slow deceleration of cylindrical billet.

indeed is what was found to happen by Grigorian [7] in his experiments with copper billets. In Figure 2 analytical approximations to Grigorian’s radial velocity and acceleration curves are shown which are sufficiently accurate for our purposes. The parameters of Grigorian’s experimentwereL=6cm,R=1~25cm,H=1~9cm,r=1~5m,z=OandU=3m/s. In order to investigate the relative importance of the initial delta function acceleration and the slower deceleration their contributions to the sound may be calculated separately. This was done by integrating expression (3) numerically for U = 3 m/s and the results are shown in Figure 3, where the pressure has been calculated at a point in the plane z = 0 and at a radial distance of 1.5 m. The contribution of the initial sharp acceleration gives rise to a sharply peaked sound pulse whereas the deceleration produces a much lower pulse spread out in time. It follows, therefore, that the final expression for the radiated pressure pulse is obtained by combining equations (3) and (10) to give H:“(M) + G(k) H’:‘(U) PRU OD I(r, z, k) eiot do. p(r, 2, t) = 2a2 H _m H:“(kR) + G(k) H:2’(kR)

(11)

Although this calculation has been done for a specific acceleration, it is clear that this type

NOISE IN IMPACT

FORMING

329

MACHINES

of sound pulse will arise more generally. If the impact velocity is increased then the height of the sound pulse produced will increase in proportion, as shown by expression (II). Useful insight into the solution (1 l), for which the integral cannot be done analytically, can be obtained by solving the much simpler but similar problem of a spherical billet of radius R whose surface expands radially with an impulsive acceleration u06(t). The result of this calculation, performed in Appendix B, is r-R t-C-

0, p(r,

1)

=

pcRv, e

C

-(clR)Ct-_(r-RNcl

t>-

7

(12)

r-R

C r I I and is shown as a dotted line in Figure 3(a) for R = 1.25 cm, u0 = 1 m/s and r = 1.5 m as before. Since the problems of the spherical and cylindrical billets are, apart from the difference in geometry, quite similar, it is interesting to compare the pulses given by equations (11) and (12). Although the spherical billet has a surface area of about two-thirds that of the cylindrical billet, it produces a higher and narrower pulse. As the pulse from the cylindrical billet undergoes a reflection at r = L, the smaller height is perhaps not too surprising. However, the width of the pulse (12), R/c z 0.04 ms, is an intrinsic feature which should be common to both problems as the pulse width is determined by the time for a sound wave to travel the diameter of the billet. This suggests that the pulse (11) should be narrower than the 0.1 ms which has been calculated and that the discrepancy is due to the numerical approximations employed. In particular, the Fourier integral (11) was cut off at 10 kHz in order to limit computation time. This was quite a reasonable approximation since at this frequency P(r, z, w) had fallen by a factor of 10 below its maximum value. However, a delta function acceleration has a constant Fourier transform A(o) = RU/2H and consequently the integrand in expression (11) converges slowly. Inclusion of the frequencies from 10 kHz to infinity would have the effect of producing a higher and narrower pulse. In fact, this trend was checked numerically but inclusion of frequencies above 10 kHz made the computation time prohibitively long.

4. EXPERIMENTAL ILLUSTRATION OF BILLET EXPANSION As mentioned previously, it is very difficult to separate experimentally the pulse produced by billet expansion from the various other effects (deceleration pulse, ringing, etc.). One can I

4-

I

“LA

I

I

I

I

I 8

I IO

_

’

2-

I

0

I 2

I 4

Time hk

Figure 4. Experimental expansion.

pressure wave. The position of the peak Xin time corresponds

to that due to billet

330

D. C. HODGSON AND J. E. BOWCOCK

say only that in the absence of these effects, billet expansion ought to give a sharp pulse at a definite time determined by the position of the microphone. In Figure 4, Grigorian’s sound measurement is shown, the peak X corresponding to the delta function acceleration of the billet surface. Since the microphone was at r = 15 m, the sound pulse from the billet should make its appearance at t = 45 ms. The measured pressure at this instant is 2.5 N/m’ and so could be accounted for almost completely by billet expansion were it not for interference from other sources. Of course this one measurement does not constitute an experimental verification of the theory. In fact it would be very difficult to verify the theory by means of experiments performed only on forging machines. It is therefore necessary to design an idealized test rig in which the effect of billet expansion can be isolated from all other noise sources. Such a test rig currently is being designed.

5. VARIATION

OF BILLET DIMENSIONS

5.1. BILLET RADIUS For the same billet height of 3.8 cm, the pressure pulse was computed for different radii by using equation (11). The result is shown in Figure 5(a). It can be seen that the pulse height is almost proportional to R2;this is to be expected and arises from two factors both of which are proportional to R: (a) the surface area of the billet; (b) the initial radial velocity of the billet.

;5 a”

0

1

2

, (b)

,

,

2 Billet ,

3 radius km) , , ,

4

3

4

,

5 ,

_

0

2 Half

billet

5

IwigM km)

Figure 5. Variation of pulse height with billet dimensions (b) billet height; R = 1.25 cm.

for r = 1.5 m. (a) Billet radius; H = 1.9 cm:

5.2. BILLET HEIGHT In addition, for a fixed billet radius of 1.25 cm, the pressure pulse was computed for different billet heights by using equation (11). It can be seen from Figure 5(b) that the pulse height is almost independent of billet height in the range of values considered. Again, this is to be expected for, as the billet height increases, the surface area increases in proportion but the initial radial velocity (as given by equation (9)) decreases inversely. In an actual forging process the billet is struck not once but several times. The radius of the billet progressively increases with each blow and the peak pressure should, according to the results of section 5.1, increase in proportion to the square of the radius as the billet is formed.

NOISE IN IMPACT FORMING MACHINES

331

6. CONCLUSIONS It has been shown that billet expansion is a general effect which always will give rise to a sharp sound pulse. The magnitude of this pulse for the case under consideration (Li = 3 m/s) is 1.4 N/m2 (97 dB re 2 x 10e5 N/m2) and could be a little larger. Thus although billet expansion contributed significantly to the peak pressure, its duration was so short that it made only a small contribution to the integrated sound energy. However, the pulse height was shown to be approximately proportional to the square of the billet radius, independent of billet height, and proportional to the impact velocity. It follows therefore that for an impact velocity of 20 m/s (which frequently occurs in practice) and a billet radius of 5 cm, the above pulse height would be multiplied by a factor of about 100: i.e., an additional peak sound pressure level of 40 dB. It is sometimes considered that it is the total sound energy averaged over a long period of time which is the relevant criterion for hearing damage. Because of its short duration, however, the pulse from billet expansion will make only a small contribution to the total sound energy. Nevertheless, current legislation in the U.S.A. limits the pulse height from impact forming machines to 140 dB and similar levels have been suggested for the U.K. Since it has been shown that billet expansion can produce a significant fraction of the peak pressure, it is an effect which must be considered in reducing noise levels from impact forming machines.

ACKNOWLEDGMENT This work is part of an S.R.C. sponsored project on “Noise Generation in Impact Forming Machines” under the general direction of Dr M. M. Sadek and Professor S. A. Tobias, the latter having suggested this mechanism for noise emission.

REFERENCES 1. M.

M. SADEK 1973 1st Convegne Techniche Su Transformaxionic

Problem;

Technologicke Zndotte DAZ Dell’ Ambiente Di Lavoro E Dali’ Nell’ Zndustria Mechanica Vito, 23-33. Sources of

noise in impact forming machines. 2. G. NISHIMURAand K. TAKAHASHI 1963 Bulletin of the Japanese Society of Precision Engineering

1,48-51. Impact sound by mutual collision of two steel balls. 3. G. H. TRENGROUSEand F. K. BANNISTER1973 American Society of Mechanical Engineers Machine Design and Vibration Conference, Cincinnati, Ohio, U.S.A. Noise due to an air ejection from clash surfaces of an impact forming machine. 4. A. E. M. OSMAN, W. A. KNIGHT and M. M. SADEK 1974 Transactions of the American Society of Mechanical Engineers Journal of Engineering for Industry 96, 233-240. Noise and vibration analysis of an impact forming machine. 5. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill Book Company, Inc. 6. J. W. COOLEY and J. W. TUKEY 1965 Mathematics of Computation 19,297-301, An algorithm for the machine calculation of complex Fourier series. 7. V. GRIGORIANPh.D. Thesis, University of Birmingham (in preparation). 8. H. LAMB 1932 Hydrodynamics Cambridge University Press. Sixth edition, see pages 522-523.

APPENDIX In order to derive equation

(3) it is necessary

A

to solve the wave equation

332

D. C. HODGSON AND J. E. BOWCOCK

subject to the boundary condition

ap

ah

Z&=-pat

to be satisfied on the different surfaces of the simplified forging machine shown in Figure 1. Thus applying equation (A2) to the various surfaces of Figure I gives on the billet surface,

for Iz] > H,

0,

aP

z = 0,

(A3)

for]z]=HandR
(A5)

Upon taking the Fourier transform of the whole problem, equations (Al), (A3), (A4) and (A5) become V’P(r, 2, w) + k2P(r, 2, w) = 0,

646)

= -p4o),

(A7)

for ]z] > H,

WI

for[.z]=HandR
rp(r,z,r)el”dt

and

-m

A(o) =

r

a(r) ei”’dt. --co

Equation (A6) must be solved in regions I and II separately and the solutions and their derivatives matched at the boundary joining the two regions: i.e., at r = L. In cylindrical co-ordinates equation (A6) becomes

azp

1 ap

av

arz+;ar+

%+k”P=O.

(AlO)

Equation (AlO) is solved easily by the separation of variables; i.e., assumingP(r,z) = R(r)Z(z) leads to two ordinary differential equations each with two linearly independent solutions: R(r) =

‘(‘)=

(Al 1)

where k2 = kP + k:. As is well known [5, see p. 3571, Hg)(k,r) and Hb2)(k,r) represent diverging and converging cylindrical waves, respectively. Both waves will be present in region I because the pulse undergoes a partial reflection at r = L but since region II is infinite in extent, the pressure pulse there will be made up from only the diverging terms.

333

NOISE IN IMPACT FORMING MACHINES

In region I, for a narrow gap, there will be almost purely cylindrical waves (in general noncylindrical wave fronts are allowed), which implies k, = 0. In fact there will be a small dependence of the pressure wave on z within region I as the wave nears the edge of the gap. The inclusion of this makes the analysis more involved and should not significantly affect the results. In the interests of simplicity therefore this dependence on z will be neglected. Hence the most general solution in region I is PI = aH($(kr) + pH$?(kr).

(Al2)

Note that this solution satisfies condition (A9) since it is independent of z. A relation between CIand j? can be determined by imposing condition (A7) : = kaH’d”(kR) + kj?Hf”(kR) = -k[srH:l’(kR)

+ bH’:‘(kR)] = +4(w),

(A13)

where primes denote differentiation with respect to the argument and the relationship HA(z) = -H,(z) has been used. where k, = v’m In region II, the solution of equation (AlO) is Hb’)(v’er)cosk,z, from equations (Al 1) and only the cos k, z solution has been used as the problem is symmetrical about the plane z = 0; the sink,z solution cannot occur as it is an odd function of z. Neither can Hz’(k,r) occur as this represents converging cylindrical waves which are not allowed in region II. Furthermore, in II there are no restrictions on k, and so it is necessary to integrate over all possible values in order to obtain the most general solution : P,,(r, z, o) = /C(k,,

o) Hb”(mr)

cos k,z dk,.

(A14)

0

Here C(k,, o) is an arbitrary function of k, and w which is to be determined by the boundary conditions. It remains now to impose the continuity of P(r,z,w) and its radial derivative across the boundary between I and II. First the derivative continuity requires that -k[crH:“(kL)

+ flH:Z’(kL)], 0,

x H:“(dnL)cosk,zdk,.

(A15)

This is a Fourier cosine transform of the form Y,

0, the inverse of which is

n F(k,)=f

s

ycosk,zdz=;--,

2y sink, H

0

z

where y is a constant. Applying this inversion formula to equation (A15) gives 2k crH:“(kL) + j?H\*‘(kL) C(kz’ w) = y dk2 _ k; H:“(,/mL)

sink, H

k,

Thus C(k,,o) has been determined in terms of c1and p. Upon substituting for C(k,,w), equation (A14) now becomes P,,(r, z, o) = c

[crH\‘)(kL) +

BH’f’(kL)IZ(r, z, k),

0546)

334

D. C. HODGSON AND J. E. BOWCOCK

where I(r, z, k) =

msink,Hcosk,z s

0

k,m

Ht)(mr)

dk,.

H:“( -L)

6417)

Now imposing the continuity of P(r, z, o) across the boundary between I and II gives aHb”(kL) + /3H:‘(kL)

=f

[orH\“(kL) + /_?H:“(kL)] I@, z, k).

6418)

Solving equation (Al 8) for /~/CC gives B -= CI

Hb”(kL) - GZ(L, z, k) H:“(kL) = G(k), -H$?(kL) + ;Z(L,

(A191

z, k) Hi2’(kL)

where G(k) is the complex reflection coefficient. Eliminating /3 from equations (A13) and (Al 9) gives ka[H\“(kR) + G(k) H\“(kR)] = pA(o). (A20) Substituting a from equation (A20) and /?/a from equation (A19) into equation (A16) results in P&, z, 0) = ‘f A(U)

H\“(kL) + G(k) H;“(kL) H:“(kR) + G(k) H\“(kR)

Z(r, z, k),

Wl)

in which everything on the right-hand side is known. In order to obtain the pressure pulse in the time domain, equation (A21) is inverted to give (9

p(r, z, t) = f

I -m

A(o)

H’,“(kL) + G(k) H:2’(kL) H:“(kR) + G(k) H’:‘(kR)

Z(r, z, k) e-‘“” do

(A22)

which is the required result,

APPENDIX B Here a derivation of equation (12) is given. The problem is to calculate the sound pulse radiated by a sphere of radius R whose surface expands radially with an impulsive acceleration a(r) = uo&t). Lamb [8] has solved a similar problem in which the impulse consists of a sideways translation rather than a radial expansion. The pressure pulse thus calculated is of a similar form to equation (12) but because of the sideways movement, the angular distribution of the pulse is not isotropic. Because of the spherical symmetry, the solution depends only on the radial co-ordinate r and time. It is necessary therefore to solve the wave equation

subject to the boundary condition

032) at r = R. As before, taking the Fourier transform of equations (Bl) and (B2) gives @3)

NOISE IN IMPACT FORMING

335

MACHINES

and

ap

(ar1

=

U34)

-pA(o).

r-R

Separating variables in equation (B3) by the standard method and imposing condition (B4) leads to the solution eik+R) w,

0)

=

p$A(o)

1

,

which, on taking the Fourier transform to give the pressure pulse in the time domain, gives e-iw[t-(r-R)/cl Ato)

ioR

do.

(B6)

If as before a(t) = ~~8(t) and hence A(w) = uo, the integral (B6) can be done very simply by contour integration. Upon noting that the integrand has a simple pole at w = -it/R, the integral can be done by closing a semi-circular contour in the upper half plane for t < (r - R)/c and in the lower half plane for t > (r - R)/c, giving the required

result:

0, Art

t)

=

PCRVOe _ (clR)Ct-(r-R)/cl

,

r

APPENDIX

radial acceleration of billet surface transform of a speed of sound (= 330 m/s) arbitrary function of k,, o normal derivative operator instantaneous semi-height of billet initial semi-height of billet Hankel functions of first and second kinds respectively and of order n 2/=T wave vector magnitude of wave vector radius of die acoustic pressure cylindrical co-ordinates initial radius of billet time normal velocity component of machine surface impact velocity velocity to which surface of spherical billet is impulsively accelerated volume of billet instantaneous radius of billet arbitrary constants Dirac delta function density of air (= 1.29 kg/m3) angular frequency

A”Fourier C

a$ h H

H’,“, H’,2’ (k,,O, kz;

k L

(r,b,f,

R t &I u vo V X

a,

8,

Y

act> P co

C : NOTATION