AN APPROXIMATEMETHOD FOR THE TRANSIENTRESPONSE OF NONLINEAR. SYSTEMS
by Patli.ck F. Cunniff, a.s •• M.s.
Thesis submitted to the Graduate Faculty of the Virginia Polyte.cbrdc Institute in candidacy for the degree of DOCTOROF PHILOSOPHY in ENGlNDRINGMIX:HANICS
January, 1962 Blackaburg, Virginia II. TABLEOP' CON-.rmrs
Page
I. TITLE PAGE. • • • • • • • • • • • • • • • l 11. TABLEOF CONTBITS•• • • • • • • 2 l. List of Symbols ••••••••••• 3 2. List of P!gurea .•••••••••• s III. INTRODUCTION.• • • • • • • • • • • • • • a
IV. THE REVIEWOF LITERATURE. • • • • • • • • 11 Y. THE INVESTIGATION••••• • ••• • • • 18 Graphical.•m.erica.l Hathod. • • • • • 18 Equ.iftlent: Numerical Method. • • • • 26 Singular Points •• . ·• ... • • • 29 Convergence of the Iteration Process. 30 Scaling Law. ., • • • • • • • .. • • • 34 RESULTS. • . . .. . • • .. . . • • • • • • 37 DISCUSSIONOP RESULTS.• • • • • • • • • 87 VIII. COHOU(JSIONS•• • • •• •. • ••••• • • 93 u. .ACICNOWLEDCJMl:ltTS•• • •• • •• • • • • • • 99 x. BIBLIOGkAPHY. • • • • • • • • • • • 100 XI. VITA. • •• • • • • • • • • • • • . . " . 102 XII. APPENDIX••• • • • • . .. • • • • • • • • 104 1,. List o,f SD!?2lp SYJ!!?2ls Ueed.f.n th!.In'ft41tis,aticm
Ci viscous damping constant between mi and 111:t-l Fi forcing tanction on fllf. fto nonlinear spring force be.tween m.1 and the foundation; kx + fax3 f21 nonlinear spring force between m2 and •1i . 3 k2"2 + .J32"2 g acce.leration due to gravity ( 386 in./ses;;.) h finite time increment, t kt spring constant be.tween~ and 11:l•l ll_f. i'th mass p w l/' - o 0( coeffici~nt of nonlinear term in the cubic spring 6( phase-plane delta for the i'th equation 0£ motion iJ' 6/x t/ J3,/rr,(· (J)/kf./11:L 2, List of Fiee.s Page Figure. 1. Graphical Representation of --=-dx V 20 dv X-+6 d.v X Figure 2. Graphical Re.presentation of -=- 20 dx _y___ l+E' Figure 3. Example Systems. • • • • • • • • • 38 Figure 4. Foundation Input •• • • • • • • • • 38 Figure S. Graphical Solution, Linear Single- Degree-of •Freedom Syatem. 0 a. Phase-Plane Construction, AB = 28.13 42 b. Analog Computer Response Curve x-t 43 Figure 6. Graphical Solution, Nonlinear Single• Degrea-of•Fre.edom System. a. Phase-Plane Construction, J).0 • 28 .13° 45 b. Analog Computer Response OUrve x-t 46 Pigure 7. Graphical Solution, Nonlinear Two•Degree- of-i'reed.om. System Without Damping. 0 a. Phase.-Plane Construction, A0, • 28.13 48 0 Phase-Plane Construction, 1>0i.. = 28 .13 49 c. Phasa•Plane Construction, AB, = 14.07° 50 Phat.ui•Plane Construction, ~0z. a 14.07° 51 e. Analog Computer Phase-Plane 52 f. Analog Computer Phase•Plane. Trajectory, x2 • v2 • • • • • • 53 g. .Analog Computer Response Curve x1 -t...... 54 h. Analog Computer Response Curve. x2 • t. • • . . • • . • • • • • 55 Figures. Graphical Solution, Nonlinear Two• Degree-of•Preedom System With Damping. a. Phase-Plane Construction, ae, • 28 .13 0 57 0 b. Phase-Plane Construction, nez.• 28 .13 58 0 c. PJ:u1se•Plane Construction, AG, • 14.07 59 Phase-Plane Construction, 6Bz. = 14.07° 60 Analog Computer Phase-Plane Trajectory, xi• v2 ••• • • • 61 f. Analog Computer Phase-Plane. Trajectory, x2 - v2 •• •••• 62 g. Analog Computer Response CUrYe x1 • t. • • • • • • .. . . • • . 63 h. Analog Computer Response Curve • • • • • • 64 Figure 9. Comparison Between '5 and !) ' Methods 67 Figure 10. Singular Points. a. Porce•Diaplacement Curves •• • 72 b. Pbaae-Plane. Trajectories in the Neighborhood of v = o, x = 1.000 75 Page c. Pbaae-.Plane Trajectories, Dam.peel Cubic Softening Spring. • • • 76 Figure ll. Sea.ling Law Examples. a. Model Input. • • • • • • • • • 81 Prototype Input. • • • ..... 81 c. Phase-Plane Construction. bG) = 28.13° 82 d. Phase. ...Plane Construction. 11e = 28.13° 86 III. INTRODUCTION The analysis of e:n.gineering structures which are sub- jected to dynamic forces is an area of study which has received considerable. attention in recent years .. In some. cases, an understanding of the behavior of a structure under its ex- pected time-varying loads is imperative so that the designed facility fulfills its :lnteri.d.ed purpose.. S'uch atructure.s might be simple beams• columns• rigid frames, electrical, and mechanical equipment, etc. In general• there are three types of motion which the de.sign engine.er might be required to investigate. due to certain prescribed load.a, namely, transient response, steady• atate. vibrations, and random vibrations. '?he motion studied usually depends upon the. expected or predicted load which is sometimes called the input of the. system. In what follows, only transient response.a of systems subjected to short time- duration loads are considered. These impulsive-type forces might arise from. sources such as e.artbquake tremors, wind gust fOl"'Ceat and pressure. waves from explosions. Of the various assumptions which the engineer mwst make when studying the dynamic response of structures, 01\e of the most important is perhaps the model representation of the t:rue structure. One method of re.presentation is to judiciously idealize the $tru,cture into concentrated mass• and to connect, each lumped mass to its neighbor by weightless springs and da.ahpota. The number of masses and the constraints or lack of them on each mass deter,n!ne the number of degrees-of. freedom. of the system. The differential equations which describe the motion of the model are either linear or non- linear, depending upon the behavior of each mass, spring, and dash.pot. The aubject of linear differential equations ia a rather well understood topic of mathematics• while nonlinear differ- ential equations is not so far ad•anced. One of the diffi ... culties with the latter subject is that the theorem of superposition does not hold. As an example of what is maant by superposition in a dynamics problem• coneider the linear equation of motion for a simple oscillator subjected to a force, P1Ct)2 my+cy + ky= t=;_(t). Suppo111e.y = Y1(t) is the solution of the differential equation., Now let a new force,. P2 (t), act on the oac:tllator for which the response is y • Y2(t). Finally, if the force, F3 • F1(t) + P2(t), acts on the oacillator 1 the theorea of SUP.-• position allows one to add the two previous solutions, that ia, Y3 = Y1(t) + Y2(t). Du.a to the inherent c011ple:d.tie.s involved in the aolu• tion of nonlinear differential equations, approximate method$ umst be applied for their solution. In the caae of transient response, these approximate methods are generally graphical, numerical, or electrical analog solutions. Soata significant reasons for developing approximate graphical and numerical techniques are. the following: a. SUChmethods are useful when computing time is not avd.lable on the computer;. b. Bven. with e~ctronic computers available, pre• l:bninr.ry &1JtitJ1&tesr·e.quire.c for computer prograad.ng can 1>.• obtained with an approximate method; c. Problems involving em.perical functions which cannot be approximated readily for a computer are sometimes solved faster by a graphical or numerical method. 'lbe purpose of this investigation is to develop a reasonably rapid, yet precise technique, for the approximate transient solution of nonlinear. nonautonomoua systems. The nonlinear components of the systems conside.red artl assumed to be cubic hardening and softening springs. An investi• gation of the. behavior of the. me.thod in the neighborhood of singular points is presented, and consideration is given to singularities in a two-d.egree-of•freedom system. In order to increase the versatility of the solutions, an associated. study of a scaling law for nonlinear multi-degree .. of•freed.011 systems is also made. IV. '?HE REVIEWOF LITERATURE Topographical methods of nonlinear ll'JeChanics are em• ployed as graphical solutions for nonlinear differential equations. The solution curves are represented on a plane which is called the phase-plane. The method of iaoclines and Lienard•a Graphical Oonstruction are two claaaical methods which are presented in the works of Minorsky (15) t Andronow and Cbaikin (1), Stoker (20), and McLachlan (16). These ae.thoda are ueeful for plotting the phaae trajectories of one d.egrh-Of•fNedom autonomous ayetema• that is, syatema for which the independan.t time variable does not enter ex• plicitly into the differential equation. Pell (18) bae extended. the Lienard Method. to include arbitrary restoring forcu in addition to arbitrary damping forces in a single• degree.-of•fre.edom. ayst•. A comprehansive review of phase.plan& methocis u grapht• cal aolutiona for transient vibration probleme hu been presented by Bi•hop (4). Single-degree-of.freedom. eyatems with damping, nonlinearity, hystereais• e.to., are treated aa well as linear 11Ult1-d.egree•of.freadom ayatema 1 such aa beama and plates. This latte.r claas of problems ia aolvad by reducing the governing differential equation of aotf.ou to a aecond..oTder differential equation in terms of the generalized. coordinates. The pbase•plane solution super-. impose.a a phase trajectory for each mode of vibration. fl'le auperposit:lon of mode.trajectories proceed.a until a auffi• cient d~ of accuracy ia obtained. Biahop also eh.ow bow tme phatle•plane can be uaed for solving probleme of e.laatic •tabili'ty. Jacob•• (10) baa developed. a 118:thod which ia an e.ffi• oient world.111tool for aolv.lng aecond..ord.er ordinary cliffer. ential equatiOl\8. By reducing a giftll equad.on of motion. to the at,ndard delta form, that is, 2 X + W ( X -t- J ) = 0 ._... o• cf The phase trajectory :f.a deacribed. by a aeries of circular area with centers located along the x-a:d.a. Takei also establishes a relationship of duality between what be calls the inverse double delta method and Jac»baen'a delta method, so that the circular arc segments are located with their centers along the v•axia. For a structure which can be idealised as a one~ of-freed.om system, Hu4aon (9) baa developed a graphical method for finding the equivalent static load wben. the system is under a dynamic load. Equivalent static toad i• that static load which will produce a deflection of the structure. equal to its peak deflection under the dynamic load. '1'he input force•titae curve is approximated by rectangular step changes, and the equivalent static load is found frOll a phase graph construction technique. Hudson presents an example problem of a typical gun-bl.a.at problem which :ts worked. out by his procedure and compare.a his results with e:xperiae.nta1 obaenations. I By letting x •••this equation can be reduced to clv x G(V;x,t) dx =- x = V At a particular time, t = t 1 , and for different values of x, accel.eration-displacement curves are. drawn. The graphical technique proceeds in a atep•by•step fashion to construct velocity-displacement curw.s for different initial or boundary conditions. -15- Ku (13) bas also extended his graphical method for the analyaia of nonlinear aystemswith more than one.-degree-of- fre.edom:. Problems which are described by more than one.- degree-of •fl:"eedom are. reduced to a higher order differential equation involving one dependent variable. Space trajectories are then drawn in phase space. A graphical method for the analysis of nonlinear single,.. degree-of-freedom systems under arbitrary tranaient condi- tions baa been developed by Bareiss and Atchison (3). '?he. equation of motion is described as mx+ cp(x) + f (x) = p(t). 1'o arrive at a graphical solution for thia equation, four graphs are. used in the procedure: mi versus 1: J x versus t; x versus t; and. p versus t. The method consists of esti• mating the mi, x, and x curves during a particular interval of time• 6. t. Prom the mean value of each function during 6.t and a transformation technique. a new mi cunte can be obtained on the p versus t plane. This new mxcurve is used for a second trial run, and the procedure continues until a resulting mi curve agrees favorably with its estimated mx curve. Ayre. (2) has ext.ended Jacobsen's delta phase-plane metbod for transient vibrations of linear multi-degree.•of• freedom systems,. As an example, Ayre. treats a two-degree-:- of-freedom elastic model subjected to a prescribed fcnmdation motion. The absolute displacements y1 and 12 of each respective mass are (ll)• Y, = ¢ z + ¢:u: J Yz -= 'i This step-by-step procedure is continued until the de.sired pliaae trajectories are obtained. The displacements y1 and y2 are then found by superposition of the two phase trajectory coordinates and multiplication by the proper constants. Li (14) has presented a graphical method for the analysis of dynamic systems described by two simultaneous first-order ordinary differential equation•. The method involves a step-by-step graphical integration in a aumner similar to Jacobsen's del.ta method. although the. centers of arc are not necessarily located. along the displacement axis. Example.a of application are taken from hydraulic 'ri.brations with nonlinear damping forces and arbitrary forcing function•• Two surge tanks connected to a conduit and a differential surge tank are examples involving two simultaneous differen• tial equations. A graphical method for solving simultaneous second- order nonlinear differential equations has been presented by n (6) .• The method can be applied to any set of equa• tiona of the form., X·=-F£ l ) where• X·· · = X· · ( dx.-' ) • l l d Xi The force• Pi, can be a function-of the velocity and dis- placement coordinates as well as the independent time variable. For example., Fi= F, ( x,jx7..)"·--;x,,) x,) ,t',.) _____I %11/ T) F:-F('. . ) - X,; Xz.. 1 .. --J Xn 1 ,,(, 1 X-"l.-1 , ... 1 X n1 T .. . Fn = Fri( x,JXi.;··--; Xn1X1; X2) ··--·; ,r,,) T). The procedure consists of the construction of phase space Clll"Yes, versus xt• 'l'b.ia ia accomplished by using the initial condition.a of the problem, the above equations, and the cums that define the inputa F:r.• Tb.e assumption ia made that the step-by-step solution of the Sf.versus xi curve.a be made up of short straight line sagments. An example problem is preaente.d which predicts the vertical ground reaction which will occur during a landing gear drop teat• The landing gear apparatus is idealized a.a a nonlinear two-degree-of•fre.edoa system.. The graphical method is useful in that the predicted ground reaction allows the engine.er to de.sign the metering pin before the landing gear is actually built and drop teated in the laboratory. Purthe.r discussion of these methods is in Appendix l. V. 1'HB INVESTIGATION 1. Grapntcal.--...ri.cal Metho41 RtMrll lawt1'J,9M of 1125'21!: Suppo•• a atruQ'tnrN, which i• aubjected. to dyn.aic forcer., ia i-educed to a model com~oaed of£ concentrated llla8aee in a straight line which are oo'mlaete4 together by we!gbtlua apringa and daahpota. The equationa of motion for this ayetem can be ~ed to the following fonat where, Si = J:-( xi f-I J xi.) )(i-1) Xi.+t) Xe'1 x,_,I t)} and the doh refer to differentiation with i>88pect to time. A &raphioal eolut:Lon for thi• aystea can be obtained on I. phaae•planea by extending the. delta pbaae•plana method (10) in oonjuction with •imple need.cal computatdona. Si!YJ\9•Desr,u-of•l£1149!! 1'ba delta phaee•plae uthod has been applied to llOll• linear aing1e-4egree•of•!reedoa ayatema. l'or such syetama, the differential. equation of motion 11ay be written aa x + e,,/(x + cf)=o. (5.1.2) A geometrical interpretation of r.q. (5.l.2) can be.obtained by introduoiDg new coordinates which are called the. pbue,. plae ooordinatea. Let x • x, and • • i/w • Equation (5.1.2) redu.oea to the following fonu -dx = - -~ v dv x+ S Tbe S has the same dimension u x and may be repre.aented along the x-axis in the phase•plane aa ahown in Fig. la. The slope of the normal to the lina PQ is (- x ~s). Since 1:hi• slope is the same as Bq. (S.l.3), the phase trajectory at point P will be normal to the direction PQ. As8U11le that I is held at its aean value for a finite time interval• '1t. Equation (5 .1.~) may uow be iutegrated ao that more information about the phase trajectory can be obtained. Jlewritiug Bq. (S.1.3) aa (x+J)dx + vdv=o, lead.a to ( x + cS )2 + v2 = constant == ( PQ )~ According to Eq. (S.1.4)• 1:he inatantaneoua phase trajectory ia a circular arc passing through P with its center at x • • S , v • o, or at Q. In order to derive a r.iationahip between the. circular arc eegmeut be.ginning at P and the time increment. 11t, note that -1_= (j) v, and dt = dx uJ v (S.l.5) Figure lb ahowa th• differential geometry of the phaae trajectory. Let th• average radius of curvature of the step X X P(tt;z.•) Normal P(:.,r) t~ PQ ' ·'rI I V --T a a, J_ I I -z. (a) (b) Figure l. Graphical Representation of dx ,, • rv . -r+r V V T V l+S' J_ X X (a) (b) . Graphical Representation of dv • • X • c!i V -W' tbla 1ea4a to de :: 1 + ( 1-})7... -= d x ,I I + (x + JY v v y vz. 'by aema of Bq. (S.1.3),. SUbad.t:ution of tbla lut exprea• aion into lq. (5.1.5) yia14e dB dt= -;;r· 81.Me the me.anvalue of J waa u.aed in the derivad.on, tld.a llaY be written - Pr01ll the aboft relationahipa, f.t i8 con.eluded that a -~• 01"dh.ary differential equation in the atandard. delta fom ean ba tntepated graphically by a auccuatve oc,n.. nrucd.01\ of circular arc •• .....,,. wboee cen.twa an loea-ncl on the xtliiC/ld.aat various atep values of x = • 8 • Poaitift tiae f.a represented by a oounterclockwiae anplar vai.atioa ot the nos,ul PQ. 'l'he 1ocu. of circular arc Nglleftta ia the approd.mat• pl'laae 'traje.ctory and, therefore, the deat..S aolud.on of the diffe-ent:la1 equation. !ha quutio'A aay now adae: Ia it poatible to plot the phase trajectory with the center of area alens the ••-'81 Ta'ket (23) has d.effloped a graphical uthod wbtoh i• applicable to autonoaou.s aecOl\4-ol'd.er ordinary 4tffeNntlal eqoa.td.ona with conatau.t coefficients other than the uet'1:ia t:ena_. Havi1',g established a.mathematical relationship of duality betwe.eu his inverse double delta m.ethod. and Jacobsen'• delta method, Takei shows how the centers of arc can be. pre• scribed along the V•lUd.a under these restrictions. As a 110re general. approach to the problem, consider Bq. (s.1.3) once again. Let Therefore, dV X -=---- ._y_ dx I+ ,S' Proa Fig. 2a, tha slope of the line perpendicular to PQ is ( - --~--) 1 which ia the phase trajectory of Eq. (S.1.7). ( I+ J' Rewrite Bq. cs.1.1) aa dv = _ _ x_ dx v- o" where o"= v - 1 0 , • Aa8\D8 that the mean values of both II x and v are used in the. 6 expreasion during a finite time intel."Val. Hence* upon integration, ( v - o" )2. + x2. = constant = ( PQ )2.• 1be instantaneous phase trajectory is a circular arc, paaaing through P with its center at x = o, v • cS" , or at point Q. An explicit expression for time ia again obtained froa the. relation, x =w v. Uaiug the differe.ntial geometry in Pig. 2b, the following relationship is obtained: I d0 dt= , · · I+ J I.U Since the aean value of the variables was assumed in deriv:l-a Bq. (S.1.9) 1 the abo'ft 6quatio11 may be written. as Positive time is represented by a clockwise angular variation of the nonnal PQ. However. the angular increment, fl e, does not remain constant for a constant tim& increment, I L\t. aa it did in the delta method. Thia ia due to the c5 term in Eq. (S.1.10) which changes with each increment of time. Hence. if a linear or nonlinear second-order ordiNU7 differential equati.on ia reduced to the o' form, it can be integrated graphically by construction of aucceaaive circular arc aegm.enta Whose center• are locat6d on the v-axi.a at various step values of v • 6 ". The locus of these circular arc aegmenta is the approximate phase trajectory. It is interesting to observe in l':tg. 2a that when I S = o0 , point ·Q coo:lncidas with point a. and the phase trajectory is parallel to the v•axi•. When c5'= ....1, point Q lies at an infitd ta distance from point R, and the. phase vajectory is parallel to the X•axi•• Ma!thodof Construction Th• construction procedure of the phase trajectory by the. delta phase.plane method for nonautonomoua aingle-degNe.- of freedoa •ystema ta now described. Aa an example, con• •ider a simple oscillator with a cubic hardening spring and a prescribed fouuclad.on velocity input. The equation of morion is •• 3 •• m X + R )( +j3 X = - mYo . Thi• ia written in the form as x +w2 (x +J)-==o) S = + tz.x3 + §z:. AallUlle that the graphical aolution has been extended to step !l.t i.e., > c5n= + z. ( X11+ Xn+,)3,+ ~z. ( Yo),.. Since the input ia 1c:nowllaa a function of the. foundation velocity, the foundation acceleration may be approximated over the finite tiae. i:nterval, b. t, by the tbeor• of tb.e mean of differential calculus. Thus, ( Yo)n=- (ro)n+\- ( rJn 1 Aa a firat trial for finding X n+i and Vn+1 t use the known 5"_' from the previous step of th& solution aa the first center of arc. By laying off the preacribed circular a:rc, ~e =w(At ), from. the. eoordinate ( v" J x n ) by means of a -25- com.pass and protractor, preliminary values of Xri+-1 and V0 +1 are. found quickly. Calculate the first trial value of 6n f1"'01l'lEq. cs.1.11). Draw a new circular arc from. ( Vn, Xn ) lfith the new ce.nter at x = - 6n ao that a new coordinate ( Yn+,, x n+i ) ia found. Again• use this coordinate to cal• culate a new 6n. This iteration process continues until the values of J,, converge. This completes one step of the solution. The initial conditions for any problem are always known while there is no preceding value of J established at the beginning of a problem. In order to calculate a trial value of Jo , the. magnitude of the phase coordinates at the end of the first step might be assumed; the initial values might be. used as the average values over the interval; or, a Maclaurin expansion might be. used to calculate the phase · coordinates at the end of the first interval,. Two·Demes-of-Freedoa The equations of motion for a two-degree-of.freedom. syatem are 2 x, + w, ( X, + £,) = 0 (5.1..13) Xz.+ w:(X2. + S2.)= 0 (S.l.14) where. o,-=d, (x,, Xz., x,, X2,t) Jz.= J2 ( X , J X 2.. , X, > X2 1 t ) . The graphical integration of these two coupled equations is obtained by extending the delta phase•plane method to two pbaae•planes• ( v., x1 ) and ( v2 , x2 ). The computational method described in the preceding section is also used. If the solution baa been extended to step !l• ( 8, >ri and ( cSz.),., are found for atep (n + l) using the known ( 8, ) 11_1 and ( Sz),,_, as the initial centera of the circular arcs. This gives trial ,ralues for ( x, >,,+,• ( V, >n+, • ( Xz.) 114, , and ( Vz >n+, which allows ( J, )" and ( Sz),, to be computed,. The procedure. is repeated as in the case of the single-degre.e-of•freedom ayatem. until ( cS,) ,, and ( Jz),, converge to their respect! ve. values. 2. §ro!ivalen~ lhJ$erica.J. >te.thod.1 · The differential equation of motion for a linear un• damped oscillator subjected to a foundation motion is •• . l •. )( + u) )( = - 'lo . O'Hara (17) has developed a numerical integration tech• nique for the solution of tbia equation 'Where the input is given by the foundation velocity. For a finite increment of time, t:,.t = h, aasume that the initial conditions for the (n + 1) interval are xll and Vn • The solution at the end of the increment is then h x.,+,=x,., cos wh + v,,SJ~ wh - ;; l [Yo(T} Jn si11 W { fl- T) d T V11+,=-X,,.si11wh + Vn cos wh - J jh[y.JT),, cos w(t,-T)dT • -27- Since the foundation acceleration is defined by Eq. (S.1.12), the approximate solution equations become Xll+t = Xn Cos wh + V11 sir, wh - (Yo)n (1- Cos {l)h) u]-,,. Vn+, = - X,., si11wh + V,, cos wh - (fa),, s/n u1h. t,J,. Tb.is same approach ia now used for the differential equation of motion in the standard delta form., that is, •• z. z.r X + w X -= -0 o. Aasuad.ng that S is held constant duri.ng the finite time increment, tit• h, the solution equations for x and vat the (n + l) interval are During any atep of the solution, the first two terms on the. right-hand side of Eqa. (S.2.2) and (5.2.3) are known as well as the constant coefficients of the third term, c),,. The procedure consists of assuming values of Xn+, and V,a, in the. S11 equation. Having this trial value of , substitute it into the right-hand side of Eqs. (S.2.2) and (5.2.3). 'l'his gives new values of x,,+,and Y'1t-,which are then used to compute a new value of &11 • 'l'he numerical pro- cedure continues until the values of S11 converge. To show that this numerical procedure is the. equivalent procedure of the graphical construction method, rearrange Sqs. (S.2.2) and (5.2.3) in the following form.: 1 - X CDS .Stn f- .~ C0-5 Xl'J+I + On - ll wfi + Vn wh uJ/2 vn+, = - Xn 5 it1w h + Vn cos w h - 5 inwh • Square each aide of Bqa. (s.2.4) 81\d (S.2.s) ad add the two naulting e.quationa. Upon aimpliftlngt one obtains !Id.a is precisely the equatf.on of a <"i.:cle wb.oaecenter is •- x,,+,= -S,,• v,,+,= o , and wbt>ae radiua squared ia 2 (xn +-fn)2+ v11 • For the two-degrM•of-fre.edoul ayetea, raw.rite lqa. (5.l.13) and (s.1.14) i~ the following fora~ !be aolution at the (n + 1) interval ia utal>liahed in a aild.lar manner aa in the caae of a eingle~-of-freed.o'il ayat•. 'tbua, ( v,)n+,= -( XJn.SlrJ f,t), h + ( Y,)ricos Wih - (&~ VI .S~f/W1h (s.2.10) (x2.)n+1= (x1.)"Cll.) c.Jz.hf ( Vz.)n5ir, Wzh- (if1.)n( 1- ~OS c.J2.h) (s.2.U) (\/2.)11+1= -(x,_)I\ siri tJi.h+ (v,.t'\Co5w~h- lc)1.)f\siflu.'hh. (s.2.12) Oboe again. all terms in the righ't•ban4 aide of tbue four equad.ona are known.excapt ( rS,) 11 ad ( J'z.)11 • The •- iterad.on procedure is followed for the convergence of ( &.>n and ( Sz>n as in the. case of St1 in the aingle-degree .. of.freedoa 3. Sine4!r Po,in,'ta•- A ae.cond-ord.er-ordinary differential equation for a aingle-degre&•of-freedom system can be. reduced to a first• order d.ifferer..tial e..;ittation. For example., cotud.der the equation dx p(x,v) d v = Q ( x,v) (5 • 3. l) A point ( Xo) Vo ) for which P( XoJ Vo ) • Q( Xo I Vo ) • o, a:lmu.ltane.ously is called a singular point. Any other point on the pti&ae-plane to which thia definition doe.a not apply ia called an ordinary point• By me.ans o~ Poincare • a criteria (21), one can determine the type of singularity for a nonlin• ear differential equation. Vortex, saddle, focal, and nodal points are four such singularities. Equations (S.l.13) and (5.1.14) can be re.written for the. two-degree.•of•freedom system•• and where 01 and S2 are no longer explicit functions of the independent 11ariable. t. The. i;.nnastigation for aingularitiea is classified into three cases for the two-.degree.-of-fre.edoa system., •30- Case I• x, = - S, , V1 -= o Xz.-=--S2 ) Vz. =O. Case II. x.,= -J, , v, -=D No reatrict:lona on xz.. and v2 • Case Ill. ><2.""'- 62..1 v2 = o llo reatrictiona Oti. x, and v, • Since the equations unr.J.er atudy are. coupled• Poincari '• crit!l'ria cannot b& applied. However, each of the above caaea ~he defini tioa of a singular point. An example problem is diacuaae4 later for the tl\rea cases.for singular points. In ad.di tion, the behavior of the graphical• numerical me.thod and l\Ullerioal equival•t procedure in tbe neighborhood of a aingu.lar point is presented. 4. stm.vraenoeof th! ~teratiop l[ocya, Oener@l l?!'D1oee,p.t Scarborough (19) haa diacuaae.d the wfficient concli- tiona for converg~e of one and two numerical equation.a when 1:be. iteration process ia applied. An ~enaion. of this procedure ia now ude to include a aet of N- numerical equations. The U8Ual range and summation convention of · ten.or calcuJ.ua (22) 1• followed. Consider the aet of equation•, where. Fi is a known func- tion of x x«:= G[x,) .....,xN], i= 1,2..) ..... ,N. (5.4.1) 'ftl& aet of equations ia satisfied by the exact values of the aet of roots xL • As a first approximation to find a set of roota, try xt. H ence, Eq.(s.4.1) givea Subtract rq. (5.4.2) from r.q. (5.4.1). 'l'bia gtvu Apply to the right•lumd aida of Eq. (5.4.3) the theorem of aeaa 't'&lue. for a funod.on of ! vari.ablu. 'fbua, ...&.---. 2 C: [ to) ( (")) (o) { ( I •••... )j £:- 6: .,...... , i.,j = , (·,; x, + e x, - x, 1 1 x"' + e x"'- x11° o e 1 • ) Add each equation of the set: of equationa aa expressed ta Bq. (s.4.s) and consider only the absolute •aluea. 'l'lma, N E. I X- - X _<•II L I )(.- X _t•JlI F-. \ i = I L L i =/ J J ,, j • (5.4.6) Now let the max:hmm value of t• terms t I F,,,I+ ..... + I F,.,,,ij, ·• • ...... , l \F,,N I+ ...... TlFN;N\} ba a proper fraction. a for all points in the region Cxt; x.). Tb.en Eq. cs.4.6) becomes Thia relation holds for the first approximation. For succeeding approxiut:lons, similar re.lationa are obtained. "' (r) I JL I ,,._,,, ? Ix, - x, b m ?, X;_ - x, . 1,-::::, Mul.t:lp1y toge'l:her all thee• inequalities aa expreaaed in Eqa. (5.4.7) and (5.4.8) and divide through by the 1 (•l, I (2.\1 commonfactors 1:-; x~- x, 1 xi: - xi. 1 • • • • • , 80 that. Since!! is a proper fraction, one may take the right• band 'll8IDber of tbia :l.nequalf.ty aa amall aa one pleuu by repeating the iteracion pr<>eua a sufficient IWlllber of tiw. Thia MW that the erron Ix~-xr>/can be made as small u one d•airea. Therefore, the iteration procua converges when the ~conditions \ F.,.I + ..... + I F~) ;f_ I (S.4.10) I F.,N \ + . . . . . + I FN,N' L. I are aatiafied for all points in the neighborhood of (i<.t>)• Two-PM£H-of•l£W9msx,_ The auffieient conditions for the convergence of Eqa. cs.2.9) through (5.2.12), which were derived for a two• de.gro-of-freadoa system, will now be establiehed. These equatiou ara rewritten in th• following fon.: (5.4.11) ( x,t,-== F, [ ( x,),,J ( v,)n) (rt] ( v;)ll?,= F;_[cx),,J {Vi\ (c[),i} (5.4.12) (x-J,,+, 5 [ ( xLt,( v,.t,{ Jz.),,] ( vJM,= [ ( x,t( v,t,(c£),,]. cs.4.14) Equation (5.4.10) is the requirement for converge.nee, namely, JF,_ +-/ J Fz. -f JI:; + d < f I J(x,),,+-,I J(x,),,1, I IJ {.rJ11.,.,I I ;;irx),,.,, I + JF,,_ JF:. 1 I~Nv,),,., 1 I -rzv:f.,.,_,I+/ { j_!i., + f ( J!5._ + dh- I lJ ( vJ,,-1,I I J( Y.,),,ttj .:l( v,J11+,I I J <'.V..) -1t-1 I • Since all the. tema in the right-hand aide of Eqa. (s.2.9) (5.2.12) are. constants ..-capt the S's, the convergence. requirements are (5.4.17) (5.4.18) where . ) J s. Sc,l)J.11 Ll!'•t It wu indicated: in the lntroduetion of this inve.at:l• ga:ticm. that w.per·poai ti on eatU:lOt ba applied in aoa.linear m.ecb.anica., An at:teapt is made b.eretn t-o lend a Cer1.iaitt llfllaSure of geurality to known solutions ot nO'Alinear transient vibrations. Rewrite 14. (5.1 ..1) with any p- 6'1Crlbed fOl"Ging functiou, , on the rf.gbt•band d.da of the equation. that is, X.·l = X l · ( t) = s,-(xi+-1)xi, Ki-,i X,-1-11x,,Xi-,). Oonatder a accotld ayatea eud.lar to the fira1: sy•t.•• By aild.1ari:ty 1 f.t t• _.t that the new eyatem is tdeal:f.aed. identically aa the first aystem ocept that the mapil'Udea of the 111Uaea, spring atld dashpot conetant•, and forcing fun.ctiorut are not -••ari],y the sane., Let the equation• of action for the second syst• be Where the pri.raea refer to differentiation with respect to the independent variable 1: • 'J.'he following reau-iottou are now·placed on the MW aysteat 1: = _..r... • ~, , , (not n 1 .JLi. = n wi , n >-J.Jt,necasa,arily a integer) Equation cs.s.2) becoaea i + wt( Si + 6,) = · In order that sJd=xi.ft), f.t is observed from a coapariaon of Eqa. (S.S.l) and (S.S.3) that the following additional re.atrictiona muet be made: - . ~(i-} -= F:,-{t). /J.i..= 5i. ) 17"4. Thia establi•he.a the requir-.nts of the scaling law for similar systems. Having the reaponae. of a given systea, the response of a similar system is obtained directly fro• the known response by using a new time scale. It is noted, however, that nothing baa been said about a linearity require• ment of the differential equations. If the inputa of the systems under consideration are represented as foundation velocities, the forcing functions, G·, reduce to F- = - ' /ov • Suppose that for a similar system - fl G, = - t . '1'be restriction between the inputs requires that -'f/z. ci-Z-o''&... -- dtz. J or However, Equating the right-hand side of Eqs. (5-,5.4) and (5,5.5)• that is. the requirement on the input in the fo:rm of a foundation velocity is established .. It states that the input of the second system has its ordinate eqll&l to !l times the ordinate of the first system and its abscissa (the time scale) divided by n,., A further generalization of the scaling law is made if, instead of Nqu.iring Lii.= Ji , a length scaling factor, .J_ • is introduced such that (5.5.7) Substitution of Eq. (s.s.7) into Eq. (5.5.2) with the other restrictiona leads to . • .L:._ , } x,. = n 's, ) Por inputs as foundation velocities, the new input ia related to the model in.put as cs.s.s> VI • ltE:l"1JLTS Tbe grapbical-maerical aethodt which uses the delta technique, baa been applied to the four systems shown in Fig. 3. In each caae, the system is initially at rest, and the input is the foundation ve.locity shown in Pig. 4. The time duration of the :lnput ia ta o.soo seconds, and the solution for each exaple extends to t = 1.000 second.a. The construction of the. phase-plane trajectories by the graphical-maerical method is obtained using a compass to lay off the required circular arc and a protractor to measure the angle of increment, Ae • An engineer•• scale .aeaaure.a the coordinate point• on the phase-plane. The aolutiona for the nonlinear ayatem.s were compared. with.their respective solutions from an analog computer at the Naval Research Laboratory, Washington• o.c. The com- puter conaiata of a Reeves electronic analog computer aaade 'by the Be.evu Instrument Corporation and a aervo-llP.\ltiplying rack made by lll.dcentury Inetrumatic. The computer resul ta for each problem are :ln the form of a phase-plane plot, x veraus v ; and a relative di•placement versus time plot, X versus t. The ayatem shown in Fig. 3b haa been solved in Example 5 by the numerical equivalent procedure of tM graphical• tmmRiCal method. Thia provided a check on the drawing _L 'j, 7 ,,:JJ;,,,_~' (a) ':/z. mL _L fZ I ,-I--~ Z. _t !J, '--~ C, • rrrl-1-,rrr- ~o (d) Figure 3. &xampla Syatema. (.)• -I) 25 • \ -!I -~i-. ""4+> \ (.) t' '\ 0 l -8.0t r--1 I I) I ;-0 • 25e :> I i ' t i s """ ,_ ""4 l I " +>., '§ --- 0 r-. 1 ------Time (sec.) o.soo Figure 4. Foundation Input. error for thie problem. A ccmpari.son bet:waan the 6 method. and the o' aethod is made in Example 6 for a cubic soften• ing· spring. Example 7 treats the problem of a:lngul.arid.ea on the. phue•plane. while Jb:am:p1.e8 applies the. derived acaling law. !115!mP\e!, Figure. 3a is a model of a linear single-degree-of-freedom ayatem with viscO'l.18damping. The equation of motion is x +u/(x+c5) -=o where, The parameter values are mg• 60 lbs.; k • 60 lb./in.; c • 0.3 lb*E~.i. A constant increment of time• flt = 0.025 seconds, was used ao that the angle of arc is 68 = 28.13° for each step of the solution. Since the model is initially at rest, the boundary conditions are y . V 0- = - 1.2.7 2 If/. 0 = - - CJ) Table 1 lists the data for this problem. The first column refers to the 40 steps required to draw the. approxl.• mate phase trajectory in Pig. Sa. The second column lists the final phase velocities• vn • while the third column represents the input of the system. The fourth column lists the final value of 811_1 as calculated from the average pha$e velocity and the input for each atep. The fifth column gives the. final relative displacements, x l'I • As a cti.ak on the x" values, Eq. (12.2.7) which ia derived in the Append.ix for linear systems, was uae.d to calculate the tl!"t.Ul values of x11 • Tb.ue are listed in the sixth column. Figure Sb is the analog computer response curve, The points plotted on this curve are the values of x11listed in Table l. Table l. &eaponae of Linear Single.o.gree-of-Preedan Systea L\e• 28.13° ll Y11(tn.) l -o.as2 -o.410 -0.574 -o.s31 2 .0.294 -0.384 -0.440 -0.117 -0.814 3 0.213 -o.Jts -o.316 ..0.824 4 0.727 -o.258 -0.209 -o.s11 -o.s6s s 0.973 -0.211 -0.128 -o.1so 6 o.967 -o.173 -o.078 o.335 o.340 7 0.122 -0.141 -o.oss 0.1,1 8 o.s13 .0.116 -0.065 1.014 1.001 9 •0.158 -o.o,s -o.087 1.054 10 -o.ss2 ..o.01s •0.114 0.871 0.848 11 -o.ss, -0.064 .0.135 0.510 12 -0.932 -o.os2 -0.140 0.063 0.044 13 -o.794 .0.043 -0.128 -o.368 14 -0.481 ...0.035 -0.098 -o.6as -o.683 lS -0.075 -0.029 -0.054 -o.825 16 0.325 -o.023 -0.011 -o.762 -o.737 17 o.632 .0.019 0.028 -0.526 18 0.779 -0.016 0.053 -0.112 -o.143 19 o.741 .0.013 0.062 0.201 20 0.531 -0.010 0.052 0.527 0.530 21 0.224 0.038 0.717 22 -0.138 0.004 o.740 0.113 23 -0.458 -0.029 o.s90 24 -0.656 -o.oss o.312 0.271 25 -o.&96 -o.066 -0.024 26 -o.s1, -0.062 -o.341 -0.361 27 -0.323 ..o.044 -0.567 28 -0.012 -0.016 -o.649 -0.628 29 o.2sa 0.014 -o.ss2 30 o.soa 0.039 -0.384 -0.337 31 o.604 o.oss -0.107 32 o.ss, 0.057 0.186 0.213 33 0.38S 0.046 0.420 34 0.130 o.oas 0.548 0.532 35 -0.142 -0.001 0.545 36 ,·0.370 -0.025 0.416 o.s,o 37 -o.so2 -o.043 0.198 38 -0.513 -o.oso -0.056 -0.091 39 -0.405 -0.045 .0.286 40 .o.208 -0.030 -o.440 -o.432 I i' I y(inchea) .000 i -1.000+ I I 0 (a). Phase-Plane Conatl'UCtion, t:. e a 28.13 • Figur. 5. Graphical Solution, Linear Single•Degrn- of-Freedom System. • Results from Graphical-Numerical Method A Exact Results d I Q I I \ i ~---- ~----·----· -···-----+·-t··•·· 'i' Time t (seconds) 1 ' " \ / I l:, / ,o I ; / \ \ / \ \ I \ I I Cl.., ' \ I I ~-- \ l (b). Analog Computer Response Curve x versus t. Figure 5. Graphical Solution, Linear Single-Degree-of-Freedom System. ...,,, 2, fl.gure 3b ia a aodel of a nonluur d:ngle~•Of• freedom system with viacO\t.8 damping. h nonlinear coa- ponent ta a cubic hardetdng spring whose force d,isplaeem.nt 3 equatiou ia +10 = k x + J X • fl\e equation of motion is i +u/(x+-c5)=o where• 'f 2 3 •• S=zr:x.v + ~xw +_&.u]Z. The parameter values are 2 m o.z.o lb- 5eC, C ""'o. 7S Jl,.-sec. 3 = j3 -= 15,4 l~./in. _; in. In. ao tut, .. (; = 0. I CfI V + 0. I Cfq X 3 + ~°z. · An increaent: of ti:ata, b. t • o.025 $4'00nd8 was ued through• 0 out, so that ne = 28.13 • The initial conditions are The analog computer response curve. is show in Fig .• 6b. A.a a coraparison between the two solutions• the values of x fotmd. by the graphical•numerical Mtbod are. plotted on 11.g. 61-. 'Dle values of x found by the graphical• numerical 111ethod. are ehown in Pig. 6a. 1.000 T ! i \ i -1.000 0 (a). Phase-Plane Construction, 6 e • 28.l.5 • Figure 6. Oraphical Solution, Nonlinear Single-Degree- of-Pre.eclom Syatem. o Results from Graphical-Numerical Method A Results from Numerical Equivalent Method \e \ o/ 0 0 g ,_o p -0.50 \ \ 10\ I (b). Analog Computer Response Cu"e x versus t. . . Figure 6. Graphical Solution, Nonlinear Single-Degree-of-Freedom System. -1=11. · Plgure 3c 1• a llOde.1 of a t:'WO.d.e~-of•f'~ .,.at• with a aabic hard•iug al)ring between maaae.aai. aDd "2• The & equat1ona aft. t kil. o, = - -k- Xz. - -A.-3--&h Xz. + u)Z. I 11 I I 1'h6 param.ete:r valWUJ -.re m,= 0.2.o tb.-:se<:.,.. j In. R'a= 11.2. 11,/,,,,.j 3 oi. == o. s-oo x z. + o. z.n x z.. - x, . Slnce LJ,• Wz • 19.65 ra4.,/aec., the angle of in.ore- aent ie the •ame on both phaae•planes, i.e., t::.e,• t1ez.• /10 • Tbia problea waa solved. for two differa.t step intervala 1 da. • ti G • 28 .130 and Ae • 14.07°. The p'baae trajectoriu for tJ.e• .28.13° are ahown in P.l.p. 7a and 7bt 1'hile those !or tJe• 11+.01° are shown in ftga. 7c and 74. The analog computer phaae trajectories ara ab.own in 11&••7a anc1 7f, ad the associated response cmrvu are aholm f.n Pip. 7g and 7h. The values of (x,)I\ Gd ( xi.)11.boa the grapldcal-nae.rical solutions are plotted in ft.gs. 7g and 7h aa a com.parieon of the resulte. x1(incbea) \ YJ_(inohae) -1.000--+-- 20 0 (a). Pbaae-Plane Conatruction.• A8 1 • 28.15 • Figure 7. Graphical Solution. Nonlinear 'l'wo-Degree- Of-Preedom Syatem Witbottt D•ping. 1.000 2Cin- - chea) -- 0 (b). Phaae-Plane Construction, !),. e z. • 28.13 • Pigure 7. Graphical Solution, Nonlinear Two-Degree-of- Preedom System Without Damping. -so-. 1.000 \. I I J \ / y,\ ' \ \ v1 Cinchea) -1.000 I 1.000 \ \ \ \ 1 \\ i \ \ \ -1.000 0 (c). Phase-Plane Construction, f::.e 1 = 14.07 • Figure 7. Graphical Solution, Nonlinear Two-DegrM-of- Preedom System Without Damping. ---+ ------ 1.000 ,/ 1.000 / ) I _ I 80 ------+------ 0 (d). Phase-Plane Construction, 6. 02 • 14 .07 • Figure 7. Graphical Solution, Nonlinear Two-Degree- of-Freedom Syatem Without Damping. •52- \ 1.000 \ l \I I i j ' \\ \ \/ \ I ! 1.000 \ ! \ \ / (e). Analog Computer Phase-Plane Trajectory :,;_-v1 • Pigure 7. Oraphical Solution, Nonlinear Two-Degree-Of• Preedom System Without Damping. ~------,/_ ------r------~~- // /,,-- 1.000 ' -~', ( ""'" ' ' I\ \ \ ' 7 1.0 -~---- -1.000 ------Cf). Analog Computer Phase-Plane Trajectory x2•V2• Pigure 7. Qraphical Solution, Nonlinear Two-Degree- of-Pree4om System Without Damping. 28.13°. 14.07.0 !.-."\, 0 \ {!/ G \ i 1.000 .o I I 4 Q ®· 0 \ \ I I \ \ \ 0 \ 0 I I I J, loOOO I Time t (seconds) 0 0 \\ 'c b c, " -1,,000 0 0 0 I 0 VI ,i::- (g). Analog Computer Response Curve~- t. 1 Figure 7. Graphical Solution, Nonlinear T~o- Degrees-o!-Freedom Without vamping. . 0 0 ·• A9z. • 28.1, • 0 A t.0 2 : l'+.0'1°. I !I / I - 0 i 1 "o~~---~-----4------+---~----.::...c,..._ ---...... ______-+----+------+-----4.------~ ,/ <1 t,.' / ; 0 1. ', / 1<> 0 \ 6 A \ Time t (seconda) ("I i\ I 0 I :,) 0 0\\ I Q \ ! Q Ci·•• f \I \I \0 VI VI• l I \ \, I f . I ' \L .....(ti).. zj'plog.Coaputer Response ~-- Jtrt. \.I (11 Figure 7 •. Graphical Sol11tion, Nonlinear Two-Degree- \ 0 " ot-Freed• Syatu Without Daapi.D.g. \ I \ij' 1'!1!12\~,4. The nonlinear uo~f-freedc:a syatem ab.ownin ftg. 3d waa eolved for the given iuput. Thia ayst• ia the smt\e aa Esanple 3 with the addition of 4-piug. '!I.MIL J equationa are (.l, 3 •• <.J 61 -= - Xi. - Xi. + ..¼.. + V, - Vz. '11 ~. w;-- m,w, R, 'Dae parame.tv values are the aaae aa :ln Bxaap1• 3, and the daping terms are lb. - .sec. • /),_-:; ec. 0. 7 S --,-- J Cz..=O.Z5 -- c, = i11. in. Hmlce, o,= -o. soo Xa - 0.10 c, x: +- y/tJ,2+ o, I "I I V, - o,o 52 = asoo X2. + o.z.99x: - X, + 0.191 Vz. - o. t9l v, • As tn Example 3, the system waa aolftd for two differ• ent thle intC""lala. The phase trajectoriee with A t3 • 21.13° are sbotm. in P.tga. 8a and Sb• while those with 6e • 14.07° are ehown ta lip. so and. 8d.., The graphical aolutiona ftom the anale>g computer are shown in ftga. '*• Sf, 8th and Sh. Once again the valuet1 of ( x, >n and ( X2) 11 ft"Olll the graphical•maer:f.cal Mthod _.. plotted on tbe computer result• in Ff.gs. Sg and Sh. x1 Cinchea) 1.000 -ss- •2 (inches) ~l-~ i /___.+---.___ \ / : "~ \ I • \ \ / \) i // / \, '. I ?\ : . / z )'!40 / \ ~-- , ------i' ,_....----,- ~/ -1.000 j 0 Pbaae-Plane Construction, 60 2 = 28.15 • 1'1.gure a. Graphical Solution, Nonlinear Two-Degree... of- Preedom Syetem With Damping. -1.000 1.000 , ' -1.000 0 (c). Phase-Plane Construction, .60 1 = 14.07 • Figure s. Graphical Solution, Nonlinear Two-De.gree- of-Freedom Systems With Damping. -60- - r- ·---- \ / \ V2(in.) -1.000 1.000 / '• / j ( I '80 l -1.000 0 {d) • Phase-Plane Construction, 6.0 2 • 14.07 • Figure 8. Graphical Solution, Nonlinear Two-Degree-of• Freedom System With Damping. Xi(i~hea) i i 1.0001 ' I ' I ---- / -1.000 (e). Analog Computer Phase-Plane Trajectory x1....-1• Figures. Graphical Solution, Nonlinear 'l'wo•Degr.e.of- Preedom System With D•ping. / ( \ / - ;-----,,~\ \ { \ \_\.i \ v2 Cinches) --1-.-000------~;::-~; ------....i-- - -,' .ooo /) ; ---~l---·--______--, -1.000~ (f). Analog Computer Phase-Plane Trajectory x2-v 2• Figure a. Graphical Solution, Nonlinear Two-Degree- of- Preedom System With Damping. 1.000 --~--i, t,.01 28.13°. \ e = 9' \ 0 a A 6.01 = 14.07. C, / c_ 0/, - .'B- ,:x" ' ' l. ' !-,.. c, I 0 .' j/ 0 "I .\ D C \ \ :, i I '! 0 00500 , \ f Time t (seconds) \ 0 0 ,I ,· /, J t:, ., l: - - "' \ 4 0 (g). Analog Computer Response Curve Xi-t. -l.000 Figure 8. Graphical Solution, Nonlinear Two-Degree-Of-Freedom System With Damping. I O'\ u) I e .6.Sz. = 28.13°. A f\ = 14. 07°. i I I i \ \ I " I I 3/ I / • I -d gl t:, " I -N I H t>O II d 6 0 '\ "'- 00500 0 1.000 0 Pt ' ). I) I Time t (seconds) I i I 'c I 0 \ ,b 0 I h (h). Analog Computer Curve x2-to -loOOO Figm-e 8. Graph1eu Solution, Nonlinear Two-Degree-of-Freedom System With.tt«,ipipg. -65- __ ,. §, The numerical equivalent procedure of the graphical- numerical method wu used to •olve Example 2. A desk calculator was ued for this solution and all nut11be.rsware carried to fi•e dec1aal places. The solution equationa ae X'I+, = X,, co.:swh + Vn si'I li>h - J,, ( t - Co.:> wh) V,-,-1-,= - X111si11 wh + Vn cos w h - c)n sin wh. Since lit • h • 0.025 aeconda throughout the solutiou• t:hu• equation.a become. ,<114-1= 0.88178 x., + o.4'1/r,,r,, V11 - o.1182.2. 6n Vn+1 = -o.4 '11Co" X., + o. 8 8 I 18 'tin - 0, 4'7 /(,,~ J".., where. 611 = O,o'/S-44-(V,,t-,+ V"')+ 0.024-94(-X,,-,.,+ x11)3+ (to/L;/-)r,. !he ruu.lta of tbi• method are plotted in ft.g. 6b only lfhere there ia a noticeable differenee from. the •aluea of 1rta.in Bam.ple 2. !i!f!!lle 61 A aimple. systea with a cubic •oftem.ng spring attached to a rigid foundation in free vibration is one whose tr&• jectoriea• lying outside a critical zone called the aepara• trix, show a no'ticeable reversal of curvature. A portion of one. of these phase trajectories was solved graphically by laying off the centers of arc along the V•ax:l.s. The. equivalent trajectory was also drawn using the x-axis for the cent6re of arc. Au equation of motion for the aingle-degree•of-freedom. system with a cubic aoftening spring is "3) X.. +w 2.( X-)( -=O so that, Figure 9 shows the •eparatrix for this syst-.. I Curve A was obtained by the method so that the centers of arc lie along the V•axis; Curve B was vbtaine.d by the o method which usu the x•axis for the centers of arc. Jen \ trix / VB II I / // / r---\_,,, Cum A_,; · /. • Calculated •O Coordinate Phase-Plane Trajectories, Cubic Softening Spring. I Figure 9. Comparison Between J and J Methods. l!aelt 7a, Let tha uonlin.ear spring for t:ha m.odel in P:tg. 3c be a cubic softening spring, ao that 'l'be equat1ou of motion for the autonom.ou.asy•t:• can be written. ae dx, v, d Xz. = - Vz.. -- -: - X2 2 dv I X,+c5 1 d Vz. +c5" C -~ 01 h = nit bz= - Xz - j 1-z..X 1+ ( j; + ;7-)tz.1_ '!he three caaea for ainpl.arid.ea are now invud.gated. v,-=0 Proa lq. (6.1.1), 8, = - __:fu_= - x, R, > -t-2.1= R,X1 • Sub.-tf.1Nt• thie reault :tnto Bq. (6.1.2) and •~ x2 • • 62.. Bence, -o ~xR-z.. I - 80 that V - 0 + ~2. • "Z. - ) - -yfiz. For this case, both maasea are in a state of equilibrium.. If xp o. there i• no energy preaent in the ayat:a., and tbia ia a state of atable equilibrl,a. If •2 • !. , 1:he energy in the ayat• ia k;j,.p~ , and thia f.a a ata'lte e>f uutable equiltbri\11l. flma, a alight disturbance of eithe'r mue would, f.n geueral, start the ayet• mori.ng--, froa the singular pointa on the pbaae•plane.. Oaae II. JCi• • 8, , •1 • O. Since the fO\mdation. dou not moftt ,- 0 • o, an4 mu• 81. ia ta a state of equilibrium.. From &q. (6.1.1), 8, = - + = - x,. !heref'ore, f21 • ki.Xi• lquat:lon (6.1.2) reduces to C 132. 3 Oz.= - -;::;;- Xz.. 11.gu.re lOa ia a plot of the nonlinear spring force u a function of relati:n displacement. If line aoA re.preaenta the linear spring force ao that k1 > ~• then the origta ia the only coordinate where f 21 • ki•i• l!or thia a:f.tuatf.on, ICJ.• x2 • O. Since •2 ia not necusarily aero, all the energy in the syet• is a function of ( Yz.)z. • Since .... •2 :la not in a state of equilibrium (v.nl .. a v2 • o>. the avail.a• ble energy of the system will move mus m1 fraa the aingll- lari ty. ;~ If x2 < - -Vj32- and Jci< k2 • line boB in Pig. 10. showa there are three possible. roots which aatiafy the requirement that t 21 = k1XJ.• L:Ucewiae, if x2 > -t _;~:, then k1 111Jat be negative as shown by line coC. The.re ia energy in tha eyst• tor any of these. roots, and mass Iii. will not remain in equilibrium. It is noted that as kt approaches aero, fn approacbaa zero. In the limit• the system reduce.a to a two mass system with a ll<>nlinear coupling spring. Case 111. x2 • • Oz. • .-2 • o. Maas 1112is not in a state of equilibrium ain.ce these. condition.a require the relative acceleration and velocity to be. zero between masau ~and",.• Equation (6.1.2) yielda, ( ~. X, "'tz.1 = _..;__'-- I+~ ma.. Since ...!!!1_ ;,, 0 m,._ where, Ki • I +k~ < k, Equation (6.l.l) yields 0.I = - The. requireraeat that f 21 a: Jlix-i ia similar to Oas• 11, exc_,t that the force of the nonlinear apring 1• ao longer equal to the linear spring force due. to the adjusted spring oodficient IC.1. Figure 10a ca be Wied in the aaae aanaer aa in Caae II. lf Ki.> 1c2• xi. • x2 • o. However, Yl 1a ·not IMlCuaaii.ly equal. to zero• and the energy in the ayat• "111 generate a proper uajectory away froa th& aingul.arlty. As in Case 11, there are tbr• ·possible. roots if -a ± 1~: , K.1 Jr.a,a\Ul if •2 > ± -{-ff:, lt1 amat be negative. the enuo present in the eptem. will likewiae geuerate a proper trajectory aw~ frca the aingularit:,. -72- A I I B X (a). Poree-Displacement Ourves. Figure 10. Singular Points. -!!J!:!Rl• 7b, 'lwo problems were chosen to examine the behavior of the graphical-numerical method and the. 'l\'Umerical equ.iva• lent procedure. in the neighborhood of a singularity. The first example ia the cubic softening spring of Example 6. Starting at x = o.700 inch.as on the. saparatrix, the saddle point (o, 1.000) waa approached using the J and J' graph• !cal.numerical methods for {).x itltervals of O.oso inches. Starting from this same point, the numerical. equivalent procedure at ae• 8 0 waa used. The results are plotted in Fig. 101>. 'lwe.nty•three steps we.re required for x > 1.000 in the numerical equivalent procedure. Initial conditions at x • 0.900 inches on the separa• trix were al.ao selected, and the numerical fiQU.ivalent pro- o cedure was applied for 60 = 2 .. It re.quired. 117 step a for v < 0. About 15 steps per hour were completed using a desk calculator. The second problem is similar to the first with the addition of damping in the model. The parameter• are ITl = o. z.o lh.-_sec..~ j k = 17.Z lb./il'l. j In, 7 lb. c = o. 6 .-sec. . _/3 = 30. B lb/in:. /11, , Carrying all nmabers to six dewimal places, the J equatiOI\ is S = o. , 9 a 8 7 o v - o. 3 c;8 '1<:,4 x 3 • The coordinates of the. three singularities are V -o- )· X _,....,-u) +- -)'--•w Figure lOc nows a few of the trajectorie.s for this system.,. A part of the aeparatrix has been found approxi- o mately by the numerical equivalent procedure where ne • 2 , and. all 'RUlllbers -. carried to six decimal places. The. coord:.inate of the singularity is v • o. x • •1.583191 inehu. To start the mass moving, the initial conditions were Ar• bivarily selected as v = 0.001000 inchea and x • -.l.583000 inches"! Thaae initial conditions are sti11·1u tb.e inraediate vicinity of the singularity and cannot be visually detected on Fig., 10c. After 25 steps in the iteration procedure.• the coordinates were v = 0.002112. • • •1.581731. From steps 26 through 57 • t:J.G• 3°, and from. ate.pa 58 through 122, Ml• 4° • Approx:J.matel.y 12 points were computed per hour. x(inchea) 1,.000 0.900 o.soo . A 0.100 I A Graphical o Method 0 Graphical 0.600 _ v( !nc_bea,) o.300 (b). Phase-Plane Trajectories in the Neighborhood of V = 0, X = 1.000. Jtigure 10. Singular Points. x(inches) f I 1.000 I I I \ I \ r I ' -1.0. \ t l.000 Separa- trix-~.,,,(· (c). Phase-Plane Trajectories, Damped Cubic Softening Spring. Pigure 10. Singular Points. 1£ats sa, Aa 1:he firet of two e:uraplu of the •caliug law, COD.• eider the liiuaar ayet• ehown in Pig. 3a. Th• equation of motion is X + Wl. ( X + 8) = - Yo '!be initial conditions are ) . '\ . / ) X Co -:::. O ; X C01 = - Y0 o . Pora aiad.lar ayat•• let the equation of motion be Where, r 2A ~- a= ..a. !'he. initial conditiou are -;(o) =o j s'(o)= - 1/co). 'l'he following reatrictiou are imposed: Al.so, 8 - Ll .. that SiMe the scaling law equates s and x, =- f7 c/ X • cf'( dt 1'he.refore. As a specific numerical problem, let the following parameters be chosen for the model. system: mg • 60 lb.; The input is selected as a step change in foundation velocity a.a shown in Fig. lla. He.nee, w • 19.65 rad./sec. and c< = o.0491. Using llG = 28.13°, the phase trajectory ia plotted in Fig •. ,llc for the first 12 steps. Suppose a prototype. system is subjected to the. founda-, tion input as shown in Pig. llb. Here n = 1/3. Ectuat:flon (6.1.4) repraae.nts its equation of motion, that is, <.. ,, ,S It + 2 A ..f2 ~- I + .fl. -= - 'lo • If t A = o< = 0. 0 4 9 I Jl.-== nw = 2. "1.. '- Since S2_ = n w , therefore, Also, A = o( or _L = _L 2. MJL z mw Therefore, C =~-3C,n - . If the prototype spring constant is selected equal to the model spring constant• the new mass ia nine. times the model mus, and the dampip.g term is three times the mod.el damping. Case II. Sele.ct Ka: nk == 1/3k. Therefore, M = m/n • 3m. and C • c. The a pring constant of the prototype ia one-.third of the. model spring constant, the mass is three times the model mass, and the damping constant remains the eam.e. Case III. Select IC= a 2k • 1/9k. -so- and C •no• '1/3c. 'fhe spring conatan.t of the prototype is one•ninth of the medel spring constant, the mass roa1ns the aaae, and the 4-pi:ng COl\Stat pi.:~•third of the model damping constant. -SJ.- Yo! 25[ ____ ! I ·--·---·---··---.J __ -···----·---·-- 0.05 t(seconds) (a). Model Input to • -.,0 • ....d -ii,, ....,4,) 0 ,0....., > C: ....0 +.) 3 -3C: ::, 0.15 t(seconds) (b). Prototype Input. Figure 11. Scaling Law Examples. x(inches) I I \ ( \vein.) ' -1.0® 1.00 \ / / // . ·--•-· -----· (c). Phase-Plane Construction, t:. e = 28.13°. Figure 11. Scaling Law Examples. l!MPJ.! 8J?, As a second example of the acaling law, consider the two-degree..of•freedaa system shown in Fig. 3c which ia sub• jected to a given. foundation velocity. The equation.a of aotion are where, 6, = - X1- - R, 3 X1- - rn..... (-• _,) s" 3 w,._ 82.= - X1- + l'Yl + m._ =-:~-...X.,_ - , <-...X 1 • m, I ~.__ ~- The initial conditions are: 0 X.1-(0) = ) Pora siad.lar sya•••the equations of motion are K = - __&_ _ B,._ 3 I K, '::>2 K, ':::,z. - M '2..?+----- ( I + I ) {3.,_ 4- -M <-:,1- fVI, M2. fl..._ = I z. The initial conditiotua are ( I :; 1 ( 0) == - 1._o( 0) S~Co)= o . The restrictions are now imposed upon the new system.. Le.t .fl. u) . I = n I ) Jl -z..= fl c.Jz.. It was noted in E)caraple Sa that the physical parameters of the prototype system could be varied in several ways. Three particular cases were developed based upon. a given selection of the spring constant. For a multi-de.gree-of- freedom eystem whatever .84le.ction is made bet~n a apring constant in the prototype and the corre~ponding spring constant in the model t£\'U.-Rt al80 be made for all apr:lngs in the prototype. For example, in the. present problem, if K1 • nk1, then K2 • nk 2 and Bz.== ny2 • The same statement ht)lds true if the masses are first selected instead of the spring constants. Por exanple, if Mi.• m.1, then M2 • m2, JS'IO.Bz. ~nyz. If thase conditions are satisfied, then c _ Ji: • .J_ _ I .. 0 z.. - Uz.. 1 .. - . •. .. fv1 - m, I - n.._ lb.ere.fore, - k-.. G 3 ~. = - s-..- s.:-= o, The forcing function is the same as in Example Sa. The amplitude of the input ie multiplied by n.and the tiM interval divided by!!.• Hence, $, Cf) = X, { t) s,(!)= X2(t). As a •pacific problem, the following parameter& are selected for the modal system., lllJ.I • 60 lb. i 112g • 30 lb.J kt. • 60 lb./in.; k2 • 30 lb ./in. ; fiz. • 6 lb./in., 3 • The input is shown in Fig. lla. 'tha angle of arc for 0 each traj•tory is 28.13 • The trajectories for the. prob• Leta are shown in Pig. lld. As in l,tample Sa, let the input for the. prototype system be given IJt Fig. llb, so that n = 1/3. The relative displacements for the prototype system are related to the model system aa s,(3t) = x,(t) $ z. (3 t) = X 2 ( t). -86- X( in.) ----- 1.000 \_ \ \ -1.000 J (in.; / / --~~------.-J.--..-r-~-~/- -1.000 / -· --· ·------ 0 (d). Phase-Plane Construction, 6.e • 28.13 • Figure 11. Scaling Law Examples. VII. DISCUSSION OF RESULTS In the preceding examples, the. construction of the phase-plane trajectories by the graphical-numerical method required a protractor, an engineer's scale, a aet of tn.• an.glee, and a com.pus. A desk calculator was used for computing the numerical value of the. J '• • A scale of l inch= 0.-400 inches was selected for each phase•plane coordinate ayatem. so that the invutigator could read a coordinate to within 0.002 inches. Bence, the construction of a circular arc ae.gment for each step required that the trial valuea of the &converge to within!. 0.002 inchu. The. protractor, which was used to measure the angle of increment,~ e • ia calibrated to o.s0 • An angle can be read approximately to within 0.2°. The solutions for the first four example problems in the preceding section we.re obtained by the graphical- numerical method. The input for each system was represe.nte.d a.a a foundation velocity from which the foundation accelera- tion was approd.matecl by the theorem of the mean of differ- ential calculus. The atep•by•step construction of the phase-plane. trajector:f.u was obtained for constant incre- ment• of time throughout each problem although the incre.• m.ent can be varied if ao desired. The linear system of Example l served two purposes. Equation (12.2..7), as derived in the Appendix, provided the exact solution of x in the problem. The values of x were compared with the results obtained from the graphical• numerical method and the analog computer. This exact solu- tion provided a cheek on the results of the analog computer. Figure Sb shows the response curve of the analog computer. The results of the graphical-numerical method are plotted at increments of 0.025 seconds. The. values of x were com• puted by Eq. (12.l.7) at intervals of o.oso seconds and are plotted on the figure where there is a noticeable. differ• ence from the response. curve. The greatest difference between the exact solution and the computer curve occurs at t • 0.500 seconds and t = 0.750 seconds; this difference is approximately o.033 inches. Otherwise, the. values compare quite favorably. The results of Example 2. shown in Pig. 6b, conform rather well with the computer results. The graphical- numerical solution was obtained rapidly since l\G = 28.13°. In all phase•plane solutions of these example.a, the step• change. in foundation velocity at t • o.s seconds was taken into account by the small horizontal line at n = 20. Exam- ples land 2 generally required two trials for each step in orde.r that the o converge. to within -+ 0.002 inches. Approximately eight points were plotted per hour for i'«ample. land six point• per hour for Example 2. The undamped nonlinear two-degree-of.freedom ayst• of Example 3 was solved by the graphical•numerical method for A e • 28.13° and 60 = 14.07°. lt ia observe.cl that in P.l.ga. 7a and 7b, where 6e = 28.13°, the phaa• trajec• torlu haft large slope. diacontiauitiea bet:ween steps at certain region• throughout the solution. To improve the de.gre,e of accuracy, the aize of the ste.p increment was re• duced to A0 • 14.07°. The improvement in the. graphical solution can be seen in Figs. 7c and 7d and by ccmpari.ng these. curves with the computer trajectories shown in Pigs. 7e and 7f. The reaponse. curves of ft.ge. 7g and 7b also show how greatly the accuracy of the solution is improved 0 · 0 for a.e a 14.07 • In the case. of A0 = 21.13 , th:ree trials were generally needed for c5,and dz. to converge in a given step, while two trial.a were generally the rule for Ae = 14.07°. Approximately five steps were completed per hour for each case. . ' ~damped caae of the.nonlinear two-degree-of- freed.oa eyatem in Example 4 ·was sol.ved by the graphical• . "'0 numerical method for /le = 28 .13° and 60 • 14.OJ • once ' again, the. elope discontinuities of the phase trajectoriea ware reduced in magnitude. for the smaller step increm.e.nt aa observed. in Figs. Sa, Sb, SC, and 8d. The computer phase trajectories in Pigs. ·seand 8f are also included for further comparison. The values of the responses ( x,)" and ( X2>n are plotted on the response curves of 0 l'iga. Se and 8f. As in Example 3, for 60 • 28.13 , three trial• were generally required for cS, and d2 to con- verge aa against two trials in the cue. of tl0 = 14.07°. Approximately four steps were completed per hour for each case. In order to find _the drawing error in the graphical• numerical procedure, the numerical equivalent method was uae.d in Example 5. The nonlinear system of Example 2 waa solved, and the numbers we.re carried to five decimal plao••• The results are plotted :ln Fig. 6b which shows the close agreement be.tween the two solutions. It is noted that only those points were plotted which showed a 1ignificant difference. A comparison of the o and 0 1 methods for constructing pbaae trajector:Lea was made in Example 6., For the cubic softening spring, a trajectory which exhibited a rewraal of curvature WQ8 selected. OUrvea A and B in Fig. 9 were plotted by locating the centers of arc along the ,,..axis and the x-axis• respectively. The rays, which were drawn £or each CU1""V'e, point in the direction of the centers of arc for each step of the construction. It :ta obac-ved that tha rays for Curve. A lie on both aides of the trajectory .. that ia, during the first eight steps, the ceter of arc approached plus infinity along the •A•axia. For the re- maimer of the conatruction. the centers of arc were located ln the ne.gative direction of the vA-axia. By integrating the differential equation of the cubic aoftening spring, the true values of the trajectory were calculated, and these have been plotted in Fig• 9. Por this particulu example, both trajectories of OUrvu A and B lie cloae to the calculated value.a. However. there. is I a disadvantage with the c5 method. As the elope of the. trajtactory t1pproachea zero, the diatance along the •-axis to the center of arc approachea infinity. fld.s can be aeen in Pig. 9 by the rays of Curve A. An investigation for aingularitiu of a two-degree-of• freed.om. system waa made in &xam.pl.e. 7a. For the cubic softening coupling spring, three cases were studied for singularities and the associated state of equilibrium. For Cue I, both maasea are in equilibrium. and remain eo if x2 • O• while the. system ia unstable if x2 • !. .J,: due to 1:he potential energy in the coupling nonlinear eprf.ng. Oases lI and III are similar to each other in tbat_the force in the nonlinear apring ia equated to a force associated with the linear spring. ln each case. a proper trajectory ia generated away from the singularity clue to the energy pre.sent in the system.. The beha'ri.or of the graphical-numerical method and. the numerical equivalent procedure in the neighborhood of a aaddla point singularity was studied in Exaapl.e 7b. The graphical construction become.a tedious, if not impoasi• ble, since the radius of arc approaehe.s zero. The mmieri- cal equivalent me.thod was used for a cubic aoftaning spring with and without d.eping. I'or both aod.ela, th& number of steps increases considerably as the trajectory approach.ea or leavu·the f11.1111Miateneighborhood of the singularity. 'l'be acalin.g law was applied in Exaaple 8 for two part1• cular cues, namely, a linear single-degree-of-freed.om aystea and a nonlinear two-degre.e ...of•freedoa system. In Bxample Sa, the scaling law for a linear damped single-d.egrae• of•freedom. system equated oi and A. Tbree possible aelec- td.ona for tha spring constants, k and K.,showed how the. muses am damping constant a could vary. Por the nonlinear ' t:wo-d.egree-of-freedoa ayatem of Example Sb, the scaling law requirad that similar selections between spring con- stants be mad~, that is, if K1 • nk1• then Ka• *2, and a• iy.3. ltke.wiae, if the masses are ff.rat aelacted in8tead of the apring conetants, that is, if Mi• lllt• then~= •2• and B • n~ • VIII• CONCLUSIONS If the equations of motion for a nonlinear one ... or two-dagree•of•freadom ayatem are reduced to the standard delta form., a.a .in Examples 1,2,3, and 4, the proposed graphical-numerical method is readily adaptable for finding the transient response to a given input. Por nonautono- m.ouasystems whose input is a foundation velocity, the corresponding foundation acceleration can be approximated at finite time intervals by the theorem of the uaa of differential calculus. The graphical-numerical method ia easy to apply, and the time re.quired to solve a problem depends upon the number of variable terms in the S equation and the degree of accuracy desired. Bowe-.er• from the experience g~ned in the example problaa, it can be stated that the proposed technique of uaiDg oo- 1 from the previows step aa the first trial of 611baa been useful for obtaining a rapid convergence of On • The following table lists tthe. approxima:tte time required to solve each of the first four exaples. !M!l?!el M!J1Ul•ElllC£!1!D:t, l!MrU! Ste.Jae UM e. At any particular ate.p in the solution, the incre- ment of time can be changed readily. f. The degree of discontinuity of the slope.a at the. commonpoint between two sueeeasive steps of the solution provides a qualitative indication of the error involved. Diaadvap.tyee; a. The method provides a step-by-atap solution. b. -lf an error of computation or drawing is made, it is carried throughout the. remainder of the solution. c. The. graphical construction becomes rather tedious as the. radius of arc a.pproaehaa zero. d. A drawing error does exist although Example 5 showed it to be. quite small. •• The. error involved is only qualitative.. The smaller the incremant of time• the smaller ia the error. The advantage• and disadvantages of the numerical equivalent procedure. are summarized. Advantages : a. Since only a desk calculator is used in this method, solutions are obtained some.what faster than by the graphieal.-numerical method. b. The coefficients of the numerical equations are the natural functions, i.e., the sine and cosine functions, for the given systems. c. Once a time increment is selected, the. coefficients of the numerical equations are. readily found and remain constant until the time increment is arbi• trarily changed. d. The time increment can be made as small as desired. Disadvantyea: a. The. error of the solution ia qualitative in that an unusually large difference between two consecu- ti va S •s indicates the equivalent of slope dis- continuities on the. pbase•plane.. b. It is not as readily adaptable to a change of increment as in the graphical-numerical method since the coefficients of the nua.e.rica.l equations change values. However, this is a minor compu• tation to make.. A comparison was made between the ~- method and the ~I u method in Example 6. Thia example showed that both methods yield the same order of accuracy. However, the .,,,,97.,. I cS aethod has the one disadvantage that aa the slope of the pbase•plane trajectory approaches aero. the distance along the. V•axi• to the center of arc approachea f.nf:bu. ty. I 1'be cS method might It& uaeful were the phase trajectory in the tS solution ti"anafo'l"ll.8 from. a concave shape. to a convex shape.. During this latter portion of the. solution cune, the arc ••P.eut•could be drawn fro• centers of are along the v•ad.• on a eeparate sheet of paper. Siugul.arities for t:wo-degree.-of•freed01a ayatema have been discussed for three special caaea. Oaae 1 ab.owed that each mass ia in a state of equf.libriua and is stable or unatable depending upon Whether there ia energy pruent in the ayat•• Cues II and Ill d.efine a singularity for one of the first order differential equations, and energy present ia the ayatem generates a proper trajectory away froa the a:tngu.lui ty in each case. the graphical,.,..maerical 11atho4 loaea mu.ch of it• value in the neighborhood of a singularity. Howeftr, tb.e maerioal equivalent procedure can be used to obtain the. coordinatea of a proper trajectory in tha neighborhood. of a singularity. The. conditions of.the scaling law have been eeta1>1Uhed. and two example problems have shown how to apply the method. The scaling law adda a certain measure of generality to the aolutiOD.S of nonlinear single• and multi-degree-of•freedoa system.a. Aa further solutions of nonlinear probl.,.. be.COIie a•ailable, the scaling law could be uae.ful fos:-atud.ying ~he. •olutiana for any maber of similar ayatema. IX. ACI The author extends his gratitude to the member• of hi• Graduate Oomm:1.tteefor their guidance and critici• of t:hia work, and in particular to Professor D. Frederick Thesis Adviser. Aa a graduate student at VPI, the author received a ~te.4 States Steel Foundatf.01\ Fellowship which enabled him to carry on hia studies full time. This vaa aoat help• fu.1 in completing the course work. The. author ia grateful to the u.s. Naval Research Laboratory where the tbuis waa written with the cooper&• tion. and encourag-.ent of Dr. a.o. Be.laheim and He-. GJJ. o•aara. '1'he author acknowledgu the assistance. of Mr. W.,P. DeWitt at1d hia staff who provided the analog oom- pu.ter solution.a. X. BIBLIOGRAPHY 1. ~•. A.•A•,illld Cbai1d.Ut lfttJ.15'111• Prinoetoa 1 Raw J'erM)' a .PrtnoetonC•l••w ._ .· . ·. •• 1t4t, 244-152.., 2. AYft• a.s.. ~ient Vf..l>rationa of . Lt.near >l&l.. · t:1-i>e.··.· ....- of•Freedoll Syat_. bV .1:l\4Pbaae•Plane lfe.tbod." .J.. ID&1i"1IMS•• us (2) (I'....,._., 1'52), 153-lSC- 3...... _ ••• a.a., aud. Atchiaon 1 s.c., "A Graphical. Jathod f• tu AU,1.yida of Nol\1.f.near Syst-.. adw Al"bitra.J.'7 tt-anaien:t Con41tiou." David Taylor ll'Jda1 Ba&tnlepo:rt 9tO ~b 1957). 4. Biahop1· a.&.D.1 ''On the Graphical Solution of Tre.ns:l.et\~ 'flwat on Prob1.w." llwll, 11!£k.t ,IUEf« lJla•• 168 (10) (1.tst.), 299-322. s. ~~•B• • n.Analyaia of Roa1iuar s.ivoa by fhaae- .·· . · ' ta Mat:bod•" J, f£laMa JMI•• 257 (1) (.tan• 19Slt), 37....48. ,. &Iman, J.• L.!I "Orapbf.cal SoJ..utiea 'l)f sfau.ltal\UUII .Secctt. 10,. 3acobMn 1 L.s., '10&1a OJ:&1Phleal Method of S01vf.ng kCOb4• Order Oldtt\alT Ditfere.ntial lqutions by Pbaae•Plane Dtaplacementa." :l&-,M!21,.ltlla•,19 (-.) (~ 1952), 543-553. 11,. Jacobaeu, and Ayre.1 a.s., .~· •--• New Yorki x..s.MC-Orav.Htll B<..."'<>kCo.~~~~;:·~ 12. I 15. 18. Pell, W.H ., "Graphical Solution.of Single•Degree-of- l'reedom Vibration Problem with Arbitrary Daiaping and Restoring Porcu. 0 J 1 5pl,, Mach.•24, (June. 1957), 311-312. 19. Scarborough, J.B., ~cal: Matbeu1;12z·J•is, Baltimore: The Johns Hopkins Press, t • • 1.i. , ' 20. Stolt~ . J .J., MeDtPf! ;tJt•5i&in ~ft ftl'?"ii!lt?"•£a!•ew~or.,~n.tarscl~~sere, 21. Jb&d., pp. 40-45. 22. Synge, J.L. and Schild, A., Tenso£ 9§!,c::ulll§•Toronto: University of Toronto Press, tf41: Takei, . K .. , "Consideration of Vibi:-ation Systems by Phaae•Pl.ane Displacem.ents. 0 Part It Si:; "ff:iRea. lnagg T ohop ~v,, Japan (B) 8 (lJ ( e 6J', lil- • The two page vita has been removed from the scanned document. Page 1 of 2 The two page vita has been removed from the scanned document. Page 2 of 2 -104- APPINDIXXll.l Of the matbods referred to in The-· R.e't1iewof ~•- t:ure, •• work (6) is the moat general because it ia. applicable to aimultaneov.a second order nonl.ineuu: Giifer,. eatial equations. '.l'hia method is.presented in detzail. ia order to compare it with the graphical•maerical method of this investigation. Additional cOblp&l\ta on the other pertihnt: graphical techniques are alao made. l!!e•• ••~ (62 The aecond order differential equations are reduced to dx.: F- - = - ___:_j_. dx· x.. ' ( The. graphical construoticm procedure on one. (x,x) plane :la pNSeuted for simplicity. Jrquation (12.l.l) beooau lf •• x, and Fare to be. plotted to the. a-.e acale, it ia naceaaary to incl uda their raapective aodul.ii ,· H x , Mx. , and M F in 'the equatiou that define.a the slope. Wilen thia :la done, 1q. (12.1.2) b6coaea The d.etd.i.: of the slope.construction for a single variable are ahown in Fig. 12. Starting with XO•Xot and To , it ia daairad to find Xp , x" , and T P • Firat, the CU1"YU de.fining F are drawn and the scale modulii, Mx , M,. , and MF, ·u-echoaeu thue determining /3 • Then tba value of Fa i• detand.ned from. theae curvu • lb.a length, F.,, ia laid off horizontally to the left when posit:l.,,., or to the right when negati ..,., of point R which ie located a diaunce Xo/J3below point o. From the figure, line OP conauucted perpendicular to OQ'ia the duind slope since dx ==' _ Q'R' = -}3 _E_ _ clx OR' x Next, pointP ia chosen on line OP. Thi• ut&bl:lshea • Xp and xP to be used in th• next: atap of the solution. The time, Tr, ia foun.4 from the nlatiOl\ 1 -, Xp - Xo Ip - I., = (.Xo +-Xe.. ) . 2... ii. ccm.atruction for Tr- T0 ia shown in Pig. 12. A line 0 ia dl°'awnfrom R. through the point S ( Xp, ;. : xp ) uatil it lntvaecta a horisontal lh.e one time unit aboft the X• axis t and TP- T0 ia measured aa shown. For muld.-degree-of•freedom ayat_., one mu.st chooae the next coordinate on one of the. planea which eatabliah• the ti.Jae incr.aent. U.ing 1:hia time increment• the other Coordinat .. can be found by const.Netiug a lh.e from. r,of.at O au.eh that the vertical distance between thi• line and th• linas OMand OP ia always equal. The intersection of thia . X ----T,, - T,. . T=-l x.. -----L----f----"-----...... ,______X: --- Xo --1'1•1 I Xr---• Fipn 12. ldlum'a Pbaa••PlalUl Oouatruction. line and the line RN fin• points. Then Point P ta utabliahecl. A coapari.•on 1-tween thia method and the graphical-ma• erlcal method ta now diacuaaed. -101- a. Edman ta method usu Xo and Xoto find Po and then constructs the increment to Xp and Xp • The proposed method uaea the average value• of x and x over the increment: for the forcu in the syst•• Thia prari.dea greater accuracy for the same time increment. b. The time increment is found after the phaae 0001:'di• nates are arbitrarily ulect:ed. Thia i• not convenient for problema which haft inputs as a function of ti•• When it ia neceaaary to select the tiae inte.nal. where., :for inatanc•• the amplitude of the input baa finite diacontinuitiea, one 11l118tlocate points on Fig. 12 and then point P. The time incNl\'l.ent ia first selected in the graphical-numerical method. '• method appears to be sufficient for the type. of problem presented in his paper. The solution extend.a over one quadrant of the (i - x) plane at very small time increment.a. A comparison be.tween the two method•, using the aame time f:ncrement 1 waa made for a aimple mass on a cubic hardening apring. For a given initial. velocity, the aaxi- llD,D cliaplac•ent waa m.eaaured by each method and oompare Li's Method (14) Li applies his method to hydraulic problems which are governed by simultaneous first-order ordinary differential equations. The equations are of the. form _Q_,L + IV\ X - N\ ( X y1 t) dt 1 Approximate solutions for x and y can be. obtained by integr- ating successively through short intervals ~t, during each of which Mand N are taken as constants. By eliminating dt in Eqa.(12.l.4) and (12.1.5), integration yields (x-.xJ.,_ + (Lt-uJ ... = constant (12.l,.6) where, (12.l.7) (12.1.8) For each circular arc, the center is located by Eq. (12.1.8) in which the values of Mand N are computed from extrapolated values of x, y, and tat the. middle of the interval t. The angle MJ for each arc is (12.l.9) This method requires two coordinates, xc and u 0 , for finding the center of arc as opposed to one coordinate, x0 =-S, in the graphical-numerical method,. For multi-degree.• of-freedom systems, Lita method of graphical extrapolation for finding the constants Mand N appears to be cumbersome and more time-consuming than the numerical computation procedure of the graphical-numerical method. Ku'• Me~hod. (12} The equation of motion is reduced to dv = G(x,11,-t) d)(.. V Starting at (x0 ,v 0 ), assume a straight line slope on the phaae ..plane to (x2.v 2) for the increment 2ll.X. This establishes (x1 , v1 ) at fl x. Now (12.1.10) Every term in Eq.(12.1.10) is known except v1 • Solve for v1 • From the relation (12.l.ll) find ~t. Thue, t 1 = t 0+ht • Use t 1 in Eq. (12.1.10) to find a new v1 • From Eq. (12.1.ll) find a new f). t. Re- peat until the values of v1 converge. This m.thod appears to require very small increments in /l x and does not seem practical to extend to multi- degree.-of-freedom syatema. Bareisa and Atchison (31 Thia graphical method of estimating the mi curve for a single-degree-of-freedom system has been aucceasfully used by the authors in their report. However, it appears that the method cannot be extended to multi-degree-of-freedom systeme. -u.e.• XII. APPENDIX.2 P.,e,£tva~on of &mzY!t.:lona£pr Linear Sing\e•Desp;ee-of- System, The equation of motion for a linear oscillator with damptngsubjected to a forcing function F(t) is Let P =w-..J.,-c,(z. where o< < I. If the initial conditions are designated as y(O) and j(O), apply the LaPlace transform to Eq. (12.2.1). Thus, The roots of the coefficient of y are Div:lding Eq. (12.2.2) by the coefficient of y and re.arranging terms using partial fractions, F.q. (12.2.2) becomes - _ Y{c,)( 4.Jo<-1-,'p) y(o)(w-<- ie) + y Co) y - Zl'p(s+w-<.-ip) - zlp(s-1-wo< rl'fJ 2,";>(s1-0o<-<'t) F (12.2.3) Apply the inverse transform. to Eq. (12.2.3) and simplify For the. case where the input is the. foundation motion, the governing differential equation of motion is . . . '- .. X -t 2. o(. W X + c.) X = - Yo • Th.e solution of Eq. (12.2.l) ie the same as the solution for Eq. n2.2.s) provided the input P/m is replaced by Yoin EQ. c12.2.4>. Hence, ) -oiw t X -:::X ( 0 e_ COS pt -t -- In order to express the general solution of the differential equation in terms of the foundation velocity, the Duhamel integral of Eq. (12.2.6) must be integrated by parts. The result is ) -o The ffal'l.aient ruponae of a structure to a preaoribed short i:f.ae. duration. loa4 ia frequently an important prob• l• for the engineer to aolve. lf the structure ia idealiad u £Concentrated masses in a straight l:lne which are. oomected together by weightless apringa and daahpots, either linear or nonlinear• the equation• of motion for the model can be written aa where. zs. • relatift d.iaplacement between mass 11:1,and -·· "t-1 wf • 1tt/a:1. kJ. • spring coruttan:t: J"i • J;_( xi.+,1 x, 1 X,-,,Xi.+,, > •• i.. t... (" x + w x =-wo. Aaame that cS ia held conetant during the finite time in• crement, . at = h. flla solution equations for diapl--•t and phaae Yel.oeity at the n+l inter,ral are X 11+, = X 11 coJ evh + V,, si-, wh - ( I- cos wh) Bearranghg_the firat equation, aqua.ring both aide.a of each equation, and adding the two equationa, tbeae equatiou an shown to represent tha equivalent solution of the graphical• numerical method. A scaling law for nonlinear multi-degree.of.freedom •yatems has been developed provided certain restrictions are made on the parameters and input of the model system and. its prototype. If these restrictions are satisfied, the known response solution of the aod~l system is appli• cable to the prototype system by seating the independent variable time. Pour examples consisting of one.~iand two-d.egree-of'• freedom. systems were solved by the graphical-numerical method. The nonlinear components of the models we.re due to cubic hardening springs, and the input was represented by a given foundation velocity. The results compared favorably with the solution curves obtained frora an analog computer at the. Naval Reaaarch Laboratory, Washington, D.C. An exam:ll"\ation of the graphical-numerical method in the neighborhood of a singularity showed that 'it become.s tediOUIJ. if not impossible, to draw the circular arcs. However, the numerical equivalent procedure provides an approximate ·solution although it may take many ate.ps to obtain the coordinates of the proper trajectory in the region of the singularity. The drawing error in the graphical-numerical method waa examined for a nonlinear single-degree-of.freedom system. The problem waa solved by the. graphical-numerical method and. the numerical equivalent procedure.. A comparison between the two solutions showed the drawing error to be quite small, indeed negligible. Tha 6 and .ss ra.d./sec. the. conditions of the scaling law are satisfied. Hence, )( ( t) = S (3 t) = S (-r-). The response curve. for the. model system obtained frotn the pbase.•plane trajectory in Fig. lle, is used to find the response. s (t) of the prototype. system. It is noted that the. physical parameters of the proto• type ayatem can be varied in sever-1 ways. In what followa •79- suppose the capital letters refer to these parameter• in the protot~ systell, and the l