Determination of Buckling Loads by Frequency Measurements Citation Chu, Tao Hung (1949) Determination of Buckling Loads by Frequency Measurements. Engineer's thesis, California Institute of Technology. doi:10.7907/chf2-y477. https://resolver.caltech.edu/CaltechTHESIS:09182025-210604666 Abstract Analytic solutions for the relation between compressive load and natural frequency are obtained for a simple column and a circular ring under uniform radial load. The relation is of the form [Formula. See abstract in scanned thesis for details]. An implicit analytic solution for a rectangular frame under symmetrical axial load and for symmetrical buckling is also obtained. The complexity of the final transcendental equation indicates that the relation between P and λ n will not be linear. The relation is checked experimentally for a simple column. The extrapolated critical load is within 2% of the computed value. The experiment on a rectangular frame shows that the relation between P and λ n can be very closely approximated by a linear one. The extrapolated critical load is 17,500 lbs. which is about 49% higher than that for the lowest symmetrical mode and about 3% higher for the lowest unsymmetrical mode. Evidently, the measured frequency is in the latter mode. The misleading result is due to the fact that these modes have frequencies very close to each other, and the unsymmetrical mode becomes more stable because of restraints at the supports and dynamic coupling between the modes. Some interesting points in this work are stressed in the conclusion and some suggestions for the additional experiments are made. Item Type: Thesis (Engineer's thesis) Subject Keywords: (Mechanical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Mechanical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Hudson, Donald E. Thesis Committee: Unknown, Unknown Defense Date: 1949 Record Number: CaltechTHESIS:09182025-210604666 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09182025-210604666 DOI: 10.7907/chf2-y477 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17688 Collection: CaltechTHESIS Deposited By: Ben Maggio Deposited On: 04 Oct 2025 13:07 Last Modified: 04 Oct 2025 13:22 Thesis Files PDF - Final Version See Usage Policy. 26MB Repository Staff Only: item control page

DETERMINATION BY OF FREQUENCY BUCKLING LOADS MEASUREMENTS Thesis by Tao Hung Chu In Partial Fulfillment of the Requirements For the Degree of Mechanical Engineer California Institute of Technology Pasadena, California ACKNOWLEDGMENTS The author wishes to make aclmowledgments to Dr. D. E. Hudson for his constant advice and guidance in this work and to Dr. G. W. Housner for his suggestions in obtaining analytic solutions, and in the preparation of this thesis He also wishes to make special aclmowledgment to Mr. H. L. Roberson for his assistance in conducting the experiments. A B S T R A C T Analytic solutions for the relation between compressive load and natural frequency are obtained for a simple column and a circular ring under uniform radial load. The relation is of the form P == Per -K A. n 2 An implicit analytic solution for a rectangular frame under symmetrical axial load and for symmetrical buc·kling is also obtained. The com- plexity of the final transcendental equation indicates that the relation between P and ).11 will not be linear. The relation is checked experimentally for a simple column. The extrapolated critical load is within 2% of the computed value. The experiment on a rectangular frame shows that the relation between P and ).,, can be very closely approximated by a linear one. The extrapolated critical load is 17,500 lbs. which is about 49%higher than that for the lowest symmetrical mode and about 3% higher for the lowest unsymmetrical mode. the latter mode. Evidently, the measured frequency is in The misleading result is due to the fact that these modes have frequencies very close to each other, and the unsymmetrical mode becomes more stable because of restraints at the supports and dynamic coupling between the modes. Some interesting points in this work are stressed in the conclusion and some suggestions for the additional experiments are made. C ON T E N T S - TITLE PART PAGE Acknowledgments Abstract Table of Contents I Introduction 1 II Analysis 3 III (1) Simple Colwnn 4 (2) Circular Ring 7 (3) Rectangular Frame 11 Experiments (1) Simple Column 16 (2) Rectangular Frame 20 Comparison of Analysis and Experiment V (1) Simple Column 24 (2) Rectangular Frame 26 Conclusions and Suggestions 30 References 32 Figures 33 I. INTRODUCTION Buckling problems in elasticity have been studied for many years. Mathematical solutions have been obtained for many idealized problems, such as prismatic bars, plates, shells, etc. of different shapes and end conditions. However, such solutions are still very li- mited, and then? remain many problems involving complicated structures, which are too cumbersome, and can not be practicably solved by analytic methods. This is especially true for a built-up structural frame in compression. In practice, a destructive experimental method is usually employed to find out the safe load. The relations between compressive load and frequency for some simple cases have been studied analytically by Charles Massanet in his article 11 Les Relations entre les Modes Normaux. de Vibration et la Stabilit~ des Systemes Elastiques 11 • (l) ➔l- Further work has been explored by Dr. Felix Buckens at the California Institute of Technology. However, the work is limited to theoretical discussions for cases where buckling problems have mathematical solutions. Massonnet has shown that, for a simple prismatic column with pinned end, the relation between axial compressive load and the square of the frequency is linear, and deviates slightly from linearity for columns with other end conditions. He also has shown that the load at zero frequency is the critical load corresponding to the associated mode of vibration. Therefore, if a few points on the curve of load * The number in the parenthesisrefers to the number in the Table of of References following the text. - 2 - versus square of frequency could be determined, then the critical load could be extrapolated for zero frequency. If this law can be ' extended to structures where no analytic solution or no simple analytic solution is possible, we shall have a non-destructive experimental method for determining the critical load. Before such a method can be applied, the accuracy of the extrapolation must first be investigated and compared with the theoretical value. It is the purpose of the present attempt, (a) to correlate buckling load and frequency for a few simple cases, (b) to check experimentally for a case where the relation is solved analytically and is known to be linear, and (c) to determine experimentally for a case where the relation is non-linear, but the critical loads can be computed, so that it may be seen how far the relation may be approximated by a linear one and how accurate the extrapolated critical load will be. - 3 - II. THEORY In the theoretical study, the following nomenclature is used: I Second moment of area of a section E Young's modulus of elasticity M Moment at a section P Axial force q Radial uniformly distributed load f Mass density per unit length 1 Length of a member r Radius Ah Frequency of the n'th mode in cycles per unit time x Coordinate along the direction of the axial load y Lateral displacement w Radial displacement u Tangential displacement 0 Angle in polar coordinates - 4 - (1) Simple Column. p p p T 1 l p p Fig. 1 When a straight prismatical column under an axial compressive load Pis deformed, the moment at a section (x,y) is rzJ'l.y - £I 'ox:.,. Theintensity of the lateral restoring force is then - £ I 0 "':l,,. '<}:( The lateral restoring force due to the axial force Pis -ae _ - p ~ 7/·:1 - - p 'c)~'- "?>LJ The inertia force is - f ?JC . For equilibrium, the sum of the forces must be zero. Let a2 ::: __E_ EI and y Then b2 = _f_ ,. and EI = X(x) .T(t) XIV ..,. aJ.x•' X where K is an arbitrary constant. b 1. T" T = J:.z - 5 - Therefore, T 11 -f (!)4r =0 (1) x' y+ a2x11 - k2x = 0 (2) The solutions of equations (1) and (2) are == A .s,"n 7 4f Bco.5 ! i -1- (3) X - C, s/n(hlx).,. c:. cos(m)() + C.J S//lh{n,'x}.J C Cosli(m':e,) (4) where I 'Po satisfy the boundary conditions: (a) at X = 0, X= O· ' (b) at X - o, X" = O; (c) at X =- 1, X = O; (d) at X - 1, X" = O; we obtain, from equation (4), the condition sin ml or where ml - n 0 n7T = 1, 2, 3, ...... Hence /_ 2. -:= ,c But from equation (3), n 1th mode. 17 2 TT ~ Oz. h v.,, 4 t~ lz ' I k is the natural frequency of the An=znfh) Therefore, An = 4llz_ • bk) z z. or Cl .2. I L "1. ( z = .2....J!.. L ._ - 4b 21 L n a. \ Z, A .,, Substituting values of 1 a 1 and 1 b 1 , we have ( 5) - 6 - From equation (5), it is seen that there is a linear relationship between An2 and When P. An= O, P= which is Euler's buckling load. When P = O, , _= n~,r "-n z l 1. -j EIp which is the natural frequency of the colwnn in free vibration. A further discussion of the equation will be found in Part r:t, where the results of the experiment on a simple column are considered. - 7 - (2) Circular Ring. 'lr I - -' --+--- - - p p Fig. 2 The effect of a constant uniform radial load upon the frequency of a circular ring of uniform cross section, has been studied by Massonnet (Ref. 1). However, he did not give the derivation of his basic equation connecting the frequency and t he radial load. A some- what simpler equation can be obtained with the following assumptions: (a) the axial load in the ring remains constant; (b) the circumferential length of the ring remains constant (Ref. 2) i.e., the circumferential strain or du d8 - w - 0 - 8 - Let Then u = Jur Je ~ h a,, cos ne SI n 'l\n "T. 1 = - L. The moment at a section of the ring due to a change in curvature (Ref. 3) is Therefore, the strain energy due to this moment is 2n >. .tu.., 7.J)z."(; d8 V. == f gZ,fo.,. ('<'QB.,_ O I 0 . ~ V ::; EI3 • • ~ T;, I -ill J( "b"tu..,. 2 7)1.-w' + 08¥ '<)87. w) ~-w-ol e 0 The effect of the axial load P is equivalent to a radially distri- buted load of intensity (Ref. 3) Therefore, the total radial l oad The potential energy of this load with a deflection 1w1 is 0 But ?.Ti b½== f ; C~~ -. lA]") ~ -w- cJ e 0 - 9 - The kinetic energy of the ring is "?. Tl T = j { [(;,7t/· + (:;t f] Y'o de 0 (1) 0 For a particular mode 'c.,_" C" wn , Q h '- "I. __!] a Litt= <ot.,_ n A • "'\. t • --~ Cl n Cos nG .sin• \/\n 1L n co~ t"\B s1" "" n = - ~~ ~; cos ne sin"l.~" t. Sa., 1 Substituting in equation (7), we have 2. "If 'bT = - 1 J ( . "l.. cos.,.n(J) <' Je f'"to Qnl\n SITI /\n 1. su, n0 + ~ oa" ' \ -i. • -i.\ 0 and also for the n 1 th mode, equation (6) gives l.lJ 1) V = ~! an sin). 0 -l J [ n'sin "e -<z~ :;\n'sin nil 0 + ( I -+ p v; 1.) sin n s] b a" s in n B sin An -l de E.I (9) - 10 - By Hamilton's Principle, f\-sr- SVJol-L =O t, Therefore, substituting equations (8) and (9), we have l t . ' [-('lo>-:(,<,)+ ~J ~n•- ,J'- (n'-t) :r1 Qn s in"Anl San ,H = 0 l, b ¾ is arbitrary, therefore ~ =\: 0 , and Since f 't;.(1+...!..)\ o h... n "L= gY..3 [Cnl...1),..-(n'--1)P!!'"J EI ( o"L-1 ) EI .". P == But r r., ' \ n y--/ 1..+ I) \_ ~ n"'-(n"L-1) " is assumed to be constant, P (10) which gives the relation between 1 q 1 and \.n2 • When ").__n = 0 , the equation gives q - (n2 - l)EI r30 which isft:, he critical load of the ring when it will buckle with the l n 1th mode (Ref. 3). Also when q == 0 , which is he natural frequency of a ring in free vibration (Ref. 4). i Equation (10) also shows that load and the square of frequency have a linear relationship. - 11 - (3) Rectangular Frame. p p eb A I, I p I l J L 2L p Fig. 4 Fig. 3 When a rectangular frame of constant cross section is loaded as shown in fig. 3, the symmetrical mode of vibration may be replaced by its equivalent system CAB shown in fig. 4. Members AB and AC are now to be considered separately and the16onditions at the point A are to be matched. p ( i) Member AB . A1 I X B L l Fig. 5 The differential equation of motion is 'o4~ + P·,>1·3 -+ f <>2.J = o E I "<) x,-1 1>.X."'1>-l'- Let For a particular 'n', - 12 - YnIV -+ Q ~ y ,, - cJ <l y n = Q or where a 2 = P/ EI , and = f' A~/EI. d4 The solution to the equation is k = ./a"~ 4a~ + z where 1. al.. k'z = , ./a~-+ 4c(" - a1.. 2 The boundary conditions are: (a) at x = 0 , Yn = 0 ,· (b) at or C4 = x =1 ' C2 -+ C4 = O , and ~ C1 c.os(k.L) + k'C 3 Co$:h<k'l) = O , where 0•• - c2 ; _Therefore, or yn r - Ol == cos(lel) cosh. (k'{) The slope at the point A is Yn' I~=- = [c, ~ (cos.be. - o< C.o5/.. /e '-:x..)-+ C1. ( -k. .s,"n kx- I,. si11/iJ'x. )_}.x.::.o 1 0 - C,k(t-Ol) (11) and the moment at the point A is EI y~· Ix.=- O =EI [c, ~(-k sin k-x. - ()(. ksinh k. 'x.) -1- Cz(-J. ccs k:t-'':01A~~g :X=O - 13 - (ii) Member AC. Fig~ 6 The differential equation of motion is _.., l. E I ·~ -+ I J = o 0 'c) 'c>:c." l _ p \: :i '1)x.."\. Let For a particular 1n 1 , --:::z ~v z;;;,. Let EI Z::n =0 = f A~I ; then the solution of the equation is ~"' =- Crs sin1x.-+ c. c.osi-x. ~ c,coshj x. + c1 sinhix q4 The boundary conditions are (a) at Therefore, C7 Zri = 0 ; c6 + Cg = 0 , C5 and or x :: 0 , or (b) at Cg Vcos 1" -+ C7 i = - /J Cs , where - b , Zn' = JS = 7h = 0 co.s ?i, ilCOE.It'll, ~" =- c~ ( 5/fl 1r-; s,,,I, !x.) -f c6 ( cos r~ - cos/, The slope at the point A is 3 0. = - ?6 C.05h l x rn' / :x. =o = [CsjC cosj x-;, ccsl. x) - 0x) c, f (~inf X -1- sin Aj x.)JX=-D ll3) and the moment at the point A is EI, z~' lx~o = E1.[c37/"(-s,-nrx-plinhb-X.)-c,f(co~zxtcosAix)l(;;:;,O :::. -ZC, j,.EI, (14) - 14 - (iii) At the Point A. The conditions to be sat isfied at · the point A are that the slopes and moments are to be equal. Equat ing ~ equations (11) and (12) to equations (13) and (H.) respect ively, we have c. kc, - o1) - c~ i c, -; ) ==- o c~(P, 4 -t _k ~2 )- zc.6i2. i' = (15) 0 (16) Also we have the conditions that z n -x.=Zb == o y n /-X = Zl = 0 I which give c,(sin.ekl- i~ .sir,hzk-'l) + c (c.os.2kl-c.o.slt zk'L) =O u,J 2 and • From equations (15), (16), (17) and(18), for the constants to be consistent, t he following relation must be satisfied: 0 (sirt zkl-t~sinli:zk'l) (cos zkL -cosh zk'l) 0 o 0 0 0 0 ==- 0 0 - 15 - or - z i z. i' k (/- o<} (coszJt - COS~ -1- f(l;&Jf ,Z~ 'I) ( s/;,,,,z!..6-/ u~J.rrl,) i '-.J.J'7j(s1°q.zJt- :':' s/nA2J'{J(cos.z1/,-c"sAzf6) == 0 Substituting the values of k, ki", q, o<, and f , we will get a trans- cendental equation in An' whose roots give the frequency of vibration of the various symmetrical modes, with the load Pas a parameter. Hence it is also an equation relating P and An2 . It will be evident from the complex nature of the equation that the relation between P and An2 will not be linear, and hence a graph of P vs. An2 will be a curve rather than a straight line. Due to the complexity of this relationship, we will leave to experimental investigations the problem of determining how far this curvalinear relation deviates from a straight line, and whether the method of frequency measurement is applicable for the determination of a critical load in this case. III. EXPERIMENTS (1) Simple Column. (A) Description of the Model. The column is made of cold rolled dural (24-ST) bar, supported on both ends on knife edges. Detailed dimensions of the column and a section of the support are shown in fig. 7 and 8. Fig . 9 shows the experimental set un of the instruments. The ends of the column are reinforced with hardened steel anvils to serve as seats for the knife edges. The top knife edge is mounted on a differential screw and nut arrangement, so that load may be applied to the column by turning thep ut. Stress in the colwnn is measured by a pair of strain gauges at about a quarter distance from the support. The gauges form one leg of a wheatstone bridge. The unbalanced voltage of the bridge circuit is measured by a selfbalancing potentiometer. The strain is then deduced by using the gauge constant supplied by the manufacturer. The potentiometer is first calibrated for a known change of resistance in the strain gauges by shunting a known resistance across the gauges . A pair of du.mnw gauges mounted on a similar bar hanging freely beside the column are used as the other corresponding leg in the bridge circuit, as a means of compensating the temperature effect. The frequency of the column is of the order of 50 cycles per second. Hence it is easily measured by a stroboscope. - 16 - - 17 - (B) Calculation of the 1st Critical Load and Stress. Length of the column 25-1/16 11 Cross-section Young's Modulus (dural) 0. 750 11 X 0.40011 6 · 2 10.6 X 10 lbs./in. Specific weight 0.100 lb./in_3 'Jf 1. EI Per - 12 _ 7r1.. X 10.6 X 106 X 0. 75 X 0.l? 12 X 25.062 = 665 lbs. 665 0.750 X 0 . 400 - 2210 lbs ./in. 2 (C) Calibration of Potentiometer and Calculation of Stress. V The Calibration data are: V 6.22 volts. V0 0.427 mv. V1 1.970 mv. Rx 245,000 ohms. R 243 ohms. - 18 - -6 V -v- = A R R = 0. 001543 6.22 243 2L~5,000 _ 243 X 6.22 - 245,000 X 0.001543 Gauge Constant = 2.00 ± 1% Strain ( e) = 2 •00 x ARR = 2.00 X 3.96 X AVV = 7.92 X 6 V 6.22 C- = E.e = 10 • 6 x 106 x• 76.22 • 9 2 x ,,.. v ~ = 13 . 5 X lo3 X .6.V. where 4 V is measured in millivolts. - 19 - (D) Experimental Results. The measured results are tabulated below: 0- A cps. \2. 0 52.2 2,730 0.021 284 48.8 2,380 0.470 0 .043 580 ·44.7 2,000 4 0.490 0 . 063 850 41.0 1,680 5 0.506 0.079 1,065 37.2 1,380 6 0.525 0.098 1,321 33.8 1,140 7 0.540 0.113 1,522 28.9 838 No. V mv. L} V 1 O.l+27 0.000 2 0.448 3 #/in2 The graphical representation of the above result s is shown in fig. 10. It may be noted here that the extrapolated critical load is 2200 lbs./in. 2 as compared to the calculated value of 2210 lbs . /in. 2 - 20 - (2) Rectangular Frame~ (A) Description of the Model. The frame is made of cold rolled steel. Each member is an- nealed before being brazed together at the four corners. frame is again annealed after completion. The whole At the corners on the center I line of each member, a shallow notch is cut to r eceive the lmife edge for l ibading. frame. Fig . 11 shows the dime1tsions and corner details of the The lmife edges are¼ x rests on two small V-blocks. by a heavy I-beam. ¼tool bits. At the bottom side they On the top, the lmife edges are carried The load is applied through another lmife edge on the beam, mid-way between the other two knife edges, so that t he total load is equally divided between the two legs of the frame. The load is applied by a universal testing machine with a maxium capacity of 30,000 lbs. Fig . 12 is a close-up view of the model aligned in the testing machine. The frequency is measured by recording the output of gauges on an oscillograph. The frame is excited by hitting one o! the vertical members with a rubber mallet. A check on the final dimension of the frame showed that the corners deviated from a right angle by an amount of about one and half degrees. This error is taken care of in the test set-up, where the bottom lmife edge seats are adjusted so that the center line of the vertical member is perpendicular to the base of the testing machine. - 21 - (B) CalcuJ.ation of Critical Loads. p 6 p b ------ I, I T p p I L p p ( b) (a) Fig . 13 The condition required for buckl:ing in the mode sho1tm in fig . 13a is {i" tan kl 2 kl 2 We have I bI = -1I -1 - r. ' 1 = 14-5/gu ' and b - 10 11 b 1 kl 2 10 14.63 , = o.6g3 = 2.1g 4 X 2.182 X 30 X 106 X 0.0044 Per = · 14.63 2 = 11,700 lbs. ➔i- The derivation of the condition may be found in 11 Theory of Elastic St.ability11 by Timoshenko, page 90 and the following pages. - 22 - The condition for buckling in the mode shown in fig . 13b is tan kl 2 kl 2 For b IT - 0.22a , Per - = kl/2 = 2.62 . 11 700 x ( 2 • 62 ) 2 ' 2.18 17 ,00O lbs. - 23 - (C) Experimental Results. The measured results are tabulated below: p-'n~ A. \."l. p➔f- \. x lbs. cps. 104/sec. 2 lbs. cps. 104/sec. 2 0 204 4.15 4,515 179 3.21 55.0 205 4.20 5,010 175 3.09 1,030 204 4.16 5,455 172 2.95 1,515 202 4.08 6,025 168 2.82 2,005 199 3.95 6,515 165 2.74 2,460 196 3.85 7,490 161 2.60 3,020 192 3.69 8,005 157 2.49 3,505 189 3.56 8,420 155 2.36 4,000 183 3.35 A graphical representation of the above data is shown in fig. 14. It should be noted that: (1) there is a rise in frequency in the earlier part of the loading; and (2) the experimental points lies close to a straight line, and the extrapolated buckling load is 17,500 lbs. which is about 49% higher than the calculated lowest critical load of ll,700 lbs., but is about 3% higher than the lowest critical load of 17,000 lbs. for unsymmetrical buckling. * Correction for the weight of the loading beam has been added. rl. COMPARISON OF ANALYSIS AND EXPERIMENT. (1) Simple Column. The experimental points, being very close to a straight line, give a good check on the analysis of the simple column. The extra- polated critical stress of 2200 lbs./in. 2 also checks well with the calculated value of 2210 lbs./in. 2 The biggest error in the measurement is A V, which is the difference of two small values. of the errors is about ± 4%. An estimate of the maximum value Hence the extrapolated result of the critical load is well within the expected experirnental error. However, it must also be remembered that the calculated value is only theoretical, and it is subjected to errors in the choice of value E, and in the measured values of I and l. It is also interesting to check the slope of the curve, K, i.e., K = But the equation (5 ), for K = 2200 X 0.3 2730 = 0.241 lb.-sec. 2 n = l, gives 4f l2 4 X 0.1 X 0,4 X 0. 75 X 25.062 32.2 X 12 = 0.195 lb.-sec. 2 The experimental value is about 23% higher. For thepurpose of better bearing of the pivots, the ends of ! - 24 - - 25 - the column are reinforced with two pieces of hardened steel. Evidently these masses lowers the frequencies of the column, for the theoretical natural frequency of the column is A= 2 I - X 25.062 10.6 X 106 X 0.43 X 0.75 X 32.2 X 12 12 X 0.1 X 0.4 X 0 .75 58.5 cps. while the measured frequency is 52.2 cps. The correction factor for the critical load due to these reinforcements (Ref. 5) is reinforcements. [1 T :~(zla)3], where a is the l ength of 1 1 For t his case the factor does not have any signific- ance. The important r esult is that the linear relations hip and t he critical load check very well with the theory. - 26 - (2) Rectangular Frame. The interesting result brought out by the experiment is that the curve of A.n2 vs. P comes out to be rather close to a straight line as far as the data go, although the theory indicates that the relation is non-linear. The fact that the extrapolated critical load of 17,500 lbs., as compared to the computed value of 17,000 lbs., for unsymmetrical buckling, indicates that the relation can be approximated with reasonable accuracy by a linear one, even though the data only extend up to about 50% of the buckling load. An increase in accuracy would be expected if the loads are carried closer to the buckling point. The data obtained might be somewhat misleading, because the critical load extrapolated is not the lowest one, 11,700 lbs., which one would naturally expect to obtain. However, the rise in frequency at the early part of the loading indicates that the frame changes its mode of vibration when load is applied. Therefore, the critical load extrapolated will correspond to the mode in which the frequencies are measured. The records show that there is a beat frequency of the order of 10 cycles per second at the beginning of loading . Hence there are two dominant frequencies present, and they are very close to each other. It will be shown later that the supports will absorb less energy for unsymmetrical modes than for symmetrical ones. As the frequencies are so close together, it may happen that the total energy required for exciting the slightly higher mode is actually lower. The unsym- Q. . rnt,rical mode is the next higher one, and its frequency appears very close to that of the lowest symmetrical mode. Hence it is this mode - 27 - in which the frequencies are mea sured. However, no analysis of the natural frequency of the system is made. It is interesting to see, however, from the experimental curve, that when the straight portion of the curve is extrapolated to zero load, a frequency of 211 cps. is found. This is probably the natural frequency of the unsymmetrical mode, and is about 7 cps. higher than the frequency measured at zero load, which probably is the frequency for the symmetrical mode, the difference being of the order of the beat fr equency. The method of excitation by hitting one of the vertical members is certainly capable of introducing the two basic modes simultaneously. With no device to indicate the mode, it is difficult to tell whether one mode is completely absent or not. may give some kind of indication. However, the beat frequency The beat frequency increases with the load, as is expected, while the amplitude of the beat decreases with the load. The presence of beats is not marked when load reaches 4,000 lbs., and the beats disappear completely from the records for loads above 6,000 lbs. This indicates that one of the modes is more stable than the other. This may be clarified by considering the con- ditions at the supports. It was noticed during loading, that local yielding at the knife edges took place at loads above 4,000 lbs. and the frame had to be excited several times before the load became steady. Inspection made after the test showed that the knife edges made indentations as deep as 1/32 11 in the notches in the frame. This indentation of the knife edge introdues friction and absorbs energy of vibration. The energy lost at the supports will be proportional to the amount of - 28 - rotation of the corners. It has been shown that, for the same corner moment, the corners rotate three times as much for the synnnetrical mode as for the unsymmetrical mode (the rotation being 2EIM and 6EIM b b respectively). So, as the lmife edges dig deeper into the notches, it becomes more difficult to excite the lowest symmetrical mode, and the unsymmetrical mode becomes the only dominant one. measured frequency is in this mode. Therefore, the The check on the critical load for this~ode also indicates that the frequency is in the same mode. When two frequencies are very close together :· as in this case, even though the initial motion may be in the symmetrical mode , the dynamic coupling between the modes ca.used by non-symmetry either in dimensions, in l oad, in density of material, or in friction at the supports, etc., may be enough to start the other mode, initiated by a small phase difference on the two sides of the frame. Evidently, what happened in this case, is that the coupling between the modes has stabilized the unsymmetrical mode. In a full scale structure, where non-symmetry in construction and loading etc. is more likely to occur than in a model, it is quite possible to have enough such dynamic coupling to throw off the fre quency from the lowest mode. Therefore, one should look out for the possibility of such misleading results. Though the measurement indicates the structure is s t able in the unsymmetrical mode of vibration, one can not, however, conclude that the structure will buckle in that mode first. Its dominance may be purely because of the method of excitation and the support conditions. To check this a further experiment will be required, where some refinement in technique may be followed. - 29 - First, a definite method of exciting the frame should be devised, such that the simplest mode will be the only dominant one. A possible method consists of stretching a wire across the middle of the vertical members to draw them toward each other, and then cutting it to start the motion. Second, the support notches should be hardened or hardened inserts should be used, in order that they may not damp out the simplest mode, and that they will not introduce too much damping force at heavy loads. . I It is difficult to say thatiwith these r efinements we shall be able to record the simplest mode with the absence of the next higher mode, because they are so close together, and one might be excited by the other though possible dynamic coupling between them. If this seems likely, the mode of vibration should be measured as well as the frequency. One method of deterrnining the modes could be to r ecord the traces from corresponding pairs of strain gauges on opposite members of the frame simultaneously, so that the phase differences could be compared. Another method could be to connect the gauges in the same bridge circuit in such a way that they would give double output for a symmetrical mode, but would nullify each ot her for an unsymmetrical mode. - 30 - V. CONCLUSIONS & SUGGESTIONS The interesting points which represent the worthwhile results of this work are: (1) For a s:im.ple pin-ended beam, the theory and experiment check very well. (2) The analytic solution for a circular ring shows that there is a linear relation between the uniform radial load and the square of frequency. (3) The analytic solution shows that for a r e ctangular frame, the re- lation of P and An2 is not linear, but the experimental result indicates that the relation is very clmse to a linear one. (4) The experimental results show that if there are modes with f requencies very close to each other, the stable one may not be the lowest mode, and thus the results may be misleading, unless the modes of vibration are measured as well as the fre quencies. The structures covered in this work are of a very simple type, and therefore no generalized conclusion can be drawn. Thus far, it may be said that the r elation of P and An2 for prismatic beams is very close to a linear one for all end conditions except the case with fixed ends, which may be th e next step to be investigated. Other experiments which might profitably be conducted involving structures of the following types: (1) Compression members in a truss. - 31 - (2) Different types of elastically unstable frameworks. (3) Columns built-up with stiffening lattice work. (4) Curved beams and arches, (5) Plates and shells. (6) Webs of plate girders in bending. When the structures become sufficiently complicated s o that many modes of vibration are possible, some definite controllable means of exciting the structure and devices of detecting the modes must be provided, or the results may become meaningless. Such tests should be checked for some complicated structures by actually carrying the structures to failure. - 32 - REFERENCES (1) C. Massonnet: 11 Les Relations entre les Modes Normaux de Vibration et la Stabilite des Systemes Elastiques. 11 - Bulletin des Cours et ¢.es Laboratoires d 1Essais des Constructions du Genie Civil et d 1 Hydraulique Fluviale, Vol. 1, Nos. 1 and 2, 1940. (2) Lord Rayleigh: Theory of Sound. (3) Timoshenko: Elastic Stability. ( ~-) Timoshenko: Vibration Problems in Engineering . (5) Karman & Biot: Mathematical Methods in Engineering . - 33 - FIGURES Title Number 7 Dimensions of the test model of a simple column. 8 A section through the top support of the simple column. 9 Photograph of the experimental set-up of the simple column. 10 Stress vs. (frequency) 2 relation of the simple column. 11 Dimensions of the test model of a ractangular frame. 12 A close-up view of the rectangular frame in the testing machine. 14 Frequency and load relation of the rectangular frame the longer members being loaded. - 34 - 1i_ --r A SECTION A-A rtG. 7 - DIIIEN8ION8 OF' THE . TEST MODEL OF' A SIMP\.E COLUMN - 35 - t J FIG. 8 .A SECTION THROUGH THE TOP 8UPP,.ORT OF' THC SIMPLE COLUMN. i I - 36 - Fig. 9 Photograph of the experimental set-up of 1 the simple column. - 38 - _________ ·,o ,. ________. .,. ,ol() - .... • I -r • - t- .•"-i FIG. 11 0tMEN6fON6 OF THE TEST · uooEL OF A RECTANGULAR FRAME. - 39 - Fig. 12 - A close-up view of the rect~ngular frame in the testing machine. Tt .. 1•0'-. ..r_t· ,.j .i . . .1::_ , .. -. ~L. -. ·F·· ,- ·I •1 I •• I • ·t l . I~·-. .. • ___ , __ - • ____ J_ a, !: . .I t .1._ ~-:~.f ~·r' ~~;-_ ~;~: •• • ' ,, • ~-~r~ - , .... . j ..•. :., - i::