ENGINEERING RESEARCH INSTITUTE DEPARTIMET OF AERONAUTICAL ENGINEERING UNIVERSITY OF MICHIGAN ANN ARBOR ELECTRONIC DIFFERENTIAL ANALYZER SOLUTION OF BEAMS tWITH NONLINEAR DAMPING py. R M. Howe Assistant Professor of Aeronautical Engineering Project 2115 Office of Ordnance Research, U. S. Army Contract No. DA-20-018-ORD-21811 ~~~~~~~AIRM~~-8 ~April, 1954

PREFACE In a previous report, the solution of linear beam-vibration problems by difference techniques using the electronic differential analyzer was described.1 This report extends the application of the same techniques to lateral-beam vibrations where nonlinear damping terms are present. Examples considered include cantilever beams with velocity-squared damping and Coulomb damping. Analyzer solutions give recorded output voltages representing lateral displacement, velocity, and bending moment as a function of time and at various stations along the beam. Approximate theoretical checks of the computer accuracy are given in several cases. The computer solutions were obtained with the electronic differential analyzer facility of the Department of Aeronautical Engineering, University of Michigan. ii

TA~BLE OF COTiLENTS Chapter Page 1 PPREACE ii LIST OF FIGURES iv INTRODUCTION 1 1.1 Equation for Lateral Vibration of Beams 1 1.2 Finite Difference Method for Approximating Derivatives 5 1.3 Principles of Operation of the Electronic Differential Analyzer 5 2 CANTILEVER BEAM WITH VELOCITY-SQUARED DAMPING 10 2.1 Beam Equation Including Velocity-Squared Damping 10 2.2 Equivalence of Damping-Coefficient Size and Amplitude of Vibration 10 2.3 Difference Equations for the Cantilever Beam with Velocity-Squared Damping 11 2.4 Analyzer Circuit for the Cantilever Beam with Velocity-Squared Damping 13 2.5 Damped First-Mode Oscillation 14 2.6 Approximate Theoretical Solution 14 2.7 Impulse Response of the Cantilever Beam with VelocitySquared Damping 17 3 CANTILEVER BEAM WITH COULOMB DAMPING 3.1 Beam Equation Including Coulomb Damping 22 3.2 Difference Equations for the Cantilever Beam with Coulomb Damping 23 3.3 Analyzer Circuit for the Cantilever Beam with Coulomb Damping 23 3.4 Impulse Response of the Cantilever Beam with Coulomb Damping 24 BIBLIOGRAPHY 26 iii

LIST OF FIGUIES Fi tres Page 1-1 Cantilever Beam 2 1-2 Cantilever Beam. Divided into Statiolns 1-3 Operational _M:,plifier 7 1-4 Servo I.ltiplier 2-1 Analyzer Circuit at the nth Station for the Cantilever Beam.ith Velocity-Scquared Dapizing 13 2-2 Dasilpecd First-Mode Oscillations of Uniformi, Cantilever Beam with Velocity-Scjuared Damp;ing 15 2-3 Variation of Logaritihmic Decremlent 6;rith filplitude of Oscillation 18 2-4 Unit Impulse Response of 5-Cell Unifori. Cantilever cBeaml vith Velocity-Squared Damping; Displacements at Stations 2, 3, 4, and 5 20 2-5 Unit Impulse Response of 5-Cell Uniform Cantilever Beam vith Velocity-Squared D.lamping; BendinCg.-Iomen.t at Stations 1, 2,, and 4 21 3-1 Analyzer Circuit at the nth Station for Cantilever Becml with Coulomb Damping 24 3-2 Impulse Response of a 5-Cell Uniform Cantilever Beam with Coulomb Damnping 25 iv

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER I INTRODUCTION Nonlinear partial differential equations are almost impossible to solve exactly in all but a few special cases. Unlike linear partial differential equations they cannot be treated by separation of variables, since for the nonlinear equations superposition of normal mode solutions does not result in another solution. Hence at the onset we are led to computing techniques in order to solve nonlinear partial differential equations. Solutions can be obtained by replacing all derivitives with finite differences and by using digital machines, or they can be accomplished by replacing derivitives with respect to all variables but one by finite differences and by using electronic differential analyzers. In this latter method the original nonlinear partial differential equation is converted to a system of simultaneous ordinary nonlinear differential equations. 1.1 Equation for Lateral Vibration of Beams In this report the nonlinear partial differential equation which we shall consider is the description of lateral (transverse) vibration of beams having nonlinear damping terms. Consider the cantilever beam shown in Figure 1-1. Let y denote the transverse motion of the neutral axis of the beam, x equal distance along the beam, and t be the time variable. The equation describing the dynamic behavior of the beam is EI(x ) 2C (-) + (x) fc ( + p(x) y f( ) (1-1) x xa2 6t 2t Here EI(x) is the flexural rigidity, p(x) is the mass per unit length, and f(x,t) is the external force per unit length. The damping force per unit length isSfc (6y//t), where fd is a nonlinear function of the transverse velocity (~y/at). We recall that the bending moment M is given by M(x,t) = EI(x) 2 (1-2) c^;2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN L e /. x_-x~~~~~~~~~~~~ I Figure 1-1. Cantilever Beam. while the shear force V is V(,t) = aM(xt) (1-5) ax For the cantilever beam of length L shown in Figure 1-1 the boundary conditions are y(o,t) = ay(ot) 0 (clamped end) (1-4) ax and M(L,t) = V(L,t) = 0 (free end) (1-5) In writing Equation (1-1) we have neglected deflection due to transverse shear or rotary inertia, which means that the transverse dimensions of the beam must be small compared with the beam length L. The effect of transverse shear can be included if necessaryl"3. Let us lump the variable characteristics of the flexural rigidity EI(x) into a dimensionless variable 0f(x), so that -------— 2.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN EI(x) = EIo 0f(x) (1-6) Here EIo could represent the maximum value of EI(x). In the same way we let p(x) = Po 0d(x) (1-7) and c(x) = co 0(x) (1-8) It is also convenient to define a dimensionless distance variable x such that the beam length in x is unity. Thus X x x = x (1-9) L from which a dx - 1;62 61; etc. (1-10) ax 3x dx L 6x' a L2 ax2 From Equations (1-6), (1-7), (1-8), and (1-10) the beam Equation (1-1) becomes f2 +2y L4o pL4 2y L4 - O ~+f0(x) + (x) fc( — + d(X) f(x,t) (1-11) ax2 6x2 EIo at EIo t2 EIo Next we introduce a dimensionless time variable t given by 1 m t L2 Epl (1-12) Since Equation (1-11) is nonlinear, the behavior of the solution will in general depend on the magnitude of the displacement y. For this reason we consider a dimensionless displacement y defined as y(x,t) = y(x,t) (1-13) Yo where y0 is a reference value of y; yo might be defined as the maximum expected value of y. In terms of Equations (1-12) and (1-13) Equation (1-11) becomes 1~~ ~~~~~~~~~~~~~ 3,,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN _ _ _ _ _ _6 ~2 y a 0f() a + Lc 0c(x) fc (L) + 0d(x) = f(x,t) (1-14) ax2 3x2 EIo yo at at2 EI0 Yo where f(x,t) = (x,t) (1-15) EIo Yo Equation (1-14) is the equation describing beam vibration with nonlinear damping which we will solve in this report. For a cantilever beam it is subject to the boundary conditions y(o,t) ay(ot) (1-16) ax and 0f(l) 2y(1t) = a f(1) a2y(lt) = 0 (1-17) ax2 ax ax2 We will denote the initial conditions as y(x,o) = Y(x) (1-18) and y(x,o) = Y(x) (1-19) Two examples will be considered. For the first example the damping force is proportional to the square of the transverse velocity. Thus f () = lay (example 1) (1-20) 8t ct at Here the sign of the damping term is always the same as that of the transverse velocity, and as a result energy is always extracted from the system. For the second example the damping force is of the Coulomb type. Thus fc() = 1, > at at -1, 0 < o at ------------ 4 ____

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Here the damping force is always constant but depends in sign on the sign of the transverse velocity. 1.2 Finite Difference Method for Approximating Derivatives Instead of considering the transverse displacement y of our beam at all points along the beam, let us consider y only at certain stations along x, as shown in Figure 1-2. Further, let the distance between x stations be Ax. Thus we define y1 as the transverse displacement at x = Ax, y2 as the displacement at x = 2Ax, yn as the displacement at x = nAx. Clearly a good approximation to ay/x In+l/2 (i.e., the partial derivitive of y with respect to x evaluated at the n+l/2 station) is simply y _ Yn+l Yn (1-22) (1-22) 6Xn 2 Ax n+l/2 Indeed, in the limit as Ax + 0 Equation (1-22) defines iy/6x at x = (n+l/2)Ax. In the same way 1I 6 - 6 -J(1-25) ax2 Ax x n+l/2 n-1/2 or 6ay! Yn+l - aYn + Yn- 1 (1-24) 6x2 (Ax)2 n Higher order derivitives are computed in the same manner. The displacement Yn at the nth station is a function only of time. Hence if we replace x derivitives in Equation (1-14) with finite differences, a system of ordinary nonlinear differential equations will result, equations which can be solved directly by the electronic differential analyzer. 1.3 Principles of Operation of the Electronic Differential Analyzer The reader unfamiliar with the theory of operation of the electronic differential analyzer is directed to other references4'5. To review briefly, we recall that the basic unit of this type of computer is the operational amplifier, which consists of a high-gain dc amplifier along with feedback impedance Zf and one or more input impedances, as shown in Figure 1-3. ------------ 5 --— 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 1 2 3 4 5 6 7 8 Figure 1-2. Cantilever Beam Divided into Stations. To a high degree of approximation the output voltage eo of an operational amplifier is equal to the input voltage divided by the ratio of feedback to input impedance, with a reversal of sign (Figure 1-3a). If several input resistors are used, the output voltage is proportional to the sum of the input voltages (Figure 1-3b). If an input resistor and feedback capacitor are used, the output voltage is proportional to the time integral of the input voltage (Figure 1-3c). The operational amplifiers shown in Figure 1-3 can be used to multiply a voltage by a constant factor, invert signs, sum voltages, and integrate a voltage with respect to time. To multiply several voltages a servomechanism which drives potentiometers is the most commonly used device. In Figure 1-4 the block diagram of a servo multiplier is shown. It consists of a number of linear potentiometers ganged together and driven by a servo motor. The reference voltage ~ VR is connected across one of the pots, and the variable tap voltage OaVR is subtracted from the voltage Z. The resulting error signal e = Z - dVR is sent through a high-gain servo amplifier and applied to the servo motor. The motor drives the variable tap in the proper direction to reduce the error to zero, i.e., to make cVR = Z. In this way the tap position on all of the ganged pots is proportional to the voltage Z. If ~ X and ~ Y are applied across each of the remaining two pots shown in Figure 1-4, it is apparent that the variable tap voltages will be XZ/VR and YZ/VR respectively. Thus the servo multiplier can generate output voltages proportional to the product of input voltages. 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN INPUT | iI DCi OUTPUT - i AMPLIFIERe2=- Zf e Zi a.) OPERATIONAL AMPLIFIER Ra a. Rf if ea Rb ib_,, OUTPUT eb ^ e2 Rc ic ec e2 =. eR —c) i2 C l ~ ~- II ----- j INPUT R il -_ R OUTPUT e, e ec Figure 1-5. Operational Amplifier. --------— 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN x o - -X Yo VR I -x -Yo SERVO ^VM HIGH GAIN Z_~xVR + MOTOR -SERVO Z AMPLIFIER VR O ------ o R REFERENCE POT -VR RF Figure 1-4. Servo Multiplier. ---------— 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For the electronic differential analyzer solutions obtained in this report REAC* Servo Unit S-101 Mod 4 servos were used. Accuracy of multiplication is about 0.1% of full scale (~ 100 volts). Drift-stabilized dc amplifiers of our own design6 were used along with computing resistors calibrated to 0.02%. Amplifier gain is about 12 x 106 and average offset referred to input is approximately 10-4 volts. By employing operational amplifiers for summation and integration, and servos for multiplication, we are able to solve the cantilever beam with velocitysquared damping. * Reeves Electronic Analog Conmputer, Reeves Instrument Corp., New York 28, New York ------------- 9 -, —--------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 2 CANTILEVER BEAM WITH VELOCITY-SQUARED DAMPING 2.1 Beam Equation Including Velocity-Squared Damping For flexural vibration of a beam with damping proportional to the square of the transverse velocity, we have the following equation from Equations (1-14) and (1-20): 62 62y L4 c y 2y X f(x) yc(x) + d(x = f(x, t) (2-1) ax2 ax2 EIo o t t at2 Since y = yo y and t = L2 Npo t, Equation (2-1) becomes O f(x) 2 + c0c(X) y + d() = f(x, t) (2-2) ax2 x2 1at at at2 where Co Yo (2-3) Po For a cantilever beam the boundary.conditions are given in Equation (1-16) and (1-17). We recall that the transverse displacement y, distance along the beam x, and time t in Equation (2-2) are all dimensionless. 2.2 Equivalence of Damping-Coefficient Size and Amplitude of Vibration In any nonlinear equation the behavior of the solution is not independent of the magnitude of the dependent variables, as with a linear system. At first thought one might therefore assume that our nonlinear beam Equation (2-2) must be solved not only for different damping constants c but also for different amplitudes of transverse vibrations. Actually, this is not true. Consider first the case where f(x, t) = 0 and where we know the solution y(x, t) for a given damping 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN constant c and for given initial conditions. If we now double the size of the initial conditions, the solution will be simply 2y(x, t) providing the damping constant is c/2. This is evident from the damping term in Equation (2-2). In the same way if we know the solution for a given force f(x, t) and damping-constant c, the solution for a force af(x, t) will be a times as big as the previous solution providing the damping constant is c/a. 2.3 Difference Equations for the Cantilever Beam with Velocity-Squared Damping As explained in Section 1.2 we will consider the transverse displacement y only at stations along the beam. In this way derivitives with respect to x can be replaced by finite differences. Following the procedure of our previous report' we introduce a new distance variable X such that the distance AX between stations is unity. If the beam is divided into N stations or cells, then X Nx and - = N. (2-4) ax aX It is also convenient to introduce a new time variable T given by T = N2I and a = N2 (2-5) at aT Equation (2-2) then becomes - 0f(X) - + Co0c(X) - + 0d(X) Y = (X )- ) - k (, T) (2-6) aX2 aX2 aT aT aT2 N4 The difference equation at the nth cell is from Equation (1-24) 2Y dy dyn dn + c n n = - mn+l + 2mn an-l + n(T) (2-7) dn dT2 dT where mn is proportional to the bending moment and is given by mn = fn (mn+l - 2mn + mn-1) (2-8) Boundary conditions for the difference technique are discussed more completely in a previous report'. For the clamped end the condition of zero displacement and slope is approximated by letting Yo = yl = 0. (clamped end at X = 1/2) (2-9) 11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For the free end the condition of zero moment and shear is approximated by letting mN = N+1 = 0 (free end at X = N + 1/2) (2-10) Note that the N-cell cantilever beam has its built-in (clamped) end at X = 1/2 and its free end at X = N + 1/2. From Equations (2-7), (2-8), (2-9), and (2-10) the complete set of difference equations for the cantilever beam with velocitysquared damping becomes 2 dy2 dy2 i2 2 2 cu dT = - ms + 2m2 - ml + O1 (T) d dT2 dT dT d2y dys dy3 d 2 +Cc3 -- - = -m4 + 2m3 - m2 + 2 (T) d3 dT2' dT dT (2-11) d2YN-2 dYN-2 dYN 2 dN-2. 2 + N-c2 =-mN1 + 2mN-2 - nN-3 + NN-2 (T) dT dT dT d2y~lz dyNl: dYNl _ OdN_1 +ccN- dT dT N-1 - N a 2N- 2 + ON-l (T.) dYN dyN dyN OdN 2 + C -- - mN-1 + ON (T) dT T dT dT where mi = fl1 Y2 m2 = Of2 (Y3 - 2y2) m3 = Of3 (y4 - 2y3 + Y2) (2-12) mN-2 = fN-2 (YN-1 - 2YN-2 + YN-3) mN-1 =:fN-l (YN - 2YN-l + YN-2)....___________________________________ 12

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Initial conditions Yn(O) and dy(0)/dT must of course be specified to define the complete problem. 2.4 Analyzer Circuit for the Cantilever Beam with Velocity-Squared Damping The electronic differential analyzer circuit for solving the equation at the nth cell is shown in Figure 1-5. The velocity dyn/dT times its absolute R <n(T) A O dnkR2 100 C(cn D R I -mnI - Yn- ^AAA 1, C I, C Yn I _ydnd___ RVO dT r R -rnn+I o-, Yn+l1 * —NV — INITIAL CONDITION CIRCUITS OMITTED FOR CLARITY ALL RESISTOR VALUES ARE MEGOHMS ALL CAPACITOR VALUES ARE MICROFARADS Figure 2-1.. Analyzer Circuit at the nth Station for the Cantilever Beam with Velocity Squared Damping. value is obtained by grounding the center tap of a servo-multiplier potentiometer and connecting dyn/dT to both ends of the pot. In the figure it is assumed that the servo reference voltage is 100 volts and that k volts on the computer equals unit y. The time scale of integration is RC seconds; thus one unit of T equals RC seconds of actual time in the computer solution. The circuit of Figure 2-1 is iterated N-l times to solve Equations (2-11) and (2-12). 3(N-1) operational amplifiers and N-l servo multipliers are required to solve the N-l simultaneous nonlinear ordinary differential equations. For a complete circuit diagram including the connections at the built-in and free ends the reader is referred to the previous report1. For the uniform cantilever beam which we will consider from now on 0cn = 0dn = 0fn = 1. An integrator time scale of 0.2 seconds (R = 0.2 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN megohms, C = 1 microfarad) was used in all cases. We let 50 volts equal a unit transverse displacement y, so that k = 50 in Figure 2-5. 2.5 Damped First-Mode Oscillation An 8-cell uniform cantilever beam with velocity-squared damping was set up on the electronic differential analyzer. The correctness of the circuit is easily established by measuring the frequency of first-mode oscillation when no damping is present (RD = 0). This should agree closely with the theoretical value for an 8-cell beam, which is 0.70 higher than the frequency for a continuous beam. First-mode oscillations were excited either by driving the beam circui with a sinusoidal voltage (8 (T) having the first-mode frequency or by applying initial conditions representing the shape of the first-mode frequency1. For the latter case the displacement ys at the free end of the beam is recorded as a function of time in Figure 2-2 for several values of the damping-constant c. Note that the damping effect is large for big amplitudes of oscillation and decreases as the amplitude falls off. This is due to the velocity-squared damping. 2.6 Approximate Theoretical Solution Let us consider the uniform cantilever beam with velocity-squared damping when the external force f(x, t) = O. In this case a4y ay ay __y 4y 0+ c y y = O. (2-13) 6x 6t 6t at We have seen in Section 2.2 that increasing the damping constant c by a factor a is equivalent to keeping the same damping constant but increasing the initial amplitude y(x, 0) by the same factor a. The resulting solution is just a times the first solution. Thus if we solve Equation (2-13) for a given damping (say c = 1) but for a number of amplitudes of oscillation, we have also covered the solutions for different damping constants c. Let us assume that the beam is vibrating periodically with frequency C and is only lightly damped. A fairly accurate approximate solution to Equation (2-13) can be written by considering the energy Ed absorbed per cycle. This will be 1 to+ 2X/ 6 ^ Ed = l + 2 c - dt 2-14) where X is the frequency of oscillation and to is the time at which the particular cycle in which we are interested starts. Let us assume next that the beam is 14

Z chP4 0 Cc+ n d1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-4I Od U - — I — e' H'I.f::r,:j!_ + ~,,+f, I~-~ >!'!'iiii:iiiilli i'~iii!!!lilli!!!!:'1!.u i'! i ~;1 ii!':'~ ~ i`.~ -.~;ii~:::~.;.-~.~ i~ii~ ~ij~!!i~ ~iiiiiiiiiii!!ii~!iii~i~ ~zi

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN oscillating at one of its normal-mode frequencies, so that the motion is approximately sinusoidal. Also we will choose one time scale such that to = 0. Let Ya(x) equal the amplitude at t = 0 and yb(x) equal the amplitude one cycle later. We will define an approximation Yab(x) cos at to y(x, t) over the entire cycle as yab(x) cos t = a(x) + b(x) cos t (2-15) Then Eka, the kinetic energy at t = 0 is Eka = a2 J1 y (x) dx (2-16) 2 0 and Ekb, the kinetic energy at t = 2At/o is _ 1 2 2O10 Ekb = 2 W2 JY2 (x) dx (2-17) We can calculate approximately the energy absorbed over one cycle from Equation (2-14), since we assumed y(x, t) = Yab(x) cos Cot for this period. Thus Ed = c3 j' Yb (x) dx [ / cos2 cot Icos cDt Idt] or Ed 2 8__ c (2-180 Ed c - Jf Yab (x) dx Let ym(x) equal the dimensionless mode shape having unit amplitude at the free end. Then f yi (x) dx = y (1) j y2 (x) dx (2-19) 0 0 Ya Ya () ym(x)dx (x) dx (1 y m2 (x) dx (2-20) and p j yab ( (x) dx = Y (1 ) d (2-21) 0 0 Clearly the difference between kinetic energies before and after the cycle of oscillation is the energy absorbed over the cycle. Thus 16

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Eka - Ekb = Ed (2-22) or from Equations (2-16) through (2-21) )1 22 ( 1 8c Y2[a(l)+Yb(l)] J y0 m (x) dx - Y (1) - L-2y^ (1) = -w y[__ )__ (__ 0 (2-25) 2 a 2 15 2 - i 2 ( 2x) d Ym (x) dx j0 For the first mode of a uniform cantilever beam fym (x) dx l - = 0.736 (2-24) J ym (x) dx 0 Thus for the first mode 2 2 Ya(l) + Yb(l) 13 Yb (1) y (1) - 3.92 c [() (2-25) 2 Equation (2-25) can be used to solve for the amplitude Yb (1) at the free end following one cycle of oscillation starting with amplitude Ya (1). If the damping is very slight, [ya (1) + yb (1)]/2 - Yb (1) and the logarithmic decrement 6 is given by Ya (1) = In [ ] - 1.96 cya (1), 5 < < 1 (2-26) Yb (1. Equation (2-26) predicts that when the damping is slight, 6 is directly proportiona to the amplitude of oscillation. In Figure 2-3 6 is plotted as a function of amplitude of oscillation from Equations (2-25) and (2-26) and compared with computer results. Evidently Equation (2-25) is quite accurate and Equation (2-26) is accurate for small damping. 2.7 Impulse Response of the Cantilever Beam with Velocity Squared Damping A number of solutions were recorded following unit impulses of one-fifth second duration applied simultaneously at each station along the beam. A five-cell uniforn cantilever beam was used for these solutions. Shown in Figure 2-4 is the 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN I I.. /.8- AERODYNAMICALLY DAMPED BEAM /.6 a4 -a-^- +4:o //.6 - Y a y a y dx4 cdt at | = / /a.4 FIRST MODE, y a() = INITIAL AMPLITUDE AT FREE END.2 Yb(I)= AMPLITUDE OF NEXT CYCLE 2 tY I )392C Ya b _) O. I L ~- FOR << I.08 Z JI.E0 ^ Il(y./yb) 1. 96 Cy0() EQ. 2.06 w ( W LL.04-Q " 0 _.02 _ CURVE FOR EQ. I, C = -------- CURVE FOR EQ. 2, C = I 0.01.008 / 0 COMPUTER SOLUTION, C0= 0.5 (0.5ya IS PLOTTED).006- / *,,,0, C=O. (0.ly~ " " ).004 I/ I "*,C =0.02 (0.02y ).002 0.001 0.01 0.1 1.0 ya(I) INITIAL AMPLITUDE OF OSCILLATION AT FREE END OF BEAM Figure 2-3 Variation of Logarithmic Decrement 6 with Amplitude of Oscillation. ------------------------- ^~1 8 —-----------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN displacement at each of the stations following the unit impulse. In Figure 2-5 recordings of the bending morments are shorm. Four different damTping cases are s ho-n. The response of nonuniformi cantilever beams or beams with other end fastenings could have been obtained with equal ease. Any arbitrary forces along the beam can be considered, as well as timle dependent boundary conditions and transverse-shear effeccts. For a complete discussion of these and other cases for lateral vibration of linear beamns the reader is directed to the previous report'. 19

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 0.08 || N 0.06 | 0.04 0.02 O.00 --- -0!~~~~~~~~-0 -0.02 -0.06 -O.08 00.4 O.02 0.0 -0.02 -0.04 | ||1 1!^::^; | H 0.02 0.0 -0.0M I —----------------- --------- ^0 ~~~~~~-4+H++f —-------------—.-.....-~ Figure 2-4. Unit Impulse Response of.-Cell Uniform Cantilever Beam with Velocity-Squared Damping Displacements at Stations 2~ 3, 4, and 5. 20

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN CHAPTER 3 CANTILEVER BEAM WITH COULOMB DAMPING 3.1 Beam Equation Including Coulomb Damping.From Equations (1-19) and (1-21) the equation for lateral vibrations of a beam with coulomb damping is given by ax2 f(x) x2 + Cd Xc(x) fc() + d(x) at2 = f(xt) (3-1) where fc(t) = 1, at (3-2) = -1, at< 0 and where cd =,Io (3-3) For a cantilever beam the boundary conditions are given in Equations (1-16) and (1-17). The lateral displacement y, distance along the beam x, and time t are all d imensionless. If we know the solution y(x,t) to Equation (3-1) for f(x,t) - O given initial conditions, and a given damping constant cd, the solution for initial conditions a times as big will be simply ay(x,t) providing the damping constant is acd. Similarly, if we know the solution y(x,t) for zero initial conditions, a given damping constant Cd, and a given external force f(x,t), then the solution for a force af(x,t) is simply ay(x,t), providing again that the damping constant is acd. Thus if we can find the beam response for a given f(x,t) and all cd values, we also know the solutions for any force af(x,t), where a is a constant factor. 22

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 3.2 Difference Equations for the Cantilever Beam With Coulomb Damping Following the procedure outlined in Section 2.3, we can rewrite Equation (3-1) as a set of simultaneous ordinary nonlinear differential equations by considering the lateral displacement only at discrete points along the beam. Thus at the nth station d n d dYn n t2 + 4 cn fc( -) = -mn+l + 2mn - mn-l + On(T) (34) n dT N4n dT where mn is proportional to the bending moment and is given by mn= fn (Yn+l - 2Yn + Yn-) (2-8) In Equation (3-4) we recall that N represents the number of cells into which the beam is divided. A new distance variable X = Nx makes AX, the distance between stations, equal to unity. The time variable T in Equation (3-4) is equal to NZt and Dn(T) = fn() (3-5) The built-in boundary condition at Xt = 1/2 implies that yo = y = O. The free condition at X = N + 1/2 implies that mn = mn+l = O. A set of N-l equations similar to (2-11) and (2-12) is obtained for the complete cantilever beam with coulomb damping. 3.3 Analyzer Circuit for the Cantilever Beam with Coulomb Damping The electronic differential analyzer circuit for solving the equation at the nth station is shown in Figure 3-1. The fc(dy/dT) function is represented by amplifier A5 in the figure. This amplifier has no feedback and is loaded with RL, a resistor selected so that the amplifier output saturates at the same voltage kv for either positive or negative outputs. A very small positive or negative input voltage (less than 4 millivolts) will produce a full output voltage kv of negative or positive sign respectively. The result is an accurate simulation of the coulomb damping force represented by fc(dy/dT), which is summed into amplifier Al in Figure 3-1. The circuit is iterated N-l times to solve the complete N-cell beam. 23

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN R -yn I - ~mn-| *-'VVWV^-|'"-AYn-I W - -mn _I AAAE [aV C CC 0.5R Odn R 0.5 k VOLTS UNITY mkN~~dn dT Figure 3-1. Analyzer Circuit at the nth Station for R I -r Cantiver Beam with Coulomb Damping R 3.4 Impulse Response of the Cantilever Beam with Coulomb Damping A 5-cell uniform cantilever beam with various amounts of coulomb damping was set up on the differential analyzer with an integrator time scale of 0.5 seconds. Response y5 at station 5 is shown in Figure 3-2 following a unit impulse of one-fifth second atio Note the dead-space effect due to the coulomb damping; the displacement y3 does not in general return to zero but ends dYn ~kv k VOLTS = UNITY up at some finite -1.displacement fonalyzer hich the elastic forces are insufficient to Cantilever Beam with Coulomb Damping. 3.4 ImPulse Response of the Cantilever Beam with Coulomb Damping overbalance the coulomb friction. It seems hardly necessary to point out that arbitrary combinations of velocity-squared, coulomb, viscous and other types of damping can readily be handled by the electronic differential analyzer. For a more complete discussion f other types of beams, time-dependent boundary conditions, theoretical accuracy of the difference technique, etc., the reader is referred to a previous report.' ---- ~ 24

Pr~~~~~~~~~~~~~~~~~~~~~~~~~r t~~~~~~~~r 4 4~~~~~i fill-eI - i II -i'c 0.O 8 i i i~~~~~~~~i I~~~ii ii i -I 1, i~~~~~0J14 L & i-1::L I i A-kit If I L ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ttii: I''ijl( I fit + ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~t i -0.08, j - I D r Site - -r!II I~~~~~~~~~~~~~~ i IU 0.00'-.-1 ii! L L!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —4L Figure 3-2. Impulse Response of a 5-Cell Uniform Cantilever Beam with Coulomb Damping. i i -Ii jj: i

BIBLIOGRAPHY 1. C. E. Howe and R. M. Howve, lpplication of Difference Techniques to the Lateral Vibration of Beams Using the Electronic Differential Analyzer, Report AIR-7, University of 1Michigan, Engineering Research Institute, OOR Contract No. DA-20-1O-ORD-21811; April, 1954. 2. Timoshenko, Vibration Problems in Engineering, D. Van Nostrand (1937). 3. C. E. Howe, R. 14. Howe, and L. L. Rauch, Application of the Electronic Differential Analyzer to the Oscillation of Beams, Including Shear and Rotary Inertia, External Memorandum UIvMMI-67 (January 1951), University of Michigan Engineering Research Institute. 4. G. A. Korn and T. M. Korn, Electronic Analog Computers, McGraw Hill (1952). 5. D. tW. Hagelbarger, C. E. Howe, and R. M. Howe, Investigation of the Utility of an Electronic Analog Computer in Engineering Yroblems, External MvemorandLtu Ul2-i-26 (April 1, 1949), University of Michigan Engineering Research Institute, AF Con. W33(038)ac-14222 (Project MX794). 6. R. M. Howe, Theory and Operating Instructions for the Air Comp Mod 4 Electronic Differential Analyzer, Report AIR-4, University of Michigan, Engineering Research Institute, OR Contract N6 onr 23223; March, 1953. 26