US3571766A - Disk-wire mechanical filter using bridging wire to achieve attenuation pole - Google Patents

Abstract

A general stopband disc-wire type mechanical filter having at least four circle mode vibration-type discs therein with a first coupling wire means connected to the perimeters of all four discs and a second coupling wire means connected only to the first and the fourth disc and bridging the two discs therebetween. The second bridging coupling wire means has a length such that it produces a 180* phase shift of energy transferred therethrough with in the passband. Circle mode-type discs resonate in phase with each other at the lower end of the passband and out of phase (with the adjacent discs) at the upper end of the passband. Consequently, the energy transfer through the bridging coupling wires is out of phase with the energy transfer through the first coupling wire means both at the lower and upper ends of the passband, thereby producing the general stopband characteristic.

Description

United States Patent 3 [72] Inventor Roger J. Teske Santa Ana, Calif. [21] Appl. No. 834,978 {22] Filed June 20, 1969 [45] Patented Mar. 23, 1971 [73] Assignee Collins Radio Company Cedar Rapids, Iowa [54] DISK-WIRE MECHANICAL FILTER USING BRIDGING WIRE TO ACHIEVE AT'IENUATION POLE Claims, Drawing Figs.

[52] US. Cl .1 333/71, 333/ [51] Int. Cl 03h 9/26 Field of Search 333/30, 70, 71

[56] References Cited UNITED STATES PATENTS 2,856,588 10/1958 Bums 333/71 2,918,634 12/1959 Bercovitz 333/71 3,135,933 6/1964 Johnson 333/71 3,142,027 7/1964 Albsrneier et a1...... 333/71 3,351,875 11/1967 Midgley 333/71 3,439,295 4/1969 Bise 333/ 72X 3,440,572 4/1969 Bise 333/71X 3,440,574 4/1969 Johnson et a! 333/71X Primary Examiner-Herman Karl Saalbach Assistant ExaminerMarvin Nussbaum Att0rneys-Henry K. Woodward and Robert J. Crawford ABSTRACT: A general stopband disc-wiretype mechanical filter having at least four circle mode vibration-type discs therein with a first coupling wire means connected to the perimeters of all four discs and a second coupling wire means connected only to the first and the fourth disc and bridging the two discs therebetween. The second bridging coupling wire means has a length such that it produces a phase shift of PATENTEDmzs I97! $571,766

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mvsur on; ROGER a. resxs I BY ATTORNEY DISK-WERE MECHANKCAL FILTER USING Blllll LGlNG Willi: TO AEHHEVE ATKENUATKQN FQLE This invention relates generally to mechanical filters and, more specifically, to a general stopband filter to a general stopband filter of the stacked disc-coupling wire type.

Disc-wire type mechanical filters have been quite well developed in the art. These type filters generally comprise a plurality of discs stacked one upon the other with their axes lying along a common line and spaced apart usually not more than one wavelength of the nominal center frequency of the filter pmsband. Most of the disc-wire type mechanical filters employ a circle mode of vibration in the discs. The circle mode-type vibration can be analogized to the action of the bottom of an oil can in that the nodes are circular and concentric around the axis of the disc. However, some of the discwire type filters employ one or more discs having a diameter mode of vibration. A diameter mode of vibration is characterized by nodes which coincide with diameters of the disc.

in the absence of special construction features a disc-wire type mechanical filter employing circle mode discs does not have attenuation poles either at the upper edge or the lower edge of the passband. While the frequency response characteristic of such a mechanical filter is quite good, there are applications where it is desirable to have sharper cutoff characteristics. Such sharper cutoff characteristics can be effected by attenuation poles at the upper and lower edges of the passband.

One means for producing an attenuation pole at the upper end of the passband is described in US. Pat. No. 3,135,933 issued .lun. 2, 1964, to Robert A. Johnson and entitles M- Derived Mechanical Filter." In this structure a coupling wire is caused to bridge a disc and to be physically attached to the discs on either side of the bridged disc. Other coupling wire means are physically attached to all three discs. Since it is a characteristic of mechanical filters that each disc vibrate 180 out of phase with the adjacent disc, at the upper edge of the passband, the energy transferred through the bridging coupling wire will be 180 out of phase with the energy transferred through the bridged disc. Cancellation of the two signals occurs to produce the attenuation pole at the upper edge of the passband.

In US. application, Ser. No. 614,621, filed Feb. 8, 1967, now US. Pat. No. 3,488,608, by Robert A. Johnson and entitled General Stopband Mechanical Disc Filter Section Employing MultiMode Disc" there is described a structure for obtaining attenuation poles both at the upper and lower edges of the passband through the use of discs operating in the diameter mode vibration. It is a characteristic of the diameter mode vibration that one or more portions of the perimeter of the discs will be vibrating in a given phase and the diametrically opposed portions of the discs will be vibrating in a phase 180 removed therefrom. Because of the coexistence of these two phases of vibrations in the same disc, it is possible to provide for cancellation of the signal both at the upper and the lower edges of the passband. Another structure employing the diameter mode-type disc which has attenuation poles both at the upper and lower edges of he passband is described in US. application Ser. No. 557,300 filed Jun. 13, 1966, now US. Pat. No. 3,439,295, by Donald L. Bise and entitled Mechanical Filter With Attenuation Poles On Both Sides Of Passband."

Another related Pat. is Mechanical Filters Employed Multilviode Resonators, U.S. Pat. No. 3,516,029, by Robert A. Johnson.

While the aforementioned general stopband type mechanical filters provide excellent response characteristics, certain problems of manufacture are present. More specifically, coupling wires attached to the perimeters of the diameter mode-type disc are quite sensitive to the detailed structure of the connection to the disc. If the welding of such coupling wire to the disc is not performed within quite close tolerances, the attenuation pole is moved up or down the frequency scale and thus can be moved either too far away from the edge of the passband, or conversely it can be moved into the edge of the passband, both of which conditions are undesirable.

It is a primary object of the present invention to provide a reliable general stopband mechanical filter which is relatively easy to manufacture.

A second purpose of the invention is to provide a simple, study and reliable general stopband mechanical filter of the disc-wire type.

' A third purpose of the invention is to provide a general stopband mechanical filter in which the general stopband characteristic is provided by a unique arrangement of coupling wires rather than through the use of diameter modes discs.

A fourth object of the invention is to provide a means for obtaining attenuation poles at both the upper and lower ends of the passband of a disc-wire mechanical filter by the unique construction and arrangement of coupling wires.

A fifth aim of the invention is to provide a general stopband mechanical filter'employing all circle mode-type discs.

A sixth aim of the invention is the improvement, generally, of mechanical filters of the disc-wire type.

in accordance with the invention there is provided a general stopband mechanical filter comprising a plurality of circle mode-type discs stacked one upon the other with their axes lying along a common line. A first plurality of coupling wire means are positioned parallel to the common axis and along the perimeters of the discs, and secured thereto to hold said disc in position and also to transfer energy from disc to disc along the filter. Second coupling wire means, which form the essence of the present invention, couple together two discs which are separated by two bridged discs. More specifically the second coupling wire means bridges two discs and couples together the discs on either side of the said two bridged discs.

The said second coupling wire means is constructed in such a manner with respect to length and diameter that it will produce a phase shift of the energy transferred therethrough, within the passband frequency. Since the circle mode discs resonate in phase at the lower edge of the passband, the energy transferred through the discs will be cancelled by the energy transferred through the bridging coupling wire. At the upper end of the passband each disc resonates 180 out of phase with the adjacent disc. Since two discs are bridged, each having 180 phase shift therein, the overall result is a phase shift of 360 through the two bridged discs. Again, the 180 phase shifted energy transferred through the bridged coupling wire will cancel the energy transferred through the bridged discs with a resultant attenuation pole at the upper edge of said passband.

in accordance with a modification of the invention the phase-inverting bridging coupling wire is caused to bridge one disc rather tan two discs. With only one disc bridged there is produced an attenuation pole only at he lower edge of the filter passband. Such attenuation pole is created at the lower edge of the filter passband. Such attenuation pole is created at the lower edge of the passband since the energy transferred through the coupling wire is phase shifted 180 whereas the energy transferred through the bridged disc has no phase shift therein. At the upper end of the filter passband the phase shift through the bridged disc is also 180 so that it tends to reinforce the energy transferred through the coupling wires. Thus there is no attenuation pole created at the upper edge of the passband.

The above-mentioned and other objects and features of the invention will be more fully understood from the following detailed description thereof when read in conjunction with the drawings in which:

H6. 1 is a perspective view of a mechanical filter in which a phase-inverting coupling wire is constructed to bridge two @1865;

HQ. 2 is a side view of a portion of FIG. 1 showing in more detail the bridging of two discs by the coupling wire;

FlG. 3 is an equivalent electrical circuit of the mechanical filter of FIG. ll;

FIGS. 3a and 4a are detailed circuit transformations employed in transforming the equivalent circuit of FIG. 3 into the equivalent circuit of FIG. 5;

FIG. 5 is the circuit of FIG. 3 after it has been transformed by the transformation techniques shown in FIGS. 3a and 5;

FIG. 6 is another embodiment of the invention wherein only one disc is bridged by the phase inverting coupling wire to thereby produce an attenuation pole only at the lower end of the filter passband;

MG. 7 is a side view of a portion of the structure of FIG. 6 to more clearly show the bridging of the single disc by the phaseinverting coupling wire;

FIG. 8 is an equivalent electrical circuit of the structure of FIG. s;

FIG. 9 is an equivalent circuit derived from the circuit of FIG. 8 and using the transformation techniques shown in FIGS. 3a and I;

FIG. III is a plot of the transfer impedance of the longitudinal mode of a bridging coupling wire as its length is varied, and more specifically shows those lengths where phase inversion with respect to the shorter coupling wires occurs and where this phase inversion does not occur;

FIG. II is a series of curves showing the transfer impedance of a bridging wire for the longitudinal mode and also for the sheer mode as its length is varied, and further shows the composite transfer impedance of the bridging wire vibrating in a complex combination of both its longitudinal and its sheer modes of vibration, as the length of the coupling wire varies;

FIG. I2 is a plot of frequency response of the circuit of FIG. 5; and

FIG. I3 is a similar plot for the circuit of FIG. 9.

This specification is organized in the following manner:

I. STRUCTURE OF FIG. I

A. General Sescription B. Phase Inversion In Coupling Wire C. Equivalent Circuit Of FIG. i and Transformations II. STRUCTURE OF FIG. '6

A. General Description B. Equivalent Circuit Of FIG. 6 and Transformations r. STRUCTURE OF FIG. 1

A. General Description Referring now to FIG. I there is shown a six-disc mechanical filter with the discs being identified by reference characters 2i), 2i, 22, 23, 24, and 25. Four coupling wires 26, 2'7, 28, and 29 are positioned along the perimeters of the six discs and parallel to the common axis thereof. Such coupling wires 26 through 29 are welded to the perimeters of the discs and function to hold the discs in position and also to transfer energy along the filter from disc to disc.

Input means 70 comprises a rod 3i of a magnetrostrictive material such as a ferrite and which is secured to the end disc 20, and upon which is wound an input winding 32, having an inherent resistence 35 therein and an inherent resonating capacitance 34 thereacross. The input signal is supplied from source 36. The output of the mechanical filter is contained within the dotted block 7I and includes a ferrite core 37 secured firmly to end disc and upon which is wound winding 38. Capacitor 39 represents the resonating capacity thereacross. A load resistor Ill represents, generally, any suitable utilization means for the mechanical filter and also includes the inherent resistance of winding 38.

The essence of the present invention is found in the coupling wire 3d and the two bridged discs 22 and 23. The bridged dims 22 and 23 each have a flat surface 48 and 4), respectively, formed thereon which permits the coupling wire 359 is secured at its two ends to the discs 21 and 24. As discussed briefly above, the length of the coupling wire is critical. It is a characteristic of a rod such as coupling wire 30 that the phase or" the energy transferred therethrough at frequencies whose wavelengths lie between a half and a full wavelength of the natural resonance frequencies of the coupling wire will be l from the phase of energy transferred through the coupling wires 26 to 29, which have lengths of less than one-half wavelength. A typical example might be as follows: At kI-Iz. a coupling wire of the material and a diameter employed in mechanical filters would have a longitudinal mode wavelength of almost 2 inches. Thus it would be necessary to have coupling wires of at least l inch in length to fall within the one-half to a full wavelength criteria. As a practical matter, a mechanical filter constructed with a l-inch bridging coupling wire would not be feasible since it would require the discs to be spaced too far apart to maintain structural strength. However, in the range of 400 kHz. or more, the coupling wire length decreases to about a quarter of an inch or less, which does permit practical spacing between the discs of the filter.

Before considering the equivalent circuit of FIG. 3 and the transformations following, reference is first made to the curves of FIGS. III and II which show in more detail the phenomena occurring in the coupling wire.

B. Phase Inversion In Coupling Wire In FIG. I0 there is shown a plot of the transfer impedance of a coupling wire as the length of the rod is varied. The impedance is plotted along the y-axis and the length along the xaxis. More specifically, the x-axis is plotted in terms of wL/ V where w and l/ are substantially constant, and the length L of the coupling rod varies.

In FIG. 10 only the longitudinal mode of vibration of the rod is considered. It can be seen that for a length of the rod less than half the acoustical wavelength of the driving signal there is no phase inversion, as is shown by curve 130. Similarly, for rod lengths greater than the acoustical wavelength of the driving signal there is no phase inversion, as is shown by curve 132. However, for bridging wire lengths which lie between one-half of an acoustical wavelength and a full acoustical wavelength of the bridging coupling wire for the given frequency to, phase inversion does occur, as represented by the curve IBI of FIG. It).

In actual practice, however, the longitudinal mode of vibration of the coupling wire is not only mode present. Also present is a sheer mode of vibration, of the coupling wires, which is introduced by the circular mode of vibration of the disc which contains a component of motion perpendicular to the axis of the coupling wire, thereby introducing said sheer mode. In FIG. 11 there is shown not only the independent effects of the longitudinal mode of vibration and the sheer mode of vibration, but also the superimposed effect of the two modes of vibration. More specifically, in FIG. II the longitudinal mode of vibration is shown by the curves I30, IBI, and I32. The sheer mode of vibration is shown by the curves I34 and 133. The resultant mode of vibration, which is in effect the longitudinal and sheer mode of vibrations superimposed one upon the other, is shown by the curves I37 and I36.

It can be stated generally that in both FIG. I0 and FIG. II a phase inversion occurs where the curve is above the x-axis and that no phase inversion occurs where the curve is below the xaxis. Thus in FIG. II phase inversion occurs between the pair of dotted lines I40 and I41 and the pair of dotted lines I42 and 143. In the actual design of a mechanical filter the area between the dotted lines M0 and 1411 is preferred to the area between dotted lines I42 and I43 for two reasons. Firstly, the length of the coupling wire for a given frequency is shorter than would be required if the response between dotted lines I42 and I43 were utilized. Secondly, the permissible variation of length between dotted lines and MI is greater than that between dotted lines I42 and M3.

As indicated in FIG. II, the dotted line Mil represents a length of the coupling wire which is equal to one-half the acoustical wavelength of such coupling wire at the driving frequency. Similarly, it can be seen that dotted line MI represents a length of coupling wire equal to about threefourths the acoustic wavelength of said coupling wire at the driving frequency.

C. Equivalent Circuit Of FIG. I

Referring now to FIG. 3 there is shown the equivalent circuit of the mechanical structured of FIG. I. In FIG. 3 the tuned circuits 52, 53, S4, 55, 56, and 57 correspond to the discs 20, 21, 22, 23, 24, and of FIG. I. The inductors L and L and the single inductor L represent the coupling wires 26, 27, 28, and 29 of FIG. I, and the inductor 60 of FIG. 3 represents the coupling wire of FIG. I.

It is to be noted that the inductor 60 is represented as a L,. in FIG. 3. Such a negative inductive value is derived from the fact that the coupling wire 30 inverts the phase of the signal transferred therethrough.

The input signal source 50 and the resistor 51 of FIG. 3 correspond to the input signal source 36 and the input resistor of FIG. 1. Similarly, the output resistor 58 of FIG. 3 corresponds to the output resistor of FIG. 1. Capacitors 34 and 39 of FIG. 1 are shown as C in FIG. 3 and the coils 32 and 38 are shown as L in FIG. 3. The circuit within dotted block 49 represents the bridged section of FIG. 1 including resonators 21, 22, 23, and 24.

In FIGS. 4 and 4a there is shown a circuit transformation useful in transforming the circuit of FIG. 3 to the form shown in FIG. 5. It can be seen in FIGS. 4 and 4a that a pi network consisting of inductors 61, 62, and 63 can be transformed into an inductor 64 in series with a negative I:1 transformer 65. In order to create a pi network of the type shown in FIG. 4 from the circuit of FIG. 3, the two inductors 75 and 76 of FIG. 3 are each split into two separate inductors consisting of two inductors in parallel, one identified as L' and the other identified as -L,, as shown in FIG. 3a.

The transformation of FIGS. 4 and 4a can then be incorporated directly in the circuit of FIG. 3 to produce the resultant equivalent circuit shown in FIG. 5. It is to be noted specifically that the inductor 60 of FIG. 3 is a negative inductor (-1 since it represents the bridging coupling wire which has been constructed to invert the phase of the signal transferred therethrough.

Thus in FIG. 5, when a second minus sign is added to inductor L,. of FIG. 3 due to the transformation, the result is that inductor 66 becomes positive.

The general purpose of transforming the circuit FIG. 3 into that of FIG. 5 is to provide an equivalent circuit both easy to analyze and to compute specific component values therefor to produce the desired frequency response characteristics. It is possible to transform the circuit FIG. 5 into other equivalent circuits for purposes of such component value computation, if desired. For example, in the aforementioned US application Ser. No. 547,947, filed May 5, 1966, by Robert A. Johnson et al. there is shown additional transformations of the type circuit shown in FIG. 5 of the instant specification.

The state of the art, however, is such that the values of the components of FIG. 5 can be obtained directly therefrom. Expressions for such component values are given below.

where R is the image impedance at where r= wlw 2yb) Once the values of the components of the circuit FIG. 5 are obtained the corresponding values of the circuits of FIG. 3 can easily be obtained. It is then a matter of design to fabricateth'e discs and coupling wires of the mechanical filter of FIG. 1 to correspond to the electrical component values of FIG. 3. More specifically, the size and shape of the discs, and the spacing therebetween, and the size and spacing of the coupling wires required to create an equivalency with the component values of FIG. 3 are well known in the art and will not be discussed in detail herein.

The resultant frequency response curve produced by the structure of FIG. 1 is shown in FIG. 12 with the two attenuation poles being identified by reference characters and 91.

II. STRUCTURE OF FIG. 6

A. General Description Referring now to FIG. 6 there is shown a modification of the invention in which the bridging coupling wire 109 bridges only one disc 112 rather than the two discs of FIG. 1. The principal difference in the operation of the structure of FIG. 6 and that of FIG. 1 is that an attenuation pole is created only at the lower end of the passband of FIG. 6, whereas attenuation poles were created at both the lower and the upper end of the ,passband in the structure of FIG. 1.

The structure of FIG. 6 is comprised of five discs, 100, 101, 102, 103, and 104 which are held in stacked relation by coupling wires 105, 106, 107, and 108 positioned along the perimeters of the disc and welded thereto as shown in the drawing. The coupling wire 109 is welded to disc 101 and 103 but bridges disc 102 over the flat surface 112 formed on the perimeter thereof.

Input means 110 and output means 111 of FIG. 6 correspond to the input means 70 and the output means 71, respectively, of FIG. 1.

In FIG. 7 there is shown a side view of the discs 101, 102, and 103 of FIG. 6 and the coupling wire 109. In FIG. 7 it can be seen how the coupling wire 109 bridges the center disc 102 over the flat surface 112 formed on the perimeter thereof.

As in the case of the structure of FIG. 1, coupling wire 109 .has a length which lies between one-half and three-fourths of the acoustical wavelength of said coupling wire at the driving frequency, which is nominally the center frequency of the filter passband. Also as in the case of the structure of FIG. 1,

the structure of FIG. 6. The circuit of FIG. 8 is quite similar to that shown in FIG. 3 except that the negative inductor 1 22 bridges only one tuned circuit 118 instead of two tuned circuits. Such difference is due to the fact that coupling wire 109 of FIG. 6, which the inductor 122 represents, bridgesonly one disc 102.

Using the transformation of FIGS. 4 and 4a the equivalent circuit of FIG. 8 is transferred into the equivalent circuit of FIG. 9. Here again, as in the transformation of the circuit of FIG. 3 to FIG. 5, the negative inductor 122 of FIG. 8 becomes the positive inductor 123 in FIG. 9 since a second minus sign is involved.

In FIGS. 8 and 9 the circuitry within dotted blocks 99 and 99' respectively, represents a three resonator bridged section.

The expression for finding the component values of the equivalent circuit of FIG. 9 are as follows:

where R istheima ge impedance where g y :l: y 1

and Where y =frequency of infinite attenuation where v= \/1 y and F:

p2: l F 2) rr The resultant frequency response characteristic produced by the structure of FIG. 6 is shown generally in FIG. 13 with the attenuation pole occurring at point 92 at the lower edge of the passband.

It is to be noted that the forms of the invention shown and described herein are but preferred embodiments thereof and that various changes can be made in the number and arrangement of coupling wires and the proportionate dimensions of the components of the filter without departing from the spirit or scope of the invention.

I claim:

1. A mechanical filter constructed to produce an attenuation pole on at least one side of the filter passband and comprising:

Tpliirality of discs stacked one upon the other along a common axis and spaced apart a distance equal to less than one-half wavelength of the nominal center frequency of said passband; first coupling wire means positioned substantially parallel to said common axis and physically secured to the perimeters of at least N a ja n me fsai dis swhcreN is e 3; and second coupling wire means positioned substantially parallel to said common axis and physically secure to the perimeters of the two end discs of said N discs and physically separated from the discs therebetween; and said second coupling wire means constructed to have a length to produce a phase shift of in the energy transferred therethrough within said passband. 2. A mechanical filter in accordance with claim 1 in which N is an odd integer.

3. A mechanical filter in accordance with claim 1 in which N=3.

4. A mechanical filter in accordance with claim 1 in which N is an even integer.

5. A mechanical filter in accordance with claim 1 in which N=4.

6. A mechanical filter means constructed to produce anati tenuation pole on at least one side of the filter passband and comprisin a plur ity of N circle mode discs stacked one upon the other along a common axis, where N 2 3; first coupling wire means positioned longitudinally along said stack of discs and physically secured tot he perimeters of said discs; and second coupling wire means positioned longitudinally along said stack of discs and physically secured only to the end discs of said N discs to effect a bridging of the discs positioned between said end discs; and said second coupling wire means having a length substantially in the range extending from 9%.! to A where his the acoustical wavelength of the natural resonant frequency of the wire material for a given frequency f in the filter passband, and producing 180 phase shift of energy in said passband. 7. A mechanical filter in accordance with claim 6 in which N=3.

8. A mechanical filter in accordance with claim 6 in which N is an odd integer.

9. A mechanical filter in accordance with claim 6 in which N is an even integer.

10. A mechanical filter in accordance with claim 6 in which N=4.

Claims (10)

1. A mechanical filter constructed to produce an attenuation pole on at least one side of the filter passband and comprising: a plurality of discs stacked one upon the other along a common axis and spaced apart a distance equal to less than one-half wavelength of the nominal center frequency of said passband; first coupling wire means positioned substantially parallel to said common axis and physically secured to the perimeters of at least N adjacent ones of said discs, where N is 3; and second coupling wire means positioned substantially parallel to said common axis and physically secure to the perimeters of the two end discs of said N discs and physically separated from the discs therebetween; and said second coupling wire means constructed to have a length to produce a phase shift of 180* in the energy transferred therethrough within said passband.

2. A mechanical filter in accordance with claim 1 in which N is an odd integer.

3. A mechanical filter in accordance with claim 1 in which N 3.

4. A mechanical filter in accordance with claim 1 in which N is an even integer.

5. A mechanical filter in accordance with claim 1 in which N4.

6. A mechanical filter means constructed to produce an attenuation pole on at least one side of the filter passband and comprising: a plurality of N circle mode discs stacked one upon the other along a common axis, where N 3; first coupling wire means positioned longitudinally along said stack of discs and physically secured tot he perimeters of said discs; and second coupling wire means positioned longitudinally along said stack of discs and physically secured only to the end discs of said N discs to effect a bridging of the discs positioned between said end discs; and said second coupling wire means having a length substantially in the range extending from 1/2 lambda to lambda , where lambda is the acoustical wavelength of the natural resonant frequency of the wire material for a given frequency f1 in the filter passband, and producing 180* phase shift of energy in said passband.

7. A mechaniCal filter in accordance with claim 6 in which N 3.

8. A mechanical filter in accordance with claim 6 in which N is an odd integer.

9. A mechanical filter in accordance with claim 6 in which N is an even integer.

10. A mechanical filter in accordance with claim 6 in which N 4.

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