Method of and apparatus for stress detection

A stress is detected by the utilization of a semiconductor equipped with a beam oscillator which is driven to generate a compound oscillation comprising a plurality of component oscillations with different component frequencies through the steps of detecting separately component amplitudes of the compound oscillation for the respective component frequencies, standardizing power spectra of the component amplitudes with theoretically obtained reference power spectra respectively, and determining a stress generated in the beam oscillator based on a mean stress value of stress values for the standardized power spectra from relations between power spectrum and stress value previously provided for the respective component amplitudes.

BACKGROUND OF THE INVENTION 
1. Field of the invention 
The invention relates to a stress detection method in which a semiconductor 
sensor having an oscillating member is used to detect stresses generated 
in the oscillating member according to amounts of dynamic forces acting on 
the oscillating member, such as pressure and the acceleration of gravity 
and a stress detection apparatus used in the stress detection method. 
2. Description of Related Art 
It has been proposed in dynamic quantity detection to utilize pertinently 
prepared semiconductor sensors. Such a dynamic quantity detecting 
semiconductor sensor is equipped with a stress generating member, such as 
a laminated film member which generates a stress according to pressure as 
a dynamic quantity acting thereon, or a beam member which generates a 
stress according to the acceleration of gravity as a dynamic quantity 
acting thereon. The dynamic quantity is determined by comparing the stress 
generated in the stress generating member with a quantitative relation 
between dynamic quantity and stress. The most importance in the dynamic 
quantity detection in which the utilization is made of the semiconductor 
sensor is how precise the stress detection is. 
One approach to detect stresses by the semiconductor sensors is to utilize 
changes in resistance of a piezo electric resistor, or changes in resonant 
oscillation frequency of a beam 
In the case where the utilization is made of changes in resistance of such 
a piezo electric resistor, a laminated film of the piezo electric resistor 
formed as the stress generating member provides a resistance value 
changing according to its strain or distortions caused due to a 
compression stress or a tensile stress generated therein by dynamic force 
acting thereon, based on which the stress generated in the stress 
generating member is determined. In this instance, since a specific 
quantitative relation is established between resistance value of the piezo 
electric resistor film and stress generated in the stress generating 
member, the stress is known from the resistance value with reference to 
the quantitative relation. 
There is, however, such a drawback in the technique that the piezo electric 
resistor shows a relatively significant change in resistance due to 
changes in ambient temperature and experiences significant changes in 
physical characteristics due to aging. Accordingly, the detected 
resistance includes a change in resistance due to distortion or strain in 
addition to changes in resistance due to changes in ambient temperature 
and aging, resulting in a significant error in stress detection. 
While the change in resistance due to ambient temperature can be 
compensated, it is necessary to provide a compensation circuit in the 
stress detection apparatus and there has not been no effective approach of 
eliminating the changes in resistance due to aging until today. 
In the case where the utilization is made of changes in resonant 
oscillation frequency, the stress generating member is provided in the 
form of a straddle mounted beam oscillator, of which a resonant 
oscillation frequency is detected during an oscillation caused by 
periodically changing external exciting force. The straddle mounted beam 
oscillator causes a resonant oscillation whose oscillation frequency 
depends upon the stress generated in the straddle mounted beam oscillator. 
Letting f.sub.r and N be the resonant oscillation frequency and the stress 
when the straddle mounted beam oscillator causes a resonant oscillation, 
the following quantitative relation is established: 
##EQU1## 
where .alpha. and .beta. are positive invariables. When the resonant 
oscillation frequency f.sub.r is known, the stress N is determined from 
the above quantitative relation. 
The utilization of the stress generating member in the form of a straddle 
mounted beam oscillator has a constraint on the extent of properly 
detectable stresses. Together, the detection of increased stress 
encounters a lack of accuracy. Specifically, since there is the 
quantitative relation between the resonant oscillation frequency and 
stress of the straddle mounted beam oscillator given by the formula (1), 
the condition of N.gtoreq.-.alpha./.beta. must be always satisfied. This 
indicates that the smallest detectable stress is greater than 
-.alpha./.beta. and it is impossible to detect a relatively large 
compression stress, consequently. Further, since a change in resonant 
oscillation frequency becomes smaller with an increase in stress generated 
in the straddle mounted beam oscillator, the resolution of stress 
detection is deteriorated with an increase in stress, resulting in large 
errors in stress detection. 
While the utilization of the semiconductor sensor enables to detect dynamic 
quantities from stresses generated in the stress generating member, 
various constraints must be imposed on the semiconductor sensor. For 
instance, the semiconductor sensor must be enclosed in a vacuum container 
so as to be isolated from air resistance acting on the stress generating 
member during practical stress detection. The necessity of vacuum 
container renders the stress detecting apparatus troublesome and expensive 
to be manufactured and causes aggravation of yield due to dispersion in 
the degree of vacuum. 
Even when a sufficiently high degree of vacuum is achieved in the vacuum 
container for the semiconductor sensor, the vacuum container experiences 
deterioration in vacuum leakage with progress of time. In cases where the 
semiconductor sensor installed in such a vacuum container is used to 
monitor the safety of an apparatus to which the semiconductor sensor is 
attached, unexpected situations occurring due to deterioration in the 
degree of vacuum of the vacuum container puts the semiconductor sensor 
unreliable and brings about a reduction in the safety of the apparatus. 
Even if the problem of air resistance and deteriorated resolution in stress 
detection have been settled, there is still another problem that, if a 
band-pass filter has a distribution of transmission factors relatively 
widely spreaded, components with different frequencies mix in during 
frequency analysis. This is because, on one hand, the utilization is made 
of amplitude spectrum strength of various different frequencies and, on 
the other hand, it is hard as a design matter to provide the filter with a 
spread in distribution of transmission factors as small as possible. 
SUMMARY OF THE INVENTION 
It is an object of the invention to provide a method of and an apparatus 
for stress detection in which a pertinently prepared semiconductor sensor 
equipped with a stress generating member which generates a stress 
according to external dynamic force, such as pressure and the acceleration 
of gravity, acting thereon. 
It is an object of the invention to provide a method of and an apparatus 
for stress detection which enables the detection of stress without an 
influence of changes in ambient temperature and/or an influence of changes 
in physical characteristics of a stress generating member due to aging, 
and provides a wide extent of detectable stress, but does not encounter 
aggravation of stress detecting resolution even when it detects increased 
stresses. 
It is another object of the invention to provide a method of and an 
apparatus for stress detection in which a pertinently prepared 
semiconductor sensor equipped with a stress generating member is used to 
detect stressed without being accompanies with an influence of vacuum 
and/or air resistance. 
It is still another object of the invention to provide a method of and an 
apparatus for stress detection in which a semiconductor equipped with a 
stress generating member is used with the effect of eliminating 
undesirable component frequencies. 
The foregoing objects of the invention are accomplished by providing a 
stress detecting method in which a semiconductor sensor provided with a 
straddle mounted beam oscillator is used and which comprises the steps of: 
driving the semiconductor sensor so as to generate a compound oscillation 
including a plurality of component oscillations having different component 
frequencies, respectively; detecting separately component amplitudes of 
the compound oscillation for the respective component frequencies; 
calculating squares of the component amplitudes to provide power spectra 
of the component amplitudes; standardizing the power spectra, 
respectively, with reference power spectra which are theoretically 
obtained power spectra for amplitudes of oscillations having the component 
oscillation frequencies of the straddle mounted beam oscillator on 
condition that at least the straddle mounted beam oscillator is held free 
from any stress; determining stress values for the standardized power 
spectra from quantitative relations between power spectra and stress 
values previously provided for the component amplitudes, respectively; and 
obtaining a mean stress value by averaging the stress values, preferably 
with weighing factors determined based on the standardized power spectra, 
and determining a stress generated in the straddle mounted beam oscillator 
based on the mean stress value. The component amplitude is detected as a 
frequency change in electrostatic capacity generated between an electrode 
fixed to the straddle mounted beam oscillator and a stationary electrode 
fixed to the semiconductor sensor vacuum container for each component 
frequency, and the frequency change is squared to provide the power 
spectrum. 
The reference power spectrum for an amplitude of an oscillation having a 
component oscillation frequency may be determined for the straddle mounted 
beam oscillator put in a vacuum. Further, the reference power spectrum for 
an amplitude of an oscillation having a component oscillation frequency 
may be determined for the straddle mounted beam oscillator assuming a 
state free from any stress in a vacuum. 
Specifically, the different component frequencies includes at least two 
component oscillations having oscillation frequencies .omega..sub.q and 
.omega..sub.r which satisfy, respectively, the following conditions: 
##EQU2## 
The component amplitude is detected as a frequency of change in 
electrostatic capacity generated between an electrode fixed to the 
straddle mounted beam oscillator and a stationary electrode fixed to the 
semiconductor sensor for the component frequency. The frequency of a 
change in electrostatic capacity is squared to provide the power spectrum. 
The smallest component oscillation frequency among the different component 
oscillation frequencies of the compound oscillation is established greater 
than a resonant oscillation frequency of the straddle mounted beam 
oscillator on which a stress value representing a stress generated in the 
straddle mounted beam oscillator assumes an upper limit.

DETAILED DESCRIPTION OF THE EMBODIMENTS 
In the analytical detection of stresses generated in a straddle mounted 
beam type oscillator of a semiconductor sensor by means of the stress 
detection apparatus of the invention, when the straddle mounted beam type 
oscillator is excited with exciting force of oscillation frequencies 
.omega..sub.1 /2.pi., . . . , and .omega..sub.n /2.pi. (n is a positive 
integer greater than 2) having amplitudes a.sub.1, . . . , and a.sub.n, 
respectively, and, as a result, generates a compound oscillation 
consisting of component oscillations having oscillation frequencies 
.omega..sub.1 /2.pi., . . . , and .omega..sub.n /2.pi., respectively, the 
amplitude of coupling vibration W(x, t) at a time t at a distance x from 
one extremity of the straddle mounted beam oscillator is expressed by the 
following equation (2): 
EQU (.differential..sup.4 
W/.differential.x.sup.4)-(N/EI).multidot.(.differential..sup.2 
W/.differential.x.sup.2)+(.rho..sub.0 A/EI).multidot.(.differential..sup.2 
W/.differential.x.sup.2)=f(t) (2) 
where A is the sectional area of the straddle mounted beam oscillator: 
.rho..sub.0 is the density of the straddle mounted beam oscillator: 
I is the second moment of area of the straddle mounted beam oscillator: 
E is the Young's modulus: 
N is the stress generated in the straddle mounted beam oscillator: 
f(t) is the exciting force. 
The exciting force f(t) is given by the following formula (3): 
EQU f(t)=a.sub.1 e.sup.i.omega.1t + . . . +a.sub.n e.sup.i.omega.nt(3) 
From the geometric condition that the straddle mounted beam oscillator is 
fixedly held at both extremities, the following formulas (4) are 
satisfied: 
EQU w(0, t)=W(L, t)=0, Wx(0, t)=Wx(L, t)=0. (4) 
where Wx(0, t)=.differential.W/.differential.x. 
Then, carrying out Fourier transformation of the equation (2) with regard 
to time t, the following formula (5) is obtained: 
EQU (d.sup.4 F(W)/dx.sup.4)-(N/EI).multidot.(d.sup.2 F(W)/dx.sup.2)-4.pi..sup.2 
.mu..sup.2 .multidot.(.rho..sub.0 A/EI).multidot.F(W)=F(f)(5) 
where .rho. is an oscillation frequency. 
F(W) and F(f) are the Fourier transformed expressions of amplitude W(x, t) 
and exciting force f(t), respectively and given by the following formulas: 
##EQU3## 
The following quantitative relation, which is obtained by solving the 
formula (5) under the condition of equations (4), is given by the 
following expression (7): 
##EQU4## 
where L is the overall length of the straddle mounted beam oscillator; 
.+-..rho. and .+-..sigma.i(i.sup.2 =-1) are the roots of a characteristic 
equation for .UPSILON. which is expressed as follows: 
EQU .UPSILON..sup.4 -(N/EI).multidot..UPSILON..sup.2 -4.pi..sup.2 .mu..sup.2 
.rho..sub.0 A/EI=0 (8) 
.rho. and .sigma. are given as follows: 
##EQU5## 
where P=N/EI, and Q=.rho..sub.0 A/EI. The Fourier transformed expression 
of amplitude W(L/2, t) at the center of the straddle mounted beam 
oscillator is given as follows: 
##EQU6## 
This Fourier transformed expression F (L/2, .mu.) describes a oscillation 
spectrum of the straddle mounted beam oscillator at the center. A power 
spectrum is given as a spare of the oscillation spectrum F(L/2, .mu.). A 
Fourier transformation of the right side of the formula (5) expressing the 
exciting force f(t) gives the following expression (11): 
##EQU7## 
where k is an integer variable between 1 and n, and .omega./2.pi. is the 
natural oscillation frequency of the straddle mounted beam oscillator. 
After substituting the formula (11) for the formula (10), the quantitative 
relation equation of oscillation spectrum F (L/2, .mu.) is obtained. 
Since .differential..multidot.(.mu.-.omega..sub.k /2.pi.) has a value only 
for .mu.=.omega..sub.k /2.pi. and, however, always assumes zero (0) for 
.mu. which is not equal to .omega..sub.k /2.pi., the oscillation spectrum 
F(L/2, .mu.) will separate for values of k (1, . . . , n). For this 
reason, letting the oscillation spectrum for .mu.=.omega..sub.k /2.pi. be 
F(L/2, .mu.), the following formula (12) is established: 
##EQU8## 
Changing the formula (12) with dimensionless values as expressed by 
equations (13)-(15), the formula (16) is established. 
##EQU9## 
where N.sub.0 =EI/L2; 
##EQU10## 
In these formulas (14) and (15), z is a stress value, indicating and 
proportional to a stress N generated in the straddle mounted beam 
oscillator, which is established by dividing the stress N by a constant 
N.sub.0 determined on the basis of a length L, a second moment of area I 
and a Young's modulus E of the straddle mounted beam oscillator, and 
y.sub.k (=.omega..sub.k /.omega.) is an angular frequency value, 
indicating and proportional to an angular frequency of the straddle 
mounted beam oscillator, which is established by dividing the angular 
frequency .omega..sub.k by a constant .omega. determined on the basis of a 
constant depending upon physical properties, a structural geometric 
condition of the straddle mounted beam oscillator. 
The power spectrum (F(L/2, .mu.)).sup.2 is standardized by a reference 
power spectrum which is established when there is no stress caused in the 
straddle mounted beam oscillator, i.e. when N is 0, and hence z=N/N.sub.0 
is 0. The standardized power spectrum G(k) is given by the following 
formula (17): 
##EQU11## 
The standardized power spectrum G(k) indicates a change rate of the power 
spectrum (F(L/2, .mu.)).sup.2 for .mu.=.omega..sub.k /2.pi. in connection 
with a stress value z. The quantitative relation between the standardized 
power spectrum G(k) and stress value z is shown as correlation curves for 
various angular frequency values (y.sub.k =.omega./.omega.), i.e. 
different component oscillations having oscillation frequencies 
.omega..sub.1 /2.omega., . . . , and .omega..sub.n /2.pi., included in a 
compound oscillation, in FIG. 8. Accordingly, the stress value z is 
obtained from the quantitative relation shown in FIG. 8 if the 
standardized power spectrum G(k) is known. 
The state where the oscillation spectrum F(L/2, .mu.), which is one of 
factors of the denominator in the formula (16), is zero (0) as expressed 
by the formula (18), means that the straddle mounted beam oscillator is in 
a resonant state. 
EQU .upsilon..sub.k sin h.upsilon..sub.k .multidot.cos .lambda..sub.k +.lambda. 
cos h.upsilon..sub.k sin .lambda..sub.k =0 (18) 
In the resonant state, the quantitative relation between angular frequency 
value (y.sub.k) and stress value z is shown as correlation curves in FIG. 
9 and is expressed by the following formula (19): 
##EQU12## 
The stress value z that can be obtained from the formula (17) and FIG. 8 
based on the standardized power spectrum G(k) is not appropriate but less 
than the stress value causing in the straddle mounted beam oscillator in a 
resonant state. Accordingly, letting z.sub.M and .omega..sub.m /2.pi. be 
the upper limit of a stress value z and the smallest oscillation frequency 
of oscillation components included in a compound oscillation of the 
straddle mounted beam oscillator, the smallest oscillation frequency 
.omega..sub.m /2.pi. must be established such that the following condition 
is satisfied: 
##EQU13## 
Together, the stress value z that can be appropriately obtained from the 
formula (17) and FIG. 8 based on the standardized power spectrum G(k) is 
greater than the stress value that the standardized power spectrum G(k) is 
at a minimum. For the minimum standardized power spectrum G(k), the 
quantitative relation between the angular frequency value (y.sub.k) and 
stress value z is shown in FIG. 10 and is expressed by the following 
formula (21): 
##EQU14## 
In this quantitative relation, the angular frequency value (y.sub.k) is 
proportional to the stress value z. Letting z.sub.m and .omega..sub.M 
/2.pi. be the lower limit of a stress value z and the greatest oscillation 
frequency of oscillation components included in a compound oscillation of 
the straddle mounted beam oscillator, the greatest oscillation frequency 
.omega..sub.M /2.pi. must be established such that the following condition 
is satisfied: 
##EQU15## 
Under these settings, while the semiconductor sensor is activated in a 
manner that the straddle mounted beam oscillator generates a compound 
oscillation consisting of a plurality of component oscillations having 
different frequencies .omega..sub.1 /2.pi., . . . , and .omega..sub.n 
/2.pi., amplitudes of the component oscillations having different 
frequencies .omega..sub.1 /2.pi., . . . , and .omega..sub.n /2.pi. are 
detected separately. A power spectrum, which has a squared value of each 
component amplitude, is standardized by the reference power spectrum which 
is the power spectrum established when there is no stress caused in the 
straddle mounted beam oscillator. In such a manner, standardized power 
spectra are established for the respective component oscillations. A 
stress value z is read on the correlation curves with reference to each 
standardized power spectrum. The mean value of the stress values thus 
obtained is calculated as a stress caused in the straddle mounted beam 
oscillator. 
The detection of a stress caused in the straddle mounted beam oscillator of 
the semiconductor sensor is hardly susceptible to changes in environment 
such as ambient temperature around the straddle mounted beam oscillator 
and changes in physical characteristics of the straddle mounted beam 
oscillator due to aging, providing a precise result in the form of a 
stress value. Furthermore, there is no decrease in detecting resolution 
which occurs generally as stress increases. Because the smallest 
oscillation frequency .omega..sub.m /2.pi. and the greatest oscillation 
frequency .omega..sub.M /2.pi. of oscillation components included in a 
compound oscillation of the straddle mounted beam oscillator are 
established so as to satisfy the given conditions (20) and (22), 
respectively, the extent of stress values detectable appropriately is 
sufficiently broadened, resulting in a wide range of stress detection. 
As apparent from the formulas (14) and (17) expressing the stress value z 
and standardized power spectrum G(k), respectively, which do not include 
n-different amplitudes a.sub.1, . . . , and an of component exciting 
amplitudes therein, the detection of a stress value representing a stress 
caused in the straddle mounted beam oscillator is free from the 
substantial effect of exciting amplitudes. 
Further, the semiconductor sensor equipped with the straddle mounted beam 
oscillator is driven such that the straddle mounted beam oscillator 
generates a compound oscillation including n-different component 
oscillations with frequencies. The respective component amplitudes of the 
compound oscillation are detected and transformed into standardized power 
spectra after transformation into power spectra. On the basis of the 
standardized power spectra, stress values are obtained and averaged as the 
stress value generated in the straddle mounted beam oscillator. 
Referring to FIGS. 1 and 2 showing a stress detection apparatus according 
to an embodiment of the invention for embodying the stress detecting 
method, a semiconductor sensor 11 has a stress generating element, 
consisting of a straddle mounted beam type of oscillator, which generates 
stresses according to amounts of dynamic forces, such as pressure and the 
acceleration of gravity, acting thereon. As shown in detail by way of 
example in FIG. 2, the semiconductor sensor 11, which is made of a silicon 
wafer given etching treatment, has a frame 12 defining an approximately 
square space 12a, a weight element 13 having a square shape geometrically 
analogous to but smaller than the square space 12a, and supporting beams 
14A and 14B integrally connecting opposite sides of the frame 12 and 
weight element 13, respectively. The supporting beams 14A and 14B generate 
a tensile stress or a compressive stress when the acceleration of gravity 
is exerted on the weight element 13 in a vertical direction. These weight 
element 13 and supporting beams 14A and 14B form the stress generating 
element for generating stresses according to amounts of external dynamic 
forces, such as pressure and gravity acceleration. Either one of the 
supporting beams 14A and 14B, for example the supporting beam 14A in this 
embodiment, is equipped with a straddle mounted beam type of oscillator 
(which is hereafter referred to simply as a straddle mounted beam 
oscillator) 15. 
FIG. 3 shows a configuration relating to the straddle mounted beam 
oscillator 15 in detail. The supporting beam 14A is formed with a 
generally rectangularly-shaped opening 16. The straddle mounted beam 
oscillator 15 extends in the lengthwise direction from one end to the 
other end of the opening 16 at the middle in the transverse direction and 
is integrally joined at both ends to the supporting beam 14A. At the 
center of the top of the straddle mounted beam oscillator 15 there is 
provided an electrode pad 17 connected to a terminal electrode 17B secured 
to the top of the frame 12 by means of a wire 17A. Since the straddle 
mounted beam oscillator 15 forms a part of the supporting beam 14A, a 
tensile stress or a compressive stress generated in the supporting beam 
14A is exerted on the straddle mounted oscillator beam 15 when gravity 
acceleration is applied to the weight element 13 in the vertical 
direction. In other words, the straddle mounted beam oscillator 15 
generates stresses according to amounts of dynamic forces. 
As shown in FIG. 3, a continuous strip of P-type zone is formed as a 
generally U-shaped electric current path 18 having an extremely low 
resistance as compared with the frame 12 and supporting beam 14A on the 
top of part of the supporting beam 14A where the straddle mounted beam 
oscillator 15 lies, part of the supporting beam 14A extending on one side 
of the opening 16 and part of the frame 12 adjacent to the supporting beam 
14A. The U-shaped electric current path 18 is attached at its extremities 
adjacent to one edge of the with electrode pads 19 and 20, respectively. 
The P-type zone continuous strip may be provided as a top layer, for 
instance, by doping impurities, such as boron, in the supporting beam 14A, 
or otherwise in a manner well known in the art. Specifically, the straddle 
mounted beam oscillator 15 has specified physical attributes, such as a 
length L of approximately 500 .mu.m, a width of approximately 20 .mu.m and 
a thickness of 3 .mu.m. In the following discussion, the length, 
Cross-sectional area, density, second moment of area and Young's modulus 
of the straddle mounted beam oscillator 15 are represented by L, A, 
.rho..sub.0, I and E, respectively. 
For practical use of the semiconductor sensor 11 thus configured and shown 
in FIG. 2, it is enclosed in an evacuated glass casing 21. The glass 
casing 21 is configured so as to provide a space 23 for deformation or 
vibration of the weight element 13 and support beams 14A and 14B. The 
glass casing 23 has a stationary electrode pad 22 attached thereto and 
facing the electrode pad 17 at the central portion of the straddle mounted 
beam oscillator 15. This face to face arrangement of the electrode pads 17 
and 22 provides an electrostatic capacity Cx therebetween according to an 
area and a distance defined by and between the electrode pads 17 and 22, 
and a dielectric constant in the vacuum. These electrodes 17B and 22 are 
connected to external terminals (not shown) by means of electrically 
conductive members. 
In the stress detection apparatus shown in FIG. 1, a magnetic field 
generating circuit 25 provides a uniform magnetic field for the straddle 
mounted beam oscillator 15 such that a magnetic flux acts across the 
straddle mounted beam oscillator 15 in the transverse direction, in other 
words, the magnetic flux is at a perpendicular angle to the current path 
18 lying on the straddle mounted beam oscillator 15. The stress detection 
apparatus has an n number of current sources 26U.sub.1 to 26U.sub.n 
yielding alternating currents U.sub.1 to U.sub.n which have frequencies 
.omega..sub.1 /2.pi. to .omega..sub.n /2.pi. and amplitudes a.sub.1 to 
a.sub.n, respectively and are superposed through a current adder 27 as a 
compound current UM including the frequencies .omega..sub.1 /2.pi. to 
.omega..sub.n /2.pi. and amplitudes a.sub.1 to a.sub.n. The compound 
current UM applied to the current path 18 of semiconductor sensor 11 
through the terminal electrodes 19 and 20 and the magnetic field M 
interact over the straddle mounted beam oscillator 15. 
The interaction between the compound current UM and magnetic field M yields 
a force perpendicular to both directions of the compound current UM and 
magnetic field M, which in turn acts on the straddle mounted beam 
oscillator 15. As a result, the straddle mounted beam oscillator 15 
generates a compound oscillation according to component frequencies of the 
compound current UM between two positions as shown, for instance, by 
broken line and solid line in FIG. 5. This compound oscillation includes 
component oscillations having the component frequencies .omega..sub.1 
/2.pi. to .omega..sub.n /2.pi.. In this instance, the straddle mounted 
beam oscillator 15 changes the amplitude of oscillation W at its middle 
according to stresses generated therein. Accordingly, the amplitude W is 
detected as the stress of the straddle mounted beam oscillator 15. As 
apparent from the above description, the magnetic field generating circuit 
25, current sources 26U.sub.1 to 26U.sub.n and current adder 27 form an 
exciting means for exciting the straddle mounted beam oscillator 15 with 
exciting frequencies .omega..sub.1 /2.pi. to .omega..sub.n /2.pi. and 
exciting amplitudes a.sub.1 to a.sub.n so as to cause a compound 
oscillation consisting of component oscillations of different frequencies. 
While the straddle mounted beam oscillator 15 generates a compound 
oscillation having component oscillations of frequencies .omega..sub.1 
/2.pi. to .omega..sub.n /2.pi., the distance between the electrode pads 17 
and 22 as shown in FIG. 6. Letting D be the distance between the electrode 
pads 17 and 22 when the straddle mounted beam oscillator 15 is free from 
any oscillation, the compound oscillation increasingly or decreasingly 
changes the distance D by a distance equal to its amplitude W, with the 
result of a change in the electrostatic capacity Cx between the electrode 
pads 17 and 22 as expressed by the following formula (23). 
EQU Cx=.epsilon..multidot.S/(D-W) (23) 
where .epsilon. is the dielectric constant in the vacuum; and 
S is the facing area of the electrodes 17 and 22. Accordingly, the 
electrostatic capacity Cx, which is equivalent to the electrostatic 
capacity between the external terminal electrode 17B and the electrode pad 
22, is detected as a substitution for the amplitude W of the straddle 
mounted beam oscillator 15 at the middle. The amplitude W is detected as 
an electrostatic capacity Cx between the electrodes 17B and 22 by means of 
an amplitude detector 28. 
As shown in FIG. 7, the amplitude detector 28 has interface terminals 30 
and 31 connected to the electrodes 17B and 22, respectively, between which 
an electrostatic capacity Cx is provided as well as between the electrodes 
17B and 22. Specifically, the amplitude detector 28 includes in its 
circuit a capacitance element 32 having a given reference electrostatic 
capacity Co and an electric power source 33 yielding a constant voltage Vo 
which are connected in series between the interface terminals 30 and 31. 
The voltage V across the capacitance element 32 has the quantitative 
relation to the electrostatic capacity Cx, reference electrostatic 
capacity Co and constant voltage Vo expressed by the following formula 
(24): 
EQU 1/Cx=(Vo/V-1)/Co (24) 
Substituting the formula (23) into the formula (24), the following formula 
(25) is obtained: 
EQU W 
=D{(1+.epsilon..multidot.S/CoD)-(.epsilon..multidot.S/CoD).multidot.(Vo/V) 
}(25) 
The amplitude detector 28 detects an amplitude W of a compound oscillation 
of the straddle mounted beam oscillator 15 at the middle and provides an 
amplitude signal SW representative of the amplitude W for each of band 
pass filters 35F.sub.1 to 35F.sub.n. 
The band-pass filters 35F.sub.1 to 35F.sub.n, which have frequency bands 
whose center frequencies are .omega..sub.1 /2.pi. to .omega..sub.n /2.pi., 
respectively, extract components SS.sub.1 to SS.sub.n of frequencies 
.omega..sub.1 /2.pi. to .omega..sub.n /2.pi., respectively, from the 
amplitude signal SW. The component amplitude signals SS.sub.1 to SS.sub.n 
filtered by the respective band-pass filters 35F.sub.1 to 35F.sub.n and 
having component frequencies .omega..sub.1 /2.pi. to .omega..sub.n /2.pi., 
respectively, represent amplitudes of component oscillations. In this 
manner, the band-pass filters 35F.sub.1 to 35F.sub.n function as 
oscillation spectrum detectors which provide component frequencies 
SS.sub.1 to SS.sub.n for square processing circuits 36Q.sub.1 to 
36Q.sub.n, respectively, which in turn square the component frequencies 
SS.sub.1 to SS.sub.n and provide signals SQ.sub.1 to SQ.sub.n 
representative of the squared results, respectively. Subsequently, the 
signals SQ.sub.1 to SQ.sub.n are transmitted to standardizing circuits 
37N.sub.1 to 37N.sub.n for standardization and also to output terminals 
J.sub.1 to J.sub.n of the square processing circuits 36Q.sub.1 to 
36Q.sub.n, respectively. 
Because the signals SQ.sub.1 to SQ.sub.n are obtained as a result of square 
processing of the component frequencies SS.sub.1 to SS.sub.n 
representative of a number of n of oscillation spectra, they show a number 
of n of power spectra obtained by the square processing of oscillation 
spectra expressed by the formula (16). Thus, the square processing 
circuits 36Q.sub.1 to 36Q.sub.n function as power spectrum calculation 
means for providing output signals SQ.sub.1 to SQ.sub.n representative of 
a number of n of power spectra. 
The standardizing circuits 37N.sub.1 to 37N.sub.n, to which squared power 
spectrum signals SQ.sub.1 to SQ.sub.n, receive data DRP.sub.1 to DRP.sub.n 
representative of reference power spectra which are the power spectra 
established when there is no stress caused in the straddle mounted beam 
oscillator in a vacuum. The respective standardizing circuits 37N.sub.1 to 
37N.sub.n standardize the power spectra represented by the squared power 
spectrum signals SQ.sub.1 to SQ.sub.n with the reference power spectrum 
data DRP.sub.1 to DRP.sub.n, respectively, and provide standardized 
spectrum signals SNP.sub.1 to SNP.sub.n representative of the standardized 
power spectra G(k) for stress value detection means 39Z.sub.1 to 
39Z.sub.n, respectively. In this instance, these reference power spectrum 
data are stored in a random access memory (RAM) 38. 
The stress value detection means 39Z.sub.1 to 39Z.sub.n receive data 
DGX.sub.1 to DGX.sub.n from a random access memory (RAM) 40, respectively, 
as well as the standardized spectrum signals SNP.sub.1 to SNP.sub.n. The 
data DGX.sub.1 to DGX.sub.n indicate the quantitative relations between 
standardized power spectra G(k) obtained from the formula (17) and stress 
values z and shown in FIG. 8 are previously stored in the random access 
memory (RAM) 40. The stress value detection means 39Z.sub.1 to 39Z.sub.n 
found stress values z.sub.1 to z.sub.n, which represent NL.sup.2 /EI (in 
which N is the stress generated in the straddle mounted beam oscillator 
15), by comparing the standardized spectrum signals SNP.sub.1 to SNP.sub.n 
with the data DGX.sub.1 to DGX.sub.n and then output stress value signals 
SZ.sub.1 to SZ.sub.n to an averaging circuit 41. 
The averaging circuit 41 calculates a mean stress value za, such as a 
geometric mean stress value and a weighted mean stress value, of the 
stress values z.sub.1 to z.sub.n which is employed as a stress value 
representative of the stress generated in the straddle mounted beam 
oscillator 15 and represented by a stress value signal SZA. 
In a practical stress detection with the stress detection apparatus 
according to the above embodiment of the invention, upper and lower limits 
z.sub.M and z.sub.m are set for stress values which the stress value 
detection means 39Z.sub.1 to 39Z.sub.n detect. Further, alternating 
currents U.sub.1 to U.sub.n having frequencies .omega..sub.1 /2.pi. to 
.omega..sub.n /2.pi. and amplitudes a.sub.1 to a.sub.n yielded, 
respectively, from the current sources 26U.sub.1 to 26U.sub.n are 
regulated such that a smallest oscillation frequency .omega..sub.m /2.pi. 
and a greatest oscillation frequency .omega..sub.M /2.pi. satisfy the 
following conditions (26) and (27), respectively: 
##EQU16## 
Subsequently, by the utilization of the formula (17) relating to the 
standardized power spectrum G(k), quantitative relations between the 
standardized power spectrum G(k) and stress value z and data DGX.sub.1 to 
DGX.sub.n are established with regard to the frequencies .omega..sub.1 
/2.pi. to .omega..sub.n /2.pi. and stored in the random access memory 
(RAM) 40. 
Under the condition where the straddle mounted beam oscillator 15 is free 
from any stress, while the magnetic field generating circuit 25 is 
activated to apply a uniform magnetic field over the straddle mounted beam 
oscillator 15, the semiconductor sensor 11 is excited with the current 
path 18 applied with a compound current UM consisting of superposed 
alternating currents U.sub.1 to U.sub.n from the respective current 
sources 26U.sub.1 to 26U.sub.n through the current adder 27. As a result, 
the straddle mounted beam oscillator 15 generates a compound oscillation 
consisting of component oscillations of frequencies .omega..sub.1 /2.pi. 
to .omega..sub.n /2.pi. according to the alternating currents U.sub.1 to 
U.sub.n having frequencies .omega..sub.1 /2.pi. to .omega..sub.n /2.pi. 
and amplitudes a.sub.1 to a.sub.n, respectively. At this time, the 
alternating currents U.sub.1 to U.sub.n has the smallest oscillation 
frequency .omega..sub.m /2.pi. and greatest oscillation frequency 
.omega..sub.M /2.pi. satisfy the conditions (26) and (27), respectively. 
During the compound oscillation of the straddle mounted beam oscillator 
15, squared power spectrum signals SQ.sub.1 to SQ.sub.n are provided by 
the square processing circuits 36Q.sub.1 to 36Q.sub.n and output through 
the output terminals J.sub.1 to J.sub.n as reference power spectra which 
correspond to the power spectra (F(L/2, .mu.)).sup.2 obtained while the 
straddle mounted beam oscillator 15 is free from any stress. Data 
DRP.sub.1 to DRP.sub.n of the reference power spectra is free from any 
stress are provided and stored in the random access memory (RAM) 38. 
Once having stored the data DGX.sub.1 to DGX.sub.n indicating the 
quantitative relations between the standardized power spectrum G(k) and 
stress value z in the random access memory (RAM) 40 and the data DRP.sub.1 
to DRP.sub.n of the reference power spectra in the random access memory 
(RAM) 38, while a specified amount of dynamic force, such as a gravity 
acceleration, is applied to the straddle mounted beam oscillator 15 so as 
to generate a stress in the straddle mounted beam oscillator 15, a uniform 
magnetic field is applied over the straddle mounted beam oscillator 15 and 
a compound current UM consisting of superposed alternating currents 
U.sub.1 to U.sub.n from the respective current sources 26U.sub.1 to 
26U.sub.n is applied to the current path 18 through the current adder 27, 
so as to cause the semiconductor sensor 11 to generate a compound 
oscillation consisting of component oscillations of frequencies 
.omega..sub.1 /2.pi. to .omega..sub.n /2.pi. in the straddle mounted beam 
oscillator 15. 
During the compound oscillation of the straddle mounted beam oscillator 15, 
the respective standardizing circuits 37N.sub.1 to 37N.sub.n standardize 
the power spectra represented by the squared power spectrum data DRP.sub.1 
to DRP.sub.n from the square processing circuits 36Q.sub.1 to 36Q.sub.n 
with the Data DRP.sub.1 to DRP.sub.n of the reference power spectra 
extracted from the random access memory (RAM) 38 and provide standardized 
spectrum signals SNP.sub.1 to SNP.sub.n as standardized power spectra. 
Together, the stress value detection means 39Z.sub.1 to 39Z.sub.n compare 
the standardized spectrum signals SNP.sub.1 to SNP.sub.n representing the 
respective standardized power spectra with the data DGX.sub.1 to DGX.sub.n 
extracted from the random access memory (RAM) 40 which represent the 
quantitative relations between the standardized power spectra G(k) and 
stress values z and find stress values z.sub.1 to z.sub.n appropriately 
corresponding to the respective standardized power spectra. Since these 
stress values z.sub.1 to z.sub.n are obtained based on the component 
frequencies SS.sub.1 to SS.sub.n of frequencies .omega..sub.1 /2.pi. to 
.omega..sub.n /2.pi. provided through the band-pass filters 35F.sub.1 to 
35F.sub.n, respectively, and hence, based on the oscillation spectra 
corresponding to the oscillation spectrum F.sub.k (L/2, .mu.) represented 
by the formula (16). The stress value detection means 39Z.sub.1 to 
39Z.sub.n output stress value signals SZ.sub.1 to SZ.sub.n to the 
averaging circuit 41 where a weighted mean stress value zw of the stress 
values z.sub.1 to z.sub.n is calculated for instance. 
Letting W.sub.1 to W.sub.n be weighing factors for the stress values 
z.sub.1 to z.sub.n, respectively, the weighted mean stress value zw is 
given by the following formula (28): 
EQU zw+(W.sub.1 z.sub.1 +. . . +W.sub.n z.sub.n)/(W.sub.1 + . . . +W.sub.n) 
(28) 
In establishing these weighing factors W.sub.1 -W.sub.n, letting G(k) (in 
which k assumes 1, . . . , n), .DELTA.G(k), H, z.sub.0 and 
.vertline..DELTA.z.sub.k .vertline. be standardized power spectra 
represented by standardized spectrum signals SNP.sub.1 to SNP.sub.n, a 
detection error in relation to the standardized power spectrum and the 
maximum value of .vertline..DELTA.G(k).vertline./G(k), a true stress value 
of each stress values z.sub.1 -z.sub.n, and the maximum error of each 
stress values z.sub.1 -z.sub.n, respectively, the following quantitative 
relations are satisfied: 
EQU .vertline..DELTA.G(k).vertline.=.DELTA.z.sub.k 
.vertline..multidot.dG(k)/dz(29) 
EQU .vertline..DELTA.z.sub.k 
.vertline.=.vertline..DELTA.G(k).vertline..multidot.[1.div.{dG(k)/dz}]=H.m 
ultidot.G(k).multidot.[1.div.{dG(k)/dz}] (30) 
As apparent from the formula (30), .vertline..DELTA.z.sub.k .vertline. 
decreases as dG(k)/dz increases provided that the stress value z is 
greater than 0 (zero), with the result of increasing detecting resolution 
concerning stress values z. On the other hand, provided that the stress 
value z is less than 0 (zero), G(k) reaches a minimum and dG(k)/dz reaches 
0 (zero), respectively, as .vertline.z.vertline. increases, resulting in 
an increase of .vertline..DELTA.z.sub.k .vertline. which causes 
aggravation of detecting resolution concerning stress values z. 
Letting .DELTA.z.sub.1 '-.DELTA.z.sub.n ' be errors included in stress 
values .DELTA.z.sub.1 -.DELTA.z.sub.n corresponding to standardized power 
spectra G(1)-G(n), respectively, the geometric mean stress value zs is 
defined as follows: 
EQU zs=z.sub.0 +(.DELTA.z.sub.1 '+ . . . +.DELTA.z.sub.n ')/n (31) 
Since there must be an upper limit for an error of a true value z.sub.0 of 
the geometric mean stress value zs, the maximum error .delta..sub.m is 
defined by the following formula (32): 
##EQU17## 
With use of the weighing factors indicated by the following monomials (33) 
for z.sub.0 +.DELTA.z.sub.1 ', . . . , z.sub.0 +.DELTA.z.sub.n ' which are 
the stress values z.sub.1, . . . , z.sub.n corresponding to the 
standardized power spectra G(1), . . . , G(n), the weighted mean stress 
value zw' is formulated as the formula (34). 
##EQU18## 
The maximum error .delta..sub.w of a true value z.sub.0 of the weighted 
mean stress value zw is formulated as follows: 
##EQU19## 
Directing concern to dimensions of the maximum errors .delta..sub.s and 
.delta..sub.w and the theorem that a geometric mean is greater than a 
harmonic mean, the following conditions must be satisfied: 
##EQU20## 
From the conditions, it is proved that the maximum error .delta..sub.s of 
the true value z.sub.0 of a geometric mean stress value zs is greater the 
maximum error .delta..sub.w of the true value z.sub.0 of a weighted mean 
stress value zw'. Accordingly, the maximum error is smaller for the 
weighted mean stress value zw' obtained from the formula (33) in which 
weighing factors are incorporated than for the geometric mean stress value 
zs in regard to each stress value z.sub.1 . . . z.sub.n. The utilization 
of weighted means stress values is contributory to improvement on the 
detecting resolution of stress values. 
In view of the above, the weighing factors W.sub.1 . . . W.sub.n are set 
for the respective stress values W.sub.1 . . . W.sub.n the averaging 
circuit 41 as follows: 
##EQU21## 
The weighted mean zw obtained in the averaging circuit 41 is substituted 
for a mean stress value za, which is employed as the stress generated in 
the straddle mounted beam oscillator 15 and output as a stress value 
signal SZA through the terminal 42. In FIG. 1, the random access memory 
(RAM) 43, which provides various exciting conditions, has five internal 
memory sections, namely A section for various amplitudes for the 
respective frequencies, B section and E section for amplitude strength 
rations, C section for various spectrum strength and D section for various 
spectrum strength ratios. 
In the detection of stress described above, the straddle mounted beam 
oscillator 15 assumes a state that there is no residual stresses left 
therein or there are left only residual stresses that are substantially 
negligible or insignificant. The stress detecting method of the invention 
is appropriately available even in the cases where there are residual 
stresses which are significant to some extent. 
In the cases where the straddle mounted beam oscillator 15 has residual 
stresses, the oscillation spectrum F.sub.k (L/2, .mu.) represented by the 
formula (16) is calculated on condition that there is no stress generated 
in the straddle mounted beam oscillator 15, i.e. on condition that the 
stress value z is 0 (zero). The oscillation spectrum F.sub.k (L/2, .mu.) 
is squared in order to provide a theoretical power spectrum (F.sub.k (L/2, 
.mu.)).sup.2 for the straddle mounted beam oscillator 15 having no 
residual stress. With regard to the straddle mounted beam oscillator 15 
which is placed in a condition that it has a residual stress but does not 
generate any stress due to an external dynamic force such as a gravity 
acceleration, power spectra which are represented by power spectrum 
signals SQ.sub.1 to SQ.sub.n from the square processing circuits 36Q.sub.1 
to 36Q.sub.n are standardized with the theoretical power spectra (F.sub.k 
(L/2, .mu.)).sup.2 provided when the straddle mounted beam oscillator 15 
has no residual stress. The standardized power spectra are defined as the 
standardized power spectra G(k, z.sub.v) for the straddle mounted beam 
oscillator 15 has only a residual stress. Each standardized power spectrum 
G(k, z.sub.v) is compared with the data DGX.sub.1 to DGX.sub.n extracted 
from the random access memory (RAM) 40 which represent the quantitative 
relations between the standardized power spectra G(k) and stress values z 
to find a stress value z appropriately corresponding to the respective 
standardized power spectrum G(k, z.sub.v) which in turn is substituted for 
the residual stress value z.sub.v of the straddle mounted beam oscillator 
15. 
Thereafter, in a condition that the straddle mounted beam oscillator 15 has 
a residual stress and generates a stress due to an external dynamic force 
such as a gravity acceleration, power spectra which are represented by 
power spectrum signals SQ.sub.1 to SQ.sub.n from the square processing 
circuits 36Q.sub.1 to 36Q.sub.n are standardized with the theoretical 
power spectra (F.sub.k (L/2, .mu.)).sup.2 provided when the straddle 
mounted beam oscillator 15 has no residual stress. The standardized power 
spectra are defined as the standardized power spectra G(k, z+z.sub.v) for 
the straddle mounted beam oscillator 15 has both residual stress and 
stress due to an external dynamic force. Each standardized power spectrum 
G(k, z.sub.v) is compared with the data DGX.sub.1 to DGX.sub.n extracted 
from the random access memory (RAM) 40 which represent the quantitative 
relations between the standardized power spectra G(k) and stress values z 
to find a stress value z appropriately corresponding to the respective 
standardized power spectrum G(k, z+z.sub.v) which in turn is substituted 
for the compound stress value z.sub.c (=z+z.sub.v) of the straddle mounted 
beam oscillator. 15. The stress value z generated in the straddle mounted 
beam oscillator 15 due to the external dynamic force is obtained by 
subtracting the residual value z.sub.v from the compound stress value 
z.sub.c. In this manner, the stress value z generated in the straddle 
mounted beam oscillator 15 due to the external dynamic force is obtained 
without being affected by a residual stress. 
The stress detection may be performed based on an analysis of the influence 
of are resistance on a distribution of amplitude of oscillations generated 
in a stress generating element of the semiconductor sensor. 
In order to quantitatively express an effect of air resistance in the 
analysis, the utilization is made of Stokes' law of resistance according 
to which the resistance acting on an object moving at a speed in a viscous 
fluid such as air is opposite to the direction of movement and 
proportional to the speed. 
Applying the Stokes' law of resistance to the straddle mounted beam 
oscillator 15 as a stress generating element of the semiconductor sensor 
shown in FIG. 2, the following formula is obtained on referring to the 
formula (3) by introducing a resistance .differential.W/.zeta.t exerted on 
the straddle mounted beam oscillator 15. 
##EQU22## 
EQU W(0, t)=W(L, t)=0 
EQU W.sub.7 (0, t)=W.sub.7 (L, t)=0 
where R is a constant. 
Solving the formula (38) for .differential.=L/2 to obtain an approximate 
solution, the following formula is obtained: 
##EQU23## 
In this instance, the following condition is satisfied: 
EQU 1+(.rho..sub.0 A/R).sup.2 .multidot..omega..sub.p.sup.2 &gt;{(.upsilon..sub.p 
sin h.upsilon..sub.p +.lambda..sub.p sin .lambda..sub.p) .div.(.rho..sub.p 
sin h.rho..sub.p .multidot.cos .lambda..sub.p .multidot.+sin 
.lambda..sub.p .multidot..lambda. cos h.upsilon..sub.p)}.sup.2 (40) 
In the condition (40), while .delta.(.mu.') is a delta function for .mu.' 
defined by (.mu.-.omega..sub.p /2.pi.) and assumes an infinity when .mu. 
is 0, .delta.(.mu.') is explained to assume 1 as the unit of spectrum in 
the following description. 
When calculating a spectrum W(L/2, t) , which is discrete, for a frequency 
.omega..sub.p /2.pi. from the formula (39), 
##EQU24## 
Since in the formula (41) Ma.sub.p /.rho..sub.0 A.omega..sub.p.sup.2 
-R.sub.1 .omega..sub.p) is canceled the standardized power spectrum, which 
is defined by the spectrum strength .vertline.F(p, z, R).vertline..sup.2 
and .vertline.F(p, 0, R).vertline..sup.2, is identical to the formula (17) 
and expressed as follows: 
EQU .vertline.F(p, z, R).vertline..sup.2 /.vertline.F(p, 0, R).vertline..sup.2 
=G(p, z) G(K) (42) 
That is, as long as the condition (40) is satisfied, the stress value z is 
found from the standardized power spectrum if the spectrum strength 
.vertline.F(p, 0. R).vertline. is known. 
The feasibility of the condition (40) as a premise for the approximate 
solution of the formula (39) must be studied hereafter. 
Discussing the constant R, when replacing part of the straddle mounted beam 
oscillator 15 having a width of b with a hemisphere having a radius of 
b/2, the resistance force f' that the hemisphere receives is given by the 
following formula (43): 
EQU f'=6.pi..rho.(b/2).multidot.(.differential.W/.zeta.t) (43) 
where .eta. is the viscosity factor of air. 
Since the friction force per unit length received by the straddle mounted 
beam oscillator 15 is given by f'/b and the viscosity factor of air .eta. 
is approximately 2.times.10.sup.-5 under the condition of room 
temperatures and an atmospheric pressure, the resistance factor R can be 
generally estimated as one expressed by the following formula (44): 
EQU R.apprxeq.3.pi..eta.=1.89.times.10.sup.-4 kg/ms (44) 
Since the straddle mounted beam oscillator 15 made of a silicon wafer after 
etching treatment has specified physical dimensions, such as a length L of 
approximately 500 .mu.m, a width b of approximately 20 .mu.m and a 
thickness h of 3 .mu.m, and the silicon has a density .rho..sub.0 of 
2.3.times.10.sup.-3 kg/m.sup.3, the Young's modulus of 1.7.times.10.sup.11 
kg/s.sup.2 .multidot.m, and a second moment of area of 
bh3/12=4.50.times.10.sup.-23 m.sup.4, .rho..sub.0 A and .omega..sub.p are 
expressed as follows: 
##EQU25## 
The left side of the condition (40) can be rewritten as follows: 
EQU 1+(.rho..sub.0 A/R).sup.2 .multidot..omega..sub.p.sup.2 
=1+1.18.times.10.sup.2 .multidot.y.sub.p (47) 
Calculating the formula (46) for various parameters y.sub.p, for instance 
60, 70, 80, 90 and 100, the results are given in the following table. 
______________________________________ 
Y.sub.p 1 + (.rho..sub.o A/R).sup.2 .multidot. .omega..sub.p .sup.2 
______________________________________ 
1 
60 4.25 .times. 10.sup.5 
70 5.78 .times. 10.sup.5 
80 7.55 .times. 10.sup.5 
90 9.56 .times. 10.sup.5 
100 1.18 .times. 10.sup.6 
______________________________________ 
The right side of the condition (40) is expressed as a function of stress 
value z and shown in FIG. 12. As apparent in FIG. 12, while the power 
spectrum for a stress value z at which each parameter y.sub.p provides a 
resonant frequency shows an abrupt divergent increase, the value of the 
right side of the condition (40) becomes smaller as the parameter becomes 
far from the resonant frequency. That is, when setting a stress detection 
limit to the parameter y.sub.p at a point at which the right side of the 
condition (40) assumes a value of the order of approximately 10.sup.3, the 
left side of the condition (40) assumes a value over the order of 10.sup.2 
and consequently, the formula is held. 
The following description will be given to provide a substantiated argument 
for compensation for an error in stress detection due to air resistance. 
Considering spectra for frequencies .omega..sub.q and .omega..sub.r 
(q.noteq.r) among the frequencies .omega..sub.p (p=1, 2, . . . , n), the 
quantitative relation is established between a spectrum strength ratio 
X.sub.R and an amplitude ratio Y.sub.R for the frequencies .omega..sub.q 
and .omega..sub.r as follows: 
##EQU26## 
In the formula, .o slashed.(z.sub.0,.omega..sub.p) is defined as follows: 
EQU .o slashed.(z.sub.0 /.omega..sub.p)=[{.upsilon..sub.p sin h.upsilon..sub.p 
(1-cos .lambda..sub.p)+.lambda..sub.p sin .lambda..sub.p (1-cos 
h.upsilon..sub.p)}.div.{(.upsilon..sub.p sin h.upsilon..sub.p 
.multidot.cos .lambda..sub.p +.upsilon..sub.p sin .lambda..sub.p 
.multidot..lambda. cos h.upsilon..sub.p)}].sup.2 (49) 
Letting X.sub.o and Y.sub.o be the spectrum strength ratio and amplitude 
strength ratio of exciting force when the straddle mounted beam oscillator 
15 is in a vacuum (R=0), the following formula is obtained on referring to 
the formula (48): 
EQU X.sub.0 =Y.sub.0 x(.omega..sub.r.sup.4 
/.omega..sub.q.sup.4)x{.female.(z.sub.0, .omega..sub.q)/.o 
slashed.(z.sub.0, .omega..sub.r)} (50) 
Solving the formulas (48) and (50) for R.sup.2, the following formula is 
obtained: 
EQU R.sup.2 =(.rho..sub.0 A).sup.2 .omega..sub.r.sup.2 .omega..sub.q.sup.2 
(X.sub.0 Y.sub.R -X.sub.R Y.sub.0)/(.omega..sub.r.sup.2 X.sub.R Y.sub.0 
-.omega..sub.q.sup.2 X.sub.0 Y.sub.R) (51) 
Substituting the formula (51) for the formula (41), 
##EQU27## 
Similarly, 
EQU .vertline.F(r, z.sub.0, R).vertline..sup.2 =(.vertline.a.sub.r.sup.R 
.vertline..sup.2 /.vertline.a.sub.r.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 X.sub.R Y.sub.0 -.omega..sub.q.sup.2 
X.sub.0 Y.sub.R)/(X.sub.0 Y.sub.R (.omega..sub.r.sup.2 
-.omega..sub.q.sup.2) X.vertline.F(r, z.sub.0, 0).vertline..sup.2(53) 
The above formulas (52) and (53) represent the spectrum strength of the 
frequencies .omega..sub.q and .omega..sub.r for the resistance factors R=0 
and R.noteq.0. 
Considering the case where the straddle mounted beam oscillator 15 has no 
residual stress (stress value z.sub.0 =0), thespectrum strength is 
obtained by substituting 0 (zero) for z.sub.0 in the formula (52), (53) as 
follows: 
EQU .vertline.F(q, 0, R).vertline..sup.2 =(.vertline.a.sub.q.sup.R 
.vertline..sup.2 /.vertline.a.sub.q.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 X.sub.R Y.sub.0 -.omega..sub.q.sup.2 
X.sub.0 Y.sub.R)/(X.sub.0 Y.sub.R (.omega..sub.r.sup.2 
-.omega..sub.q.sup.2)x.vertline.F(q, 0, 0).vertline..sup.2(54) 
EQU .vertline.F(r, 0, R).vertline..sup.2 =(.vertline.a.sub.r.sup.R 
.vertline..sup.2 /.vertline.a.sub.r.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 X.sub.r Y.sub.0 -.omega..sub.q.sup.2 
X.sub.0 Y.sub.R)/)X.sub.0 Y.sub.R (.omega..sub.r.sup.2 
-.omega..sub.q.sup.2 x.vertline.F(r, 0, 0).vertline..sup.2(55) 
Since the spectrum strength are expressed by rewriting the formula (13) as 
described below, they are theoretically determined when the basis of 
practical physical characteristics (.rho..sub.0, E), dimensions (L, A, I) 
and exciting conditions (a.sub.q.sup.R, a.sub.q.sup.0, .omega..sub.q, 
.omega..sub.r) are unconditionally fixed. 
In the practical a procedure for the compensation for an error in stress 
detection due to air resistance, if the straddle mounted beam oscillator 
15 has a residual stress generated, for instance, during manufacturing, 
the residual stress is used for the true stress value z.sub.0. 
(i) In the case where the amplitude strength ratios Y.sub.R and Y.sub.0 are 
equal to each other: 
(1) The amplitude strength ratio Y.sub.0 is determined based on exciting 
amplitude strength .vertline.a.sub.q.sup.0 .vertline..sup.2 and 
.vertline.a.sub.r.sup.0 .vertline..sup.2 which are arbitrarily 
established, and spectrum strength .vertline.F(q, 0, 0).vertline..sup.2 
and .vertline.F(r, 0, 0).vertline..sup.2 are theoretically determined 
based on the amplitude strength ratio Y.sub.0. 
(2) While the straddle mounted beam oscillator 15 is left in a vacuum 
(resistance factor R=0), it is excited with the arbitrarily determined 
exciting amplitude strength .vertline.a.sub.q.sup.0 .vertline..sup.2 and 
.vertline.a.sub.r.sup.0 .vertline..sup.2. During the excitation, practical 
spectrum strength .vertline.F(q, z.sub.0, 0).vertline..sup.2 and 
.vertline.F(r, z.sub.0, 0).vertline..sup.2 are observed and a spectrum 
strength ratio X.sub.0 =.vertline.F(q, z.sub.0, 0).vertline..sup.2 
/.vertline.F(r, z.sub.0, 0).vertline..sup.2 is determined. 
(3) From the results, the standardized spectrum strength ratios G(q, 
z.sub.0)=.vertline.F(q, z.sub.0, 0).vertline..sup.2 /.vertline.F(q, 0, 
0).vertline..sup.2 and G(r, z.sub.0)=.vertline.F(r, z.sub.0, 
0).vertline..sup.2 /.vertline.F(r, 0, 0).vertline..sup.2 are calculated, 
on the basis of which the residual stress value z.sub.0 is found from the 
quantitative relations between standardized power spectrum and stress 
value z shown in FIG. 8. In order to increase the reliability of residual 
stress value z.sub.0, the values z.sub.0.sup.q and z.sub.0.sup.r may be 
obtained from the standardized spectrum strength ratios G (q, z.sub.0) and 
G (r, z.sub.0), respectively, as weighted means. 
(4) While the straddle mounted beam oscillator 15 is left in the air 
(resistance factor R.noteq.0), the exciting amplitude strength 
.vertline.a.sub.q.sup.R .vertline..sup.2 and .vertline.a.sub.r.sup.R 
.vertline..sup.2 are determined such that the strength ratios 
.vertline.a.sub.q.sup.R .vertline..sup.2 /.vertline.a.sub.r.sup.R 
.vertline..sup.2 and .vertline.a.sub.q.sup.0 .vertline..sup.2 
/.vertline.a.sub.r.sup.0 .vertline..sup.2. During exciting the straddle 
mounted beam oscillator 15 with the exciting amplitude strength 
.vertline.a.sub.q.sup.R .vertline..sup.2 and .vertline.a.sub.r.sup.R 
.vertline..sup.2, practical spectrum strength .vertline.F(q, z.sub.0, 
0).vertline..sup.2 and .vertline.F(r, z.sub.0, 0).vertline..sup.2 are 
observed and a spectrum strength ratio X.sub.R =.vertline.F(q, z.sub.0, 
R).vertline..sup.2 /.vertline.F(r, z.sub.0, R).vertline..sup.2 is 
determined. 
(5) Based on the resultant spectrum strength .vertline.F(q, 0, 
0).vertline..sup.2 and .vertline.F(r, 0, 0).vertline..sup.2 and practical 
spectrum strength ratios X.sub.0 and X.sub.R, calculations of the formulas 
(54) and (55) provide the following results: 
EQU .vertline.F(q, 0, R).vertline..sup.2 =(.vertline.a.sub.q.sup.R 
.vertline..sup.2 /.vertline.a.sub.q.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 X.sub.R -.omega..sub.q.sup.2 
X.sub.0)/X.sub.0 (.omega..sub.r.sup.2 -.omega..sub.q.sup.2) 
x.vertline.F(q, 0, 0).vertline..sup.2 (65) 
EQU .vertline.F(r, 0, R).vertline..sup.2 =(.vertline.a.sub.r.sup.R 
.vertline..sup.2 /.vertline.a.sub.r.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 X.sub.R -.omega..sub.q.sup.2 
X.sub.0)/X.sub.R (.omega..sub.r.sup.2 -.omega..sub.q.sup.2) 
x.vertline.F(r, 0, 0).vertline..sup.2 (57) 
(6) If an external stress represented by a stress value z is applied to the 
straddle mounted beam oscillator 15, the straddle mounted beam oscillator 
15 is placed under a stress represented by a stress value z+z.sub.0. While 
the straddle mounted beam oscillator 15 is excited with the same exciting 
amplitude strength .vertline.a.sub.q.sup.R .vertline..sup.2 and 
.vertline.a.sub.r.sup.R .vertline..sup.2 as in the procedure (4), 
practical spectrum strength .vertline.F(q, z+z.sub.0, R).vertline..sup.2 
and F(r, z+z.sub.0, R).vertline..sup.2 are observed and standardized with 
the spectrum strength .vertline.F(q, 0, R).vertline..sup.2 and 
.vertline.F(r, 0, R).vertline..sup.2, respectively. On the basis of the 
standardized power spectra G(q, z+z.sub.0) and G(r, z+z.sub.0), the 
residual stress value z+z.sub.0 is found from the quantitative relations 
between standardized power spectrum and stress value z shown in FIG. 8. By 
subtract subtracting the residual value z.sub.0 from the stress value 
z+z.sub.0, the stress value z is obtained. In this instance, the values 
z+z.sub.0 may be averaged in the weighted mean method. 
Although the above example is given for the case where the amplitude 
strength ratios Y.sub.R and Y.sub.0 are equal to each other, it is more 
general to define the amplitude strength ratio Y.sub.R is linearly 
proportional to the amplitude strength ratio Y.sub.0, i.e. Y.sub.R 
=.alpha.Y.sub.0 (.alpha. is a positive integer). 
(ii) In the case where the spectrum strength ratios X.sub.R and X.sub.0 are 
equal to each other: 
(1) The amplitude strength ratio Y.sub.0 is determined based on exciting 
amplitude strength .vertline.a.sub.q.sup.0 .vertline..sup.2 and 
.vertline.a.sub.r.sup.0 .vertline..sup.2 which are arbitrarily 
established, and spectrum strength .vertline.F(q, 0, 0).vertline..sup.2 
and .vertline.F(r, 0, 0).vertline..sup.2 are theoretically determined 
based on the amplitude strength ratio Y.sub.0. 
(2) While the straddle mounted beam oscillator 15 is left in a vacuum 
(resistance factor R=0) , it is excited with the arbitrarily determined 
exciting amplitude strength .vertline.a.sub.q.sup.0 .vertline..sup.2 and 
.vertline.a.sub.r.sup.0 .vertline..sup.2. During the excitation, practical 
spectrum strength .vertline.F(q, z.sub.0, 0).vertline..sup.2 and 
.vertline.F(r, z.sub.0, 0).vertline..sup.2 are observed and a spectrum 
strength ratio X.sub.0 =.vertline.F(q, z.sub.0, 0).vertline..sup.2 
/.vertline.F(r, z.sub.0, 0).vertline..sup.2 is determined. 
(3) From the results, the standardized spectrum strength ratios G(q, 
z.sub.0)=.vertline.F(q, z.sub.0, 0).sup.2 /.vertline.F(q, 0, 
0).vertline..sup.2 and G(r, z.sub.0)=.vertline.F(r, z.sub.0, 
0).vertline..sup.2 /.vertline.F(r, 0, 0).vertline..sup.2 are calculated, 
on the basis of which the residual stress value z.sub.0 is determined in 
the same manner applied to the case where the amplitude strength ratios 
Y.sub.R and Y.sub.0 are equal to each other. The weighted mean processing 
may be introduced in the procedure. 
(4) While the straddle mounted beam oscillator 15 is left in the air 
(resistance factor R.noteq.0), the exciting amplitude strength 
.vertline.a.sub.q.sup.R .vertline..sup.2 and .vertline.a.sub.r.sup.R 
.vertline..sup.2 are determined such that the spectrum strength ratio 
X.sub.R =.vertline.F(q, z.sub.0, R).vertline..sup.2 /.vertline.F(r, 
z.sub.0, R).vertline..sup.2 is equal to the practical spectrum strength 
ratio X.sub.0 =.vertline.F(q, z.sub.0, 0).vertline..sup.2 /.vertline.F(r, 
z.sub.0, 0).vertline..sup.2. Then, the amplitude strength ratio Y.sub.R 
=.vertline.a.sub.q.sup.R .vertline..sup.2 /.vertline.a.sub.r.sup.R 
.vertline..sup.2 is calculated 
(5) Based on the arbitrarily exciting amplitude strength 
.vertline.a.sub.q.sup.R .vertline..sup.2 and .vertline.a.sub.r.sup.R 
.vertline..sup.2 and their amplitude strength ratio Y.sub.R, the 
theoretical spectrum strength .vertline.F(q, 0, 0).vertline..sup.2 and 
.vertline.F(r, 0, 0).sup.2, and the exciting amplitude strength 
.vertline.a.sub.q.sup.R .vertline..sup.2 and .vertline.a.sub.r.sup.R 
.vertline..sup.2 and their amplitude strength ratio Y.sub.R according to 
the practical spectrum strength ratios X.sub.0 and X.sub.R, calculations 
of the formulas (54) and (55) provide the following results: 
EQU .vertline.F(q, 0, R).vertline..sup.2 =(.vertline.a.sub.q.sup.R 
.vertline..sup.2 /.vertline.a.sub.q.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 Y.sub.0 -.omega..sub.q.sup.2 
Y.sub.R)/Y.sub.R (.omega..sub.r.sup.2 -.omega..sub.q.sup.2) 
x.vertline.F(q, 0, 0).vertline..sup.2 (58) 
EQU .vertline.F(r, 0, R).vertline..sup.2 =(.vertline.a.sub.r.sup.R 
.vertline..sup.2 /.vertline.a.sub.r.sup.0 
.vertline..sup.2)(.omega..sub.r.sup.2 Y.sub.0 -.omega..sub.q.sup.2 
Y.sub.R)/Y.sub.0 (.omega..sub.r.sup.2 -.omega..sub.q.sup.2) 
x.vertline.F(r, 0, 0).vertline..sup.2 (59) 
(6) If an external stress represented by a stress value z is applied to the 
straddle mounted beam oscillator 15, the straddle mounted beam oscillator 
15 is placed under a stress represented by a stress value z+z.sub.0. While 
the straddle mounted beam oscillator 15 is excited with the same exciting 
amplitude strength .vertline.a.sub.q.sup.R .vertline..sup.2 and 
.vertline.a.sub.r.sup.R .vertline..sup.2 as in the procedure (4), 
practical spectrum strength .vertline.F(q, z+z.sub.0, R).vertline..sup.2 
and .vertline.F(r, z+z.sub.0, R).vertline..sup.2 are observed and 
standardized with the spectrum strength .vertline.F (q, 0, 
R).vertline..sup.2 and .vertline.F(r, 0, R).vertline..sup.2, respectively. 
On the basis of the standardized power spectra G(q, z+z.sub.0) and G(r, 
z+z.sub.0), the residual stress value z+z.sub.0 is found from the 
quantitative relations between standardized power spectrum and stress 
value z shown in FIG. 8. By subtract subtracting the residual value 
z.sub.0 from the stress value z+z.sub.0, the stress value z is obtained. 
In this instance, the values z+z.sub.0 may be averaged in the weighted 
mean method. 
Although the above example is given for the case where the spectrum 
strength ratios X.sub.R and X.sub.0 are equal to each other, it is more 
general to define the spectrum strength ratio X.sub.R is linearly 
proportional to the spectrum strength ratio X.sub.0, i.e. X.sub.R 
=.beta.X.sub.0 (.beta. is a positive integer). 
Periodically conducting the procedures (4)-(6) for stress detection error 
compensation prevents aggravation of reliability in stress detection which 
possibly occurs due to air leakage in a long use of the stress detection 
device. 
FIG. 13 shows a stress detection device in accordance with another 
embodiment of the invention in which compensation is made for an error in 
stress detection. In order for the stress detection device to perform 
stress detection error compensation, there are added a random access 
memory (RAM) 44 and subtraction operation circuit 46. The subtraction 
operation circuit 46 carries out a subtraction calculation on any two 
power spectra signals among the power spectra represented by the power 
spectrum signals SQ.sub.1 to SQ.sub.n provided by the square processing 
circuits 36Q.sub.1 to 36Q.sub.n through the output terminals J.sub.1 to 
J.sub.n. The result of the subtraction calculation is stored in the random 
access memory (RAM) 44 and further in the random access memory (RAM) 38. 
Once the exciting frequencies .omega..sub.1 /2.pi. to .omega..sub.n /2.pi. 
have been established, exciting amplitude strength .vertline.a.sub.q.sup.0 
.vertline..sup.2 and .vertline.a.sub.r.sup.0 .vertline..sup.2 are 
established based on any two exciting frequencies .omega..sub.q /2.pi. and 
.omega..sub.r /2.pi. voluntarily selected and stored in the internal 
memory section A of the random access memory (RAM) 43. After having stored 
the amplitude strength ratio Y.sub.0 (=.vertline.a.sub.q.sup.0 
.vertline..sup.2 /.vertline.a.sub.r.sup.0 .vertline..sup.2) in the 
internal memory section B of the random access memory (RAM) 43, the 
theoretical spectrum strength .vertline.F (q, 0, 0).vertline..sup.2 and 
.vertline.F (r, 0, 0).vertline..sup.2 by solving the formula (13) for the 
selected frequencies .omega..sub.q and .omega..sub.r and exciting 
amplitude strength .vertline.a.sub.q.sup.0 .vertline..sup.2 and 
.vertline.a.sub.r.sup.0 .vertline..sup.2 and stored in both internal 
memory section C of the random access memory (RAM) 43 and random access 
memory (RAM) 38. On the other hand, the spectrum strength ratio X.sub.0 
(=.vertline.F(q, 0, 0).vertline..sup.2 /.vertline.F(r, 0, 
0).vertline..sup.2) is stored in the internal memory section D of the 
random access memory (RAM) 43. 
While holding the semiconductor sensor 11 in a vacuum (resistance factor 
R=0) so as to isolate the straddle mounted beam oscillator 15 from 
external actions, an oscillation is caused in the straddle mounted beam 
oscillator 15 with the exciting frequencies and amplitudes stored in the 
random access memory (RAM) 43. At this time, a residual stress value 
z.sub.0 represented by the stress value signal SZA and output through the 
terminal 42 is stored in the random access memory (RAM) 45. 
Thereafter, while leaving the semiconductor sensor 11 in a condition 
(resistance factor R.noteq.0) other than a vacuum, the exciting amplitude 
strength .vertline.a.sub.q.sup.R .vertline..sup.2 and 
.vertline.a.sub.r.sup.R .vertline..sup.2 are established so that their 
amplitude strength ratio is equal to the amplitude strength ratio Y.sub.0 
and stored in both internal memory section E of the random access memory 
(RAM) 43. Any two power spectra represented by spectrum signals SQ.sub.q 
to SQ.sub.r provided by the square processing circuits 36Q.sub.1 to 
36Q.sub.n are operated in the subtraction operation circuit 46 to provide 
the spectrum strength ratio X.sub.R of spectrum strength .vertline.F(q, 
z.sub.0, R).vertline..sup.2 and .vertline.F(r, z.sub.0, R).sup.2 which in 
turn is stored in the random access memory (RAM) 44. 
With use of the exciting amplitude strength .vertline.a.sub.q.sup.0 
.vertline..sup.2 and .vertline.a.sub.r.sup.0 .vertline..sup.2, the 
theoretical spectrum strength .vertline.F(q, 0, 0).vertline..sup.2 and 
.vertline.F(r, 0, 0).vertline..sup.2 and the spectrum strength ratios 
X.sub.0 and X.sub.R having stored in the A memory section, C memory 
section and D memory section of the random access memory (RAM) 43, and the 
random access memory (RAM) 44, respectively, calculations of the formulas 
(56) and (57) are carried out in the air resistance compensation circuit 
47 to determine spectrum strength .vertline.F(q, 0, R).vertline..sup.2 and 
.vertline.F(r, 0, R).vertline..sup.2 which are substituted for the 
theoretical spectrum strength .vertline.F(q, 0, 0).vertline..sup.2 and 
.vertline.F(r, 0, 0).vertline..sup.2 in the random access memory (RAM) 38. 
In this instance, the spectrum strength .vertline.F(q, 0, 
R).vertline..sup.2 (p assumes values other than q and r) which is 
determined by solving the formula (41) may be stored in the random access 
memory (RAM) 38. 
When exciting the semiconductor sensor 11 with the exciting amplitude which 
is stored in the memory section E of the random access memory (RAM) 43, 
the straddle mounted beam oscillator 15 experiences the stress of a value 
z due to an external action in addition the residual stress of a value 
z.sub.0, a stress value signal SZA representing the compound stress of a 
value z+z.sub.0 is provided at the terminal 42. The stress value z is 
obtained by feedbacking a stress value -z.sub.0 previously stored in the 
random access memory (RAM) 45 in the averaging processing circuit 41. 
In another way of stress detection, after arbitral selection of two 
exciting frequencies out of the exciting frequencies .omega..sub.1 /2.pi. 
to .omega..sub.n /2.pi., exciting amplitude strength 
.vertline.a.sub.q.sup.0 .vertline..sup.2 and .vertline.a.sub.r.sup.0 
.vertline..sup.2 are established on the basis of the selected exciting 
frequencies .omega..sub.q /2.pi. and .omega..sub.r /2.pi. and stored in 
the internal memory section A of the random access memory (RAM) 43. 
Together, after having stored the amplitude strength ratio Y.sub.0 
(=.vertline.a.sub.q.sup.0 .vertline..sup.2 /.vertline.a.sub.r.sup.0 
.vertline..sup.2) in the internal memory section B of the random access 
memory (RAM) 43, the theoretical spectrum strength .vertline.F(q, 0, 
0).vertline..sup.2 and .vertline.F(r, 0, 0).vertline..sup.2 by solving the 
formula (13) for the selected frequencies .omega..sub.q and .omega..sub.r 
and exciting amplitude strength .vertline.a.sub.q.sup.0 .vertline..sup.2 
and .vertline.a.sub.r.sup.0 .vertline..sup.2 and stored in both internal 
memory section C of the random access memory (RAM) 43 and random access 
memory (RAM) 38. On the other hand, the theoretically obtained spectrum 
strength ratio X.sub.0 (=.vertline.F (q, 0, 0).vertline..sup.2 
/.vertline.F (r, 0, 0).vertline..sup.2) is stored in the internal memory 
section D of random access memory (RAM) 43. 
While holding the semiconductor sensor 11 in a vacuum (resistance factor 
R=0) so as to isolate the straddle mounted beam oscillator 15 from 
external actions, an oscillation is caused in the straddle mounted beam 
oscillator 15 with the exciting frequencies and amplitudes stored in the 
random access memory (RAM) 43. At this time, a residual stress value 
z.sub.0 represented by the stress value signal SZA and output through the 
terminal 42 is stored in the random access memory (RAM) 45. 
Thereafter, while leaving the semiconductor sensor 11 in a condition 
(resistance factor R.noteq.0) other than a vacuum, the exciting amplitude 
strength .vertline.a.sub.q.sup.R .vertline..sup.2 and 
.vertline.a.sub.r.sup.R .vertline..sup.2 are established so that their 
amplitude strength ratio is equal to the amplitude strength ratio Y.sub.0 
and stored in both internal memory section F of the random access memory 
(RAM) 43. With use of the exciting amplitude strength 
.vertline.a.sub.q.sup.0 .vertline..sup.2 and .vertline.a.sub.r.sup.0 
.vertline..sup.2, the theoretical spectrum strength .vertline.F(q, 0, 
0).vertline..sup.2 and .vertline.F(r, 0, 0).vertline..sup.2 and the 
spectrum strength ratios X.sub.0 and X.sub.R having stored in the A memory 
section, C memory section and B and F memory sections of the random access 
memory (RAM) 43, respectively, calculations of the formulas (58) and (59) 
are carried out in the air resistance compensation circuit 47 to determine 
spectrum strength .vertline.F(q, 0, R).vertline..sup.2 and .vertline.F(r, 
0, R).vertline..sup.2 which are substituted for the theoretical spectrum 
strength .vertline.F(q, 0, 0).vertline..sup.2 and .vertline.F(r, 0, 
0).vertline..sup.2 in the random access memory (RAM) 38. In this instance, 
the spectrum strength .vertline.F(q, 0, R).vertline..sup.2 (p assumes 
values other than q and r) which is determined by solving the formula (41) 
may be stored in the random access memory (RAM) 38. 
When exciting the semiconductor sensor 11 with the exciting amplitude which 
is stored in the memory section F of the random access memory (RAM) 43, 
the straddle mounted beam oscillator 15 experiences the stress of a value 
z due to an external action in addition the residual stress of a value 
z.sub.0, a stress value signal SZA representing the compound stress of a 
value z+z.sub.0 is provided at the terminal 42. The stress value z is 
obtained by adding the stress value -z.sub.0 previously stored in the 
random access memory (RAM) 45 in the averaging processing circuit 41. 
In the previous embodiments, in order to make compensation for changes due 
to residual stress and/or air resistance, amplitude spectrum strength of 
different frequencies are used. During frequency analysis with respect to 
the amplitude spectrum strength, band-pass filters must have a narrow 
distribution of transmission factors for preventing power spectra of 
frequencies different from an intended frequency being mixed in. 
Fabrication of such a band-pass filter often accompanies technological 
difficulties. Further, as shown in FIG. 8, the standardized spectrum 
strength shows a divergent increase according to the values of y.sub.p and 
z in some cases. This is because that as reaching a resonant condition 
defined by the values of y.sub.p and z, the straddle mounted beam 
oscillator 15 encounters an oscillation of which the amplitude shows a 
divergent increase. Practically, the straddle mounted beam oscillator 15 
generates an oscillation with an increased amplitude but does not show a 
divergence in amplitude. Consequently, if trying to determine the stress 
value z from a divergent part of the spectrum strength curve G(p, z, R), 
the detected stress value contains a large error. 
In view of the above discussion, in the stress detection according to the 
invention, the exciting frequencies may be established such that the 
frequency difference is sufficiently large with respect to the 
distribution of transmission factors. Further, the available region of 
spectrum strength curve G(p, z, R) may be restricted to values equal to or 
less than 1. 
Specifically, as shown in FIG. 14, in the quantitative relation between 
standardized power spectrum G(p, z, 0) and stress value z, the 
standardized power spectrum G(p, z, 0) is given by an increasing function 
for parameters y.sub.p greater than 44.75 and by a decreasing function for 
parameters y.sub.p less than 44.75. The parameter y.sub.p of 44.75 is 
nothing else but the value of the factor (sin h.sqroot.y.sub.k /8 
cos.sqroot.y.sub.k /8+ sin.sqroot.y.sub.k /8 cos h.sqroot.y.sub.k /8) in 
the formula (17) is 0 (zero). When considering the case where the 
standardized power spectrum G(p, z, 0) is equal to or less than 1, two 
exciting frequencies are established so as to satisfy the following 
conditions: 
##EQU28## 
The exciting frequency .omega..sub.q thus established conditions G(p, z, 
0).ltoreq.1 for z.ltoreq.0. Similarly, the exciting frequency 
.omega..sub.r thus established conditions G(p, z, 0).gtoreq.1. for 
z.gtoreq.0. This enables the detection of stress value z on condition of 
G(p, z, 0).ltoreq.1. That is, the stress value z may be found from the 
characteristic curve (FIG. 14) for a standardized power spectrum G(p, z, 
0).ltoreq.1. In this manner, the foregoing of divergent increase in 
amplitude is eliminated. 
In this instance, the frequencies .omega..sub.q and -.omega..sub.r are 
established to be sufficiently larger than the decentralization of 
transmission factor of the band-pass filter. As a result, the band-pass 
filter does not mix standardized power spectra G(p, z, 0) of frequencies 
of .omega..sub.q and .omega..sub.r. 
While the region of standardized power spectra G(q, z, 0) less than 1 (one) 
is advantageous from a viewpoint of stress detection error, because 
amplitudes of oscillations existing in that region are relatively small, 
the stress detection must be performed with an increased accuracy. For 
this reason, one of the frequencies .omega..sub.q /2.pi. and .omega..sub.r 
/2.pi., for instance the frequency .omega..sub.q //2.pi. is established to 
a value relatively close to the resonant frequency, and another frequency 
.omega..sub.r /2.pi. to a value sufficiently far from the resonant 
frequency. This results in an increase in amplitudes of the whole 
oscillations and, however, a small amplitude of a component oscillation of 
the frequency .omega..sub.q /2.pi.. When the standardized power spectra 
G(q, z, 0) is less than 1, the standardized power spectra G(r, z, 0) is 
greater than 1. That is, when detecting a stress through the spectrum of 
the frequency .omega..sub.q, the frequency .omega..sub.r causes an 
oscillation with a large amplitude similar to the resonant oscillation in 
the straddle mounted beam oscillator 15. On the other hand, when the 
standardized power spectra G(r, z, 0) is less than one (1), the frequency 
.omega..sub.q causes an oscillation with a large amplitude similar to the 
resonant oscillation in the straddle mounted beam oscillator 15 in the 
region of standardized power spectra G(q, z, 0) greater than 1. 
Specifically, when establishing the exciting frequencies .omega..sub.q 
/2.pi. and .omega..sub.r /2.pi. so as to satisfy the conditions (60), the 
stress value is determined based on the standardized spectrum strength for 
one of exciting frequencies .omega..sub.q /2.pi. and .omega..sub.r /2.pi. 
which is kept sufficiently far from the resonant frequency in spite of 
positive stress or negative stress. 
Because the exciting frequencies .omega..sub.q /2.pi. and .omega..sub.r 
/2.pi. which are established through casual selection without any limit to 
the difference therebetween, and because it is troublesome to derive a 
quantitative relation for particular stress value z when determining 
stress value z from a quantitative relation between standardized power 
spectrum and stress which is obtained on the basis of theoretical values, 
it is preferred to establish the exciting frequencies .omega..sub.q and 
.omega..sub.r so as to provide a symmetry of standardized power spectra 
G(q, z, 0) and G(r, z, 0) with respect to stress values z. The symmetry of 
standardized power spectra G (q, z, 0) and G(r, z, 0) is achieved by 
appropriately selecting .omega..sub.p for the characteristic curves of 
standardized power spectrum G(p, z, 0) shown in FIG. 14. For example, as 
shown in FIG. 15, a symmetry of standardized power spectra G(q, z, 0) and 
G(r, z, 0) is provided when selecting the exciting frequencies 
.omega..sub.q and .omega..sub.r given by the following formulas (61): 
##EQU29## 
The symmetry is provided for various combinations of parameters y.sub.p. 
The available combinations of parameters y.sub.p is formulated for the 
parameters yq.sub.p &gt;44.75 and y.sub.r &lt;44.75 as follows: 
EQU y.sub.q.sup.2 =3.913.times.10.sup.3 -9.353.times.10.sup.-1 y.sub.r.sup.2( 
62) 
As long as band-path filters have distributions of transmission factors, it 
is generally impossible to perfectly separate amplitude spectra by 
exciting frequencies .omega..sub.q and .omega..sub.r in order to detect 
the strength of the separated amplitude spectra. However, when the 
difference between the exciting frequencies .omega..sub.q and 
.omega..sub.r is sufficiently large, the amplitude spectra are almost 
perfectly separated by the exciting frequencies .omega..sub.q and 
.omega..sub.r. Specifically, when the exciting frequencies .omega..sub.q 
and .omega..sub.r are given by a quantitative relation of .omega..sub.q 
=n.omega..sub.r (where n is sufficiently larger than 1). In other words, 
the parameters y.sub.q.sup.2 and y.sub.q.sup.2 are formulated with 
reference to the formula (62) as follows: 
##EQU30## 
For example, the parameters y.sub.r.sup.2 and y.sub.q.sup.2 are 6.2 and 
62.3, respectively, for the constant n=10. With an increase in the 
constant n, the parameters y.sub.r.sup.2 and y.sub.q.sup.2 become closely 
to 0 and 62.6. This means that the exciting force comprises a direct wave 
component and a sine wave component. The formulas (63) are rewritten as 
follows: 
##EQU31## 
By means of the exciting frequencies .omega..sub.q and .omega..sub.r as 
given by the formula (64), the amplitude spectra are given a symmetry and 
almost perfectly separated. 
In practical stress detection in the stress detecting method according to 
the first embodiment of the invention in which the straddle mounted beam 
oscillator 15 has a residual stress z.sub.0, standardized power spectra 
G(q, z, 0) and G(r, z, 0) are detected while the straddle mounted beam 
oscillator 15 is excited on a condition free from any external action. If 
the standardized power spectrum G(r, z, 0) is equal to or less than one 
(1), the stress value z is read from the characteristic curve and given a 
minus sign as a negative value which indicates a compression residual 
stress. On the other hand, if the standardized power spectrum G(q, z, 0) 
is equal to or less than 1, the stress value z is read from the 
characteristic curve as a positive value which indicates a tensile 
residual stress. 
Thereafter, standardized power spectra G(q, z, 0) and G(r, z, 0) are 
detected while the straddle mounted beam oscillator 15 is excited under an 
external condition. If the standardized power spectrum G(r, z, 0) is equal 
to or less than 1, the stress value z is read from the characteristic 
curve and given a minus sign as a negative value which indicates a loaded 
compression stress generated in the straddle mounted beam oscillator 15. A 
true stress is obtained by subtracting the compression residual stress 
from the loaded compression stress. Similarly, if the standardized power 
spectrum G(q, z, 0) is equal to or less than 1, the stress value z is read 
from the characteristic curve as a positive value which indicates a loaded 
tensile stress. A true stress is obtained by subtracting the tensile 
residual stress from the loaded tensile stress. 
In practical stress detection in the stress detecting method according to 
the second embodiment of the invention in which the straddle mounted beam 
oscillator 15 is put in a vacuum (resistance factor R=0) so as to be free 
from air resistance, because the degree of vacuum gradually decreases due 
to leakage, the spectrum strength for a frequency .omega..sub.p is given 
by the following formula (65) with reference to the formula (41): 
##EQU32## 
This formula (65) is available under satisfaction of the condition (40) 
necessary to provide the approximate solution (39) of the formula (38). 
For this requirement, as shown in FIG. 17, quantitative relations given by 
the right side of the condition (40) are provided for parameters y.sub.r 
and y.sub.q which achieve the symmetry of spectrum strength curves G(p, z, 
R). It is apparent from FIG. 17 that the right side of the condition (40) 
assumes a value less than 1 in spite of negative or positive stress values 
z. In this instance, the resistance factor R is given by the formula (44). 
The straddle mounted beam oscillator 15 made of a silicon wafer has 
specified physical dimensions, a length L of approximately 500 .mu.m, a 
width b of approximately 20 .mu.m and a thickness h of 3 .mu.m, has a 
density .rho..sub.0 of 2.3.times.10.sup.-3 kg/m.sup.3, the Young's modulus 
of 1.7.times.10.sup.11 kg/s.sup.2 .multidot.m, and a second moment of area 
I of bh3/12=4.50.times.10.sup.-23 m.sup.4. The .rho..sub.0.sup.A. 
.omega..sub.p are given as follows: 
##EQU33## 
The following formula is obtained for the left side of the condition (40): 
EQU 1+(.rho..sub.0 A/R).sup.2 =1+1.18.times.10.sup.4 y.sub.p (1/s)(67) 
Resultingly, while the condition (40) is held if the parameter is 0, the 
left side of the condition (40) has a value one order in magnitude larger 
than the right side. For example, considering a combination of parameters 
y.sub.r =10 and y.sub.q =61.8, the right side of the condition (40) 
becomes three orders in magnitude larger than the right side, there is no 
problem of using the approximate solution given the formula (38). 
In the study of the term .rho..sub.0 A.omega..sub.p.sup.2 -R.sub.1 
.omega.p.sub.2 included in the approximate solution given by the formula 
(38), when estimating the value of the following formula (68), 
##EQU34## 
the following formula is given: 
EQU (R.sup.2 /.rho..sub.0 A.omega..sub.p).sup.2 =8.45.times.10.sup.-3 
/y.sub.p.sup.2 (69) 
Since the value of .rho..sub.0 A.omega..sub.p becomes nearly equal to the 
resistance factor R for the parameter y.sub.p of 9.19.times.10.sup.-2, the 
term R.sub.1 .omega..sub.p is not negligible. However, the term R.sub.1 
.omega..sub.p is not negligible is negligible when the value of 
.rho..sub.0 A.omega..sub.p is sufficiently larger than the resistance 
factor R for the parameter y.sub.p greater than 1. 
From the result of the above study, in the method in which exciting 
frequencies .omega..sub.q and .omega..sub.r are selected so as to provide 
the symmetry of standardized power spectra G(p, z, 0) with respect to 
specified stress values z and the stress value is determined within the 
region where the standardized power spectra G(p, z, 0) are equal to or 
less than 1, air resistance can be ignored for parameters y.sub.r other 
than almost 0. 
As described above, when the spectrum for, for instance, a frequency 
.omega..sub.q /2.pi. is less than 1, the spectrum for another frequency 
.omega..sub.r /2.pi. becomes greater than 1, eliminating an occurrence of 
detection error due to an divergent increase in amplitude. Furthermore, 
while the amplitude of a component oscillation with a frequency of 
.omega..sub.q /2.pi. becomes smaller, the amplitude of a component 
oscillation with a frequency of .omega..sub.r /2.pi. is large, aggravation 
of sensitivity in amplitude detection is prevented in, in particular, an 
electrostatic capacity detection method. 
By means of providing standardized spectrum strength symmetrical with 
respect to standardized stress value like an even function, the 
quantitative relation between standardized spectrum strength and stress 
value is abridged, and the stress detecting apparatus can be made simple 
consequently. 
The frequencies .omega..sub.q and .omega..sub.r established to be 
relatively far different from each other are almost perfectly separated 
without the effect of crosstalk in the spectra. 
While, the degree of vacuum of a container in which the stress detecting 
apparatus is placed is one of serious constraints, the selection of the 
frequencies .omega..sub.q and .omega..sub.r satisfying the given condition 
allows to ignore the effect of air resistance even in the air and makes 
compensation of errors due to air resistance unnecessary. 
If the straddle mounted beam oscillator is held in a vacuum, the stress 
detecting apparatus is allowed to be configured more simply, which makes 
the following condition (70) available. 
##EQU35## 
This condition corresponds to a state where the constant n is infinity and 
is the most advantageous one for spectrum separation. 
With reference to the formulas (70), the standardized spectrum G (z) can be 
expressed as follows: 
##EQU36## 
It is to be understood that although the present invention has been 
described with regard to preferred embodiments thereof, various other 
embodiments and variants may occur to those skilled in the art, which are 
within the scope and spirit of the invention, and such other embodiments 
and variants are intended to be covered by the following claims.