2. Chapter 9 The Vibrations of Drumheads and Soundboards

3. Glockenspiel Bar Experiment by striking the bar at various places Where is the fundamental the loudest? Where is it weakest? We should expect the fundamental to be basically one-half wavelength with antinodes at the ends, since they are free.

4. Position of Nodes Touching a node makes no change in the sound (Why?) Notice that the felt support for the bar is at a node. To get at other modes of vibration, recall that each mode adds a node. Modes 2 and 3 might look like…

5. Mode 2 Confirmed by tapping predicted nodes and antinodes Add one-half wavelength to fundamental

6. Presumed Mode 3 Touching two fingers on either side of center should kill modes 1 and 2, leaving mode 3. Tapping at the center should produce 3096 Hz of mode 3. Mode 3 is not excited by these simple tests

7. Mode Three There are two perpendicular nodal lines Mode 3 is a twisting mode

8. Finding Modes Motion on one side of a node is opposite from the other side of the node. Tapping at the node does nothing to stimulate that mode. Tapping near antinode gives maximum stimulation of that mode.

9. Mode Shapes Length Modes Width Modes Mode 1 Mode 2

10. Estimating Mode Frequency By direct measurement upper octave bars are 70% the length of lower octave. What frequency do we get by cutting a bar in two? Note that 0.7 * 0.7  0.5 Each 0.7 is an octave so we have two octaves Glockenspiel has length 2.5 times width Width frequencies are about six times length frequencies.

11. Mode Families Length and width modes are different families. Mixed modes can exist. (2,1) mode

12. Wooden Plates Wooden plates have a grain or preferred direction. Stiffness is much lower against the grain than with it. Wood can flex better at the grain boundaries. Frequencies of the width modes are decreased compared to the uniform plate.

13. Mode 3 revisited For wooden plates where l is about 3 times w (or uniform plates where l = w) this is the lowest frequency. Important in violin making

14. Classes of Plates Free Edge – antinodes always appear at the edges Glockenspiel Cymbals, gongs, bells Tuning forks Clamped Edge – ends are merging into nodes rather slowly Soundboards of pianos and harpsichords Hinged Edge – ends come more rapidly into nodes Violin family (purfling)

15. First Four Modes of Guitar

16. First Four Modes of Rectangular Wood

17. Clamped vs. Hinged Edge More bending at the edges of a clamped plate produces higher frequency modes than the hinged edge. Frequency differences between clamped and hinged are less important for the higher modes.

18. 1-D Example

19. Purfling Thin hardwood inlaid strips in violins give the edge a hinge-like quality. If the violin hasn’t been played in awhile, the purfling gets stiff. Loud playing of chromatic scales can loosen it up again

20. Violin Parts

21. Membrane Thickness Variations in the thickness of a membrane can alter the natural frequencies it produces. Drumhead is to a Circular Plate As Flexible String is to a Bar

22. Analogy Flexible string and drumhead don’t have much stiffness They need to be stretched at the edges to produce tension. Drumhead under tension acts like a plate with hinged edges.

23. Normal Modes of a Vibrating Membrane Normal Modes

24. Frequency Comparison Mode DrumheadPlate 11.0001.000 21.5932.092 32.1353.427 42.2953.910 52.6536.067 Notice that the mode frequencies are much farther apart for the plate

25. Frequency Comparison

26. Drumhead/Plate Comparisons Above mode 5 the plate has nearly a constant interval between mode frequencies. (Straight line graph) Interval for the drumhead grows smaller at higher modes. Graph turns almost horizontal.

27. Tuning a Plate – a Model Adding mass will decrease the frequency Add small amounts of mass to the plate Positioned near a node has no effect on that mode Positioned near an antinode has maximum effect on that mode Rayleigh found…  fractional change in frequency  2 X the fractional change in mass  Also several lumps of wax should have the same effect as the sum of their individual effects.

28. Rayleigh’s Condition in Symbols  f = change in frequency  M = change in mass

29. Example A plate of iron has a diameter of 10 cm and a thickness of 0.025 cm and is clamped around the rim. Mode one has a frequency of 250 Hz The volume is = 1.96 cm 3 Using the density of iron (7.658 grams/ cm 3 ) the mass is 15 g.

30. Adding Mass Place.5 grams of wax at the center Antinode for mode 1 By Rayleigh Fractional change of frequency = -2(.5gm/15 gm) = -0.067 Mode 1 has its frequency changed by 250*.067 = -16.7 Hz and is now 233.3 Hz (just above A 3 #). Note decrease

31. Mode Frequency Differences Mode 2 has a frequency of 2.092 times mode one frequency or 523 Hz (C 5 ) Frequency difference before wax was applied 523 – 250 = 273 Hz The wax does not affect mode two since the center of the plate is a mode two node New frequency difference after wax is 523 – 233.33 = 289.7 Hz

32. Moving the Added Mass Move wax to midway between center and edge Here mode 2 has an antinode Now mode 2 has its frequency decreased by 6.7 % to 488 Hz Mode 1 also affected at this position of the wax, but only 1% since this is not an antinode (makes frequency 247.5 Hz) Frequency difference is 240.5 Hz Much less of a change by moving the mass.

33. Fixing the Frequency Difference Trial and error could be used to find a position where the frequency difference between the first two modes is one octave (here, 250 Hz).

34. Effect of Thinning the Plate Changing the plate thickness affects the plate stiffness Since f  (S/M) ½, thinning the plate decreases the mass (raising the frequency) M  means f  Thinning the plate also lowers the stiffness (lowering the frequency) S  means f 

35. Trade-off Rayleigh finds that the change caused by stiffness in one direction is about three times the effect caused by mass in the other direction.  f/f  4 * (  )

36. Building a Sounding Plate The craftsman finds the places where he can add wax to get the frequencies he wants. Wax adds mass without affecting stiffness. The change in stiffness dominates in the other direction Cut away wood at the positions of the wax. The amount of wood mass removed is half the mass of the wax. Note: these ideas don’t apply to membranes (drumheads). Adding mass to those raises the frequency.

37. Building a Sounding Plate

38. Kettledrums Calfskin or Plastic Membrane Hemispherical Copper Shell

39. Mode Ratios (as before) Mode DrumheadPlate 1 1.0001.000 2 1.5932.092 3 2.1353.427 4 2.2953.910 5 2.6536.067 Why aren’t these ratios whole numbers?

40. Deviations from Whole Integer Mode Ratios The shell itself is a trapped volume of air Normal play mode is to strike about half way from center to edge, thus enhancing mode 2 But even striking near the center gives very little mode 1 The reason is the vent hole that tends to damp mode 1

41. Mode Component Ratios ComponentRatioComponentRatio P 1.000 U 2.494 Q 1.502 V 2.800 R 1.742 W 2.852 S 2.000 X 2.979 T 2.245 Y 3.462 Careful tuning can get S exactly twice P, and X is not far off Also note that Q and X form an harmonic sequence f X  2f Q

42. Other Sequences Recall that your ear will assign the fundamental, even if it is not there, provided that an harmonic sequence is present. For a fundamental of C 2 = 65.4 Hz f P = 130.8 Hz = 2*C 2 f Q = 196.72 Hz = 3*C 2 f s = 261.6 Hz = 4*C 2 f U = 326.22 Hz = 4.99 C 2 f X = 389.65 Hz = 5.96 C 2