78 BELL SYSTEM TECHNICAL JOURNAL 



labeled 6th Xy, etc., change very little and are nearly horizontal straight lines. 

 Here again they appear to be simple harmonics of some common low fre- 

 quency. Also it will be noted that the coupling between the Zx flexures and 

 the Zx shear is quite appreciable and in general decreases as the difference in 

 order of the two modes becomes greater. This plot of the various flexure 

 frequencies tells us a great deal about the behavior of progressively higher 

 order of flexure type motion. The important effect to be noticed is that for 



w 

 high orders, and a fixed ratio of — , the flexure may be treated as though it 



were harmonic so far as frequency is concerned. Some variations to this 

 rule will be observed and special cases will be discussed. So far we have 

 discussed the case of flexure modes of relatively low order. In the case of 

 high frequency shear modes of motion, we would expect that the order of 

 flexure which would interfere with this type of motion would be rather high. 



Figure 6.14 shows a plot of these flexure modes as observed in an .4r-cut 

 plate. These are shown by dashed lines. The dots indicate actual meas- 

 ured resonances. This figure also shows the various other resonant fre- 

 quencies observed in this type of plate as discussed in section 6.2. The solid 

 lines labeled mnp represent the type of shear motion shown in Fig. 6.5. 

 Here again we may observe certain statements made before with respect to 

 the coupling between shear and flexure type motions. Notice in this case 

 that the coupling between an even order flexure and an odd order shear is 

 high and increases as the orders more nearly approach each other. For 

 example, the 38th flexure mode is coupled to the fundamental shear labeled 

 niiHipi has very little coupling to the second order shear mifiipi, and again 

 is strongly coupled to the third shear niinzpi and correspondingly higher 

 coupling to the fifth shear. When we speak of higher order shears, such as 

 W2W3«6, they are not higher order in the sense of harmonics, but do differ by 

 a small amount in frequency. In the case of a plate where I is not great 

 compared to t, these differences will be greater. 



In actual practice in the case of AT plates, we are usually concerned 

 mainly with the fundamental high frequency shear and high even order flex- 

 ures along the length. This case is shown in Fig. 6.15 which gives experi- 

 mental results of measurements on actual AT plates. It will be noticed 

 that the flexure frequencies show a rather regular displacement as the ratio 

 of the length of the plate to its thickness is changed. In this case only the 

 odd order modes of shear and the even modes of flexure are shown. It will 

 be observed that as the ratio of the length to thickness decreases, the cou- 

 pling between these modes is quite high. This some state of affairs is illus- 

 trated again in the case of the third harmonic of high frequency shear and is 

 shown in Fig. 6.16. The near vertical dashed lines represent even order 



