by G. J. Adams [ B&W Loudspeakers, Worthing, Sussex, England]
-------- Laser interferometry can, for the first time, make accurate
measurements of high-frequency motion of a speaker without the need for
any attachment to the speaker cone.
Having described the laser interferometry testing system for making accurate high-frequency loudspeaker cone measurements, I will now proceed to the computer's role in analysis of the breakup patterns, measurements of distortion and calculation of the sound-pressure response.
In addition to assisting with the measurement and storage of vibration data and the control of the point of illumination, the computer also serves a vital role in processing and combining the data and displaying it in a form which is easily interpreted. The flexibility of a computer equipped with a graphics facility lends itself very well to this sort of display problem. There are several computer programs that can be written to display part or all of the data in a number of different aspects or formats. The particular display program which is executed, and hence the display format that is chosen, will depend on the type of vibrational mode of interest.
Fig. 8--Conventional loudspeaker cones exhibit two main types of cone
resonance. One type can arise from travelling waves running in both circumferential
directions (a) which are excited by inhomogeneities in the cone material
or by uneven tension in the suspensions due to misalignment. These travelling
waves may result in the formation of standing waves with a number of nodal
diameters. The example shown (b) has two nodal diameters.
Fig. 9--The second type of cone resonance can arise from travelling waves
running in a radial direction (a) which are excited by the motion of the
voice coil. These may result in the formation of standing waves with nodal
circles. The example shown (b) has two nodal circles.
Fig. 10--Because the coupling between bending and stretching notions on
the cone is least at the outer edge of the cone, bending waves travelling
in the radial direction first appear at the outer edge only. As the frequency
of the driving signal is increased the inner boundary of the bending waves
moves towards the cone center until eventually the whole of the cone surface
is covered in bending waves.
Conventional loudspeaker cones exhibit two main types of vibrational mode [17]. The first type can be thought of as arising from travelling waves running in both circumferential directions, as shown in Fig. 8 (a). These travelling waves may result in the formation of standing waves with nodal lines in the radial direction as illustrated in Fig. 8 (b).
This type of standing-wave or breakup pattern has no rotational symmetry about the cone axis and is therefore described as being asymmetric. Distortion of the cone away from its static circular shape, as shown in Fig. 8 (b), causes very little change in the cone circumference for any given radius. The stiffness opposing the bending waves travelling in the circumferential directions is thus mainly influenced by the force required to bend the material and is only slightly influenced by the force required to compress or stretch it. Asymmetric modes generally occur at low frequencies such that the distance between adjacent segments of the cone which are moving in anti-phase is small compared to the wavelength of sound in air. Thus, in essence, the air in front of the cone is merely displaced from one cone segment to the next and back again etc., resulting in very little contribution by the mode to the total sound output. The main component of the latter arises from an axial motion which is superposed on that of the alternate segments moving in opposite directions [15].
The second type of vibrational mode proves to have a much greater effect on the sound output than does the asymmetric mode. Travelling waves running in the radial direction from the cone center outwards are partly reflected at the outer suspension back towards the center, as shown in Fig. 9 (a). These travelling waves may result in the formation of standing waves with concentric nodal circles centered on the cone axis, as shown in Fig. 9 (b). This type of standing-wave pattern is symmetric about the cone axis and is thus described as being axi-symmetric. Distortion of the cone away from its static shape, as shown in Fig. 9 (b), results in a change in the circumference of the cone at a given radius. The stiffness opposing bending waves travelling in the radial direction is thus, in addition to being influenced by the force required to bend the material, considerably influenced by the forces required to compress or stretch the material. The latter are generally much greater than the forces required for bending, and, as a result, the total bending stiffness opposing the propagation of bending waves in the radial direction is much greater than that opposing bending waves travelling in the circumferential directions. The coupling between bending and stretching motions in a cone is the reason why the forming of a flat circular piece of paper into a cone shape makes it more rigid. A mathematical analysis of the mechanical behavior of a cone [27] shows that this coupling leads to a phenomenon which is not observed on a flat diaphragm: Bending waves can only propagate in the radial direction above a certain frequency fib given by
[eq1]
where E is the Young's modulus of the cone material, a is its density, p is the semi-apex angle of the cone and Rb is its outer radius. The Young's modulus is a measure of the stiffness of the material, e.g., the Young's moduli for paper and aluminum are approximately 2 and 70 GNm^-2 respectively. From Equation (1) the values of f_tb for a paper and an aluminum cone of 16 cm diameter having semi-apex angles of 60° are approximately 1.8 kHz and 5 kHz respectively.
These figures indicate the frequencies at which axi-symmetric cone breakup commences.
The degree of coupling between bending and stretching motions at any point on the cone can be shown to be inversely proportional to the radius at which the point lies, i.e., the coupling is greatest near the center of the cone and is least at the outer edge. The result of this variation along the cone radius is that as the frequency of the driving signal is increased above f_tb, bending waves first appear at the outer edge of the cone only, while the remainder of the cone still vibrates approximately uniformly, as shown in Fig. 10. As the frequency is further increased, the dividing circle between the cone part which is exhibiting bending and the cone part which is moving approximately uniformly, reduces in radius until eventually, at some frequency f_tb, the whole of the cone surface is exhibiting bending.
Fig. 11--The typical form of the axial sound-pressure amplitude/frequency
response of a straight-sided loudspeaker cone mounted in an infinite baffle.
The dashed curve shows the response of the same cone which would be obtained
if it were perfectly rigid.
Fig. 12--This “3-dimensional" type of graph generated by the computer
shows the variation of the amplitude of the measured cone acceleration
along one radius of a loudspeaker cone as a function of frequency.
Fig. 13--A perfectly rigid loudspeaker cone has the same amplitude of
acceleration at all points on the surface when driven by a constant-amplitude
sinusoidal force for frequencies above the fundamental resonance of the
system. The 3-dimensional display thus appears as a flat block for a perfectly
rigid diaphragm.
Fig. 14--The axial sound-pressure amplitude/frequency response of the
loudspeaker used for the measurements shown in Fig. 12. Fig.
16--As Fig.
14, but after the application of the damping ring.
Fig. 15--The vibration measurements taken under the same conditions as
for Fig. 12, but following the application of a ring of damping compound
to a selected area of the cone.
The sound output of the cone above fib is largely determined by the motion of the inner region of the cone which is moving approximately uniformly. The outer cone region which is exhibiting bending waves contributes only a little to the sound output because the distance between adjacent (annular) parts moving in antiphase is small compared to the wavelength of sound in air. The outer region thus "short circuits itself." The motion of the inner region is approximately uniform, i.e., the whole of the region moves in phase, but in general its amplitude increases with increasing distance from the voice coil [28]. As the frequency is increased above fib, the radius of the inner region decreases and therefore the area of the cone which is effective in radiating sound decreases.
Thus, one might expect the sound out put to decrease with increasing frequency above fib. However, because the outer cone region is in resonance, the inner region behaves approximately as if it were decoupled from the outer region.
The elective cone mass as "seen" by the driving voice coil thus decreases by the same proportion as does the effective radiating area. For a constant driving force from the voice coil, the amplitude of vibration of the inner region increases as a result of the decrease in effective mass such that the sound output re mains at approximately the same level as below f15. In practice, the amplitude of vibration of the inner region can often increase with increasing distance from the voice coil resulting in an increase in the sound output above fib. This phenomenon is associated with the "trap ping" of energy at the boundary be tween the inner and outer regions of the cone. See "The Trapping of Acoustical Energy by a Conical Membrane and its Implications for Loudspeaker Cones" by L. J. van der Pauw, Jour. of the Acoustical Society of America, vol. 68, pp. 1163-1168, 1980. The sound output eventually falls with increasing frequency because, in addition to the cone becoming completely covered in "non-radiating" bending waves, the excitation of the cone diminishes as a result of the magnitude of the mechanical impedance of the voice coil becoming greater than that of the cone.
Figure 11 shows a typical form of the axial sound-pressure output/frequency response of a cone loudspeaker mounted on a large baffle. Above the fundamental resonance frequency f5 of the loudspeaker driver, the response re mains approximately flat until fib when the output increases due to axi-symmetric cone breakup. Between fib and La, the response is roughly constant with a "fine structure" [17] of resonances caused by the incomplete cancellation of the sound outputs from the parts moving in antiphase on the outer region of the cone. The irregular roll-off of the response at high frequencies is due to resonances in the plane of the cone similar to those which occur in a flat plate. For the sake of interest, Fig. 11 also shows the frequency response (dashed curve) that would be obtained if the cone were perfectly rigid. This demonstrates that cone breakup has the effect of extending the response at high frequencies.
Careful choice of the cone geometry and the material parameters of the cone can result in a smooth and useful extension of the frequency response above that of a perfectly rigid cone. The "loudspeaker problem" is often stated as being the elimination of cone breakup; however, a more relevant and useful objective would be the correct control of cone breakup to achieve a smooth frequency response having a wide bandwidth.
The brief discussion given above of the mechanical behavior and sound-out put/frequency response of a loudspeaker cone is based an interesting the theoretical analysis of a straight-sided cone having an unsupported outer edge [17].
In practice, the outer edge is supported, very often by a suspension made from a different material to that of the cone. In addition, the cone is seldom straight-sided but is usually curved (or flared) to improve "dispersion", and the front end of the voice-coil bobbin is covered by a dust cap. All of these features modify the cone behavior and the sound-output/ frequency response. However, the simplified analysis is still very useful be cause it tells us roughly what sort of mechanical cone behavior we can expect in practice. This information is a helpful guide as to the number and spacing of the points on the cone surface at which the motion should be measured. The theoretical analysis showed that the vibrational modes which have most effect on the sound-pressure/frequency response are symmetrical about the cone axis. Thus, a very useful initial investigation of the cone vibration of some particular loudspeaker driver would be to mea sure the transfer function between the input voltage to the voice coil and the cone velocity at a number of points spaced along just one radius of the cone. If the cone motion is symmetrical in the frequency range of interest, then these data will be sufficient to enable the cone breakup pattern to be drawn for the whole of the cone at any frequency with in this range. Of course, the symmetry or otherwise of the cone motion can only be verified by repeating the measurements along several different radii of the cone (as shown in Fig. 3). This should be the next step after the initial investigation if a more thorough analysis is required.
An example of the data obtained from vibration measurements taken along only one radius is shown in Fig. 12. This is a "3-dimensional" type of graphical representation of the cone motion measured at 13 points evenly spaced along a radius of the cone of a 100-mm diameter mid-range loudspeaker. Points on the dust cap and the outer suspension were included in the measurements be cause of their often significant contributions to the total sound output. The graph shown in Fig. 12 has effectively three axes and shows the variation of the amplitude of the acceleration at a point on the cone as a function of both the frequency of the driving signal and the position of the point along the radius.
The acceleration /frequency responses were obtained form the velocity impulse responses measured at each point as described in the previous installment.
The choice of acceleration, rather than velocity amplitude, for the vertical axis of the graph was made because a perfectly rigid cone would have a constant acceleration amplitude at all points on the surface for frequencies above the fundamental resonance of the system. The display for a perfectly rigid diaphragm would thus appear as a flat block as shown in Fig. 13.
Fig.
17--The 3-dimensional display of the variation of the amplitude of acceleration,
as a function of frequency, along a diameter of a hard dome tweeter. The
increase in the amplitude of vibration of the center of the dome in the
region of 6 kHz is quite remark able, being about 30 dB.
Fig. 18--A vertical cross section of Fig. 17 at 5.7 kHz showing the variation
of the vibration amplitude across the dome at resonance.
Fig. 19--The velocity waveforms measured at a point on a loudspeaker cone
for sine-wave excitation of various frequencies. The distortion of the
waveform observed at these low frequencies is due to non-linearities in
the moving-coil drive mechanism and in the cone suspension.
Fig. 20--The static transfer characteristic between the voice-coil input
voltage and the cone displacement (for The same driver as used for Fig.
19) measured using the displacement output of the interferometer.
Fig. 21--The measured 2nd- and 3rd-harmonic distortion of the axial sound-pressure
output of a 26-mm diameter soft dome tweeter. The high level of distortion
observed around 2.5 kHz was unusual for this driver and prompted an investigation
of the dome's mechanical behavior.
Fig.
22--The fundamental amplitude of the vibration velocity over the surface
of the dome measured for a sine-wave input voltage at a frequency of 2.5
kHz. The contour lines are drawn by the computer through points on the
surface which have the same amplitude. This "contour map" demonstrates
very clearly the asymmetric nature of the dome motion at 2.5 kHz. Fig.
23--The amplitude of the 2nd harmonic distortion of the vibration velocity
over the dome surface for an excitation frequency of 2.5 kHz.
Fig. 24--As Fig. 23, but showing the amplitude of the 3rd harmonic.
The displays shown in Figs. 12 and 13 have been made more easy to visualize by drawing lines through points on the acceleration responses which correspond to the same frequency. By using a spline interpolation curve to join these points, the display acquires a smooth "surface" or "landscape" appearance.
If the acceleration/frequency responses are displayed in the same order and with the same relative spacing as the points on the cone to which they correspond, then the display shows the amplitude of acceleration along the radius as a function of frequency.
The measured axial sound-pressure amplitude/frequency response of the loudspeaker used for the measurements shown in Fig. 12 is shown in Fig. 14.
The frequency response is smooth with the exception of the resonance around 6 kHz. Examination of the cone vibration measurements in Fig. 12 shows that around this frequency the area of the cone approximately midway between the center and the edge shows a variation in acceleration amplitude which is similar to that of the sound pressure. If mechanical damping can be introduced into this area of the cone without significantly affecting the other material parameters of the cone (notably the density and the Young's modulus), then this resonance should be reduced in amplitude and the sound-pressure /frequency response thereby improved. Figure 15 shows the vibration measurements made on the same driver after the application of a ring of damping compound to the area of the cone in question. The variation of the acceleration amplitude in this area can be seen to have been reduced by this action. The beneficial effect of this "selective damping" of the cone is shown in the sound-pressure/frequency response of the modified driver given in Fig. 16.
Another example of the 3-dimension al type of display is given in Fig. 17 which shows the variation of the acceleration amplitude as a function of frequency along one diameter of a 12-mm diameter hard plastic dome tweeter. The dome of this tweeter shows a remark able resonance at around 5.7 kHz which is of sufficient amplitude to be seen by the naked eye under normal white light! The resonance is shown by Fig. 17 to be due to a considerable increase (about 30 dB) in the amplitude of the center of the dome around 5.7 kHz. The variation of the vibration amplitude along the diameter can be seen more clearly in Fig. 18 which shows, in effect, a vertical cross-section of the 3-dimensional graph corresponding to a frequency of 5.7 kHz. This type of display is easily obtained from the vibration data stored in the computer and is useful for the de tailed analysis of the cone or dome vibration once the frequencies of interest have been identified from the 3-dimensional graph.
Distortion Measurements
Because the laser interferometer sys tem provides an instantaneous measurement of the vibration at the point of illumination on the cone surface, the sys tem can be used to measure the distortion of the cone vibration at any point on the cone surface.
The loudspeaker can, for example, be excited with a sine-wave voltage, and the velocity waveform at the point of interest observed on an oscilloscope and/ or analyzed with a distortion meter or spectrum analyzer. Distortion of the cone motion can arise for a number of reasons. At low frequencies, where the cone motion is greatest, distortion may be introduced by the non-linear stiffnesses of the cone suspensions, by the variation of the force factor BI of the coil and magnet with cone displacement, and by the variation of the inductance of the voice coil with cone displacement. Figure 19 shows the velocity waveforms measured at a point on the cone of a 16-cm diameter driver. All the waveforms shown were measured at frequencies below that at which cone breakup commences, and thus the distortion is likely to be a result of one, or a combination of, the sources listed above.
The ability of the laser system to mea sure displacement enables the transfer characteristic between the voice-coil voltage and the cone displacement to be determined. Figure 20 shows the measured characteristic for the same driver as used for the measurements shown in Fig. 19. The departure of the measured characteristic away from the ideal straight-line characteristic is a result of the variation of the suspension stiffnesses and the BI factor with displacement.
At high frequencies, where the cone exhibits vibrational modes, distortion may be introduced by the non-linear bending or stretching of the cone material. Normally this would occur only at high input voltage levels such that the cone material is strained beyond its elastic limit. However, significant distortion of the cone motion can sometimes be ob served at quite low input levels if the cone exhibits a vibrational mode of large amplitude. Figure 21 shows the measured 2nd- and 3rd-harmonic distortions of the axial sound-pressure output of a 26-mm diameter soft dome tweeter. The 2nd-harmonic is particularly high in the region of 2.5 kHz which is not typical for this driver. Using the laser, the velocity waveforms at 40 points spread evenly over the dome surface were measured and stored in the computer for a sine-wave rms input voltage of 1V at a frequency of 2.5 kHz. The amplitudes of the fundamental and the 2nd and 3rd harmonics of the velocity waveform at each point were found from the Fourier transform of an integral number of cycles of the waveform. Figures 22, 23, and 24 show the amplitudes of the fundamental and the harmonics as a function of position on the dome surface. A "contour map" type of display has been used here; the contour lines are drawn through points on the dome surface which have the same amplitude. Figure 22 shows very clearly that there is an area of the dome which is vibrating at a much greater amplitude than the rest of the dome. As a result of this greater amplitude, the dome motion is more distorted in this area, as is shown in Figs. 23 and 24. Close examination of the dome surface revealed a slight "dent," about 3 mm wide by 10 mm long, located in the same area as the observed increase in motion. This small imperfection in the dome shape was thus responsible for an asymmetric mode of vibration at 2.5 kHz which was not normally present in this type of driver. Because of the small area of the imperfection, the fundamental sound output at 2.5 kHz was barely affected leaving only the increase in distortion as an indicator of its presence.
Although the last example is not particularly relevant to the general problems of loudspeaker driver design, it illustrates how the laser interferometer can be used to identify those areas of the loudspeaker diaphragm which are the major contributors to the distortion output at any particular frequency. This information can often suggest modifications to the loudspeaker driver which will reduce the distortion.
Fig. 25--The sound-pressure/frequency response of a direct-radiator loudspeaker
driver mounted in a baffle can be calculated from the velocity impulse
responses measured at a number of points spread over the vibrating surface
of the driver. Curve (1) shows the measured axial sound-pressure amplitude/frequency
response of a 26-mm diameter dome tweeter mounted in a baffle, and curve
(2) shows the same response calculated from the velocity impulse responses
measured at 33 points.
Fig. 26--The contribution which the motion of the dome suspension makes
to the axial sound-pressure can be seen by comparing the calculated response
shown in Fig. 25 curve (2) to the response calculated using only those
impulse responses measured on the main part of the dome, curve (3).
Calculation of Sound-Pressure Response
Knowledge of the amplitude and phase (relative to the sinusoidal driving signal) of the motion at a number of points spread over the surface of the loudspeaker cone enables the sound power output, the directivity diagram, and the axial sound-pressure of the loudspeaker to be calculated corresponding to the amplitude and frequency of the driving signal. If the velocity impulse response is measured at each point (as described earlier), these factors can be calculated as a function of frequency. The calculation process is essentially one of summation; the cone surface is treated as a number of elemental areas which vibrate as tiny rigid pistons such that the total sound-pres sure response can be obtained from a sum of the sound-pressure responses of all the individual elements. Each elemental area is assigned an amplitude and phase of vibration which correspond to those measured at a point somewhere within the area. The summation must take into account the areas of the elements and their distances from the point at which the sound pressure is being calculated. The calculation of the sound-pressure /frequency response from measured cone vibration data is discussed in more detail in [29].
Because of the nature of the calculation process, it is a simple matter to determine the contribution which certain areas make towards the total sound power or sound pressure. Thus, for ex ample, the contribution made by the dust cap to the total sound-pressure response could be calculated and com pared to the calculated total response.
This information might be of considerable interest if, for instance, the motion at points on the dust cap is found to be distorted at some frequencies.
By way of example of the results that can be obtained, Fig. 25 shows the measured and calculated axial sound pressure/frequency responses of a 26-mm diameter dome tweeter mounted in a baffle. The calculated response was obtained from the velocity impulse responses measured at 33 points spread over the dome and the outer (roll) suspension. At some frequencies the motion of the suspension of this tweeter was found to be more distorted than the motion of the main part of the dome. The contribution which the motion of the suspension made to the total sound-pres sure response was thus of considerable interest. Figure 26 shows the calculated total frequency response (from Fig. 25) and the frequency response calculated using only those impulse responses measured on the main part of the dome i.e., the contribution of the suspension is omitted.
Looking Ahead
The laser interferometry measurement system discussed in this article is a considerable advance in the instrumentation available for the analysis and study of loudspeaker cone behavior. In the short time this system has been in use, the B&W research department has learned quite a lot about cone behavior. However, one fact has clearly emerged; there is still a lot more to learn! The behavior of the free-edged, straight-sided cone studied by Frankort [17] is complicated in it self, but real loudspeaker cones exhibit even more complicated behavior due to resonances in the cone suspensions and in the voice-coil bobbin and its dust cap.
The theoretical analysis of this more complicated behavior has recently become possible by employing the finite-element technique implemented on high-speed digital computers. Several authors [30, 31, 32] have already used this technique to study the linear behavior of complete loudspeaker diaphragm assemblies. However, there is still much work to be done to explain all the ob served phenomena of real loudspeakers Some workers in the loudspeaker field may argue that the time spent on formulating a theoretical loudspeaker model is wasted because prototypes of the actual loudspeaker can be made so quickly and cheaply that design is best accomplished by experiment. While this is partly true, I believe that the problems of loudspeaker driver design will become better understood by the additional study of an accurate theoretical model, rather than by experimentation only. One approach, which falls between those of pure theory and pure experiment, would be to build up a mathematical model of the loudspeaker driver under investigation by taking measurements of the transfer functions between the voice-coil input voltage and the motion at a number of points on the moving assembly (including the voice-coil bobbin and the cone suspensions, etc.). From this set of transfer functions, it should be possible to compute a model [33, 34] which approximates the real system. Material or geometrical changes to the loudspeaker driver could then be simulated by adjustment of the parameters of the model, and the effect on the loudspeaker performance calculated [35].
While we confidently expect the future to hold advances and improvements in laser and computer technology, the future of the loudspeaker driver is less certain. Might we still be paying tribute to Rice and Kellogg in another 50 years time?
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(adapted from Audio magazine, Sept. 1981)
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Measuring Speaker Motion With A Laser--PART ONE (Aug. 1981)