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Stress pulse dispersion in helical springs
Journal of Sound and Vibration (1971) 18 (2), 247-259
STRESS PULSE DISPERSION
IN HELICAL
SPRINGS
W. G. B. BRITTON AND G. 0. LANGLEY Department of Physics, Chelsea College of Science and Technology, London, S. W.3, England (Receiaed 29 March 197 1) This paper presents the results of investigating the effect of curvature and pitch on stresswave propagation in helical springs. It supplements a previous paper in which the work carried out on a single spring of fixed curvature and pitch was described and the experimental results compared with theory. A short duration, wide-band pulse technique, described in the first paper, was used to investigate the nature of the propagation. Theoretical dispersion curves, appropriate to the springs used, werecomputedfor comparison with theexperimental results which were interpreted according to Kelvin’s principle of stationary phase.
1. INTRODUCTION In a previous paper [I] the authors presented experimental results and associated theoretical curves for the dispersion of elastic waves in a curved mechanical waveguide. The particular waveguide used was a cylindrical bar of radius of cross-section, a, wound as a helical spring of mean radius R and pitch angle c(. The spring used was made of 0.25 in (6.35 mm) diameter steel rod and had 12 turns. The central elastic line was a helix with CC= 3.9” and a/R = 0.106. TABLE 1
Spring
alR
2a *to.01 cm
2R *O.lO cm
Pitch kO.lOcm
CY’
No. of turns
A
l/8
064
5.08
0.79
2.8
40
: D E F G
l/8 l/4 l/4 l/4 l/16
064 064 064 064 064
5.08 254 254 254 10.16
2.54 1.27 0.79 1.27 1.91 0.74
9.0 4.6 5.7 9.0 13.4 1.4
40 36 80 70 50 20
The experimental results presented in this paper are obtained from the springs listed in Table 1. The wide-bandwidth pulse method previously described [I, 21 was used. All the springs were made of hard stainless steel in which attenuation of stress waves in the frequency range considered is negligible. It is seen that springs of three different curvatures were considered corresponding to values a/R = l/4, l/8 and I/ 16. Three different values of pitch angle, a (listed in Table l), were used for each of the cases a/R = l/4 and a/R = l/S whereas there was only one spring with a/R = l/ 16. The experimental conditions were chosen as previously [l] to generate either predominantly-longitudinal or predominantly-flexural waves and in each case the results were compared with theory. However, some of the records show the presence of a stress component 247
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W. G. B. BRITTON AND G. 0. LANGLEY
with dispersion characteristics consistent with theoretical predictions for a torsional mode, thus confirming earlier results [1] which were attributed to a torsional mode. 2. THEORY 2.1.
GENERAL
A general discussion on the derivation of characteristic or dispersion equations for curved elastic waveguides was given in the previous paper; there is a more detailed discussion in reference [2]. Theoretical curves shown in this paper are computed from the approximate theories of Timoshenko [3], Morley [4] and Wittrick [5], as before. Considering first the flexural mode only, the Timoshenko, Morley and Wittrick theories deal with flexure in a straight rod, in the plane of a circular ring and in a helical spring respectively: the Morley and Wittrick theories also predict the existence of modes other than flexural modes and these will be considered later. Each of these approximate theories allows for displacement, rotation and shearing of the elements of the waveguide and are therefore expected to give results which are in good agreement with exact-theory for the lowest flexural mode. The Wittrick theory, however, assumes small curvature (a/R < 0.1). The Morley theory puts no restriction on curvature but the equations assume a simplified form if the curvature is taken to be slight. For the case of a straight rod (a/R = 0) the equations of both the Morley and Wittrick theories which relate to flexure reduce to that of the Timoshenko theory. Before considering the theory for a helical spring, we will discuss the case of a plane ring which can be regarded as a helical spring of zero pitch angle, CL.In this case of Q = 0 we can consider separately (i) flexure in the plane of the ring and (ii) flexure perpendicular to the plane of the ring: the two modes are not coupled. 2.2.
THE CIRCULAR RING: FLEXURE IN THE PLANE OF THE RING
The Morley-theory equations for vibrations in the plane of a ring have been solved for values of a/R = l/4, l/8 and 1/16, assuming material of Poisson’s ratio, v = 0.29. The phase velocity curves are similar to those shown in Figure 2 of the previous paper [I] and are not included here. However, the arrival-time curves (see section 2.4) are shown in Figure 1. These curves are presented in a non-dimensional form by plotting Tp/Ta against t/T,, where T, is the predominant period of a group of waves which produce the main effect at a distance x from the excitation point at a time t after the initiation of the pulse, To = x/co and T, = a/c,: c,, is the velocity of extensional waves of infinite wavelength in a straight rod. Three modes of propagation are predicted by the Morley theory, longitudinal, flexural and radial-shear. The radial-shear mode which has a short predominant period for all values of t/To, was not investigated experimentally and the arrival-curves are not shown here. The flexural-mode arrival-curves shown in Figure 1 are for a/R = l/4,1 /8 and zero (straight rod) : the corresponding curve for a/R = l/16 lies too close to the straight-rod curve to be drawn separately on the diagram. However, as shown in the diagram, there is appreciable separation of the longitudinal-mode arrival-curves for a/R = l/4, l/8 and l/16. It must be noted that restrictions made on the deformation of the cross-section of the rod lead to an elementarytype theory for the longitudinal modes. It is therefore expected that for these modes agreement between experiment and theory would only be good in a region where the wavelength is large compared with the lateral dimensions of the waveguide. It was stated previously that curvature could be considered slight for values of a/R -c 0-I. It is found, however, that even for a value of a/R as high as 118 calculations based on the
Morley slight-curvature approximation agree, within the accuracy of the graphs in this paper, with the results obtained from the more complicated equations of the full Morley theory. The Morley equations assuming slight curvature are, as expected, identical with the equations obtained for vibrations in the plane of a circular ring by putting 0:= 0 in the Wittrick theory.
STRESS PULSES IN HELICAL
249
SPRINGS
240 2x)200180160140120100-
I
2
3
4
Figure 1. The Morley theory arrival-curves. Curves 1,2 and 3 are associated with the lowest flexural mode in rings of a/R = l/4, l/8 and 0, respectively. Curves 4, 5 and 6 are associated with the longitudinal mode in rings of a/R = l/4, l/8 and l/16, respectively.
2.3.
THE CIRCULAR
RING:
VIBRATIONS
PERPENDICULAR
TO THE PLANE
OF THE RING
Theoretical dispersion curves for vibrations perpendicular to the plane of a ring have been obtained by solving the appropriate Wittrick equations for a helical spring of zero pitch angle. The corresponding arrival-curves for a/R = l/S and l/4, and Poisson’s ratio, v = 0.29, are shown in Figure 2, together with the Timoshenko-theory curve for a straight rod. The two bending-moment mode curves of the Wittrick theory are omitted since these highfrequency modes were not investigated experimentally. It will be remembered that the Wittrick theory is for slight curvatures only (a/R cc O-1). However, for the reasons indicated in section 2.2, we expect the theory to be sufficiently accurate for our purposes for values of a/R < l/8 : it is also expected that the arrival-curves for a/R = l/4 in Figure 2 are a good indication of the nature of the dispersion in this case. It is seen from Figure 2 that for a ring of a/R = l/4, the predominant period of the torsional mode is less than that of the flexural mode for nearly all values of t/To and that a predominant period of approximately 3OT, is predicted for a large range of arrival times. Since T, is about 0.62 ps in this case, the oscillations predicted have a predominant period of around 19 ps. 2.4.
THE HELICAL
SPRING
The general characteristic equation due to Wittrick is extremely complicated to solve but roots have been computed for a/R = O-106, tc = 0 and cc= 3.9” using the London University ATLAS computer. This value of a/R is for the spring originally investigated and the phasevelocity curves were plotted in the previous paper [I] together with the arrival-curves for cr = 0. Considering first tc = 0, it was shown that, within the scale of the diagrams, the two flexural mode arrival-curves were coincident. Numerical calculations indicate however that
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W. G.
B. BRITTON AND G. 0. LANGLEY
Figure 2. The Wittrick theory arrival-curves for vibrations perpendicular to the plane of a ring. Curves 1, 2 and 3 are associated with the lowest flexural modes in rings of a/R = l/4,1/8 and 0, respectively. Curves 4 and 5 are associated with the torsional mode in rings of a/R = l/4 and l/8, respectively.
240 -
220 -
200 -
180 -
160 -
140 h? k
120 -
100 -
80 -
60 -
2
3
4
t/r,
Figure 3. Wittrick theory arrival-curves for the faster flexural mode in a helical spring. Curve 1 is associated with a ring of a/R = 0.106 and c( = 0. Curve 2 is the flexural mode curve for a straight rod shown for comparison. 0, Values calculated for a spring of a/R = 0.106 and a = 3.9”.
STRESS PULSES IN HELICAL
SPRINGS
251
flexural waves in the plane of the ring have a slightly higher phase velocity than flexural waves perpendicular to the plane of the ring with the same wavelength. The phase velocity curves for a = 0 and a = 3.9” shown in the previous paper indicate that an increase of pitch angle a has the effect of separating the two flexural modes. This separation is, however, not very sensitive to changes in a. The arrival-curve for flexure in the plane of a ring for a/R = 0.106 is shown in Figure 3 together with the Timoshenko arrival-curve for a straight rod. The points surrounded by circles were calculated for the faster flexural mode for a spring of a/R = O-106, a = 3.9”. It is seen that the points for a = 3.9” lie very close to the curve for a = 0. It is apparent from these results that the effect of pitch angle a on the flexural dispersion characteristics is small compared with the effect of changes in a/R. In view of the insensitivity to a of the arrival curves, the Morley theory and Wittrick theory for a = 0 have been used to calculate arrival-curves for the springs investigated in this work. However, in interpreting experimental results it is remembered that an increase in pitch angle will tend to separate the two flexural modes of propagation. 2.5.
THE METHOD OF STATIONARY
PHASE
The experimental results have been interpreted according to Kelvin’s principle of stationary phase. The method has been fully described by Davies [6] and outlined by the authors [l]. The following will be sufficient to indicate the meaning of the symbols used in the diagrams. Consider an initial stress pulse of infinitely short duration, initiated at x = 0 at time t = 0, where x denotes distance along the central elastic line of an infinitely long waveguide. This pulse can be considered to have a uniform distribution of all possible frequencies. If it is assumed that at the origin the sinusoidal components are of equal amplitude and phase, it is possible, according to Kelvin’s method of stationary phase, to deduce the mean period T, of a group of waves which produce the main effect at a point x at time t. It can be shown [l, 61 that for a given mode of propagation, a theoretical expression can be found for T, as a function of wavelength X provided that the variation with wavelength of the phase velocity, c and the group velocity, c, is known. The variables can be plotted in a non-dimensional form by plotting Tp/Ta against t/T,, as discussed in section 2.2 of this paper. The (TJT,, t/T,) curves are called arrival-curves. The method of stationary phase assumes pulses of infinitely short duration. However, it has been found experimentally, for straight rods, that this method of analysis leads to results which are in agreement with other methods and with the predictions of the exact theory. 3. EXPERIMENTS 3.1.
EXPERIMENTAL
PROCEDURE
The experimental set-up was fully discussed in the previous paper [l] and only an outline of the experimental procedure is given here. Short stress pulses were initiated at one end (x = 0) of the waveguide by applying electrical pulses at a fixed repetition rate to a small coil surrounding a magnetostrictive nickel wire which was in contact with the waveguide at x = 0. The mechanical disturbances introduced by this generating transducer were recorded at a point a distance x along the waveguide by means of a small lead-zirconate surface strain gauge. The electrical signals induced in the recording gauge were applied after amplification to the Y-plate of a cathode ray oscilloscope. Electrical pulses, synchronous with the electrical pulses, supplied to the generating transducer, were used to trigger the X-sweep of the oscilloscope. In this way a stationary (stress, time) pattern was obtained on the oscilloscope screen similar to those shown in Plates 14. The recording circuit contained components which reduced extraneous signals to a minimum and also a variable filter which could be used as a low-pass, band-pass or high-pass
252
W. G. B. BRITTON AND G. 0. LANGLEY
filter. This filter was particularly useful in analysing a trace which was due to the presence of two or more modes of widely different predominant period arriving at the gauge simultaneously : judicious use of the filter made it possible to investigate the dispersion characteristics of each mode separately. The generating transducer was always oriented so that the longitudinal mode or one of the two flexural modes of the spring would predominate. However, in practice, an appreciable amount of energy was carried by modes other than the predominant mode and the variable filter was used as described above. The interpretation of a record becomes more difficult when it is due to two or more modes of similar period arriving simultaneously at the strain gauge. The length of a strain gauge limits the shortest wavelength it can faithfully record. For a O-25-in (6.35 mm) diameter rod a strain gauge of length 0.125 in (3.175 mm), as used in this work, permits the investigation of dispersion in a region corresponding to 0 < a/h < 0.2. [I ] 3.2.
ANALYSIS OF THE STRESS PATTERN
Direct measurement of T, for a group of waves at a point x along the waveguide is made by measuring the time interval corresponding to the horizontal distance between the successive points where the trace crosses the zero strain line in the same sense [I, 21. If the time, t, from the initiation of the pulse is equal to T, at the first cross-over point and is T2 at the next point of cross-over in the same sense, then T,, = Tz - T, and the corresponding arrival time is taken to be t = (T, + T,,,,). 4. EXPERIMENTAL
RESULTS
4. I. INTRODUCTION
The experimental results obtained for a spring of fixed curvature and pitch (a/R = 0.106, CI= 3.9”) were shown in the previous paper [l] to be in good agreement with theory. The springs listed in Table 1 were used to investigate the effect of curvature and pitch on the dispersion characteristics of various modes of propagation. The results obtained for the longitudinal mode and the two flexural modes will be described under separate headings. 4.2.
LONGITUDINAL STRESS PULSES
The generating transducer was oriented to excite preferentially the longitudinal mode. A pulse of 20 ps duration was introduced into the waveguide and the resulting (strain, time) profile was detected at various points along the waveguide by means of surface strain gauges. The nature of longitudinal stress wave dispersion in helical springs is very different from that in a straight rod. The stress pattern for the case of a helical spring shows marked dispersion at large predominant periods, that is at low values of a/h. This region is relatively free from dispersion in the case of a straight rod. A more complete discussion of the effect and examples of the type of oscilloscope trace obtained have been given previously [ 1,2]. Figure 4 shows a plot of experimental points obtained from the oscilloscope traces. The variable filter was used, in analysing these records to remove from the traces frequencies outside the range corresponding to a/X ==z O-2. The high frequency components thus removed were found qualitatively to have similar dispersion characteristics to those of high frequency longitudinal waves in a straight rod. The results shown in Figure 4 were obtained from springs A (a/R = l/S, c( = 2*8”), G (a/R = l/16, cc= 1.4”) and C (a/R = l/8, cc= 9”). It is seen that for each spring the experimental points are in close agreement with the Morley theory extensional-wave arrival-curve corresponding to a circular ring of the same value of a/R. It will be remembered that for values of a/R up to l/8, the Wittrick theory curves lie very close to those of the Morley theory (see section 2.2). These experimental results confirm that, whereas the dispersion characteristics of longitudinal waves change appreciably with changes in a/R, changes in u have little
Plate 1. The flexural pulse profile at 150 cm along a straight, 0.25-in (6.35 mm) diameter ., rod. 1‘he time marker s are 100 ps apart.
Plate 2. The flexural pulse profile at 207.7 cm along helical spring A. The generating transd cer was orientec i to preferentially excite the faster flexural mode. The time markers are 100 ~LSapart.
(facing p. 252)
PIat e 3. The flexural pulse profile (faster mode) at 208.1 cm along helical spring B. The time markers are 100 ps apart.
Plate: 4. The flexural pulse profile (faster mode) at 210.1 cm along helical spring C. The time markers are 100 ps apart.
STRESS PULSES IN HELICAL SPRINGS
253
1::
15
2
2.5
Figure 4. Curves 1 and 2 are the Morley theory curves for the longitudinal mode in rings of a/R = I /8 and 0, A, Experimental points obtained by exciting springs G, A and C respectively along their central elastic line. 1/I6 respectively. 0,
effect : the experimental points from springs A and C which have appreciably different pitch angle lie along the Morley curve for a/R = l/8 and the points from spring G lie along the Morley curve for a/R = l/ 16. 4.3. FLEXURAL MODE: FASTER WAVE The faster flexural wave has vibrations which are predominantly transverse to the axis of the spring. This mode corresponds to vibrations in the plane of the ring for a spring with a = 0. The generating transducer was orientated so that this faster flexural mode would predominate and again stress pulses of 20 ps duration were injected into the springs. Figure 5 shows the results obtained from springs A (a/R = l/8, CI= 2.8”) and D (a/R = l/4, x = 5-7”) together with those obtained from a straight rod which are included for comparison.
80 -
60 -
4020 -
I
I
4
,
I
I
2
3
4
t/T,
Figure 5. Curves 1,2 and 3 are the Morley theory curves for flexure in the plane of a ring of a/R = l/4,1 /S and 0 respectively. 0, 0, Experimental points obtained by exciting springs D and A respectively in a direction perpendicular to the axis of the spring. A, Experimental points obtained from a straight rod.
254
W. G. B. BRITTON AND G. 0. LANGLEY
It is seen that the dispersion is sensitive to changes in a/R but whereas the results for spring A (a = 2.8”) are in close agreement with the Morley arrival-curve, for spring D (a = 5.7”) there is some scatter of experimental points around the corresponding Morley curve. This type of scatter is indicative of the presence of at least two modes of similar dispersion characteristics. It will be remembered that an increase in a separates slightly the dispersion curves for the two flexural modes in a spring. It appears that for spring D (a/R = l/4, a = 5.7”) both modes are present in appreciable quantity.
Figure 6. The curve is the Morley theory curve for flexure in the plane of a ring of a/R = l/8. 0, ?!,? Experimental points obtained by exciting springs A and B, respectively, in a direction perpendicular to the axis of the spring.
Figure 6 shows results obtained from springs A (a/R = l/8, u = 2.8”) and B (a/R = l/8, a = 4.6”). There is no appreciable scatter of experimental points about the Morley curve for this value of a/R even for spring B (oz= 4.6”). However, for spring C (a/R = l/8, a = 9”), which is of the same curvature, the pitch angle is sufficiently high to show the scatter of points characteristic of the presence of more than one mode. (See Figure 7.) In Figure 8 the points obtained from springs E (a/R = l/4, a = 9”) and F (a/R = l/4, tc = 13.4”) show scatter and appreciable deviation from the arrival-curve for a/R = l/4 which was obtained from the Morley theory for pronounced curvature. This effect may again be attributed to a complex waveform consisting of two or more modes. From the above results it appears that the nature of the flexural wave dispersion is very sensitive to changes in a/R but not to changes in a. However, as cc increases it becomes difficult to excite only the faster flexural mode: there is also some evidence that the larger the value of a/R, the smaller tc must be to avoid generating more than one flexural mode. The presence of more than one flexural mode has a marked effect on the appearance of the oscilloscope trace. It is worth examining the stress patterns obtained from a straight rod and from springs A, B and C (for all of which a/R = l/8) shown in Plates l-4. In the case of a
255
STRESS PULSES IN HELICAL SPRINGS
I
I
240
J
22iJ
200
180
160
140 Lp ;=_
120
100
8C
,-
6CP-
40
20 I I
\
,
I
2
3
I 4
f/T,
Figure 7. The curve is the Morley theory curve for flexure in the plane of a ring of a/R = l/8. points obtained by exciting spring C in a direction perpendicular to the axis of the spring.
0,
Experimental
240
160
Figure 8. The curve is the Morley theory curve for flexure in the plane of a ring of a/R = l/4. o, 0, Experimental points obtained by exciting springs E and F, respectively, in a direction perpendicular to the axis of a spring.
256
W. G. B. BRITTON
AND G. 0.
LANGLEY
straight rod the stress amplitude at x = 150 cm (Plate 1) rises to a certain level and remains at this level for the rest of the observed range of arrival times: the ripples in the tail of the trace are due to reflections from the far end of the rod. The type of pattern observed is consistent with the presence of only one flexural mode and the dispersion characteristics are those predicted by the Timoshenko theory. In the record from spring A (a = 2.8, x = 207.7 cm), shown in Plate 2, we see that a small high-frequency flexural pulse precedes the main pulse. The shape of the Morley-theory arrival-curve for flexure (Figure 1) indicates that in the initial portion of a recorded flexural signal, two high-frequency components would arrive simultaneously: the superposition of these components would lead to a precursor pulse of the type recorded in Plate 2. In the main pulse the amplitude rises to a maximum and a slight decrease of amplitude is noted at the largest arrival times observed. This is characteristic of interference between two modes of similar period travelling at slightly different velocities in the slower, low frequency, region. The record (Plate 3) from spring B (a = 4.6”, x = 208.1 cm) again shows the initial high-frequency components and a maximum stress amplitude in the main pulse followed by a more rapid decrease of amplitude with increased arrival time than shown in Plate 3. In Plate 4, taken from spring C (a = 9”) at x = 210.1 cm, we again see the initial high-frequency stress components. The stress pattern then rises to a maximum and falls to a minimum before rising to a second maximum. A comparison of the three spring records shows that as a increases the effects of interference are seen earlier in the trace. This is precisely what is expected from theoretical considerations. As indicated in section 2.4 and illustrated in Figure 4 of our previous paper [I] the two flexural mode dispersion curves for a spring separate increasingly in the low-frequency range with increasing a. This is discussed further in section 5. 4.4.
FLEXURAL
MODE:
SLOWER
WAVE
The slower flexural wave has vibrations which are predominantly parallel to the axis of the spring. In the case of a circular ring (a = 0), this mode involves vibrations which are perpendicular to the plane of the ring. The generating transducer was orientated so that the slower flexural mode would predominate and pulses of 20 ys duration were injected into the spring. The stress pattern at some distance from the point of initiation of the stress pulse is not of the familiar flexural type. As described in our previous paper [l], the traces show the presence of at least two modes of different predominant periods arriving simultaneously. In the initial region of the pattern a low-frequency oscillation is superimposed on the main pulse. The tail of the pulse shows features which are characteristic of the superposition of two vibrations of similar period. In order to analyse the low-frequency component of the initial portion of the trace, the variable filter mentioned in section 3.1 was adjusted to remove the high-frequency components from the pattern. The results obtained from the oscilloscope traces are presented as arrival-curve plots in Figures 9 and 10. The points shown in Figure 9 were obtained by analysing records taken from spring B (a/R = 1IS, a = 4.6”). The theoretical curves 1 and 2 in Figure 9 were obtained from the Wittrick theory for a circular ring with a/R = l/8 and curve 3 is the Timoshenko arrival-curve for flexure in a straight rod. The points surrounded by squares in Figure 9 were obtained from the high-frequency components of the initial portions of the traces. It is seen that these points are in good agreement with the torsional-mode arrival-curve. The experimental points, surrounded by circles, from the tail of the pulse show general agreement with the arrival-curves for the flexural mode (a/R = l/8, a = 0) but there is some scatter. It will be remembered that no scatter was observed when investigating propagation of the faster flexural mode in spring B. Figure 10 shows the experimental points obtained from spring E (a/R = l/4, a = 9”). It is seen that there is again scatter of the flexural-mode points and a greater deviation from the
STRESS PULSES IN HELICAL SPRINGS
257
160
140 h? 120 ;=_ 100
60
40
20
I
2
3
4
Figure 9. Curves 1 and 2 are the Wittrick theory curves for flexure perpendicular to the plane of a ring for a/R = l/8. Curves 1 and 2 are associated with the flexural and the torsional modes, respectively. Curve 3 is the Timoshenko curve for flexure in a straight rod, shown for comparison. 0, 0, Experimental points obtained by exciting spring B in a direction parallel to the axis of the spring.
220 -
2cO-
180 -
160 -
140 h" La
120 -
100 -
80-
60-
Figure 10. Curves 1 and 2 are the Wittrick theory curves for flexure perpendicular to the plane of a ring of a/R = l/4. Curves 1 and 2 are associated with the flexural and the torsional modes respectively. Curve 3 is the Timoshenko curve for flexure in a straight rod. 0, Experimental points obtained from spring E.
258
W. G. B. BRITTON AND G. 0. LANGLEY
corresponding theoretical arrival-curve than there was in the results for a/R = l/8 shown in Figure 9. These results are similar to those obtained when investigating the faster flexural mode in spring E and can be interpreted as being due to the simultaneous presence of the faster and slower flexural modes. It will also be remembered that the Wittrick theory is for slight curvature and that the flexural arrival-curve for a/R = l/4, cc= 0 in Figure 10 is only an indication of the effect of curvature on the dispersion. Some of the experimental points shown in Figure 10 lie close to the torsional mode-curve predicted by the Wittrick theory for a/R = l/4, a = 0. It is seen that for this value of a/R, theory predicts a torsional mode which has a smaller predominant period than the flexural mode over the whole range of arrival times shown. According to the torsional mode-curve, there is an oscillation ofpredominant period T, equal to approximately 3OT, : this corresponds to an oscillation with a period Tp of about 19 vs. Such an oscillation, which extended over a large part of the stress pattern, was observed experimentally [2]. 5. DISCUSSION The investigations described in this paper are an extension of those carried out on a single spring, with parameters a/R = O-106, a = 3*9”,which were reported in our previous paper [I]. The earlier work showed that experimental results for this spring were in good agreement with the theories of Morley and Wittrick and that the difference in the theoretical dispersion curves for CL= 0 and c( = 3.9” was slight for the case a/R = 0.106. The springs listed in Table 1 were used to investigate the effects of curvature and pitch angle on the dispersion characteristics of various modes of stress-wave propagation. In each case the experimental results have been compared with the arrival curves for the appropriate value of a/R, calculated from the Morley theory or from the Wittrick theory. In view of the insensitivity to changes in CLof the theoretical curves, the arrival curves obtained from the Wittrick theory were all calculated for the case a = 0. Considering first the results for extensional (longitudinal) waves, shown in Figure 4, it is seen that they are in good agreement with the Morley theory curves for a circular ring. A change in CLfor a fixed value of a/R has a negligible effect on the dispersion characteristics of the longitudinal mode. Two flexural modes were investigated. The faster flexural mode involves vibrations which are mainly perpendicular to the axis of the spring while the slower mode has vibrations which are mainly parallel to the axis of the spring. The results for the faster flexural mode are shown in Figures 5-8. As in the case of the longitudinal mode, there is general agreement between experiment and theory. However, for a given value of a/R, an increase in czleads to an increase in the scatter of experimental points about the theoretical arrival-curve. In the case a/R = l/B, for example, the experimental points lie close to the theoretical curve for a = 2.8” and cz= 4.6” but appreciable scatter is observed for tc = 9”. The scatter and deviation from the theoretical curve is even greater for a/R = l/4 with cc= 9” and cz= 13.4”. These results are attributed to the presence of the slower flexural mode in appreciable quantity in addition to the faster mode. According to the Wittrick theory, the dispersion curves for the two flexural modes in a spring are coincident for CC = 0 but there is a slight separation in the curves for a finite value of cr: the separation increases with increasing a. This slight difference in phase velocity for any particular value of wavelength will lead to interference between the two modes and the scatter of experimental points is attributed to this effect. The interference can clearly be seen in the oscilloscope traces such as those shown in Plates 2-4. It is even more difficult to generate a nearly-pure mode in the case of the slower flexural wave. A scatter of experimental points is observed in Figure 9, for a/R = l/S, a = 4*6“, which
STRESSPULSES IN HELICAL SPRINGS
259
indicates the simultaneous presence of the slower and faster flexural wave. It will be remembered that it was possible in the same spring to generate the faster flexural wave without introducing an appreciable amount of the slower wave. The presence of the two modes is seen to be even more pronounced in Figure 10 which shows the results obtained from a spring with parameters a/R = l/4 and CL= 9”. The presence of a torsional mode was detected during investigations on the slower flexural mode. The results presented in Figures 9 and 10 are in good agreement with theory for this mode and show that a change in curvature has a pronounced effect on the dispersion characteristics of the torsional mode. It is worth noting that the lowest order torsional mode in a straight cylindrical rod is non-dispersive [6]. It is seen that the experimental results show that curvature and pitch angle affect the dispersion characteristics of stress wave propagation in a helical spring according to the predictions of the Wittrick theory. The results can be interpreted with respect to four of the six modes predicted by the theory: the two highest, bending-moment modes were not investigated. It will be remembered that the Wittrick theory assumes slight curvature. However, in the case of the faster flexural wave for CC= 0, comparison can be made with the Morley theory which places no restriction on curvature. The dispersion curves of the two theories for the values of a/R used here are found to lie close to each other. The stress pulses introduced into the waveguides in the investigations described here were all of approximately 20 ps duration. The bandwidth of such a pulse is sufficiently narrow to excite only the lowest order modes. However, it has been found that a pulse of duration 8 ~LS leads to the appearance of higher frequency modes in appreciable amplitude. A study of the higher modes, using pulses of shorter duration, would be a useful extension of this work. REFERENCES 1.
2. 3. 4. 5. 6.
W. G. B. BRITTON and G. 0. LANGLEY 1968 Journal of Sound and Vibration 7, 417-430. Stress pulse dispersion in curved mechanical waveguides. G. 0. LANGLEY 1968 Ph.D. Thesis, University of London. Stress pulses in curved mechanical waveguides. S. P. TIMOSHENKO1922 Philosophical Magazine 43, 125-131. On the transverse vibrations of bars of uniform cross-section. L. S. D. MORLEY 1961 Quarterly Journal of Mechanics and Applied Mathematics 14, 155-172. Elastic waves in a naturally curved rod. W. H. WITTRICK 1966 International Journal of Mechanical Sciences 8, 2547. On elastic wave propagation in helical springs. R. M. DAVIES 1948 Transactions of the Royal Society A 240, 375-457. A critical study of the Hopkinson pressure bar.
17
STRESS PULSE DISPERSION
IN HELICAL
SPRINGS
W. G. B. BRITTON AND G. 0. LANGLEY Department of Physics, Chelsea College of Science and Technology, London, S. W.3, England (Receiaed 29 March 197 1) This paper presents the results of investigating the effect of curvature and pitch on stresswave propagation in helical springs. It supplements a previous paper in which the work carried out on a single spring of fixed curvature and pitch was described and the experimental results compared with theory. A short duration, wide-band pulse technique, described in the first paper, was used to investigate the nature of the propagation. Theoretical dispersion curves, appropriate to the springs used, werecomputedfor comparison with theexperimental results which were interpreted according to Kelvin’s principle of stationary phase.
1. INTRODUCTION In a previous paper [I] the authors presented experimental results and associated theoretical curves for the dispersion of elastic waves in a curved mechanical waveguide. The particular waveguide used was a cylindrical bar of radius of cross-section, a, wound as a helical spring of mean radius R and pitch angle c(. The spring used was made of 0.25 in (6.35 mm) diameter steel rod and had 12 turns. The central elastic line was a helix with CC= 3.9” and a/R = 0.106. TABLE 1
Spring
alR
2a *to.01 cm
2R *O.lO cm
Pitch kO.lOcm
CY’
No. of turns
A
l/8
064
5.08
0.79
2.8
40
: D E F G
l/8 l/4 l/4 l/4 l/16
064 064 064 064 064
5.08 254 254 254 10.16
2.54 1.27 0.79 1.27 1.91 0.74
9.0 4.6 5.7 9.0 13.4 1.4
40 36 80 70 50 20
The experimental results presented in this paper are obtained from the springs listed in Table 1. The wide-bandwidth pulse method previously described [I, 21 was used. All the springs were made of hard stainless steel in which attenuation of stress waves in the frequency range considered is negligible. It is seen that springs of three different curvatures were considered corresponding to values a/R = l/4, l/8 and I/ 16. Three different values of pitch angle, a (listed in Table l), were used for each of the cases a/R = l/4 and a/R = l/S whereas there was only one spring with a/R = l/ 16. The experimental conditions were chosen as previously [l] to generate either predominantly-longitudinal or predominantly-flexural waves and in each case the results were compared with theory. However, some of the records show the presence of a stress component 247
248
W. G. B. BRITTON AND G. 0. LANGLEY
with dispersion characteristics consistent with theoretical predictions for a torsional mode, thus confirming earlier results [1] which were attributed to a torsional mode. 2. THEORY 2.1.
GENERAL
A general discussion on the derivation of characteristic or dispersion equations for curved elastic waveguides was given in the previous paper; there is a more detailed discussion in reference [2]. Theoretical curves shown in this paper are computed from the approximate theories of Timoshenko [3], Morley [4] and Wittrick [5], as before. Considering first the flexural mode only, the Timoshenko, Morley and Wittrick theories deal with flexure in a straight rod, in the plane of a circular ring and in a helical spring respectively: the Morley and Wittrick theories also predict the existence of modes other than flexural modes and these will be considered later. Each of these approximate theories allows for displacement, rotation and shearing of the elements of the waveguide and are therefore expected to give results which are in good agreement with exact-theory for the lowest flexural mode. The Wittrick theory, however, assumes small curvature (a/R < 0.1). The Morley theory puts no restriction on curvature but the equations assume a simplified form if the curvature is taken to be slight. For the case of a straight rod (a/R = 0) the equations of both the Morley and Wittrick theories which relate to flexure reduce to that of the Timoshenko theory. Before considering the theory for a helical spring, we will discuss the case of a plane ring which can be regarded as a helical spring of zero pitch angle, CL.In this case of Q = 0 we can consider separately (i) flexure in the plane of the ring and (ii) flexure perpendicular to the plane of the ring: the two modes are not coupled. 2.2.
THE CIRCULAR RING: FLEXURE IN THE PLANE OF THE RING
The Morley-theory equations for vibrations in the plane of a ring have been solved for values of a/R = l/4, l/8 and 1/16, assuming material of Poisson’s ratio, v = 0.29. The phase velocity curves are similar to those shown in Figure 2 of the previous paper [I] and are not included here. However, the arrival-time curves (see section 2.4) are shown in Figure 1. These curves are presented in a non-dimensional form by plotting Tp/Ta against t/T,, where T, is the predominant period of a group of waves which produce the main effect at a distance x from the excitation point at a time t after the initiation of the pulse, To = x/co and T, = a/c,: c,, is the velocity of extensional waves of infinite wavelength in a straight rod. Three modes of propagation are predicted by the Morley theory, longitudinal, flexural and radial-shear. The radial-shear mode which has a short predominant period for all values of t/To, was not investigated experimentally and the arrival-curves are not shown here. The flexural-mode arrival-curves shown in Figure 1 are for a/R = l/4,1 /8 and zero (straight rod) : the corresponding curve for a/R = l/16 lies too close to the straight-rod curve to be drawn separately on the diagram. However, as shown in the diagram, there is appreciable separation of the longitudinal-mode arrival-curves for a/R = l/4, l/8 and l/16. It must be noted that restrictions made on the deformation of the cross-section of the rod lead to an elementarytype theory for the longitudinal modes. It is therefore expected that for these modes agreement between experiment and theory would only be good in a region where the wavelength is large compared with the lateral dimensions of the waveguide. It was stated previously that curvature could be considered slight for values of a/R -c 0-I. It is found, however, that even for a value of a/R as high as 118 calculations based on the
Morley slight-curvature approximation agree, within the accuracy of the graphs in this paper, with the results obtained from the more complicated equations of the full Morley theory. The Morley equations assuming slight curvature are, as expected, identical with the equations obtained for vibrations in the plane of a circular ring by putting 0:= 0 in the Wittrick theory.
STRESS PULSES IN HELICAL
249
SPRINGS
240 2x)200180160140120100-
I
2
3
4
Figure 1. The Morley theory arrival-curves. Curves 1,2 and 3 are associated with the lowest flexural mode in rings of a/R = l/4, l/8 and 0, respectively. Curves 4, 5 and 6 are associated with the longitudinal mode in rings of a/R = l/4, l/8 and l/16, respectively.
2.3.
THE CIRCULAR
RING:
VIBRATIONS
PERPENDICULAR
TO THE PLANE
OF THE RING
Theoretical dispersion curves for vibrations perpendicular to the plane of a ring have been obtained by solving the appropriate Wittrick equations for a helical spring of zero pitch angle. The corresponding arrival-curves for a/R = l/S and l/4, and Poisson’s ratio, v = 0.29, are shown in Figure 2, together with the Timoshenko-theory curve for a straight rod. The two bending-moment mode curves of the Wittrick theory are omitted since these highfrequency modes were not investigated experimentally. It will be remembered that the Wittrick theory is for slight curvatures only (a/R cc O-1). However, for the reasons indicated in section 2.2, we expect the theory to be sufficiently accurate for our purposes for values of a/R < l/8 : it is also expected that the arrival-curves for a/R = l/4 in Figure 2 are a good indication of the nature of the dispersion in this case. It is seen from Figure 2 that for a ring of a/R = l/4, the predominant period of the torsional mode is less than that of the flexural mode for nearly all values of t/To and that a predominant period of approximately 3OT, is predicted for a large range of arrival times. Since T, is about 0.62 ps in this case, the oscillations predicted have a predominant period of around 19 ps. 2.4.
THE HELICAL
SPRING
The general characteristic equation due to Wittrick is extremely complicated to solve but roots have been computed for a/R = O-106, tc = 0 and cc= 3.9” using the London University ATLAS computer. This value of a/R is for the spring originally investigated and the phasevelocity curves were plotted in the previous paper [I] together with the arrival-curves for cr = 0. Considering first tc = 0, it was shown that, within the scale of the diagrams, the two flexural mode arrival-curves were coincident. Numerical calculations indicate however that
250
W. G.
B. BRITTON AND G. 0. LANGLEY
Figure 2. The Wittrick theory arrival-curves for vibrations perpendicular to the plane of a ring. Curves 1, 2 and 3 are associated with the lowest flexural modes in rings of a/R = l/4,1/8 and 0, respectively. Curves 4 and 5 are associated with the torsional mode in rings of a/R = l/4 and l/8, respectively.
240 -
220 -
200 -
180 -
160 -
140 h? k
120 -
100 -
80 -
60 -
2
3
4
t/r,
Figure 3. Wittrick theory arrival-curves for the faster flexural mode in a helical spring. Curve 1 is associated with a ring of a/R = 0.106 and c( = 0. Curve 2 is the flexural mode curve for a straight rod shown for comparison. 0, Values calculated for a spring of a/R = 0.106 and a = 3.9”.
STRESS PULSES IN HELICAL
SPRINGS
251
flexural waves in the plane of the ring have a slightly higher phase velocity than flexural waves perpendicular to the plane of the ring with the same wavelength. The phase velocity curves for a = 0 and a = 3.9” shown in the previous paper indicate that an increase of pitch angle a has the effect of separating the two flexural modes. This separation is, however, not very sensitive to changes in a. The arrival-curve for flexure in the plane of a ring for a/R = 0.106 is shown in Figure 3 together with the Timoshenko arrival-curve for a straight rod. The points surrounded by circles were calculated for the faster flexural mode for a spring of a/R = O-106, a = 3.9”. It is seen that the points for a = 3.9” lie very close to the curve for a = 0. It is apparent from these results that the effect of pitch angle a on the flexural dispersion characteristics is small compared with the effect of changes in a/R. In view of the insensitivity to a of the arrival curves, the Morley theory and Wittrick theory for a = 0 have been used to calculate arrival-curves for the springs investigated in this work. However, in interpreting experimental results it is remembered that an increase in pitch angle will tend to separate the two flexural modes of propagation. 2.5.
THE METHOD OF STATIONARY
PHASE
The experimental results have been interpreted according to Kelvin’s principle of stationary phase. The method has been fully described by Davies [6] and outlined by the authors [l]. The following will be sufficient to indicate the meaning of the symbols used in the diagrams. Consider an initial stress pulse of infinitely short duration, initiated at x = 0 at time t = 0, where x denotes distance along the central elastic line of an infinitely long waveguide. This pulse can be considered to have a uniform distribution of all possible frequencies. If it is assumed that at the origin the sinusoidal components are of equal amplitude and phase, it is possible, according to Kelvin’s method of stationary phase, to deduce the mean period T, of a group of waves which produce the main effect at a point x at time t. It can be shown [l, 61 that for a given mode of propagation, a theoretical expression can be found for T, as a function of wavelength X provided that the variation with wavelength of the phase velocity, c and the group velocity, c, is known. The variables can be plotted in a non-dimensional form by plotting Tp/Ta against t/T,, as discussed in section 2.2 of this paper. The (TJT,, t/T,) curves are called arrival-curves. The method of stationary phase assumes pulses of infinitely short duration. However, it has been found experimentally, for straight rods, that this method of analysis leads to results which are in agreement with other methods and with the predictions of the exact theory. 3. EXPERIMENTS 3.1.
EXPERIMENTAL
PROCEDURE
The experimental set-up was fully discussed in the previous paper [l] and only an outline of the experimental procedure is given here. Short stress pulses were initiated at one end (x = 0) of the waveguide by applying electrical pulses at a fixed repetition rate to a small coil surrounding a magnetostrictive nickel wire which was in contact with the waveguide at x = 0. The mechanical disturbances introduced by this generating transducer were recorded at a point a distance x along the waveguide by means of a small lead-zirconate surface strain gauge. The electrical signals induced in the recording gauge were applied after amplification to the Y-plate of a cathode ray oscilloscope. Electrical pulses, synchronous with the electrical pulses, supplied to the generating transducer, were used to trigger the X-sweep of the oscilloscope. In this way a stationary (stress, time) pattern was obtained on the oscilloscope screen similar to those shown in Plates 14. The recording circuit contained components which reduced extraneous signals to a minimum and also a variable filter which could be used as a low-pass, band-pass or high-pass
252
W. G. B. BRITTON AND G. 0. LANGLEY
filter. This filter was particularly useful in analysing a trace which was due to the presence of two or more modes of widely different predominant period arriving at the gauge simultaneously : judicious use of the filter made it possible to investigate the dispersion characteristics of each mode separately. The generating transducer was always oriented so that the longitudinal mode or one of the two flexural modes of the spring would predominate. However, in practice, an appreciable amount of energy was carried by modes other than the predominant mode and the variable filter was used as described above. The interpretation of a record becomes more difficult when it is due to two or more modes of similar period arriving simultaneously at the strain gauge. The length of a strain gauge limits the shortest wavelength it can faithfully record. For a O-25-in (6.35 mm) diameter rod a strain gauge of length 0.125 in (3.175 mm), as used in this work, permits the investigation of dispersion in a region corresponding to 0 < a/h < 0.2. [I ] 3.2.
ANALYSIS OF THE STRESS PATTERN
Direct measurement of T, for a group of waves at a point x along the waveguide is made by measuring the time interval corresponding to the horizontal distance between the successive points where the trace crosses the zero strain line in the same sense [I, 21. If the time, t, from the initiation of the pulse is equal to T, at the first cross-over point and is T2 at the next point of cross-over in the same sense, then T,, = Tz - T, and the corresponding arrival time is taken to be t = (T, + T,,,,). 4. EXPERIMENTAL
RESULTS
4. I. INTRODUCTION
The experimental results obtained for a spring of fixed curvature and pitch (a/R = 0.106, CI= 3.9”) were shown in the previous paper [l] to be in good agreement with theory. The springs listed in Table 1 were used to investigate the effect of curvature and pitch on the dispersion characteristics of various modes of propagation. The results obtained for the longitudinal mode and the two flexural modes will be described under separate headings. 4.2.
LONGITUDINAL STRESS PULSES
The generating transducer was oriented to excite preferentially the longitudinal mode. A pulse of 20 ps duration was introduced into the waveguide and the resulting (strain, time) profile was detected at various points along the waveguide by means of surface strain gauges. The nature of longitudinal stress wave dispersion in helical springs is very different from that in a straight rod. The stress pattern for the case of a helical spring shows marked dispersion at large predominant periods, that is at low values of a/h. This region is relatively free from dispersion in the case of a straight rod. A more complete discussion of the effect and examples of the type of oscilloscope trace obtained have been given previously [ 1,2]. Figure 4 shows a plot of experimental points obtained from the oscilloscope traces. The variable filter was used, in analysing these records to remove from the traces frequencies outside the range corresponding to a/X ==z O-2. The high frequency components thus removed were found qualitatively to have similar dispersion characteristics to those of high frequency longitudinal waves in a straight rod. The results shown in Figure 4 were obtained from springs A (a/R = l/S, c( = 2*8”), G (a/R = l/16, cc= 1.4”) and C (a/R = l/8, cc= 9”). It is seen that for each spring the experimental points are in close agreement with the Morley theory extensional-wave arrival-curve corresponding to a circular ring of the same value of a/R. It will be remembered that for values of a/R up to l/8, the Wittrick theory curves lie very close to those of the Morley theory (see section 2.2). These experimental results confirm that, whereas the dispersion characteristics of longitudinal waves change appreciably with changes in a/R, changes in u have little
Plate 1. The flexural pulse profile at 150 cm along a straight, 0.25-in (6.35 mm) diameter ., rod. 1‘he time marker s are 100 ps apart.
Plate 2. The flexural pulse profile at 207.7 cm along helical spring A. The generating transd cer was orientec i to preferentially excite the faster flexural mode. The time markers are 100 ~LSapart.
(facing p. 252)
PIat e 3. The flexural pulse profile (faster mode) at 208.1 cm along helical spring B. The time markers are 100 ps apart.
Plate: 4. The flexural pulse profile (faster mode) at 210.1 cm along helical spring C. The time markers are 100 ps apart.
STRESS PULSES IN HELICAL SPRINGS
253
1::
15
2
2.5
Figure 4. Curves 1 and 2 are the Morley theory curves for the longitudinal mode in rings of a/R = I /8 and 0, A, Experimental points obtained by exciting springs G, A and C respectively along their central elastic line. 1/I6 respectively. 0,
effect : the experimental points from springs A and C which have appreciably different pitch angle lie along the Morley curve for a/R = l/8 and the points from spring G lie along the Morley curve for a/R = l/ 16. 4.3. FLEXURAL MODE: FASTER WAVE The faster flexural wave has vibrations which are predominantly transverse to the axis of the spring. This mode corresponds to vibrations in the plane of the ring for a spring with a = 0. The generating transducer was orientated so that this faster flexural mode would predominate and again stress pulses of 20 ps duration were injected into the springs. Figure 5 shows the results obtained from springs A (a/R = l/8, CI= 2.8”) and D (a/R = l/4, x = 5-7”) together with those obtained from a straight rod which are included for comparison.
80 -
60 -
4020 -
I
I
4
,
I
I
2
3
4
t/T,
Figure 5. Curves 1,2 and 3 are the Morley theory curves for flexure in the plane of a ring of a/R = l/4,1 /S and 0 respectively. 0, 0, Experimental points obtained by exciting springs D and A respectively in a direction perpendicular to the axis of the spring. A, Experimental points obtained from a straight rod.
254
W. G. B. BRITTON AND G. 0. LANGLEY
It is seen that the dispersion is sensitive to changes in a/R but whereas the results for spring A (a = 2.8”) are in close agreement with the Morley arrival-curve, for spring D (a = 5.7”) there is some scatter of experimental points around the corresponding Morley curve. This type of scatter is indicative of the presence of at least two modes of similar dispersion characteristics. It will be remembered that an increase in a separates slightly the dispersion curves for the two flexural modes in a spring. It appears that for spring D (a/R = l/4, a = 5.7”) both modes are present in appreciable quantity.
Figure 6. The curve is the Morley theory curve for flexure in the plane of a ring of a/R = l/8. 0, ?!,? Experimental points obtained by exciting springs A and B, respectively, in a direction perpendicular to the axis of the spring.
Figure 6 shows results obtained from springs A (a/R = l/8, u = 2.8”) and B (a/R = l/8, a = 4.6”). There is no appreciable scatter of experimental points about the Morley curve for this value of a/R even for spring B (oz= 4.6”). However, for spring C (a/R = l/8, a = 9”), which is of the same curvature, the pitch angle is sufficiently high to show the scatter of points characteristic of the presence of more than one mode. (See Figure 7.) In Figure 8 the points obtained from springs E (a/R = l/4, a = 9”) and F (a/R = l/4, tc = 13.4”) show scatter and appreciable deviation from the arrival-curve for a/R = l/4 which was obtained from the Morley theory for pronounced curvature. This effect may again be attributed to a complex waveform consisting of two or more modes. From the above results it appears that the nature of the flexural wave dispersion is very sensitive to changes in a/R but not to changes in a. However, as cc increases it becomes difficult to excite only the faster flexural mode: there is also some evidence that the larger the value of a/R, the smaller tc must be to avoid generating more than one flexural mode. The presence of more than one flexural mode has a marked effect on the appearance of the oscilloscope trace. It is worth examining the stress patterns obtained from a straight rod and from springs A, B and C (for all of which a/R = l/8) shown in Plates l-4. In the case of a
255
STRESS PULSES IN HELICAL SPRINGS
I
I
240
J
22iJ
200
180
160
140 Lp ;=_
120
100
8C
,-
6CP-
40
20 I I
\
,
I
2
3
I 4
f/T,
Figure 7. The curve is the Morley theory curve for flexure in the plane of a ring of a/R = l/8. points obtained by exciting spring C in a direction perpendicular to the axis of the spring.
0,
Experimental
240
160
Figure 8. The curve is the Morley theory curve for flexure in the plane of a ring of a/R = l/4. o, 0, Experimental points obtained by exciting springs E and F, respectively, in a direction perpendicular to the axis of a spring.
256
W. G. B. BRITTON
AND G. 0.
LANGLEY
straight rod the stress amplitude at x = 150 cm (Plate 1) rises to a certain level and remains at this level for the rest of the observed range of arrival times: the ripples in the tail of the trace are due to reflections from the far end of the rod. The type of pattern observed is consistent with the presence of only one flexural mode and the dispersion characteristics are those predicted by the Timoshenko theory. In the record from spring A (a = 2.8, x = 207.7 cm), shown in Plate 2, we see that a small high-frequency flexural pulse precedes the main pulse. The shape of the Morley-theory arrival-curve for flexure (Figure 1) indicates that in the initial portion of a recorded flexural signal, two high-frequency components would arrive simultaneously: the superposition of these components would lead to a precursor pulse of the type recorded in Plate 2. In the main pulse the amplitude rises to a maximum and a slight decrease of amplitude is noted at the largest arrival times observed. This is characteristic of interference between two modes of similar period travelling at slightly different velocities in the slower, low frequency, region. The record (Plate 3) from spring B (a = 4.6”, x = 208.1 cm) again shows the initial high-frequency components and a maximum stress amplitude in the main pulse followed by a more rapid decrease of amplitude with increased arrival time than shown in Plate 3. In Plate 4, taken from spring C (a = 9”) at x = 210.1 cm, we again see the initial high-frequency stress components. The stress pattern then rises to a maximum and falls to a minimum before rising to a second maximum. A comparison of the three spring records shows that as a increases the effects of interference are seen earlier in the trace. This is precisely what is expected from theoretical considerations. As indicated in section 2.4 and illustrated in Figure 4 of our previous paper [I] the two flexural mode dispersion curves for a spring separate increasingly in the low-frequency range with increasing a. This is discussed further in section 5. 4.4.
FLEXURAL
MODE:
SLOWER
WAVE
The slower flexural wave has vibrations which are predominantly parallel to the axis of the spring. In the case of a circular ring (a = 0), this mode involves vibrations which are perpendicular to the plane of the ring. The generating transducer was orientated so that the slower flexural mode would predominate and pulses of 20 ys duration were injected into the spring. The stress pattern at some distance from the point of initiation of the stress pulse is not of the familiar flexural type. As described in our previous paper [l], the traces show the presence of at least two modes of different predominant periods arriving simultaneously. In the initial region of the pattern a low-frequency oscillation is superimposed on the main pulse. The tail of the pulse shows features which are characteristic of the superposition of two vibrations of similar period. In order to analyse the low-frequency component of the initial portion of the trace, the variable filter mentioned in section 3.1 was adjusted to remove the high-frequency components from the pattern. The results obtained from the oscilloscope traces are presented as arrival-curve plots in Figures 9 and 10. The points shown in Figure 9 were obtained by analysing records taken from spring B (a/R = 1IS, a = 4.6”). The theoretical curves 1 and 2 in Figure 9 were obtained from the Wittrick theory for a circular ring with a/R = l/8 and curve 3 is the Timoshenko arrival-curve for flexure in a straight rod. The points surrounded by squares in Figure 9 were obtained from the high-frequency components of the initial portions of the traces. It is seen that these points are in good agreement with the torsional-mode arrival-curve. The experimental points, surrounded by circles, from the tail of the pulse show general agreement with the arrival-curves for the flexural mode (a/R = l/8, a = 0) but there is some scatter. It will be remembered that no scatter was observed when investigating propagation of the faster flexural mode in spring B. Figure 10 shows the experimental points obtained from spring E (a/R = l/4, a = 9”). It is seen that there is again scatter of the flexural-mode points and a greater deviation from the
STRESS PULSES IN HELICAL SPRINGS
257
160
140 h? 120 ;=_ 100
60
40
20
I
2
3
4
Figure 9. Curves 1 and 2 are the Wittrick theory curves for flexure perpendicular to the plane of a ring for a/R = l/8. Curves 1 and 2 are associated with the flexural and the torsional modes, respectively. Curve 3 is the Timoshenko curve for flexure in a straight rod, shown for comparison. 0, 0, Experimental points obtained by exciting spring B in a direction parallel to the axis of the spring.
220 -
2cO-
180 -
160 -
140 h" La
120 -
100 -
80-
60-
Figure 10. Curves 1 and 2 are the Wittrick theory curves for flexure perpendicular to the plane of a ring of a/R = l/4. Curves 1 and 2 are associated with the flexural and the torsional modes respectively. Curve 3 is the Timoshenko curve for flexure in a straight rod. 0, Experimental points obtained from spring E.
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W. G. B. BRITTON AND G. 0. LANGLEY
corresponding theoretical arrival-curve than there was in the results for a/R = l/8 shown in Figure 9. These results are similar to those obtained when investigating the faster flexural mode in spring E and can be interpreted as being due to the simultaneous presence of the faster and slower flexural modes. It will also be remembered that the Wittrick theory is for slight curvature and that the flexural arrival-curve for a/R = l/4, cc= 0 in Figure 10 is only an indication of the effect of curvature on the dispersion. Some of the experimental points shown in Figure 10 lie close to the torsional mode-curve predicted by the Wittrick theory for a/R = l/4, a = 0. It is seen that for this value of a/R, theory predicts a torsional mode which has a smaller predominant period than the flexural mode over the whole range of arrival times shown. According to the torsional mode-curve, there is an oscillation ofpredominant period T, equal to approximately 3OT, : this corresponds to an oscillation with a period Tp of about 19 vs. Such an oscillation, which extended over a large part of the stress pattern, was observed experimentally [2]. 5. DISCUSSION The investigations described in this paper are an extension of those carried out on a single spring, with parameters a/R = O-106, a = 3*9”,which were reported in our previous paper [I]. The earlier work showed that experimental results for this spring were in good agreement with the theories of Morley and Wittrick and that the difference in the theoretical dispersion curves for CL= 0 and c( = 3.9” was slight for the case a/R = 0.106. The springs listed in Table 1 were used to investigate the effects of curvature and pitch angle on the dispersion characteristics of various modes of stress-wave propagation. In each case the experimental results have been compared with the arrival curves for the appropriate value of a/R, calculated from the Morley theory or from the Wittrick theory. In view of the insensitivity to changes in CLof the theoretical curves, the arrival curves obtained from the Wittrick theory were all calculated for the case a = 0. Considering first the results for extensional (longitudinal) waves, shown in Figure 4, it is seen that they are in good agreement with the Morley theory curves for a circular ring. A change in CLfor a fixed value of a/R has a negligible effect on the dispersion characteristics of the longitudinal mode. Two flexural modes were investigated. The faster flexural mode involves vibrations which are mainly perpendicular to the axis of the spring while the slower mode has vibrations which are mainly parallel to the axis of the spring. The results for the faster flexural mode are shown in Figures 5-8. As in the case of the longitudinal mode, there is general agreement between experiment and theory. However, for a given value of a/R, an increase in czleads to an increase in the scatter of experimental points about the theoretical arrival-curve. In the case a/R = l/B, for example, the experimental points lie close to the theoretical curve for a = 2.8” and cz= 4.6” but appreciable scatter is observed for tc = 9”. The scatter and deviation from the theoretical curve is even greater for a/R = l/4 with cc= 9” and cz= 13.4”. These results are attributed to the presence of the slower flexural mode in appreciable quantity in addition to the faster mode. According to the Wittrick theory, the dispersion curves for the two flexural modes in a spring are coincident for CC = 0 but there is a slight separation in the curves for a finite value of cr: the separation increases with increasing a. This slight difference in phase velocity for any particular value of wavelength will lead to interference between the two modes and the scatter of experimental points is attributed to this effect. The interference can clearly be seen in the oscilloscope traces such as those shown in Plates 2-4. It is even more difficult to generate a nearly-pure mode in the case of the slower flexural wave. A scatter of experimental points is observed in Figure 9, for a/R = l/S, a = 4*6“, which
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indicates the simultaneous presence of the slower and faster flexural wave. It will be remembered that it was possible in the same spring to generate the faster flexural wave without introducing an appreciable amount of the slower wave. The presence of the two modes is seen to be even more pronounced in Figure 10 which shows the results obtained from a spring with parameters a/R = l/4 and CL= 9”. The presence of a torsional mode was detected during investigations on the slower flexural mode. The results presented in Figures 9 and 10 are in good agreement with theory for this mode and show that a change in curvature has a pronounced effect on the dispersion characteristics of the torsional mode. It is worth noting that the lowest order torsional mode in a straight cylindrical rod is non-dispersive [6]. It is seen that the experimental results show that curvature and pitch angle affect the dispersion characteristics of stress wave propagation in a helical spring according to the predictions of the Wittrick theory. The results can be interpreted with respect to four of the six modes predicted by the theory: the two highest, bending-moment modes were not investigated. It will be remembered that the Wittrick theory assumes slight curvature. However, in the case of the faster flexural wave for CC= 0, comparison can be made with the Morley theory which places no restriction on curvature. The dispersion curves of the two theories for the values of a/R used here are found to lie close to each other. The stress pulses introduced into the waveguides in the investigations described here were all of approximately 20 ps duration. The bandwidth of such a pulse is sufficiently narrow to excite only the lowest order modes. However, it has been found that a pulse of duration 8 ~LS leads to the appearance of higher frequency modes in appreciable amplitude. A study of the higher modes, using pulses of shorter duration, would be a useful extension of this work. REFERENCES 1.
2. 3. 4. 5. 6.
W. G. B. BRITTON and G. 0. LANGLEY 1968 Journal of Sound and Vibration 7, 417-430. Stress pulse dispersion in curved mechanical waveguides. G. 0. LANGLEY 1968 Ph.D. Thesis, University of London. Stress pulses in curved mechanical waveguides. S. P. TIMOSHENKO1922 Philosophical Magazine 43, 125-131. On the transverse vibrations of bars of uniform cross-section. L. S. D. MORLEY 1961 Quarterly Journal of Mechanics and Applied Mathematics 14, 155-172. Elastic waves in a naturally curved rod. W. H. WITTRICK 1966 International Journal of Mechanical Sciences 8, 2547. On elastic wave propagation in helical springs. R. M. DAVIES 1948 Transactions of the Royal Society A 240, 375-457. A critical study of the Hopkinson pressure bar.
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