Ll?“ m. A 51‘s, Michigan State .‘ .pznflh'q'h‘ Egg Ln... ‘- Int-run: This is to certify that the thesis entitled ON THE MANEUVERING AND MODELING OF FLEXIBLE STRUCTURES presented by SLIM CHOURA has been accepted towards fulfillment of the requirements for Masters degree in Mechanical Engineering 1 % Major professor Date February 26, 1987 0-7639 MS U is an Afiirmative Action/Equal Opportunity Institution MSU LIBRARIES .— RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ON THE HANEUVERINC AND MODELING OF FLEXIBLE STRUCTURES By Choura,Slim A.IHESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1987 ON THE HANEUVERING AND MODELING OF FLEXIBLE STUCTURES 3? Choura Slim A rigid hub-single flexible beam system is the focal point of this study. A time-dependent torque is applied at the hub to maneuver the tip position in space and time. Two approximate models are derived to describe the flexibility in the system. Approximate analytical and numerical solutions are obtained for a rectangular-pulse angular velocity using an approximate flexure model. It is shown that it is easy to eliminate the effect of any mode by a proper choice of the constant angular velocity. ACKRWLEIHENTS I greatefully acknowledge the efforts and guidance given by my supervisor Professor M.A.Medick of the Department of Mechanical Engineering, Michigan State University. Special thanks must also be extended to Professor S.Jayasuriya of Michigan State University, who offered much advice and encouragment. I would like to thank my father and mother who supported me through all my education. I thank also my wife Nadra and son Mohamed Amin for their patience and support during my research. TABLE OF CONTENTS LIST OF FIGURES AND PLOTS CHAPTER ONE: INTRODUCTION . CHAPTER TWO: EQUATIONS OF MOTION (i) Rayleigh beam (ii) Bernoulli-Euler beam CHAPTER THREE: SIMPLE FLEXURE CHAPTER FOUR: RECTANGULAR PULSE ANGULAR VELOCITY CHAPTER FIVE: NUMERICAL ANALYSIS Rectangular-pulse angular velocity Stability CHAPTER SIX: NUMERICAL RESULTS CHAPTER SEVEN: DISCUSSION OF RESULTS SUMMARY AND CONCLUSIONS APPENDIX A APPENDIX B REFERENCES Page iv 11 15 20 24 25 28 34 36 37 43 47 TABLE OF FIGURES AND PLOTS Page Figure 1: Schematic representation of the system 2 Figure 2: Deflected configuration of a particle point A Figure 3: Impulse representation 24 Figure 4: Hub-beam system in three dimensions 29 Plot 1 : Tip displacement using the series solution 30 Plot 2 : Tip displacement using the finite difference scheme 31 Plot 3 : Torque applied at the hub 32 Plot 4 : Comparison between exact and approximate 33 representations of the body force iv CHAPTERONE INTRONCIION In literature, a number of people are interested in the modeling and control of flexible structures. Jasinski [6], Sung [11] and Viscomi [13] developed the equations of motion of a slider crank mechanism. They started from kinematics to generate a set of two coupled equations in flexure and extension. Nachman [8] and Cannon [3] took simple rotating beams and developed the equations of motion by neglecting the effect of extension. In the first part of this thesis, our interest was to obtain and examine the fundamental equations of motion for simple rotating beams. These equations describe the coupling between flexure and extension, and the torque applied at the hub. Consequently, these equations are of use in the control and positioning of the beam in space and time. In the second part we were interested in one maneuvering problem using an approximate simple flexure model. The maneuvering test was confined to a certain range of angular velocities for which the model ‘was valid” The model used was valid within 10% maximum deflection with respect to the length of the beam. Our physical system was a rigid-hub flexible beam mechanism (fig.1). A time-varying torque was applied at the hub to maneuver the beam tip position. This investigation was confined to planar motion. The experimental beam was made of aluminum and had a rectangular-cross section. The maneuvering problem consisted of rotating the hub through a finite angle with desired physical states 1 of its tip position at the end time. For instance, a rest position of the beam is desired for accurate operations. TORQUE fig.1: Hub—beam system cummrwo WATIONS OF MOTION The basic differential equations governing the axial and transverse displacements of the robot arm relative to its rotating undeformed position are developed by energy methods. The plane mechanism (fig.1) consists of a rigid hub of radius ro and mass moment of inertia Ih’ and a flexible beam of length L, cross- section area A (bxh), and constant material properties. The hub angular rotation is 0(t) measured counterclockwise from the x-axis. The transverse and longitudinal displacements v and u respectively, are measured with respect to the undeformed position of the beam, i.e. an observer is located at the origin 0 and rotates with the hub. It is assumed that plane sections remain plane during deformation. The effects of shear defamation and rotary iertia are assumed negligible. We shall consider two approximations to the physical problem. (1) W The extensional deformation can be simulated by measuring the distance PP" (see fig.2) where P" corresponds to the projection of P' on the x'-axis. According to Cannon [3] these approximations are valid for a maximum deflection less than 10% of the beam length. If a small chunck of material is to be taken, then from (fig.2) in its undeformed configuration it lies on the x'-axis. A particle P 3 located on the neutral axis (N.A.) moves to a new position I" . Any particle point Q at a distance s perpendicular to the neutral axis from P at its undeformed configuration moves to Q' at a distance 3' normal to the neutral axis from P' . £151; Deflected configuration of a particle point If the expansion and compression of any section are small during defamation, then it is reasonable to assume that s' and s are equal. Then the position vector 1 - BM, (2.1) can be written as A A 1(x,s,t)- (ro-t- x + u - c,)e1 + (v + c2)e2 (2.2) The angle ¢ is assumed small and expressed as: o a tan¢ a sin¢ ~ av/ax (2.3.1) and cos¢ a 1.0 , (2.3.2) Equation (2.3.1) is valid if the vector RP'O is approximately parallel to the tangent at the neutral axis through P'. Then the position vector I is expressed as: A A I (x,s,t) - (r°+ x + u - abet/awe1 + (v + s)e2 (2.4) The velocity vector is obtained by differentiating (1.4) once with respect to time i (x,s,t) - { au/at - sag/axat - (v + S); };1 A + { (ro+ x + u)§ - adv/6x5 + av/at }e2 (2.5) Where the differentiations of e1 and e2 with respect to time are 3 e2 and -3 e1 respectively. The energy methods are then employed. The total kinetic energy of the system is: T ' Tbeam + Thub (2’6) where b/2 L'. . I r - 1 dx (h ds) (2.6.1) T - 0.5pI beam -b/2 o 2 hub ' °°5Ih9 (2.6.2) h is the height of the beam cross section and L' corresponds to the deformed arc length of the beam. L' is assumed to be equal to the undeformed beam length L. The potential energy depends on the deformations u and v. The total potential energy is: L 2 L 2 2 2 V - 0.5EAI (flu/6x) dx +0.5EII (av/6x ) dx (2.7) 0 0 where the first term corresponds to the total strain energy due to compression or tension, and the second term represents the total bending energy. The work done by the torque,r, at the hub is expressed as: W - 1.0 (2.8) Next, Hamilton's principle is employed where the variation is taken into the extensional and flexural deflections u and v and the rotating angle 0. t52 I (6T - 5V + 6W) dt - 0 (2.9) t1 The cross section area and the moment of inertia come from the following expressions b/2 A - I h ds - bh (2.10) ~b/2 b/2 2 s . I - I hs ds - hb /12 (2.11) -b/2 The The coupled equations of motion are then: 2 2 2 ” 2 2 EA au/ax + pA [(ro+x+u)3 + v0 + 25 av/at - au/at ] - 0 (2.12.1) a 4 4 2 2 2 2 2 E1 av/ax - pI av/ax at + p13 av/ax 2 so 0 2 2 + pA [-v3 + (ro+x+u)0 +20 au/at + av/at ] - 0 (2.12.2) L r(t) - { pIL + 1h + I 2 2 2 [pAv + pA(ro+x+u) + pI(3v/6x) ]dx }0 O L 2 { I [2pAv8v/at + 2pA(ro+x+u)au/6t + pIav/ax 8v/8t6x]dx }a O L 2 2 s 2 2 2 { I [~pA8u/8t v + pIav/axat + pA(r°+x+u)av/at ]dx } 0 (2.12.3) corresponding boundary conditions are: u(0,t) - 0 (2.12.4) au/ax(L,t)- 0 (2.12.5) v(0,t) - 0 (2.12.6) av/ax(0,t) - 0 (2.12.7) 63/3x2(L,t) - 0 (2.12.8) 63/6x8(L,t) - 0 (2.12.9) 328v/6x(L,t) - 3° - a3/axac2(1.,t) - o (2.12.10) where (2.12.10) is a natural boundary condition. The Rayleigh beam model is valid for the case where the width of the beam is not small. (ii) fiegggulli-Euler Approximation In this case, the width is considered to be small, and in each section all particles have the same velocity. Hence the new position vector x for a particle point on the neutral axis becomes ; (x,t) - (ro+ x + u)e1 + ve2 (2.13) The derivation of the governing equations of motion is accomplished in the same manner as in case (i).The final equations are: 2 2 .2 .. 2 2 EA au/ax + pA[ (ro+x+u)o + v0 + 25 av/at - au/at ] - 0 (2.14.1) 4 4 2 .. . 2 2 EI av/ax + pA[ -v3 4- (r°+x+u)0 + 20 au/at + av/at ] - 0 (2.14.2) L 2 2 .. 1(t) - { Ih + I; [pAv + pA(ro+x+u) ]dx }0 L +{ I [ 2pAv6v/at + 2pA(ro+x+u)'6u/6t ]dx }3 0 L 2 2 2 2 +{ I I: -pAau/at v + pA(r°+x+u)av/at ]dx } (2-14-3) 0 The corresponding boundary conditions are: u(0,t) - O 8u/8x(L,t) - 0 v(0,t) - O 8v/6x(0,t) - O 2 2 av/ax (L,t) - O S 3 mm»: (L,t) - o (2 (2 (2 (l (2 (2 .14.4) .14.5) .14.6) .14.7) .l4.8) .14.9) Both approximations include two coupled equations in flexure and extension, and a torque dependent on u,v,0, and the system's parameters. The rigid body motion can be deduced from equations (2.14.1) through (2.14.3) by dividing equation (2.14.1) and (2.14.2) by Young/s modulus E, then letting u and v go to zero and E go to infinity. The first two equations (2.14.1) and (2.14.2) become identities and (2.14.3) becomes of the form 1(t) - It? L 2 where I - I + pA (ro+x) dx t h 0 (2.15) (2.16) 10 The decoupled equations can be derived by going back to the position vector (2.13), set u to zero and get the simple flexure model in the same manner as case (i). The simple extension is obtained by setting v to zero. In equation (2.14.1) terms involving v and its derivative are treated as internal forces exciting the extensional vibration of the beam and vice versa in equation (2.14.2). The terms pA(ro + x)32 and pA(ro + x)? represent body forces in the first and second field equations respectively . CHAPTERTHREE SIHPLEFLEXURE It is difficult to get an analytical solution to the coupled equations of motion (2.14). In this Chapter, we shall deduce a simple approximate model of flexure. The equations of motion are obtained by setting 11 to zero in equation (2.13), writing the kinetic energy and potential energy expressions and using Hamilton's principle. This leads to the equations of motion 4 4 2 .. 2 2 E1 av/ax + pA[ -v§ + (ro+x)o + av/at ] - 0 (3.1.1) L 2 2 g. L o 1(t) - { Ih + pAI [v + (ro-t-x) ]dx }0 + { 2pAI [vav/at]dx }0 O 0 L 2 2 #{ pAI [(ro+x)av/at ]dx } (3.1.2) 0 The boundary conditions are 2 2 s s v(0,t) - 8v/6x(0,t) - av/ax (L,t) - av/ax (L,t) - 0 (3.1.3) To generalize this problem, nondimensional equations of motion will be used. The nondimensional variables and parameters are described as the following 11 12 re - ro/L 3* - 5/1 a/pL’ ee 2 - 0/(E/pL ) 2 I - I/AL r - r/AEL 3 1 - Ih/pAL (3. (3. (3. (3. (3. (3. (3. (3. (3. .1) .2) .3) .4) .5) .6) .7) .8) .9) Equations (3.1”1) and (3.1.2) then reduce to the following nondimensional boundary value problems * “ * *‘ * 2 * *ee* 2 2 I a v /8x - v 5* + (r°+x )o + a v*/ac* - o 1 1*(t*) - { I: + J [ v*2 + (r:+x*)2]dx* } + { 2I:[ v*av*/ac*] dx* }5* (3.3.1) 13 1 + { I [ (r:+x*)azv*/ax*2 ]dx* } (3.3.2) o The associated boundary conditions are: v*(o,c*) - 0 (3.3.3) av*/ax*(o,c*) - 0 (3.3.4) 32v*/ax*2(l,t*) - 0 (3.3.5) 63v*/8x*3(l,t*) - 0 (3.3.6) Without the body force term, equation (3.3.1) is seperable. Let * * * * * v (x .t ) - fix )q(t) (3.4) Then the eigenvalue problem is: 4 §"" - p O - 0 (3.5.1) 0(0) - ¢'(0) - 0"(1) - 0"‘(l) - 0 (3.5.2) This eigenvalue problem is the same for a fixed cantilever beam. The solution of equations (3.5.1)-(3.5.2) is 2 2 * * * ¢n(x ) - coshfinx - cosfinx - en(sinhflnx - sinflnx ) (3.6.1) where cosh)?n + cos]?n e - n sinhfin + sinfln coshfin cosfln - -1 (3.6.2) n-l,2,3,... (3.6.3) Equations (3.6.1) and (3.6.3) are expressions of the problem eigenfunctions and eigencondition respectively. By expanding the body force (r:+x*).0.* about the eigenfuntions On's, an infinite set of ordinary differential equations results: .. *2 *2 «k qn + (wn - a )q - -an0 where * 2 ”a: wn - finj I n- l,2,3,... (3.7) (3.8) n- l,2,3,... (3.9) The expansion of the body force does not lead to a uniform convergence of the series to (r:+x*).0‘*. This is because the boundary condition at x* - 0 is not consistent with the eigenfunctions ¢n(x*) which vanish at * x - 0. However, the series represents a good approximation to the body * force function away from x - 0. CHAPTERFOUR RECTANGULAR PUISE ANGULAR VEIDCITY In this case the angular velocity is a nonzero constant during the maneuvering time and zero after the final time. Mathematically, the angular velocity and acceleration are written as: 3*(t*) - 5* [ u(c*) - u(c*-t*) ] (4.1) m e 3*(c*) - a; [ 5(c*) - 6(t*-t:) ] (4.2) where H and 5 are the heaviside step and Dirac-delta functions respectively. Using zero initial conditions, i.e. the beam is initially at rest, the set of ordinary differential equations (3.7) are written as: H; + (w:2 - 3*2(t*))qn - -an3*(c) (4.3.1) with qn(0) - dq/dt*(0) - 0 (4.3.2) By using an asymptotic approach, the nonhomogeneous term can be translated into the initial conditions. First (4.3.1) is integrated from time zero to e .15 16 C 6 qun(t*) dt* + wlzloqnu‘t) dt* *2 ‘ * * s 2 * * -Dm Io[h(t ) - H(t - te)] qn(t ) dt 6 * * * * * - -an&m [6(t ) - 8(t - ‘6’] dt 0 which can be reduced to (as 6 goes to zero) + 0* qn(0 ) ‘ 'anom Therefore the new problem becomes as: .. ~* *2 5*2 * 0 0 < * < * qn + (wn - m >qn(t ) - t te qn(0) - 0 * * dqn/dt (0) - -an&m The solution is a 5* sins t* q (t*) - - n m n n * * O t: (4.7.1) * * a 9 sins t _ n m n e *+ * *- rn(te - te) - qn(te ) - K (4.7.2) n * *+ * * *- * drn/dt (ce - te) - dqn/dt (ce ) + anbm - a 5*[1 - coss t*] (4.7.3) n m n e Therefore: 2 * a 3 sins t * n m * * * rn(t ) - [ - ~ ]coswh(t - te) n * + anbm 1 * i * * * * s 4 8 - * ( - cossnte) s nwn(t - te) t > te ( . ) w n Therefore the formal solution of the initial boundary value problem (3.3) is : v*(x*,c*) -z 1{qn(t*)[fl(t*) - H(t* - t:)] “- +rn(t*)[H(t* - c:)] }on(x*) (4.9) 18 where qn(t*), rn(t*) and §n(x*) are defined in equations (4.6) , (4.8) and (3.6.1) respectively. Equation (4.9) is a generalized or approximate solution because of the nonuniform convergence of the body force series. This equation is valid as long as the Bernoulli-Euler approximation is not violated. The dynamics of the beam during maneuver is not very important. However, the dynamics after the final time are the focal point of the maneuvering problem. The latter consists of examining the behavior of rn(t*) at time t: and thereafter. A very interesting point to make is that the term by term in the series solution (4.9) can be made zero by making sn an integer multiple of 2s/t: , i.e. 2k * k 1 2 3 4 1o sn - 1r/te - , , ,... ( . ) 2 2 2 2 1 2 solving for 3* - (w* -42 a /c* ) / (4.11) m n e 11' * The relation between an and the final time te is * * 5m - a/te (4.12) where a is the maneuvering angle. For example, if b; is to be made such that there is no contribution from the first mode, then *2 22 212 i; - (w1 - 4k « /c: ) / - a/t: k - 1.2.3,... (4.13) * The nondimensional natural frequency w1 can be written as: w‘f - 2m“: (4.14) 19 where T: is the first nondimensional natural period. Introduce (4.14) into (4.13) and solve for t:, then 22 212 c: - If (4: k + a ) / /2« k - 1.2.3,... (4.15) Therfore t: corresponds to an infinite set of critical final times such that the first term does not have any contribution to the free vibration response. MGM. ANALYSIS In this section, a general numerical scheme shall be constructed to solve the simple flexure problem. This scheme will be useful for any time-dependent angular velocity and acceleration. The nondimensional equation of motion (3.3.1) shall be used. The first term in this equation is approximated by a central difference on space and averaged. on time between the previous and future times. The second and third terms are evaluated at the present time, and the fourth is approximated by a central difference. The number of steps in space and time are N and M respectively, where M is the number of time steps up to the final time t*, i.e. e (5.1.1) ZIP 3(- At* - (5.1.2) :ZIr-t o The finite difference approximation of the nondimensional flexure equation is given by 1* { 1 *j +1 *1 ' 1 _‘ _ AX* 2 20 Nib + (r:+1Ax*)7i*(jAc*) - v"J Yugo?) 1 +1 -1 +i {x3j +v"j -2v*J}-o (5.2) *2 1 At 1 i (5.2) can be arranged as the following: 1+1 3+1 *‘ 1+1 1+1 3+1 v* - 4v* + ( 6 + ZAX )v* - 4v* + v* 2 - ' - 1+2 1+1 I*At* i i 1 i 2 *‘ * * 1 *‘ * -* * - 2 Ax ( 2v J - v j' ) + 2 Ax [ -(ro+ iAx )3 (jAt ) * *2 1 1 * I At I j -1 -l -l -l + v* 3*2(jAt*) ] - { v*J - 4v*J + 6v*J - 4v“.J 1 1+2 i+l i i-l + v } (5.3.1) where the terms of the left side of the equal sign are the future time unknowns, and i andj are defined as integers corresponding to space and time respectively. The approximated boundary and initial conditions are: v - 0 (5.3.2) 22 *J v - 0 (5.3.3) 1 J .1 J v* + v* - 2v* - 0 (5.3.4) N+1 N-1 N J J J J v* - 3v* + 3v* - v* - 0 (5.3.5) N+2 n+1 N N-l o v* - 0 (5.3.6) 1 o -1 v* - v* - 0 (5.3.7) 1 1 *‘1 2 a: where v is a fictitious point at t - -At . The combination of 1 equations (5.3.1)-(5.3.5) constitute a set of linear algebraic equations which are to be solved simultaneously at each time step. For progarmming convenience the linear equations are written in a matrix form: - B (5.4.1) where *3 *M *M *M *M T Xk'[22 2324 EN] (5-4-2) M - 1+1: j, j-1 23 *‘ * * * . *3 2 * 5 - 2Ax [ -(r° + I Ax ) 5*(jAc*) + g 3* (jAt ) ] * i I AX*‘ *J *j ' 1 + 2 [ 21 - y ] - 9 (5.4.3) 2 1*At* 1 1 where i - 2,3,...,N j - 0,1,2,3,...,M,M+1,... and j-l j-l j-l j-l *j-l * * * * Q ‘ 3 1+2 ' “3 1+1 + 63 1 ’ “3 1-1 + 3 1-2 (5'“'“) The boundary conditions (5.3.4) and (5.3.5) are accounted for in (5.4.4) when i is equal N-l or N. The matrix M is of the following format a -4 1 O 0 O 0 ......... 0 O 0 0 0 -4 a —4 1 0 0 0 ......... 0 0 0 O 0 l -4 a -4 l O 0 ......... 0 0 O 0 O 0 l -4 a -4 l 0 ......... 0 0 0 0 0 fl - . . . . . . . . . . . . (5.4.5) 0 0 0 0 0 0 0 ......... l —4 a -4 l 0 0 0 0 O 0 0 ......... 0 l -4 a-l -2 0 0 0 0 0 0 O ......... 0 O 1 -2 a-5 24 where o - 6 + 2 A" (5.4.6) and M is an (N-1)x(N-l) matrix. The matrix 11 accounts for the boundary conditions at the fixed and free ends. tan u - - V The above scheme is applied to a rectangular-pulse-angular velocity. The expression (4.2) can be approximated by 9* * In/At: 3-01.... .M-‘M+l...time 03(- * ' * * ~2- t-OAt . .. .t te+At. I i . fig(3): Impulse approximation 25 If At* is chosen as small as possible, then the impulses at t*-0 * 2 and tint—te can be approximated by a step function of duration At and 5* magnitude _2_ . Therefore, the angular acceleration is expressed as: At* '; if j - o 3*(3Ac*) - 4 o if 0 < j < M (5.5) -D* /At* if j - u m L 0 if j > M The angular velocity is defined as: ' 0 if j - 0 3*(jot*) - < a; if 0 < j < M (5.6) 0 if j - M k 0 if j > M The numerical scheme above is constructed using an implicit method. We shall study the stability of this scheme: Stability J A Let 3* - VowjeiKJ (5.7) 1 where j - J -l 26 Substitute (5.7) into equation (5.2) Vow { wJ e(i+2)KJ _ 4wJ e(i+l)l(.j + (6 + 2a)wj eiKJ - 4WJ e(i'l)Kj + VJ e(1-2)Kj } " 20‘Vo ( 2‘41 €in - wj'l e1K1 ) + Zak."r2 At*2 VowJ ein m 'Vow'l {wj e(i+2)l:._. ._:0ZOZ eo+me ¢0+mn 2“40+MN ¢0+m: zo:.<2.x0m_md< ZOEjOm mmEmw ll 0 OP.0l 00.0 070 NOIlISOd clll 'lVNOlSNElWlGNON Plot 1: Tip displacement using the series solution 31 m2: I_=0ZOZ ¢0+mm 001%? 00.0%.»; +0.“.MN 001m“: 0 P Pi PAID I I > I I 1 I I I I I I ZO_.—<2_X0mm&< mozwmmEEQ thE .II I I I I I 1 I I I 0_..0l 00.0 9.0 NOLLISOd clLL WVNOISNEWICINON Plot 2: Tip displacement using the finite difference scheme 32 ¢0+mm 00.1.? P m:>:._. :_:DZOZ vofimm LVOWMN 00.05: 0 A l l l l l l IIITITTI ZOr—(ZXOmnE/x mozwmwmma m:._ZE ll 0. I O. 0 0. 9+30l'300801 ‘lVNOISNEWIGNON Plot 3: Torque applied at the hub 33 1.0 APPROXIMATE vs EXACT REPRESENTATIONS 0F r, +x NONDIMENSIONAL POINT LOCATION 1 .I .I -I 1 J .I .I .1 3 1.5 x+ ‘31 = (x); Plot 4: Comparison between exact and approximate representations of the body force CHAPTER SEVEN DISCUSSION OF RESULTS The rectangular velocity has discontinuities at t*- 0 and t*- t:. These discontinuities produce impulsive loadings at the begining and ending of the maneuver. These impulses occur in the field equation of motion as body forces. The sudden change of the angular velocity from zero to I; at t*- 0 and I: to zero at t*- t: produce defamation in the beam. In plots (1) and (2) there are two phases: forced-motion phase and free-motion phase. In the forced—motion phase, the number of oscillations from zero to t: is equal to k (see equation (4.15)). For a - s/2, one can show that c: - If (k2 + 1 )V2 -- 21? (7.1) For instance if k - 6, there will be six oscillations in the forced- motion phase. The free motion depends on the physical state of the beam at the final time. It also depends on the velocity jump which has an absolute value of (r: + x*)6; . This jump depends on the point location in space, and also on the magnitude of the maximum angular velocity. Plots (1) and (2) correspond to the case where the first mode has no contribution to the overall response in the free vibration phase. Plot (1) was obtained by taking eight terms of the series solution (4.9). It was verified that 8 terms are adequate to represent the 34 35 series solution. Plot (2) was obtained by using the finite difference scheme described in Chapter 5. Plots (1) and (2) are in good agreement. The minor difference occuring at t*- t: and thereafter is due to an inaccurate estimation of the maximum angular velocity IZ'Uy the series solution (4.9). The main error was generated from the representation of the body force by a nonunifom convergent series.. Plot (4) describes very well the difference between the ideal function (r: + x*) and the partial sum of the series. It was verified by taking more terms that the series is convergent to the function in plot (4). The numerical solution is a representation of the exact solution. It was verified that the curve in plot(2) is recovered for different step sizes. Plot (3) was obtained using the numerical scheme. It represents the applied torque necessary to produce a constant angular velocity. Note the two impulses which produce the sudden changes in angular velocity at t*- 0 and t*- t: The vibrations, in the first and second phases, indicate that a torque has to be applied at the hub in order to kill the rotational vibrations of the rigid hub due to the nonzero * moment at x - 0. SUMMARY AND CONCLUSIONS - Two approximate mathematical models were derived in order to describe the overall flexibility of a rotating beam. - An approximate analytical solution was found and then used to estimate the necessary maximum angular velocity required to delete the effect of any desired mode in the free vibration phase. - A general numerical scheme was developed, and shown to be unconditionally stable. - The analytical and numerical solutions are in good agreement in killing the first made in the free vibration phase. 36 fin-laid Shirt OE" APPENDIXA C************************************************************C C C C THIS PROGRAM IS USED TO SOLVE THE NONDIMENSIONAL C C SIMPLE FLEXURE FOR ANY TIME-DEPENDENT ANGULAR C C VELOCITY ( THE ANGULAR PULSE ANGULAR VELOCITY C C IS AN ILLUSTRATION IN THIS PROGRAM) C C C C************************************************************C PROGRAM BACK DOUBLE PRECISION W(20,20),A(19,l9),T,SI,THT2 DOUBLE PRECISION Vl(19),V2(l9),DX,DT,Cl,C2,D,SIH,TORQUE #,V3(19),TE,R0,B(19),PI,ALPHA,Sl(19),S2(19),S3(l9),THTT #,Y,THTM OPEN(100,FILE-'AMINIS') C DEFINITION OF PARAMETERS C SI: NONDIMENSIONAL MOMENT OF INERTIA OF THE BEAM C R0: NONDIMENSIONAL HUB RADIUS C Y: CONSTANT 0F INTEGRATION C SIH: NONDIMENSIONAL MASS MOMENT 0F INERTIA OF THE HUB C TE: NONDIMENSIONAL FINAL TIME AT WHICH ROTATION IS STOPPED C KK: THE NUMBER OF TIME STEPS UP TO TE 37 38 DT: NONDIMENSIONAL TIME INCREMENT DX: NONDIMENSIONAL SPACE INCREMENT ALPHA: FINAL ANGLE THTM: NONDIMENSIONAL MAXIMUM ANGULAR VELOCITY SI-1.0288E-7 20-0.125 Yh((R0+1)**3.0-R0**3.0)/3.0 SIM-0.091145833 ' D-(l.8751041)**2.0*SQRT(SI)*0.041630544 TE-ACOS(-l.0)/(2.0*D) xx-sooo P-RK DT-TE/P ox-1.0/20.0 PI-ACOS(-l.0) ALPHA-PI/2.0 THTM-ALPHA/TE C**** ZERO ALL THE MATRIX ENTRIES, DEFINE INITIAL CONDITIONS 112 111 D0 111 Kl-l,l9 DO 112 K2-l,l9 A(Kl,K2)-0.0 CONTINUE CONTINUE D0 101 I-1,19 V1(I)-0.0 V2(I)-0.0 101 135 c**** c**** 39 CONTINUE T-0.0 rurz-o TORQUE-0.0 WRITE(100,135)T,THT2,V2(19),TORQUE FORMAT(F16.8,1X,F12.8,lX,F12.8,lX,F16.14) T-DT DO 102 J-1,6000 DEFINE THE ANGULAR VELOCITY AND ACCELERATION IF(J.GE.1.AND.J.LT.(KK))THT2-THTM IF(J.EQ.(KK))THT2-0.0 IF(J.GT.(KK))THT2-0.0 IF(J.EQ.1)THTT-THTM/(DT) IF(J.NE.1)THTT-0.0 IF(J.EQ.(KK))THTT--THTM/(DT) DEFINE THE NONHOMOGENEOUS VECTOR B 02-0x**4.0/(sr) 00 103 1-1,19 IF(I.EQ.l)G-Vl(3)-4*V1(2)+6*V1(l) IF(I.EQ.2)G-V1(4)-4*Vl(3)+6*V1(2)—4*Vl(1) IF(I.GT.2.AND.I.LT.18)G-Vl(I+2)-4*V1(I+1)+6*V1(I) # -4*Vl(I-l)+Vl(I-2) IF(I.EQ.18)G--2*V1(19)+5*V1(18)-4*V1(17)+V1(l6) IF(I.EQ.l9)G-Vl(l9)-2*V1(18)+V1(17) B(I)--2*C2*(RO+(I+1)*DX)*THTT 4O # +2*C2*THT2*THT2*V2(I)-2*C2*(V1(I)-2*V2(I))/DT**2.0-G 103 CONTINUE C**** RESET THE NONZERO ENTRIES IN THE MATRIX A C1-6+2*DX**4.0/(SI*DT**2.0) N-1 D0 113 K1-3,17 A(K1,N)-1.0 A(Kl,N+1)--4.0 A(K1,N+2)-C1 A(K1,N+3)--4.0 A(Kl,N+4)-l.0 N-N+1 113 CONTINUE A(1,1)-C1 A(1,2)--4.0 A(1,3)-1.0 A(2,1)--4.0 A(2,2)-Cl A(2,3)--4.0 A(2,4)-1.0 A(18,16)-1.0 A(18,17)--4.0 A(18,18)—C1-1.0 A(18,19)--2.0 A(19,17)-1.0 A(19,18)--2.0 A(19,19)-C1-5.0 41 C**** CALCULATE THE FUTURE TIME DISPLACEMENT VECTOR CALL DLINEQ(V3,B,A,W,l9,20,IERR) C**** CALCULATE THE TORQUE NECESSARY TO PRODUCE THE PRESCRIBED C**** ANGULAR VELOCITY USING SIMPSON'S METHOD 00 161 I-1,19 Sl(I)-ABS(V3(I))**2.0 82(I)-2.0*(V3(I)-V2(I))*V3(I)/DT S3(I)-(R0+(I+1)*DX)*(V3(I)+V1(I)-2*V2(I)) #/DT**2.0 161 CONTINUE SUMl-DX/3.0*(2.0*Sl(1)+4*Sl(2)+2*Sl(3)+4*Sl(4)+2*Sl(5)+4*Sl(6) #+2*Sl(7)+4*Sl(8)+2*Sl(9)+4*Sl(10)+2*Sl(11)+4*Sl(12)+2*Sl(13) #+4*Sl(14)+2*Sl(15)+4*Sl(16)+2*81(17)+4*Sl(18)+Sl(19)) SUM2-DX/3.0*(2.0*82(1)+4*82(2)+2*52(3)+4*82(4)+2*82(5)+4*82(6) #+2*82(7)+4*82(8)+2*82(9)+4*82(10)+2*82(11)+4*82(12)+2*82(13) #+4*82(14)+2*82(15)+4*82(16)+2*S2(17)+4*82(18)+S2(19)) SUM3-DX/3.0*(2.0*S3(1)+4*83(2)+2*S3(3)+4*83(4)+2*83(5)+4*S3(6) #+2*53(7)+4*s3(8)+2*s3(9)+4*s3(10)+2*s3(11)+4*s3(12)+2*s3(13) #+4*S3(14)+2*S3(15)+4*S3(16)+2*S3(17)+4*83(18)+SB(19)) TORQUE-((SIH+Y+SUM1)*THTT #+SUM2*THT2+SUM3)*1E+6 C**** PRINT THE RESPONSE WRITE(lOO,179)T,THT2,V3(19),TORQUE 179 FORMAT(F16.8,1X,F12.8,1X,F12.8,lX,Fl6.l4) 42 C**** REINITIALIZE THE DISPLACEMENT VECTORS D0 106 I-l,19 V1(I)-V2(I) V2(I)-V3(I) 106 CONTINUE T-T+DT 102 CONTINUE STOP END 0******+****************** END op PROGRAM ********+****************c 6******************************************************************C C C C DEFINITION OF VARIBLES C c -------------------- c C T: NONDIMENSIONAL TIME C C THT2: NONDIMENSIONAL ANGULAR VELOCITY C C THTT: NONDIMENSIONAL ANGULAR ACCELERATION C C TORQUE: NONDIMENSIONAL APPLIED TORQUE C C Vl,V2,V3: NONDIMENSIONAL DISPLACEMENTS AT DIFFERENT TIME C C A: MATRIX C C B: NONHOMOGENEOUS VECTOR C C C C******************************************************************C APPENDIX B APPENDIX.B C**************************************************************C C C C THIS PROGRAM CALCULATE THE TIP RESPONSE USING 8 TERMS C C FROM THE SERIES . C C C C**************************************************************C PROGRAM SERIES DIMENSION V(8),VV(8),BE(8) #,PHI(8),W(8),WN(8),A(8),EE(8) DOUBLE PRECISION A,V,W,WN,T,BE,PHI,VV,EE,THTO #,Z,VT,VVT,DT,SI,ALPHA,TO,PI,RO 0PEN(lOO,FILE-'DELTA1') PI-ACOS(-l.0) C**** DEFINE THE NONDIMENSIONAL WAVE LENGTH BE(1)-l.8751041 BE(2)-4.69409ll BE(3)-7.8547574 BE(4)-10.9955407 BE(5)-14.1371684 BE(6)-l7.2787595327 BE(7)-20.420352251 43 c**** c**** c**** c**** c**** it 102 44 BE(8)-23.56l944900414 DEFINITION OF PARAMETERS ALPHA: ANGLE OF ROTATION SI: NONDIMENSIONAL MOMENT OF INERTIA R0: NONDIMENSIONAL HUB RADIUS TO: NONDIMENSIONAL FINAL TIME ALPHA-PI/2.0 r-o.0 SI-l.0288E-04 no-o.125 00 102 3—1,8 EE(J)-(DCOSH(BE(J))+DCOS(BE(J)))/(DSINH(BE(J))+DSIN(BE(J))) PHI(J)-DCOSH(BE(J))-DCOS(BE(J))-EE(J)*(DSINH(BE(J)) -DSIN(BE(J))) A(J)-2.0*(EE(J)*R0+1.0/BE(J))/BE(J) W(J)-BE(J)**2.0*SQRT(SI) CONTINUE CV-1.0/(SQRT(1.0+16.0*l.0)) THTO-CV*W(1) TO-ALPHA/THTO or-r0/5000.0 PRINT *,nr 00 101 I-1,6000 00 105 J-1,8 WN(J)-SQRT(W(J)**2.0-THTO**2.0) I .'_-- ms- 45 C**** DEFINE THE DISPLACEMENT AND VELOCITY DURING THE FORCED C**** VIBRATION MOTION IF(T.LE.T0)THEN V(J)--A(J)*THTO*DSIN(WN(J)*T)/WN(J)*PHI(J) VV(J)-°A(J)*THTO*DCOS(WN(J)*T)*PHI(J) ENDIF a. H Li)“: ALL‘h‘aJA'J 1 . C**** DEFINE THE DISPLACEMENT AND VELOCITY DURING THE FREE C**** VIBRATION MOTION IF(T.GT.TO)THEN V(J)-(-A(J)*THTO*DSIN(WN(J)*TO)*DCOS(W(J)*(T-TO))/WN(J) #+A(J)*THTO*(1.0-DCOS(WN(J)*TO))/W(J)*DSIN(W(J)*(T-TO)))* #PHI(J) VV(J)-(A(J)*THTO*DSIN(WN(J)*TO)*W(J)*DSIN(W(J)*(T-TO))/WN(J) #+A(J)*THTO*(1.0-DCOS(WN(J)*TO))*DCOS(W(J)*(T—TO)))*PHI(J) ENDIF 105 CONTINUE C**** SUM OF THE FIRST EIGHT TERMS IN THE SERIES VT-V(1)+V(2)+V(3)+V(4)+V(5)+V(6)+V(7)+V(8) VVT-VV(1)+VV(2)+VV(3)+VV(4)+VV(5)+VV(6)+VV(7)+VV(8) WRITE(100,104)T,VT,VVT 104 FORMAT(F18.8,3X,F12.8,3X,F12.8) T—T+DT lOl CONTINUE PRINT *.A(1).A(2).A(3).A(4).A(5).A(6).A(7).A(8) 46 STOP c************************ END op PROGRAM **********+*************c C****************************************************************C C C DEFINITION OF VARIABLES T: NONDIMENSIONAL TIME V(I): NONDIMENSIONAL TIP DISPLACEMENT USING THE ITH TERM VV(I): NONDIMENSIONAL TIP VELOCITY USING THE ITH TERM W(J): NONDIMENSIONAL NATURAL FREQUENCY OF THE JTH MODE PHI(J): NONDIMENSIONAL EIGEN-FUNCTION OF THE JTH MODE A(J): NONDIMENSIONAL CONSTANT FROM THE EXPANSION OF THE THE BODY FORCE ABOUT THE EIGENFUNCTIONS PHI(J)'S C C C****************************************************************C REFERHICES [1] [2] [3] [4] [5] [6] REFERENCES Book, W.J., and, Majette, M., 1983. CONTROLLER DESIGN FOR FLEXIBLE DISTRIBUTED PARAMETER MECHANICAL ARMS VIA COMBINED STATE SPACE AND FREQUENCY DOMAIN TECHNIQUES. Journal of Dynamics Systems, Measurment, and Control, December 1983, Vol. 105: 245-254. Book, W.J., Maizza-netta, 0., and Whitney, D.E., 1975. FEEDBACK CONTROL OF TWO BEAM, TWO JOINT SYSTEMS WITH DISTRIBUTED FLEXIBILITY. Transaction of the ASME, December 1975: 424-431. Cannon, R.H. Jr, and Eric Shmitz, 1984. INITIAL EXPERIMENTS ON THE END-POINT CONTROL OF A FLEXIBLE ONE-LINK ROBOT. The International Journal of Robotics Research, Vol.3, No.3, Fall 1984: 62-75. Cannon R.H. ,Jr. , 1984. ROBOTS WITH A LIGHT TOUCH. Control Systems Magazine, May 1984: 14-16. Churchill,R.V. , and Brown,J.W. , FOURIER SERIES AND BOUNDARY VALUE PROBLEMS, McGraw-Hill 1978, Chapters 2-6. Jasinski,P.W., Lee,.H.C, and Sandor,G.N., 1971. VIBRATIONS 0F ELASTIC CONNECTING ROD OF A HIGH-SPEED SLIDER CRANK MECHANISM. Transactions of the ASME, May 1971: 636-644. 47 48 [7] Meirovitch,L. , ANALYTICAL METHODS IN VIBRATIONS, MacMillan, 1967, chapters 5 and 10. [8] Nachman,A., 1986. BUCKLING AND VIBRATIONS OF A ROTATING BEAM. Journal of Sound and Vibration, 1986, Vol 109, No 3, 435-443. [9] Potter, M.C., MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES, Prentice Hall, 1978, Chapter 7. [10] Richtmyer,R.D., and Morton,l(.W. DIFFERENCE METHODS FOR INITIAL- VALUE PROBLEMS, Wiley, 1957, Chapter 11. [ll] Sung,C.K. , Thompson,B.S. , and MacGrath,J.J. , A VARIATIONAL PRINCIPLE FOR THE LINEAR COUPLED THERMOELASTODYNAMIC ANALYSIS OF MECHANISM SYSTEMS. Transactions of the ASME, December 17, 1984. [12] Timoshenko , S . P , Young , D . H , and Weaver ,W. , VIBRATION PROBLEMS IN ENGENEERING, Wiley, 1974, 4th Edition, Chapter 5. [13] Viscomi,B.V. , and Ayre,R.S., 1971. NONLINEAR DYNAMIC RESPONSE OF ELASTIC SLIDER-CRANK MECHANISM. Journal of Engineering for Industry, February 1971: 251-261. [14] Volterra, E. , and Zachmanaglou, E.C. , DYNAMICS OF VIBRATIONS. Merril, 1965, Chapter 4. "IIIIIIIIIIIIIIIIIJIIISIIIIIIIIII“ 3 1293