Method and apparatus for hardness measurement
BRIEF SUMMARY OF THE INVENTION 
     The invention relates to a method for determining the hardness of solid bodies in which a rod rendered resonant and having a defined contact surface of hard material is applied with a predetermined contact force to a workpiece, the hardness of which is to be measured. The frequency of the mechanical vibrations causing the rod to be resonant is held constant and is equal to the resonant frequency ωo of the rod when it is not mechanically coupled to the workpiece. 
     As disclosed in U.S. Pat. No. 3,572,097 issued to C. Kleesattel dated Mar. 23, 1971, a vibrating rod is coupled to a workpiece and a characteristic parameter of this mechanically resonant device (e.g. the elasticity modulus or mechanical impedance) is varied until the frequency of vibration of the vibrating rod coupled to the workpiece is equal to ωo. The change in the characteristic parameter is then used to determine the hardness of the workpiece. 
     A disadvantage of this known method is that relatively complicated devices are necessary in order to vary the corresponding characteristic parameters and determine the correspondingly changed resonant frequency. U.S. Pat. No. 3,153,338 of C. Kleesattel dated Oct. 20, 1964; U.S. Pat. No. 3,955,404 of W. Bickel, et al. dated May 11, 1976, and U.S. Pat. No. 4,523,467 of R. Diederichs et al. dated June 18, 1985 disclose the measurement of hardness by using the difference between the resonant frequency of the probe when it is freely vibrating and when it is coupled to the workpiece. In the known devices, the hardness is usually measured in Vickers hardness values, i.e. the probe is provided with a diamond tip at its end. The softer the material of the workpiece, the greater is the area of indentation and the resulting change in resonant frequency. 
     The heretofore described method has been found disadvantageous in that the frequency measurement has to be relatively exact and the electronic circuits are complicated. 
     A principal object of this invention, therefore, is the provision of a method and apparatus making it unnecessary to cause a variation of the characteristic resonance parameters, or requiring a complicated measurement of frequency differences, without impairing the accuracy of measurement. 
     In contrast with the known methods, the present invention is not based on measuring the frequency difference of the vibrating rod, but is based on determining the amplitude of vibrations of the vibrating rod when the rod is mechanically coupled to the workpiece. This does not mean, as might at first sight be supposed, that the amplitude of oscillation is determined at the resonant frequency of the vibrating rod when it is coupled to the workpiece. Instead, the amplitude of vibrations of the rod is measured at the resonant frequency of the rod when it is oscillating free of such contact. It has been found, according to the invention, that the square of the amplitude of vibration is almost exactly proportional to the hardness of the workpiece. 
     Details and further advantages of this invention will be explained in more detail with reference to the following description when read in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic representation of the construction of the known hardness measuring instrument which can be used in the present invention; 
     FIG. 2 shows a part of the instrument per FIG. 1, the vibrating rod being coupled to a workpiece; 
     FIG. 3 is a graph showing the hardness measured on various workpieces and also calculated from the measured amplitudes, and 
     FIG. 4 is a graph for illustrating the invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to the figures, FIG. 1 shows the hardness probe 1 connected via a receiver 2 and a computer and control unit 3 to a measurement indicator 4. Probe 1 is also connected to a high frequency generator 5 which is actuated via the computer and control unit 3. Reference numeral 6 denotes the workpiece, the hardness of which is to be determined. 
     The probe 1 comprises in part a vibrating rod 10, a sleeve 11 partly surrounding the rod 10, a spring 12 and a casing 13 enclosing the aforementioned parts. 
     The rod 10 is resiliently held in the sleeve 11 by a hard rubber ring 14. Rod 10 and sleeve 11 can be fastened to the ring 14 by adhesive bonding. The lateral distance between sleeve 11 and casing 13 is determined by two spherical bushings 15 and 16. Along the longitudinal direction, the sleeve 11 is movable in the direction of the spring 12, starting from a predetermined distance relative to the top surface 17 of the casing 13. The predetermined distance is determined by a tubular inner cylinder 18 of casing 13, against which the sleeve 11 abuts when the rod 10 vibrates freely, i.e., free of contact with a workpiece, as seen in FIG. 1. 
     Two piezoelectric transducers 19 and 20 are secured to the rod 10. Transducer 19 serves as receiving transducer and is coupled to the receiver 2. Transducer 20 serves as a vibration exciting transducer and receives corresponding electrical high frequency signals from the high frequency generator 5. The bottom end of rod 10 bears a Vickers diamond 21, the indentation of which produced in the workpiece is to be determined indirectly. The diamond 21 provides a defined contact surface engaging the workpiece. 
     Probes of this kind are known and are used in conventional hardness gauges supplied by the assignee of this patent application under the name &#34;Microdur&#34;. The main difference between the invention and the heretofore known devices is, firstly, the difference in the operation of the probe and, secondly, the electronic circuit coupled to the probe 1 which is not designed for determining frequency differences but for determining amplitudes of vibration. 
     In the prior art devices, as disclosed, for instance, in U.S. Pat. No. 3,153,338, which patent is incorporated herein by reference, the hardness is determined as follows: 
     The transducer 20 excites the rod 10 to cause it to vibrate along its longitudinal axis. The vibrations are sensed and received by the transducer 19 and sent via the receiver 2 and unit 3 (or another circuit device) to the high frequency generator 5. Generator 5 then varies the vibration frequency until it coincides with the natural frequency of the rod. This frequency ωo is then measured. 
     Next, as shown in FIG. 2, rod 10 is pressed upon the workpiece 6 whose hardness is to be measured until the lower radial surface of cylinder 22 of casing 13 comes into contact with the workpiece 6. As a result, the diamond 21 penetrates into the surface of the workpiece 6 and the rod 10 no longer vibrates freely. The resonant frequency ω1 of the rod now coupled to the workpiece 6 is then redetermined using the previously described feedback circuit. The softer the material of the workpiece 6, the greater is the area of indentation and the resulting frequency shift Δω=ω1-ωo is a measure of the hardness of the workpiece. 
     In the known hardness gauges, therefore, the frequency difference Δω has to be determined very accurately and then converted in a relatively complicated manner to corresponding hardness values. 
     When using the method according to U.S. Pat. No. 3,572,097, the resonant frequency ω1 of the rod 10 when coupled to the workpiece 6 is varied until the value of Δω is equal to zero. The resonant frequency is varied e.g. by means of an additional coil (not shown in FIG. 1) through which current flows and which is wound around the rod 10. The polarization current induced in the rod changes the elasticity modulus and thus changes the natural frequency of the rod. The polarization current at which the value Δω is equal to zero is then used to determine the hardness. 
     By contrast, the device according to the present invention shown in FIG. 1 operates as follows: 
     Firstly, as before, the natural frequency of the freely vibrating rod 10 is determined as in the prior art. Next, the feedback signal between the receiver 2 and the high frequency generator 5 is interrupted, e.g. by the computer and control unit 3, so that the generator 5 produces only vibrations at the frequency ωo. Next, as shown in FIG. 2, diamond 21 is pressed into the surface of workpiece 6 and the amplitude of vibration is measured at the frequency ωo by the receiver 2 and the corresponding hardness HV is determined in the computer and control unit 3. 
     It has been found that the relation between the measured amplitudes B and the hardness HV can be described fairly accurately as follows: 
     
         HV=C·B.sup.2 ·Eo.sup.2                   (1) 
    
     wherein C is an instrument parameter and Eo is a modulus derived respectively from the modulus of elasticity E1 and E2 of the workpiece 6 and the diamond 21, and from the Poisson&#39;s ratios ν1 and ν2 of the workpiece and the diamond: ##EQU1## 
     Advantageously the value of C is found by measuring the amplitude in a workpiece of known hardness, such as a calibration block, and solving the equation (1) with respect to value C. The Poisson&#39;s ratio ν2 and the modulus of elasticity E2 of the diamond are available from a table. The Poisson&#39;s ratio ν1 and the modulus of elasticity E1 of the workpiece are determined by measuring the longitudinal and transverse wave velocities (see &#34;Ultrasonic Testing of Materials&#34; by Krautkramer (book), 2nd edition, Springer Verlag, New York, N.Y., p. 580). 
     FIG. 3 shows the measured hardness of various grades of steel. The relatively amplitude B/Bo is shown along the abscissa and the corresponding Vickers hardnesses (values HV) are plotted along the ordinate. 
     The value Bo denotes the voltage amplitude generated by the high frequency generator 5. In the embodiment, the value Bo was 20 V peak-to-peak for the freely-vibrating rod and the frequency value ωo was 78 kHz. 
     The force exerted by the spring 12 when the rod (FIG. 2) was placed on workpiece 6 was about 9.81N (1 Kp). The corresponding indentation caused in the workpiece 6 by the diamond 21 was optically evaluated in a known manner, using a microhardness gauge, model 3212 made by Zwick GmbH &amp; Co., Ulm, West Germany, the hardness being calculated from the diagonal of the base area of the permanent indentation. Curve 100 shows the optically measured hardness in dependence on the relative amplitude associated with the indentation. 
     The workpieces were four hardness comparison plates available to the inventors. Measurements with the aforementioned Zwick hardness gauge gave the following values: 120, 246, 563 and 879 HV 1/30. 
     The values of E1/(1-ν1 2 ) for the hardness comparison plates used were: 2.34×10 11  ; 2.327×10 11  ; 2.269×10 11  and 2.233×10 11  Pa. The value obtained for E2/(1-ν2 2 ) was 6×10 11  Pa. From these values, the value Eo was calculated, after which the corresponding hardness was calculated from the measured amplitudes and the constant C in equation (1). Curve 200 in FIG. 3 shows the result. The constant C was calculated using the hardness comparison plate having 563 HV 1/30. ##EQU2## 
     As a comparison between curves 100 and 200 shows, the optically measured values and the values calculated from the amplitudes agree closely, practically through the entire range. 
     The following shall show that the relation found experimentally and defined by equation (1) can also be deduced by calculation: 
     When a cylindrical vibration rod free at both ends is excited by a periodic external force of frequency ω, a standing wave is formed. The solution u (x,t) of the wave equation for this case is: 
     
         u(x,t)=[A(ω)×sin (ωt)+B(ω)×cos (ωt)] cos (κx) 
    
     
         ω=2πf, f=frequency of oscillation 
    
     
         κ=2π/λ, λ=wavelength 
    
     The expression in the square brackets represents the solution of the differential equation for a damped harmonic oscillator forced to vibrate at the frequency ω. At every measuring point x, therefore, the rod behaves like a harmonic oscillator. 
     The quantities A(ω) and B(ω) depending on the excitation frequency are called the absorption amplitude and the dispersion amplitude respectively. 
     When the vibrating rod is in contact with a workpiece having a large mass, the following relation can be obtained from the articles by G. M. L. Gladwell and C. Kleesattel, &#34;The Contact-Impedance Meter-2&#34;, Ultrasonics, October 1968, pages 244-251: 
     
         Δf=f1-fo≃D×Eo×√F 
    
     wherein 
     fo=resonance frequency of freely-oscillating rod, 
     D=a constant depending only on the modulus of elasticity of the rod, the geometry of the rod and the indentation member, and 
     F=area of indentation by Vickers diamond. 
     This relation can be converted to: 
     
         ω1.sup.2 =ωo.sup.2 +D×(ω1+ωo)×Eo√F            (3) 
    
     The absorption and the dispersion amplitudes, in the case of contact, have the following forms: 
     
         A(ω)=Bo&#39;×(Γ×ω)/((ω1.sup.2 -ω.sup.2).sup.2 +(Γ×ω).sup.2) 
    
     
         B(ω)=Bo&#39;×(ω1.sup.2 -ω.sup.2)/((ω1.sup.2 -ω.sup.2).sup.2 +(Γ×ω).sup.2) 
    
     Γ=half-width of resonance frequency during contact, Bo&#39;=amplitude of the periodic external force, which is proportional to the voltage amplitude Bo of the high frequency generator 5. 
     If, during contact, the rod 10 is excited at the frequency ω=ωo=2πfo, a simple relation for the dispersion amplitude B(ω) can be derived in view of the fact that the following conditions apply to steel in the hardness range 100≦HV≦1,000 when in contact: 
     
         Γ&lt;200 Hz and Δf&gt;300 Hz. 
    
     In the present embodiment where fo=78 kHz, therefore: 
     
         (Γωo).sup.2 &lt;&lt;(ω1.sup.2 -ω.sup.2).sup.2 
    
     and 
     
         A(ω)&lt;&lt;B(ω). 
    
     At the frequency ω=ωo, the absorption amplitude A(ω) is already about two orders of magnitude smaller than the dispersion amplitude B(ω) and therefore negligible. 
     B(ω) can be simplified to: 
     
         B(ωo)˜Bo&#39;/(ω1.sup.2 -ωo.sup.2)     (4) 
    
     If one inserts equations (3) and (4) into the equation for the Vickers hardness: 
     
         HV=test force/surface area of indentation=K/F, 
    
     after solving equation (3) with respect to F, one obtains: 
     
         HV=C×Eo.sup.2 ×B(ωo).sup.2 
    
     with C=K×(D 2  /Bo&#39; 2 )×(ω1+ωo) 2 . 
     Since Δf/fo&lt;2% for the above-mentioned hardness range of steel, (ω1+ωo) 2  and consequently C can be regarded as constant. 
     In FIG. 4, the amplitudes measured are shown once more graphically for the present embodiment. The exciting frequency ω is plotted on the abscissa and the amplitudes sensed by the transducer 1 (FIG. 1) are plotted on the ordinate. The resonance curve marked 300 is obtained for the freely-vibrating rod 10 when ω varies. The corresponding resonant frequency ωo is 78 kHz. 
     After the rod 10 makes contact with the workpiece 6 (see FIG. 1), resonance curve 300 becomes displaced. In the example shown in FIG. 4, the curve marked 400 is obtained when the vibrating rod is coupled to the workpiece. The corresponding resonant frequency ω1 in the illustrated example is 79 kHz. Numeral 401 denotes the calculated absorption curve A(ω) associated with the measured curve and numeral 402 denotes the corresponding dispersion curve calculated. 
     Since the actual frequency at which the rod 10 is energized is only 78 kHz when coupled to the workpiece, the receiver 2 (FIG. 1) receives only a relatively small voltage. In the example illustrated E/Emax≈0.02. The voltage calculated from this ratio substantially corresponds to the dispersion amplitude B(ωo) because the corresponding proportion of the absorption amplitude is negligible at 78 kHz. 
     Basically, therefore, in the method according to the invention, the dispersion amplitude B(ωo) of the vibrating rod is determined when coupled to the workpiece, and the resulting value is used as a measure of the hardness of the workpiece.