The Principle of Virtual Displacements in Structural Dynamics

Transcription

1 The Prncpe of Vrtua Dspacements n Structura Dynamcs 1 Stran Energy n Eastc Sods CEE 541. Structura Dynamcs Department of Cv and Envronmenta Engneerng Due Unversty Henr P. Gavn Fa 18 Consder an eastc object n equbrum subjected to statc forces and dspacements. F R S r f F1 R 1 R n F n σ ε F1 R R F F R R j Fj F j R j f r F n R n F and f are rea externa forces n equbrum, actng at ponts, or over a porton of a surface S, R and r are rea dspacements, admssbe wth respect to the support condtons, coocated wth F and f, σ are rea nterna stresses, dstrbuted wthn the sod voume V, n equbrum wth F & f, ɛ are rea nterna strans, dstrbuted wthn the sod voume V, compatbe wth R and r, 1.1 Externa Wor The wor of externa forces ncreasng from to F and f and pushng through dspacements from to R and r s R r W = F (R ) dr + f(r ) dr ds (1) where the forces F and f depend on dspacements R and r R and r are dummy varabes of ntegraton S

2 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn 1. Interna Stran Energy Stran energy s a nd of potenta energy arsng from stress and deformaton of eastc sods. In nonnear eastc sods, the stran energy of stresses ncreasng from to σ and worng through strans from to ɛ s ɛ U = σ dɛ dv () where V s the voume of the sod σ = { σ xx σ yy σ zz τ xy τ yz τ xz } ɛ = { ɛ xx ɛ yy ɛ zz γ xy γ yz γ xz } ɛ s a dummy varabe of ntegraton V 1.3 The Prncpe of Rea Wor In an eastc sod, the wor of externa forces, W, s stored entrey as eastc stran energy, U, wthn the sod. U = W (3) In near eastc sods: Dspacements and rotatons ncrease neary wth forces and moments F = D... and... M = κθ The wor of an externa force F actng through a dspacement D on the sod s W = 1 F D = 1 D = U The wor of an externa moment M actng through a rotaton Θ on the sod s W = 1 MΘ = 1 κθ = U

3 Prncpe of Vrtua Dspacements 3 σ(ε) σ du du= ε d ε (σ.ε) dv F(R ) W R W= F R ε R F dr R 1.4 Stran energy n sender structura eements In sender structura eements (bars, beams, or shafts) the nterna forces, moments, shears, and torques can vary aong the ength of each eement; so do the dspacements and rotatons. The stran energy of spatay-varyng nterna forces F (x) actng through spatay-varyng nterna dspacements D(x) wthn near eastc sods s U = 1 F (x) dd(x) dx dx = 1 F (x)d (x) dx (4) The stran energy of spatay-varyng nterna moments M(x) actng through spatay-varyng nterna rotatons Θ(x) wthn near eastc sods s U = 1 M(x) dθ(x) dx dx = 1 M(x)Θ (x) dx (5) In sender structura eements, the reaton between nterna forces and moments F and M, and nterna dspacements and rotatons v and φ, depend on the nd of oadng. Axa Bendng Shear Torson N x (x) = E(x)A(x)u (x) M z (x) = E(x)I(x)v (x) V y (x) = G(x)A s (x)v s(x) T x (x) = G(x)J ( x)φ (x)

4 4 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn Insertng these expressons nto the genera expressons for nterna stran energy above, force deformaton stran energy (U) Axa N x (x) u 1 (x) N x (x)u 1 (x)dx Bendng M z (x) v 1 (x) z (x)v M (x)dx 1 Shear V y (x) v s(x) 1 V y (x)v s(x)dx 1 Torson T x (x) φ 1 (x) T x (x)φ 1 (x)dx N x(x) dx 1 E(x)A(x) M z(x) dx 1 E(x)I(x) V y(x) dx 1 G(x)A s(x) T x(x) dx 1 G(x)J(x) E(x)A(x)(u (x)) dx E(x)I(x)(v (x)) dx G(x)A s (x) (v s(x)) dx G(x)J(x)(φ (x)) dx E(x) G(x) A(x) I(x) A(x)/α J(x) N x (x) M z (x) V y (x) T x (x) u(x) u (x) s Young s moduus s the shear moduus s the cross sectona area of a bar s the bendng moment of nerta of a beam s the effectve shear area of a beam s the torsona moment of nerta of a shaft s the axa force wthn a bar s the bendng moment wthn a beam s the shear force wthn a beam s the torque wthn a shaft s the axa dspacement aong the bar s the axa dspacement per unt ength, du(x)/dx, the axa stran v(x) s the transverse bendng dspacement of the beam v (x) s the sope of the dspacement of the beam v (x) s the rotaton per unt ength, the curvature, approxmatey d v(x)/dx v s (x) v s(x) φ(x) φ (x) s the transverse shear dspacement of the beam s the transverse shear dspacement per unt ength, dv s (x)/dx s the torsona rotaton (twst) of the shaft s the torsona rotaton per unt ength, dφ(x)/dx

5 Prncpe of Vrtua Dspacements 5 Vrtua Wor n Eastc Sods The Prncpe of Vrtua Dspacements Now consder a second set of oads, δf, δf, n equbrum and apped subsequenty to the oads F and f. The oads δf and δf gve rse to dspacements δr and δr coocated wth forces F and f, and nterna stresses δσ and strans δɛ. The dspacements δr and δr are consstent wth the support condtons of the system. In other words, the dspacements δr and δr are admssbe wth respect to nematc constrants. Ca δf and δf a set of any arbtrary vrtua forces n equbrum. Ca δr and δr a set of vrtua dspacements, coocated wth forces F and f, and resutng from forces δf and δf (and therefore nematcay admssbe). Forces F and f are hed constant as oads δf and δf are apped. Stresses σ, n equbrum wth forces F and f, are therefore aso hed constant as oads δf and δf are apped. Forces F and f do not ncrease wth dspacements δr and δr. Strans δɛ ncrease as oads δf and δf are apped. δ f F R δr S r δr f δf j δσ δε σ δw= (σ.δε) I dv ε ε+δε F R R + δr δw= F δr E The prncpe of vrtua dspacements states that the vrtua externa wor of rea externa forces (f and F ) movng through coocated admssbe vrtua dspacements (δr and δr) equas the nterna vrtua wor of rea stresses (σ) n equbrum wth rea forces (f and F ) wth the vrtua strans (δɛ) compatbe wth the vrtua dspacements (δr and δr), ntegrated over the voume of the sod. V δw I = δw E σ δɛ dv = f δr ds + S F δr ()

6 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn.1 Axa oad effects n sender structura eements In sender sod eements, nonunform transverse dspacements (dv(x) ) nduce ongtudna shortenng, de(x). v + δv de+d δe δv + d δv de v v + dv δv de v v + dv v v dx dx Fgure 1. Transverse deformaton v (x) and ongtudna shortenng de(x). A reaton between dv and de can be derved from the Pythagorean theorem and s quadratc n dv and de. (dx de) + (dv) = (dx) (de)(dx) (de) = (dv) de dx 1 (v ) Wth addtona vrtua dspacements δv(x) a reaton for the ncrementa vrtua shortenng dδe may aso be derved from the Pythagorean theorem. (dx de dδe) + (dv + dδv) = (dx) (de)(dx) (de)(dδe) + (dδe)(dx) (de) (dδe) = (dv) + (dv)(dδv) + (dδv) Subtractng (de)(dx) (de) = (dv) and dvdng by (dx) eaves de ( ) dδe dδe dx dx + dδe dx = (v )(δv ) + (δv ) dx Negectng hgher order terms (assumng vrtua dspacements are nfntesma), eaves dδe dx (v )(δv ) (7) The vrtua wor of a dstrbuted axa compresson P (x) (apped externay, for exampe, by gravtatona acceeraton) actng through vrtua shortenng dspacements δe(x) ntegrated aong a sender eement s, then, δw G = P (x) dδe dx dx = P (x) v (x) δv (x) dx (8) Ths resut can aso be obtaned by ntegratng aong the arc-ength of the deformed eement as s done n Tedesco, McDouga, and Ross s textboo, Structura Dynamcs: Theory and Appcatons.

7 Prncpe of Vrtua Dspacements 7 3 The Prncpe of Vrtua Dspacements for Dynamc Loadng The prncpe of vrtua dspacements appes to both statc and dynamc forces. Eastc forces (x)r(x, t) are present n structura systems respondng to statc or dynamc oads. Forces arsng from dynamc effects ony ncude vscous dampng forces c(x)ṙ(x, t) and nerta forces m(x) r(x, t). Eastc forces, vscous dampng forces, and nerta forces can be deveoped wthn sender structura eements n response to axa, bendng, shear, and torsona deformatons. rea vrtua force deformaton nterna vrtua wor (δw I ) Axa N x (x, t) δu (x, t) Bendng M z (x, t) δv (x, t) Shear V y (x, t) δv s(x, t) Torson T x (x, t) δφ (x, t) Geometrc P (x) δe(x, t) N x (x, t) δu (x, t) dx M z (x, t) δv (x, t) dx V y (x, t) δv s(x, t) dx T x (x, t) δφ (x, t) dx P (x) δe(x, t) dx EA(x) u (x, t) δu (x, t) dx η a A(x) u (x, t) δu (x, t) dx ρa(x) ü(x, t) δu(x, t) dx EI(x) v (x, t) δv (x, t) dx η a I(x) v (x, t) δv (x, t) dx ρa(x) v(x, t) δv(x, t) dx GA s (x) v s(x, t) δv s(x, t) dx η s A s (x) v s(x, t) δv s(x, t) dx ρa(x) v(x, t) δv(x, t) dx GJ(x) φ (x, t) δφ (x, t) dx η s J(x) φ (x, t) δφ (x, t) dx ρj(x) φ(x, t) δφ(x, t) dx P (x) v (x, t) δv (x, t) dx In ths tabe: The nterna vrtua wor of vscous effects s derved assumng near vscous stress - stran-rate reatons: σ = η a ɛ and τ = η s γ. As w be seen ater n the course, the dampng propertes of rea structura materas are more compcated. Rotatory nerta effects are negected n the vrtua wor of nerta forces n bendng beams.

8 8 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn 4 Generazed Coordnates A dynamc response r(x, t) may be represented as the nner product of spatay dependent quanttes and tme dependent quanttes r(x, t) = ψ (x) q (t) The functons ψ (x) are caed shape-functons, and the functons q(t) are caed generazed coordnates. In order for the above expanson to yed reastc and accurate soutons, the shape functons must at east satsfy the essenta boundary condtons. (The shape functons must be nematcayadmssbe.) Shape functons whch aso satsfy the natura boundary condtons w yed more accurate soutons. Aso, f the shape functons are dmensoness, the generazed coordnate have the same unts as the response, whch permts a usefu nterpretaton of the generazed coordnates. Further, f the shape functons are nematcay admssbe, then vrtua dspacements defned as varatons n r(x, t) wth respect to the set of coordnates q (t) are aso nematcay admssbe δr(x, t) = j r(x, t) q j (t) δq j(t) = j ψ j (x) δq j (t) Dervatves of r wth respect to x and t can be evauated: r(x, t) = q (t)ψ (x) ṙ(x, t) = q (t)ψ (x) r(x, t) = q (t)ψ (x) r (x, t) = q (t)ψ (x) ṙ (x, t) = q (t)ψ (x) r (x, t) = q (t)ψ (x) r (x, t) = q (t)ψ (x) ṙ (x, t) = q (t)ψ (x) r (x, t) = q (t)ψ (x) Interna vrtua wor can aso be expressed n terms of generazed vrtua dspacements, for exampe for eastc bendng a beam, δw I = = = j EI(x) v (x, t) δv (x, t) dx EI(x) [ ψ (x) q (t) j EI(x) ψ j (x) ψ (x) dx And for nerta axa forces and vrtua dspacements n a beam, δw I = ρa(x) v(x, t) δv(x, t) dx ψ j (x)δq j (t) dx q (t) δq j (t) (9) = ρa(x) ψ (x) q (t) ψ j (x)δq j (t) dx j = [ ρa(x) ψ j (x) ψ (x) dx q (t) δq j (t) (1) j

9 Prncpe of Vrtua Dspacements 9 Externa vrtua wor can be expressed n terms of generazed vrtua dspacements (that s, the varatons n the generazed coordnates), δq j (t). δw E = f(x) δv(x, t) dx + F δv(x, t ) = f(x) ψ j (x) δq j (t) dx + F ψ j (x ) δq j (t) j j = [ f(x) ψ j (x) dx δq j (t) + [ F ψ j (x ) δq j (t) (11) j j And the externa vrtua wor of axa compresson movng through vrtua end shortenng s, δw E = P (x) v (x, t) δv (x, t) dx = P (x) ψ (x) q (t) ψ j(x) δq j (t) dx j = [ P (x) ψ j(x) ψ (x) dx q (t) δq j (t) (1) j By settng the nterna vrtua wor equa to the externa vrtua wor, and factorng out the ndependent and arbtrary varatons δq j, equatons (9), (1), (11), and (1), resut n {[M{ q(t)} + [K E {q(t)} [K G {q(t)} {f(t)}} {δq(t)} = Notng that each varaton ψ j δq j s be arbtrary, and the set of varatons j = 1,,... must be ndependent, not ony must the dot product equa zero, but each term wthn the nner product must be zero. Therefore, the term on the eft of the nner product must evauate to the zero-vector. Ths s an mportant concept n the prncpe of vrtua wor and n the cacuus of varatons. It s appcaton resuts n the matrx equatons of moton, [M{ q(t)} + [K E {q(t)} [K G {q(t)} = {f(t)} where the j, term of the mass matrx s, M j = ρa(x) ψ j (x) ψ (x) dx the j, term of the eastc stffness matrx s, K Ej = EI(x) ψ j (x) ψ (x) dx the j, term of the geometrc stffness matrx s, K Gj = P (x) ψ j(x) ψ (x) dx and the j-th eement of the forcng vector s the nner product of the forcng wth the j-th shape functon, f j = f(x) ψ j (x) dx + F ψ j (x ) From the above reatons, t s cear that M j = M j (the mass matrx s symmetrc), K j = K j (the stffness matrces s symmetrc), and that [M and [K are postve defnte, provded that the set of shape functons are neary ndependent.

10 1 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn 5 Exampes 5.1 Exampe 1: a snge generazed coordnate In ths exampe, the essenta boundary condtons are v(t, ) = and v (t, ) =, so any shape functon used n ths probem must aso satsfy ψ () = and ψ () =. In ths frst exampe, we w consder a snge (dmensoness) shape functon, such as, ψ(x) = (x/l), ψ(x) = (x/l) 3, or ψ ( x) = 1 cos(πx/(l)). Just to eep ths smpe for now, we choose ψ(x) = (x/l) 3. Forces and assocated vrtua dspacements are tabuated beow. δ v(t,x) EI, m F(t) v(t,x) M P c f(t,x) x= x=a x=b x=l Eement Rea Interna Force Vrtua Interna Dspacement M c M v(l, t) = Mψ(L) q(t) = M q(t) δv(l, t) = ψ(l)δq(t) = δq(t) c v(a, t) = cψ(a) q(t) = c(a/l) 3 q(t) δv(a, t) = ψ(a)δq(t) = (a/l) 3 δq(t) EI, m EI, m F(t) f(t,x) P v(a, t) = ψ(b)q(t) = (b/l) 3 q(t) EIv (x, t) = EIψ (x)q(t) = EI x/l 3 q(t) m v(x, t) = mψ(x) q(t) = m(x/l) 3 q(t) Rea Externa Force F (t) f(x, t) P δv(b, t) = ψ(b)δq(t) = (b/l) 3 δq(t) δv (x, t) = ψ (x)δq(t) = x/l 3 δq(t) δv(x, t) = ψ(x)δq(t) = (x/l) 3 δq(t) Vrtua Dspacement δv(a, t) = ψ(a)δq(t) = (a/l) 3 δq(t) δv(x, t) = ψ(x)δq(t) = (x/l) 3 δq(t) v (x, t)δv (x, t) = 9x 4 /L q(t) δq(t)

11 Prncpe of Vrtua Dspacements 11 Equatng the wor of rea nterna forces movng through nterna vrtua dspacements, wth rea externa forces movng through coocated vrtua dspacements, M q δq + c((a/l) 3 ) q δq + ((b/l) 3 ) q δq + L = F (t)(a/l) 3 δq + EI((x/L 3 )) dx q δq + L b f(x, t)(x/l) 3 dx δq + L L m((x/l) 3 ) dx q δq P (9x 4 /L ) dx q δq Evauatng the defnte ntegras, factorng out the (arbtrary) vrtua coordnate δq, specfyng that the dstrbuted dynamc force s unform wth ntensty f o, and groupng terms, the equaton of moton for ths system s (M + 1 ) ( ) ( ) a b ml q(t) + c q(t) EI L L L 9 ( ) 3 5L P a 3 1 L 4 b 4 q(t) = F (t) + f L 4 L 3 o (t) Note that ths equaton of moton s dmensonay homogeneous (as t shoud be). The natura frequency of ths system s ω n = ( ) b L + 1 EI 9 P L 3 5L M + 1mL In ths equaton the term (9P q(t))/(5l) s moved to the eft hand sde of the equaton, as t s a functon of poston q(t). The coeffcent (9P )/(5L) s caed the geometrc stffness of ths system. The negatve sgn on ths term shows that the axa compressve force P s destabzng for ths system. Under the condton ( ) b + 1 EI L L 9 3 5L P = the natura frequency woud go to zero, and the system woud buce. So the crtca axa bucng oad for the system s ( ) b P cr = + 1 EI ( ) 5L L L 3 9 Dynamca responses of compex systems requre compex mathematca descrptons. The smpe approxmaton v(x, t) = (x/l) 3 q(t) used here coud be passabe for a smpe cantever beam. But n ths exampe f the sprng stffness were much hgher than EI/L 3 the dynamc response at x = b woud have a very sma amptude compared to responses the domans x < b and x > b. Ths nd of response s not captured by the approxmaton ψ(x) = (x/l) 3. In fact, the nature of the free dynamc response n systems such as the one n ths exampe depend on the reatve vaues of the physca parameters, EI/L 3, Mg/L, mg, P/L,, etc. More compex mathematca modes are requred to descrbe the dynamc responses of compex systems such as ths.

12 1 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn 5. Exampe : the same exampe wth two generazed coordnates In ths exampe, the dspaced shape s expressed as the sum of two (ndependent and nematcay admssbe) shape functons, ψ 1 (x) and ψ (x) v(x, t) = [ ( ) 3 x ( ) [ 1 x 3 ( ) x 3 ( x q 1 (t) q (t) L L L L) shape functons: ψ 1 (x/l), ψ (x/l) ψ 1 (x/l) = (3/)(x/L) - (1/)(x/L) 3 ψ (x/l) = 8(x/L) 3-7(x/L) x/l Generazed coordnates assocated wth dmensoness shape functons have the same physca dmensons as the response varabes, whch s generay desrabe. Shape functons that resembe the actua dynamc responses correspond to more reastc dynamc modes. Actua dynamc responses must adhere to essenta and natura boundary condtons. So as a frst requrement, shape functon approxmatons must adhere to the essenta boundary condtons. Shape functons that aso adhere to the natura boundary condtons correspond to more reastc modes. Mass, and stffness matrces derved from sets of neary ndependent shape functons are postve defnte (assumng the system has no rgd body modes). Mass and/or stffness matrces derved from sets of mutuay orthogona shape functons are numercay we condtoned. Because of ths, modes derved from sets of mutuay orthogona shape functons are more precse over a broader frequency range. In ths exampe, ψ 1 (x) corresponds to the statc defecton of a cantever beam wth a pont oad at x = L; ψ (x) has an nfecton pont and a zero-crossng. The appcaton of the prncpe of vrtua dspacements n whch the responses are an expanson of n (admssbe and neary ndependent) shape functons resut n n dmensona matrx equatons of moton. Exampes of mass and stffness matrces for hgher dmensona approxmatons are gven n equatons (9), (1), (11), and (1). Ths probem s sghty more compex as t nvoves a sprng, a damper, and a concentrated mass.

13 Prncpe of Vrtua Dspacements 13 Appyng the prncpe of superposton, expressons for the nterna and externa vrtua wor correspondng to each of these varous components may be taen ndvduay. Interna Vrtua Wor from the dstrbuted mass of the beam, m δw I = j [ m(x) ψ j (x) ψ (x) dx q (t) δq j (t) Interna Vrtua Wor from the pont mass of the beam, M δw I = j [ Mδ(x L) ψ j (x) ψ (x) dx q (t) δq j (t) Interna Vrtua Wor from the Beam, EI δw I = j [ EI(x) ψ j (x) ψ (x) dx q (t) δq j (t) Interna Vrtua Wor from the Sprng, δw I = j [ δ(x b) ψ j (x) ψ (x) dx q (t) δq j (t) Interna Vrtua Wor from the Damper, c δw I = j [ cδ(x a) ψ j (x) ψ (x) dx q (t) δq j (t) Externa Vrtua Wor from the dynamc pont Force, F (t) δw E = j [ F (t)δ(x a) ψ j (x) dx δq j (t) Externa Vrtua Wor from the dynamc dstrbuted Force, f(t) δw E = j [ L f(x, t) ψ j (x) dx b δq j (t) Externa Vrtua Wor from the constant Axa force, P δw E = j [ P (x) ψ j(x) ψ (x) dx q (t) δq j (t)

14 14 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn Each j, term wthn the square bracets corresponds to the j, term of a mass, dampng, or stffness matrx. In these dervatons, δ(x a) s the Drac deta functon, whch has the defnng property, g(x)δ(x a) dx = g(a) The evauaton of the assocated dervatves and ntegras can be easy carred out usng symboc manpuaton pacages e Mathematca, Mape, or Wofram-α. : ogn-teer-1 Sun Sep 14:11:35 ## EVALUATE INTEGRALS... : mape \ˆ/ Mape 17 (X8 4 LINUX)._ \ / _. Copyrght (c) Mapesoft, a dvson of Wateroo Mape Inc. 17 \ MAPLE / A rghts reserved. Mape s a trademar of < > Wateroo Mape Inc. Type? for hep. ## MASS MATRIX TERMS... ## INPUT THE SHAPE FUNCTION EQUATIONS... > p1 := (3/)*(x/L)ˆ - (1/)*(x/L)ˆ3; 3 3 x x p1 := L L > p := 8*(x/L)ˆ3-7*(x/L)ˆ; 3 8 x 7 x p := L L ## EVALUATE DERIVITIVES... > m11 := nt(m*p1*p1,x=..l); > m1 := nt(m*p1*p,x=..l); > m := nt(m*p*p,x=..l); > M11 := eva(m*p1*p1,x=l); 33 m L m11 := m L m1 := m L m := M11 := M > dp1 := dff(p1,x); 3 x 3 x dp1 := L L > M1 := eva(m*p1*p,x=l); > M := eva(m*p*p,x=l); M1 := M M := M > ddp1 := dff(dp1,x); > dp := dff(p,x); > ddp := dff(dp,x); 3 3 x ddp1 := L L 4 x 14 x dp := L L 48 x 14 ddp := L L ## STIFFNESS MATRIX TERMS... > EI11 := nt(ei*ddp1*ddp1,x=..l); 3 EI EI11 := L > EI1 := nt(ei*ddp1*ddp,x=..l); 3 EI EI1 := L > EI := nt(ei*ddp*ddp,x=..l); 9 EI EI := L

15 Prncpe of Vrtua Dspacements 15 > 11 := eva(*p1*p1,x=b); ## EXTERNAL FORCING TERMS... > 11 := smpfy(11); > 1 := eva(*p1*p,x=b); 4 b (3 L - b) 11 := L > 1 := smpfy(1); 4 b (3 L - b) (8 b - 7 L) 1 := L > := eva(*p*p,x=b); > F1 := eva(f*p1,x=a); > F1 := smpfy(f1); > F := eva(f*p,x=a); > F := smpfy(f); F a (3 L - a) F1 := L F a (8 a - 7 L) F := L > := smpfy(); 4 b (7 L - 8 b) := L ## DAMPING MATRIX TERMS... > c11 := eva(c*p1*p1,x=a); > f1 := nt(fo*p1,x=b..l); > f1 := smpfy(f1); > f := nt(fo*p,x=b..l); fo (3 L - 4 L b - b ) f1 := L > c11 := smpfy(c11); 4 c a (3 L - a) c11 := L > f := smpfy(f); fo (L - 7 L b + b ) f := L > c1 := eva(c*p1*p,x=a); > c1 := smpfy(c1); 4 c a (3 L - a) (8 a - 7 L) c1 := L > c := eva(c*p*p,x=a); > c := smpfy(c); 4 c a (8 a - 7 L) c := L ## GEOMETRIC STIFFNESS TERMS... > P11 := nt(p*dp1*dp1,x=..l); > P1 := nt(p*dp1*dp,x=..l); > P := nt(p*dp*dp,x=..l); P P11 := L 41 P P1 := ---- L 188 P P := L

16 1 CEE 541. Structura Dynamcs Due Unversty Fa 18 H.P. Gavn The resutng equatons of moton n terms of generazed coordnates, q 1 (t) and q (t) are ml + M ml + M ml + M 9 15 ml + M q 1 (t) q (t) + ca4 L (3L a) 4 (3L a)(8a 7L) (3L a)(8a 7L) (8a 7L) q 1 (t) q (t) + b4 L (3L b) 4 (3L b)(8b 7L) (3L b)(8b 7L) (8b 7L) q 1 (t) q (t) + EI L q 1 (t) q (t) P L q 1 (t) q (t) = a (3L a) 3L 4 4Lb 3 b 4 L 3 8L 3 a (8a 7L) L4 7Lb 3 +b 4 L 3 3L 3 F (t) f o (t) (13) Wth the scang of the dmensoness shape functons, ψ 1 (L) = ψ (L) = 1, q 1 (t) and q (t) are the vaues of v(l, t) correspondng to ψ 1 (x) and ψ (x). Wth the dmensoness formuaton of the shape functons, every term n ths equaton has unts of force. Ths exampe s an ntroducton to a methodoogy that w be nvoed esewhere n the course.