When the driving point impedance or admittance of a finite structure is plotted as a function of frequency, a series of maxima and minima are added to the curves for the corresponding infinite structure. The normalized
www.SolutionManual.info
98 3. Two-Dimensional Systems: Membranes, Plates, and Shells
10
., ~ lt.o
.§
~ t-
1-~
1- I-
F 1- 1-1-
I
1-~
1- I-
F t-
I- 1- 3 F1
0
.L-.
~)_ I I
I I I I I
2 3
~c5E =0.01 c5E = 0.1
) / c5l = t.oL
/ - L
I { ((
I I I I I I
4 s 6 7
I
I !
' ~
!
I !
)~ ~I /_
'If :'if '-
1/ 1\/ i/
r ~ ~
I I I I I
8 9 10 11 12
FIGURE 3.21. Normalized driving-point impedance at one end of a bar with free ends. Note that the horizontal axis is proportional to v']; the normalized impedance, which includes a factor 1//, decreases with frequency even though the actual driving-point impedance increases as v'J. Three different values of damping are shown (Snowdon, 1965).
driving-point impedance at one end of a bar with free ends is shown in Fig. 3.21. The heavily damped curve (c5 = 1) approximates Eq. (3.29) for an infinite bar, while the lightly damped curve (c5 = 0.01) has sharp maxima and minima.
The minima in Fig. 3.21 correspond to normal modes of the bar, whereas the maxima occur at frequencies for which the bar vibrates with a nodal line passing through the driving point. Impedance minima (admittance maxima) correspond to resonances, while impedance maxima {admittance minima) correspond to antiresonances.
The driving-point admittance at two different locations on a rectangular plate with simply supported edges is shown in Fig. 3.22. In these graphs, resonances corresponding to normal modes of the plate give rise to maxima on the curves. Note that some normal modes are excited at both driving points but some are not (when a node occurs too near the driving point).
100 90 80 70
0 60
"'
--"' bO 50 ..9 0 40
N
30 20 10
I
Drive <-I,--->
. --t=:.::""l t
pomt l~_Jf
- -
}\
1\\t -~
-\ - -
I ['._ 1 1\
~\I
\
0 W ~ 60 W1001W 1~160
Frequency (Hz) (a)
References 99 Drive <----I, ----+
pomt -~t L.: __ rf
100 90 80 70
0 60
"'
1 I
--"' bO 50 ..9 0
40
N
I 1
I
I
30
20 I I
10
0 W ~ 60 W 1001W1~160
Frequency (Hz) (b)
FIGURE 3.22. Driving-point admittance of a rectangular plate with simply- supported edges: (a) driven at the center and (b) driven off center (after Cremer et al., 1973).
References
Caldersmith, G.W. (1984). Vibrations of orthotropic rectangular plates. Acus- tica 56, 144-152.
Caldersmith, G., and Rossing, T.D. (1983). Ring modes, X-modes and Poisson coupling. Catgut Acoust. Soc. Newsletter, No. 39, 12-14.
Caldersmith, G., and Rossing, T.D. (1984). Determination of modal coupling in vibrating rectangular plates. Applied Acoustics 17, 33-44.
Chladni, E.F.F. (1802). "Die Akustik," 2nd ed. Breitkopf u. Hartel, Leipzig.
Cremer, L. (1984). "The Physics of the Violin." Translated by J.S. Allen, M.I.T.
Press, Cambridge, Massachusetts.
Cremer, L., Heckl, M., and Ungar, E.E. (1973). "Structure-Borne Sound,"
Chapter 4. Springer Verlag, Berlin and New York.
Fliigge, W. (1962). "Statik und Dynanik der Schalen." Springer-Verlag, Berlin.
French, A.P. (1971). "Vibrations and Waves," p. 181 ff. Norton, New York.
Hearmon, R.F.S. (1961). "Introduction to Applied Anisotropic Elasticity."
Oxford Univ. Press, London.
Hutchins, C.M. (1977). Another piece of the free plate tap tone puzzle. Catgut Acoust. Soc. Newsletter, No. 28, 22.
www.SolutionManual.info
100 3. Two-Dimensional Systems: Membranes, Plates, and Shells
Hutchins, C.M. (1981). The acoustics of violin plates. Scientific American 245 (4), 170.
Kalnins. A. (1963). Free nonsymmetric vibrations of shallow spherical shells.
Proc. 4th U.S. Gong. Appl. Mech., 225-233. Reprinted in "Vibrations: Beams, Plates, and Shells" (A. Kalnins and C.L. Dym, eds.), Dowden, Hutchinson and Ross, Stroudsburg, Pennsylvania 1976.
Kalnins, A. and Dym, C.L. (1976). "Vibration: Beams, Plates, and Shells,"
Benchmark Papers in Acoustics 8, Dowden, Hutchinson and Ross, Strouds- burg Pa.
Kinsler, L.E., Frey, A.R., Coppens, A.B., and Sanders, J.V. (1982). "Fundamen- tals of Acoustics," 3rd ed., Chapter 4. Wiley, New York.
Leissa, A.W. (1969). "Vibration of Plates," NASA SP-160, NASA, Washington, D.C., reprinted by Acoustical Society of America, Woodbury NY, 1993.
Leissa, A.W. (1973). "Vibration of Shells," NASA, Washington, D.C., reprinted by Acoustical Society of America, Woodbury NY, 1993.
Leissa, A.W., and Kadi, A.S. (1971). Curvature effects on shallow spherical shell vibrations. J. Sound Vibr. 16, 173-187.
Love, A.E.H. (1888). On the free vibrations and deformation of a thin elastic shell. Phil. Trans. Roy. Soc. London Ser.A, 179, 543-546.
Mcintyre, M.E., and Woodhouse, J. (1984/1985/1986). On measuring wood properties, Parts 1, 2, and 3, J. Catgut Acoust. Soc. No. 42, 11-25; No. 43, 18-24; and No. 45, 14-23.
Morse, P.M. (1948). "Vibration and Sound," Chapter 5. McGraw-Hill, New York. Reprinted 1976, Acoustical Soc. Am., Woodbury, New York.
Morse, P.M., and lngard, K.U. (1968). Chapter 5. "Theoretical Acoustics,"
McGraw-Hill, New York. Reprinted 1986, Princeton Univ. Press, Princeton, New Jersey.
Rayleigh, Lord (1894). ''The Theory of Sound," Vol. 1, 2nd ed., Chapters 9 and 10. Macmillan, New York. Reprinted by Dover, New York, 1945. Vol. 1, pp.395-432
Reissner, E. (1946). On vibrations of shallow spherical shells. J. Appl. Phys. 17, 1038-1042.
Reissner, E. (1955). On rod-symmetrical vibrations of shallow spherical shells.
Q. Appl. Math. 13, 279-290.
Rossing, T.D. (1982a). "The Science of Sound," Chapters 2 and 13. Addison~
Wesley, Reading, Massachusetts.
Rossing, T.D. (1982b). The physics of kettledrums. Scientific American 247 (5), 172-178.
Rossing, T.D. (1982c). Chladni's law for vibrating plates. American J. Physics 50, 271-274.
Snowdon, J.C. (1965). Mechanical impedance of free-free beams. J. Acoust. Soc.
Am. 37, 24Q-249.
Soedel, W. (1993). "Vibrations of Plates and Shells," second edition. Marcel Dekker, New York.
Ver, I.L., and Holmer, C.I. (1971). Interaction of sound waves with solid struc- tures. In "Noise and Vibration Control" (L.L. Beranek, ed.). McGraw-Hill, New York.
References 101 Waller, M.D. (1938). Vibrations of free circular plates. Part I: Normal modes.
Proc. Phys. Soc. 50, 7Q-76.
Waller, M.D. (1949). Vibrations of free rectangular plates. Proc. Phys. Soc.
London B62, 277-285.
Waller, M.D. (1950). Vibrations of free elliptical plates. Proc. Phys. Soc. London B63, 451-455.
Waller, M.D. (1961). "Chladni Figures: A Study in Symmetry." Bell, London.
Warburton, G.B. (1954). The vibration of rectangular plates. Proc. Inst. Mech.
Eng. A168, 371-384.
Wood Handbook (1974). Mechanical properties of wood. In "Wood Hand- book: Wood as an Engineering Material." U.S. Forest Products Laboratory, Madison, Wisconsin.
www.SolutionManual.info
4
Coupled Vibrating Systems