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The design of resonators with specified reactances
Journal of Sound and Vibration(1971)19 (l), 29-38
THE DESIGN OF RESONATORS WITH SPECIFIED REACTANCES G. M. L. GLADWELL AND P. C. MAINALI Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada (Received 2 June 1971) The transfer matrix method is used to analyse the vibration of a thin straight rod made up of a number of linearly tapered cones. This analysis is used to design a bar with a specified end reactance in a certain range of frequency. Results are given for three specified
reactances and it is shown how they may be used in the design of a contact impedance meter and a stiffness measuring device. 1. INTRODUCTION
With the many powerful numerical techniques at our disposal it is relatively easy to analyse the vibration of almost any mechanical vibration. What is more difficult, and often more important, is to design a mechanical system so that it will vibrate in a specified way. This paper explores this problem for the very simplest vibrating system-a longitudinally vibrating thin straight rod. The property that the resonator is designed to have is a specified input reactance, that is, a specified ratio of force/velocity at one end of the bar. It is required to design a bar. The bar is assumed to be straight, thin, and of circular crosssection. The quantity to be found is the radius of the cross-section, or rather the way in which this radius varies along the length of the bar. This radius could be taken to be a continuously varying function, but both for numerical and for machining purposes it is preferable to consider it to be made up of various sections, in each of which it is constant, or varies linearly. If it is assumed to be piecewise constant then there will be discontinuities at the joints between sections. Such discontinuities result in unwanted stress concentrations. We therefore suppose that the radius is piecewise linear and continuous along the length. This is equivalent to assuming that the bar is made up of a number of linearly tapering (frustrate) cones. First we describe a simple computer program based upon the transfer matrix technique [l], which analyses the vibration of such a bar. This is then linked to an optimization technique to produce designs for various specified reactances. Finally it is shown how the bars so designed may be used in the design of a surface hardness tester. 2. ANALYSIS The equation governing the small longitudinal vibration U(X,t) of a thin straight rod is (1)
where A is the area of cross-section, p the density, and E is Young’s modulus. If the rod has a linearly tapered conical section, then A = D*, where D is proportional to the diameter and D = D,(l - SX),
x = x/L.
Do refers to the value of D at x = 0, and S represents the shape or taper of the element. 29
30
G. M. L. GLADWELL
AND P. C. MAINALI
Merkulov [2] has solved equation (1) for this case and, using his analysis, we may obtain the transfer matrix linking the dimensionless force and displacement at one end of the rod to the corresponding quantities at the other. We introduce the following quantities : L = standard length, e.g. the total length of the bar, A, = standard cross-section, & = standard Young’s modulus, Ai = cross-section of the element at end 1, 1= length of the element. Then we may define dimensionless quantities: RL = I/L,
RA = AI/A,,,
C = RLI(RA . RE).
RE = E/E,,
For vibration at frequency w we assume U(X,t) = U(X) exp (iwt), and the force amplitude at the section x is then
au
F= EAdx.
We define dimensionless displacement, force and frequency, respectively, by v= U/L,
h=wl/a;
G = F/(Ao Ed,
the last can be rewritten as X = (wL/a) . (l/L) = J2. RL,
a = 2/E/p.
The dimensionless frequency Sz is independent of the length of the element, whereas h depends on 1. The dimensionless transfer matrix relation is now found to be
where T, , = {cos X - S sin X/h}/(1 - S), T,, ={CsinX/h}/(l
-S),
T,,={S2(cos~-sinh/X)-h(l-S)sinh}/C, T22 = (1 - S) cos X + S sin h/h. When the cone is a uniform bar, that is when S = 0, T reduces to the standard transfer matrix C sin h/A cos x T = (-hsinA)/C COSX I ’
C
cm -9 1*
When the reduced frequency, 52, is small the matrix may be approximated by T* =
0
1 [
1 C/(1 - S) + 92 -(3 - S)/6(1 - S) 1 -{S2/3 + (1 - S)>/C
-(3 - 2S)/6
This approximation has the advantage that the two matrices are independent of frequency and may therefore be computed once only and stored. 3. NATURAL FREQUENCIES The preceding analysis may be programmed to give the displacements and forces at the joints between the elements into which the bar is divided. The natural frequencies are the frequencies at which the bar can vibrate freely under certain end conditions. The natural
31
RESONATORS WITH SPECIFIED REACTANCES
frequencies may be found by a variant of Holzer’s method-satisfying the boundary conditions at one end of the bar and iterating the frequency until the boundary conditions at the other end are satisfied. For the computation it was found to be convenient to define a function ENDFORCE. The variations of this function with frequency for a particular bar under free-free and clamped-free conditions are shown in Figures 1 and 2. In order to determine whether a given frequency is above or below a certain natural frequency we calculate the value ENDFORCE for the frequency and observe its sign. __
End
-
I
End 2
o.&+--J~ 0.30 G, = 0.0 v, = I.0
/
I
200 f
G,=ENDFORCE
300
(Hz)
Figure 1. The function ENDFORCE are in metres.
for a typical free-free
bar. All dimensions
End
on the insert diagram
I
End 2
‘&$y=Jb l( =o.o
G2= ENDFORCE
G, =-I,0
-1.21
1
100
I
I
200 f
300
I
(Hz)
Figure 2. The function ENDFORCE are in metres.
for a typical clamped-free
bar. All dimensions on the insert diagram
The following strategy was found to provide an efficient iterative scheme.
(1) Fundamental frequency,
fi
Starting frequency,5 was the fundamental natural frequency of a uniform diameter bar of the same length as the given bar. Step increment, df, was 5 % of the starting frequency.
(2) Second
natural frequency, f2
Starting frequency, f = 1.lfi Step increment Af = 0.OSf, (3) rth natural frequency, f, (r = 3,4,. . .) Starting frequency, f =fr-1 + O*l(f,_, -fi_J. Step increment Af = O*l(f,_, - frJ.
32
G. M. L. GLADWELL
4. IMPEDANCES
AND P. C. MAINALI
AND
REACTANCES
The impedance of a bar is defined as the ratio of force to velocity at a cross-section. Thus 2 = force/velocity = r + ix, where r is the mechanical resistance and x is the mechanical reactance. In the absence of damping r is zero and Z is purely imaginary. Thus the (input) impedance at end 2 of a free-free bar is
z* = where we have used the notation of section 2. The input impedance may be calculated by using the transfer matrix analysis of section 2; it is basically the ratio of the ENDFORCE to the velocity at the end. 5. DESIGN
CONSTRAINTS
The main purpose of the design is to find a bar with a certain specified end reactance. However, if the bar is to be used it must satisfy other general design criteria. The bar must not be very thin at one section and very thick elsewhere for this might lead to high stress concentrations. (Notice that the absolute magnitude of the diameter of the cross-section does not enter the analysis. It has been assumed throughout that the diameter is small compared to the length. This means, for instance, that if the diameter of every cross-section is decreased in the same ratio then the patterns of the displacement and force along the bar will remain unchanged even though their absolute magnitudes may be changed.) We introduce the following dimensionless parameters for the designation of the shape of the bar: d = diameter at ith cross-section I standard diameter D,, ’ I,, = length ofjth element
J
standard length L
’
Also we constrain di and 1; to lie within the limits 0.4 < di < 3.0,
0.125 < I; < 0.875.
(2)
Thus maximum diameter < 2 = 7 5 minimum diameter ’ 0.4 * and maximum length of element < 7. minimum length of element The n length ratios must of course satisfy the further constraint z;+I;+...+I;=l. 6. OPTIMlZATION
The design was carried out by using the general purpose optimization routine described by Vlach [3]. This programme is designed to find the minimum value of an arbitrary nonlinear function, F( Q(l), Q(2), . . . Q(N)), in the case when the parameters Q(i) are subject to simple constraints : QL(i> G Q(i) G QU(i>.
RESONATORS
WITH SPECIFIED REACTANCES
33
In the present problem the parameters Q(i) are the n dimensionless lengths of the elements and the n + 1 values of the dimensionless diameters at the ends of the elements, and the constraints are those given in equation (2). The bars were to be designed to have one of three specified reactances, x,,(f), for one of two frequency ranges, (a)fi
(3)
where the frequencies fi were distributed throughout the interval of interest. The iteration proceeds as follows. (i) Choose starting values of the parameters Q(i). The simplest choice is a uniform bar divided into equal elements. (ii) Calculate the dimensionless natural frequency of a bar having this shape. (iii) Adjust the total length of the bar so that the dimensional natural frequency is fi. (Notice that in the small diameter theory the natural frequencies of two bars of the same shape are inversely proportional to their lengths.) (iv) Calculate F and go into the optimization routine. This routine will calculate new values of the parameters, that is, a new shape of the bar, and the programme will go back to stage (ii) unless a minimum value of F has been found, or at least the criteria built into the optimization routine have been satisfied. 7. MOTIVATION The contact impedance meter [4] is a surface hardness tester based upon the measurement of the change in resonance frequency of a mechanical resonator, a bar in longitudinal resonance, brought into contact with the surface, as shown in Figure 3. The change in the resonance frequency of the resonator is a function of the elastic compliance of the excited surface which, in turn, is a function of d, the diameter of the contact area, and the properties (i.e. the Young’s modulus and Poisson’s ratio) of the material. In applications the contact impedance meter is used on a material for which the Young’s modulus and Poisson’s ratio are known, it being used, instead of a conventional microscope, to determine the diameter of the indentation made by the sensor tip. It may be shown [4] that the contact compliance is inversely proportional to the contact diameter and that, when damping and other similar effects are ignored, the equation governing the resonance frequency of the system of sensor rod, indenter tip and test piece is d=
kfx(f).
When the rod resonates out of contact with the surface, x(f) is zero, and so is d. If the rod is uniform then
x(f) = &tan(df -fJ/fA
(4)
where Z, is a constant and fi is the fundamental frequency of the bar. This means that provided the resonance frequency of the whole system does not differ much from f,, so that
Aflh = (f-AM is small, then 3
x(f) +z,~Aflfi
34 so
G. M. L. GLADWELL
AND P. C. hlAINAL1
that d + k(f, + Aj-)Z,nAflfi
i KAjI
(5) This means that the indentation diameter is directly proportional to the change in resonance frequency of the system. Test piece +/--surface
Figure 3. The contact impedance meter.
However, equation (5) holds only for very small values of Af; it is based upon the approximation tan 8 = 8, valid for small 8, and upon the neglect of the term in (Af)z in the product. If we could find a rod such that near its natural frequency, fi, its end reactance satisfied x(f) = k’df/.f=
k’(f-fi)/.L
then equation (4) would become d= kk’.f (f -I-d/f = K(f -fd and the diameter would truly be proportional to the change in resonance frequency. This is one application of the suggested design. If, instead of equation (5), we were to have
x(f >= k’(f -fd, so that the reactance would be directly proportional to the frequency difference, Af, then the change in natural frequency of the rod when a spring of reactance x, is attached to its end would be directly proportional to x,. In this case the change in frequency could be used to measure x,. The third case which we consider is so that
x(f I= k’ml_fi d = Kl/Af
or
d2
a
Of.
In this case the area of contact would be proportional to A$ 8. RESULTS In the first test for the procedure we took an arbitrary bar consisting of two elements, as shown in Figure 1, and, by designing bars with reactances which approximated the reactance of the given bar at various frequencies, tried to reconstruct the original bar. It was found that a number of widely differing bars could be found by choosing different sets of points for the sum F. Some of the results are shown in Table 1. The first two lines refer to the specified bar. The radii have been scaled but it will be seen that 0.75 : 1.35 : 2.10 = 5 : 9 : 14. Figure 4 shows the percentage variations of the reactances of the designed bars from those of the specimen bar. The following results were found for bars designed for the three reactances specified in section 6.
35
RESONATORS WITH SPECIPIED REACTANCES
Reactance 1: x(f) = k(f-f,)/f,. Four bars were designed, there being a two-element and a three-element bar for each of the frequency ranges, (a) and (b), listed in section 6 Each had a continuously varying cross-section. The value of k was arbitrarily chosen so that the straight line passed through the point on the reactance curve of the specimen bar of Table 1 at the frequency f = 1*025f,. Table 2 shows the dimensions of the four bars. The 1
TABLE
Design data for various bars
Dimensions of each element , RL RDl RD2
Total
S.N.
No. of elements
Element No.
length Cm) 0.8
2-F 1
2
0.736 35
2
2
0.762 07
3
2
0.743 21
4
2
0.716 13
o-75
0.375 0.625
1
2
1.35 0.628 0,731 0.633 0.806 0645 0.769 0.656 0.817
0.371 0.629 0,490 0510 0.530 0.470 0.687 0.313
1
2 1 2 1 2 1 2
0.75 2.10 0.729 0.730 0,738 O-669 0,758 0.722 0900 1.075
i Specimen bar.
E = 0.7339 x lo9 g/cm’;
p = 2.8 g/cm*. Fundamental
resonance
frequency
specified :
fi=113Hz.
2'0 A
A
t
A
A
A A
A
.
A
-3.0
-
-4.0
--
.
c 0 0 0
-5.0
I 5.0
1 IO.0
I 15.0 (Af/(,
I 200
I 250
30.0
x 100
Figure 4. Percentage variation of the reactance of the designed bars from that of the given bar. Frequency points: o, 1.25,2.5, 3.15 and 5; Q2.5, 5, 7.5 and 10; ?? ,2.5, 5, 10 and 20; A, lo,20 and 30.
36
G. M. L. GLADWELL
AND P. C. MAINALI
TABLE 2
Design of bars for cases l(a) and(b) [x(f) a: Of/f,] Dimensions No. of elements
Total length
1
2
0.668 8487
2
3
0.668 2887
3
2
0596 9551
4
3
0.682 3232
S.N.
Specified resonance frequency
of each element
/
Y
RL 0.493 0.506 0.237 0.275 0,487 0.479 0.520 0.331 0.337 0.331
RDl
3820 6181 3806 1217 4977 9963 0036 3269 3471 3269
0.795 0.714 0.795 0.754 0.714 1.032 0.741 1.106 0.741 1.770
RD2
7374 1030 7374 9202 1030 808 0860 153 8814 151
0.714 0.868 0.754 0.714 0,868 0.741 1.262 0.741 1.770 1.326
Design range
1030 9063 9202 i 0 < Af’ < 0.1 1030 9063 0860 619 8814 i 0 < Ajj” < 0.3 151 925
: fi = 113.769Hz.
o+pTfZJJ 0 507 0 493 L=0.660 a5 m Two element
2
4
6
8
100x(Af/f,)
I
bar
IO
I-= 0.668
29 m
Three-element
bar
Figure 5. Design for a bar having a straight line reactance in the rangef,
L=0.602 32 m Three-element
bar
L=0596 96m Two-element
bar
Figure 6. Design for a bar having a straight line reactance in the range f, < f < 1*3f. 0, Two-element bar with shifted resonance frequency; A, three-element bar with shifted resonance frequency; ---, uniform diameter bar; o, frequency pointsfi used for design,
RESONATORS
37
WITH SPECIFIED REACTANCES
,.z@g~ 0.527 0 473 L =0.496 29m Two-element bar
P L =0956
25 m
Three-element bar
:,;.t 5
15
IO
20
25
30
(Of/r, ) x 100
Figure 7. Design for a bar with a reactance proportioned to (f-fr)/fin the rangef,
Two-element bat
IOOX
(Af/q)
Three-element bar
Figure 8. Design for the bar with a reactance proportional to ( f - fi)/fin the rangef, 4 f< 1. lf, . 0, TWOelement rounded bar with shifted frequency; A, three-element rounded bar with shifted frequency; ---, uniform diameter bar; o, frequency pointsf; used for design.
7 2 z 5
40.0-
8 n"
0.20
0
cAr/f,)
c
i0
x IOO
Figure 9. The contrast between the specified reactance bar for case 3 (curve B) and the reactance of a typical bar (curve A).
38
G. M. L. GLADWELL AND P. C. MAINALI
dimensions of the bars are given by the optimization routine to seven decimal digits. It is clearly impossible to machine a bar to such accuracy and it is therefore necessary to round off the dimensions. This, however, will change the resonance frequency of the system and affect the variation of the impedance from the straight line issuing from the original resonance frequency. But if we measure the variation from a straight line issuing from the new resonance frequency we find that it is almost as small as that for the unrounded case. Figures 5 and 6 show the dimensions of the bars where the parameters are rounded to three decimal places, and the percentage errors from the specified reactances. Reactance 2: x(f) = k’(f-fi)/f: This is the case used for the contact impedance meter. Figures 7 and 8 show the results. Reactance 3 : x(f) = k’ddf /J It was found to be impossible to design a bar to have this reactance variation. The reason for this is shown in Figure 9. The reactance curve of any bar curves upward from a natural frequency, while the curve given by the reactance 3 relationship curves downward. 9. CONCLUSIONS The efficient optimization routines which are now available make it feasible to compute the design of simple resonators so that they have specified reactances in some specified frequency range. REFERENCES E. C. F’ESTELand F. A. LECKIE 1970 Matrix Methods in Elastomechanics. New York: McGrawHill. 2. L. G. MERKULOV 1957 Akustische Zhurnal 3, 230. Translated in Soviet Physics-Acoustics 4, 246-255. Design of ultrasonic concentrations. 3. J. VLACH 1971 In Computer Aided Engineering. (G. M. L. Gladwell, ed.) pp. 593ff. Waterloo, Ontario: Solid Mechanics Division, University of Waterloo. Comparison of some minimization algorithms. 4. C. KLEESATTELand G. M. L. GLADWELL 1968 Ultrasonics 6, 175-180,244-251; 7, 57-62. The contact impedance meter. 1.
THE DESIGN OF RESONATORS WITH SPECIFIED REACTANCES G. M. L. GLADWELL AND P. C. MAINALI Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada (Received 2 June 1971) The transfer matrix method is used to analyse the vibration of a thin straight rod made up of a number of linearly tapered cones. This analysis is used to design a bar with a specified end reactance in a certain range of frequency. Results are given for three specified
reactances and it is shown how they may be used in the design of a contact impedance meter and a stiffness measuring device. 1. INTRODUCTION
With the many powerful numerical techniques at our disposal it is relatively easy to analyse the vibration of almost any mechanical vibration. What is more difficult, and often more important, is to design a mechanical system so that it will vibrate in a specified way. This paper explores this problem for the very simplest vibrating system-a longitudinally vibrating thin straight rod. The property that the resonator is designed to have is a specified input reactance, that is, a specified ratio of force/velocity at one end of the bar. It is required to design a bar. The bar is assumed to be straight, thin, and of circular crosssection. The quantity to be found is the radius of the cross-section, or rather the way in which this radius varies along the length of the bar. This radius could be taken to be a continuously varying function, but both for numerical and for machining purposes it is preferable to consider it to be made up of various sections, in each of which it is constant, or varies linearly. If it is assumed to be piecewise constant then there will be discontinuities at the joints between sections. Such discontinuities result in unwanted stress concentrations. We therefore suppose that the radius is piecewise linear and continuous along the length. This is equivalent to assuming that the bar is made up of a number of linearly tapering (frustrate) cones. First we describe a simple computer program based upon the transfer matrix technique [l], which analyses the vibration of such a bar. This is then linked to an optimization technique to produce designs for various specified reactances. Finally it is shown how the bars so designed may be used in the design of a surface hardness tester. 2. ANALYSIS The equation governing the small longitudinal vibration U(X,t) of a thin straight rod is (1)
where A is the area of cross-section, p the density, and E is Young’s modulus. If the rod has a linearly tapered conical section, then A = D*, where D is proportional to the diameter and D = D,(l - SX),
x = x/L.
Do refers to the value of D at x = 0, and S represents the shape or taper of the element. 29
30
G. M. L. GLADWELL
AND P. C. MAINALI
Merkulov [2] has solved equation (1) for this case and, using his analysis, we may obtain the transfer matrix linking the dimensionless force and displacement at one end of the rod to the corresponding quantities at the other. We introduce the following quantities : L = standard length, e.g. the total length of the bar, A, = standard cross-section, & = standard Young’s modulus, Ai = cross-section of the element at end 1, 1= length of the element. Then we may define dimensionless quantities: RL = I/L,
RA = AI/A,,,
C = RLI(RA . RE).
RE = E/E,,
For vibration at frequency w we assume U(X,t) = U(X) exp (iwt), and the force amplitude at the section x is then
au
F= EAdx.
We define dimensionless displacement, force and frequency, respectively, by v= U/L,
h=wl/a;
G = F/(Ao Ed,
the last can be rewritten as X = (wL/a) . (l/L) = J2. RL,
a = 2/E/p.
The dimensionless frequency Sz is independent of the length of the element, whereas h depends on 1. The dimensionless transfer matrix relation is now found to be
where T, , = {cos X - S sin X/h}/(1 - S), T,, ={CsinX/h}/(l
-S),
T,,={S2(cos~-sinh/X)-h(l-S)sinh}/C, T22 = (1 - S) cos X + S sin h/h. When the cone is a uniform bar, that is when S = 0, T reduces to the standard transfer matrix C sin h/A cos x T = (-hsinA)/C COSX I ’
C
cm -9 1*
When the reduced frequency, 52, is small the matrix may be approximated by T* =
0
1 [
1 C/(1 - S) + 92 -(3 - S)/6(1 - S) 1 -{S2/3 + (1 - S)>/C
-(3 - 2S)/6
This approximation has the advantage that the two matrices are independent of frequency and may therefore be computed once only and stored. 3. NATURAL FREQUENCIES The preceding analysis may be programmed to give the displacements and forces at the joints between the elements into which the bar is divided. The natural frequencies are the frequencies at which the bar can vibrate freely under certain end conditions. The natural
31
RESONATORS WITH SPECIFIED REACTANCES
frequencies may be found by a variant of Holzer’s method-satisfying the boundary conditions at one end of the bar and iterating the frequency until the boundary conditions at the other end are satisfied. For the computation it was found to be convenient to define a function ENDFORCE. The variations of this function with frequency for a particular bar under free-free and clamped-free conditions are shown in Figures 1 and 2. In order to determine whether a given frequency is above or below a certain natural frequency we calculate the value ENDFORCE for the frequency and observe its sign. __
End
-
I
End 2
o.&+--J~ 0.30 G, = 0.0 v, = I.0
/
I
200 f
G,=ENDFORCE
300
(Hz)
Figure 1. The function ENDFORCE are in metres.
for a typical free-free
bar. All dimensions
End
on the insert diagram
I
End 2
‘&$y=Jb l( =o.o
G2= ENDFORCE
G, =-I,0
-1.21
1
100
I
I
200 f
300
I
(Hz)
Figure 2. The function ENDFORCE are in metres.
for a typical clamped-free
bar. All dimensions on the insert diagram
The following strategy was found to provide an efficient iterative scheme.
(1) Fundamental frequency,
fi
Starting frequency,5 was the fundamental natural frequency of a uniform diameter bar of the same length as the given bar. Step increment, df, was 5 % of the starting frequency.
(2) Second
natural frequency, f2
Starting frequency, f = 1.lfi Step increment Af = 0.OSf, (3) rth natural frequency, f, (r = 3,4,. . .) Starting frequency, f =fr-1 + O*l(f,_, -fi_J. Step increment Af = O*l(f,_, - frJ.
32
G. M. L. GLADWELL
4. IMPEDANCES
AND P. C. MAINALI
AND
REACTANCES
The impedance of a bar is defined as the ratio of force to velocity at a cross-section. Thus 2 = force/velocity = r + ix, where r is the mechanical resistance and x is the mechanical reactance. In the absence of damping r is zero and Z is purely imaginary. Thus the (input) impedance at end 2 of a free-free bar is
z* = where we have used the notation of section 2. The input impedance may be calculated by using the transfer matrix analysis of section 2; it is basically the ratio of the ENDFORCE to the velocity at the end. 5. DESIGN
CONSTRAINTS
The main purpose of the design is to find a bar with a certain specified end reactance. However, if the bar is to be used it must satisfy other general design criteria. The bar must not be very thin at one section and very thick elsewhere for this might lead to high stress concentrations. (Notice that the absolute magnitude of the diameter of the cross-section does not enter the analysis. It has been assumed throughout that the diameter is small compared to the length. This means, for instance, that if the diameter of every cross-section is decreased in the same ratio then the patterns of the displacement and force along the bar will remain unchanged even though their absolute magnitudes may be changed.) We introduce the following dimensionless parameters for the designation of the shape of the bar: d = diameter at ith cross-section I standard diameter D,, ’ I,, = length ofjth element
J
standard length L
’
Also we constrain di and 1; to lie within the limits 0.4 < di < 3.0,
0.125 < I; < 0.875.
(2)
Thus maximum diameter < 2 = 7 5 minimum diameter ’ 0.4 * and maximum length of element < 7. minimum length of element The n length ratios must of course satisfy the further constraint z;+I;+...+I;=l. 6. OPTIMlZATION
The design was carried out by using the general purpose optimization routine described by Vlach [3]. This programme is designed to find the minimum value of an arbitrary nonlinear function, F( Q(l), Q(2), . . . Q(N)), in the case when the parameters Q(i) are subject to simple constraints : QL(i> G Q(i) G QU(i>.
RESONATORS
WITH SPECIFIED REACTANCES
33
In the present problem the parameters Q(i) are the n dimensionless lengths of the elements and the n + 1 values of the dimensionless diameters at the ends of the elements, and the constraints are those given in equation (2). The bars were to be designed to have one of three specified reactances, x,,(f), for one of two frequency ranges, (a)fi
(3)
where the frequencies fi were distributed throughout the interval of interest. The iteration proceeds as follows. (i) Choose starting values of the parameters Q(i). The simplest choice is a uniform bar divided into equal elements. (ii) Calculate the dimensionless natural frequency of a bar having this shape. (iii) Adjust the total length of the bar so that the dimensional natural frequency is fi. (Notice that in the small diameter theory the natural frequencies of two bars of the same shape are inversely proportional to their lengths.) (iv) Calculate F and go into the optimization routine. This routine will calculate new values of the parameters, that is, a new shape of the bar, and the programme will go back to stage (ii) unless a minimum value of F has been found, or at least the criteria built into the optimization routine have been satisfied. 7. MOTIVATION The contact impedance meter [4] is a surface hardness tester based upon the measurement of the change in resonance frequency of a mechanical resonator, a bar in longitudinal resonance, brought into contact with the surface, as shown in Figure 3. The change in the resonance frequency of the resonator is a function of the elastic compliance of the excited surface which, in turn, is a function of d, the diameter of the contact area, and the properties (i.e. the Young’s modulus and Poisson’s ratio) of the material. In applications the contact impedance meter is used on a material for which the Young’s modulus and Poisson’s ratio are known, it being used, instead of a conventional microscope, to determine the diameter of the indentation made by the sensor tip. It may be shown [4] that the contact compliance is inversely proportional to the contact diameter and that, when damping and other similar effects are ignored, the equation governing the resonance frequency of the system of sensor rod, indenter tip and test piece is d=
kfx(f).
When the rod resonates out of contact with the surface, x(f) is zero, and so is d. If the rod is uniform then
x(f) = &tan(df -fJ/fA
(4)
where Z, is a constant and fi is the fundamental frequency of the bar. This means that provided the resonance frequency of the whole system does not differ much from f,, so that
Aflh = (f-AM is small, then 3
x(f) +z,~Aflfi
34 so
G. M. L. GLADWELL
AND P. C. hlAINAL1
that d + k(f, + Aj-)Z,nAflfi
i KAjI
(5) This means that the indentation diameter is directly proportional to the change in resonance frequency of the system. Test piece +/--surface
Figure 3. The contact impedance meter.
However, equation (5) holds only for very small values of Af; it is based upon the approximation tan 8 = 8, valid for small 8, and upon the neglect of the term in (Af)z in the product. If we could find a rod such that near its natural frequency, fi, its end reactance satisfied x(f) = k’df/.f=
k’(f-fi)/.L
then equation (4) would become d= kk’.f (f -I-d/f = K(f -fd and the diameter would truly be proportional to the change in resonance frequency. This is one application of the suggested design. If, instead of equation (5), we were to have
x(f >= k’(f -fd, so that the reactance would be directly proportional to the frequency difference, Af, then the change in natural frequency of the rod when a spring of reactance x, is attached to its end would be directly proportional to x,. In this case the change in frequency could be used to measure x,. The third case which we consider is so that
x(f I= k’ml_fi d = Kl/Af
or
d2
a
Of.
In this case the area of contact would be proportional to A$ 8. RESULTS In the first test for the procedure we took an arbitrary bar consisting of two elements, as shown in Figure 1, and, by designing bars with reactances which approximated the reactance of the given bar at various frequencies, tried to reconstruct the original bar. It was found that a number of widely differing bars could be found by choosing different sets of points for the sum F. Some of the results are shown in Table 1. The first two lines refer to the specified bar. The radii have been scaled but it will be seen that 0.75 : 1.35 : 2.10 = 5 : 9 : 14. Figure 4 shows the percentage variations of the reactances of the designed bars from those of the specimen bar. The following results were found for bars designed for the three reactances specified in section 6.
35
RESONATORS WITH SPECIPIED REACTANCES
Reactance 1: x(f) = k(f-f,)/f,. Four bars were designed, there being a two-element and a three-element bar for each of the frequency ranges, (a) and (b), listed in section 6 Each had a continuously varying cross-section. The value of k was arbitrarily chosen so that the straight line passed through the point on the reactance curve of the specimen bar of Table 1 at the frequency f = 1*025f,. Table 2 shows the dimensions of the four bars. The 1
TABLE
Design data for various bars
Dimensions of each element , RL RDl RD2
Total
S.N.
No. of elements
Element No.
length Cm) 0.8
2-F 1
2
0.736 35
2
2
0.762 07
3
2
0.743 21
4
2
0.716 13
o-75
0.375 0.625
1
2
1.35 0.628 0,731 0.633 0.806 0645 0.769 0.656 0.817
0.371 0.629 0,490 0510 0.530 0.470 0.687 0.313
1
2 1 2 1 2 1 2
0.75 2.10 0.729 0.730 0,738 O-669 0,758 0.722 0900 1.075
i Specimen bar.
E = 0.7339 x lo9 g/cm’;
p = 2.8 g/cm*. Fundamental
resonance
frequency
specified :
fi=113Hz.
2'0 A
A
t
A
A
A A
A
.
A
-3.0
-
-4.0
--
.
c 0 0 0
-5.0
I 5.0
1 IO.0
I 15.0 (Af/(,
I 200
I 250
30.0
x 100
Figure 4. Percentage variation of the reactance of the designed bars from that of the given bar. Frequency points: o, 1.25,2.5, 3.15 and 5; Q2.5, 5, 7.5 and 10; ?? ,2.5, 5, 10 and 20; A, lo,20 and 30.
36
G. M. L. GLADWELL
AND P. C. MAINALI
TABLE 2
Design of bars for cases l(a) and(b) [x(f) a: Of/f,] Dimensions No. of elements
Total length
1
2
0.668 8487
2
3
0.668 2887
3
2
0596 9551
4
3
0.682 3232
S.N.
Specified resonance frequency
of each element
/
Y
RL 0.493 0.506 0.237 0.275 0,487 0.479 0.520 0.331 0.337 0.331
RDl
3820 6181 3806 1217 4977 9963 0036 3269 3471 3269
0.795 0.714 0.795 0.754 0.714 1.032 0.741 1.106 0.741 1.770
RD2
7374 1030 7374 9202 1030 808 0860 153 8814 151
0.714 0.868 0.754 0.714 0,868 0.741 1.262 0.741 1.770 1.326
Design range
1030 9063 9202 i 0 < Af’ < 0.1 1030 9063 0860 619 8814 i 0 < Ajj” < 0.3 151 925
: fi = 113.769Hz.
o+pTfZJJ 0 507 0 493 L=0.660 a5 m Two element
2
4
6
8
100x(Af/f,)
I
bar
IO
I-= 0.668
29 m
Three-element
bar
Figure 5. Design for a bar having a straight line reactance in the rangef,
L=0.602 32 m Three-element
bar
L=0596 96m Two-element
bar
Figure 6. Design for a bar having a straight line reactance in the range f, < f < 1*3f. 0, Two-element bar with shifted resonance frequency; A, three-element bar with shifted resonance frequency; ---, uniform diameter bar; o, frequency pointsfi used for design,
RESONATORS
37
WITH SPECIFIED REACTANCES
,.z@g~ 0.527 0 473 L =0.496 29m Two-element bar
P L =0956
25 m
Three-element bar
:,;.t 5
15
IO
20
25
30
(Of/r, ) x 100
Figure 7. Design for a bar with a reactance proportioned to (f-fr)/fin the rangef,
Two-element bat
IOOX
(Af/q)
Three-element bar
Figure 8. Design for the bar with a reactance proportional to ( f - fi)/fin the rangef, 4 f< 1. lf, . 0, TWOelement rounded bar with shifted frequency; A, three-element rounded bar with shifted frequency; ---, uniform diameter bar; o, frequency pointsf; used for design.
7 2 z 5
40.0-
8 n"
0.20
0
cAr/f,)
c
i0
x IOO
Figure 9. The contrast between the specified reactance bar for case 3 (curve B) and the reactance of a typical bar (curve A).
38
G. M. L. GLADWELL AND P. C. MAINALI
dimensions of the bars are given by the optimization routine to seven decimal digits. It is clearly impossible to machine a bar to such accuracy and it is therefore necessary to round off the dimensions. This, however, will change the resonance frequency of the system and affect the variation of the impedance from the straight line issuing from the original resonance frequency. But if we measure the variation from a straight line issuing from the new resonance frequency we find that it is almost as small as that for the unrounded case. Figures 5 and 6 show the dimensions of the bars where the parameters are rounded to three decimal places, and the percentage errors from the specified reactances. Reactance 2: x(f) = k’(f-fi)/f: This is the case used for the contact impedance meter. Figures 7 and 8 show the results. Reactance 3 : x(f) = k’ddf /J It was found to be impossible to design a bar to have this reactance variation. The reason for this is shown in Figure 9. The reactance curve of any bar curves upward from a natural frequency, while the curve given by the reactance 3 relationship curves downward. 9. CONCLUSIONS The efficient optimization routines which are now available make it feasible to compute the design of simple resonators so that they have specified reactances in some specified frequency range. REFERENCES E. C. F’ESTELand F. A. LECKIE 1970 Matrix Methods in Elastomechanics. New York: McGrawHill. 2. L. G. MERKULOV 1957 Akustische Zhurnal 3, 230. Translated in Soviet Physics-Acoustics 4, 246-255. Design of ultrasonic concentrations. 3. J. VLACH 1971 In Computer Aided Engineering. (G. M. L. Gladwell, ed.) pp. 593ff. Waterloo, Ontario: Solid Mechanics Division, University of Waterloo. Comparison of some minimization algorithms. 4. C. KLEESATTELand G. M. L. GLADWELL 1968 Ultrasonics 6, 175-180,244-251; 7, 57-62. The contact impedance meter. 1.