1 differential equations of mechanical systems

... or (11 .10 .15 ) or (11 .10 .16 ) (11 .10 .17 ) and (11 .10 .18 ) From Eq. (11 .10 .12 ) the partial velocities of G 1 and G 2 and the partial angular velocities of B 1 and B 2 are (see also Eqs. (11 .4.23)): ... be defined as: (11 .11 .10 ) Then, from Eq. (11 .11 .1) , we have: (11 .11 .11 ) which is consistent with Eq. (11 .11 .9). It happens that Eqs. (11 .11 .5) and (11 .11 .10 ) are potential energy functions for gravity ... vector as in Figure 11 .11 .1. The generalized force due to gravity is then: (11 .11 .4) Let P be a potential energy defined as: (11 .11 .5) Then, from Eq. (11 .11 .1) , F h is: (11 .11 .6) which is consistent

Ngày tải lên: 10/08/2014, 20:20

... Dynamics of Mechanical Systems (12 .4 .15 ) (12 .4 .16 ) (12 .4 .17 ) (12 .4 .18 ) (12 .4 .19 ) By substituting from Eqs. (12 .4.8), (12 .4.9), and (12 .4 .10 ) and Eqs. (12 .4 .14 ) through (12 .4 .19 ) into Lagrange’s equations, ... degree of freedom (see Problem P12.2 .11 ). Use Kane’s equations to determine the equations of motion. Section 12 .3 Lagrange’s Equations P12.3 .1 to P12.3 .10 : Repeat Problems P12.2 .1 to P12.2 .10 by ... governing dynamical equations of the system: (12 .5 .12 ) where the coefficients m rj , n rj , and k rj are: (12 .5 .13 ) (12 .5 .14 ) (12 .5 .15 ) (12 .5 .16 ) (12 .5 .17 ) (12 .5 .18 ) 12 .6 Closure The computational

Ngày tải lên: 10/08/2014, 20:20

... and (14 .3 .12 ) By substituting from Eqs. (14 .3 .11 ) and (14 .3 .12 ) into (14 .3 .1) we obtain: (14 .3 .13 ) Because the right side of Eq. (14 .3 .13 ) is a constant, the stability (or instability) of the ... inspection, we see that the ω i (i = 1, 2, 3) of Eqs. (14 .4 .11 ), (14 .4 .12 ), and (14 .4 .13 ) are solutions of Eqs. (14 .4.8), (14 .4.9), and (14 .4 .10 ). To test for the stability of this solution, let small ... (14 .4.6) and (14 .4.7) into (14 .4.5), we obtain: (14 .4.8) (14 .4.9) (14 .4 .10 ) where I 11 , I 22 , and I 33 are the central principal moments of inertia. Equations (14 .4.8), (14 .4.9), and (14 .4 .10 ) form

Ngày tải lên: 10/08/2014, 20:20

... i (i – 1) cosφφ φφ i (i – 1) sinφφ φφ i (i – 1) cos2φφ φφ i (i – 1) sin2φφ φφ i 10 1 010 0 0 0 0 2900 1 ? ?10 0 1 ? ?10 3 270 0 ? ?1 ? ?10 0 –2 –20 4 18 0 ? ?10 10 –30 30 5 18 0 ? ?10 10 –40 40 6 270 0 ? ?1 ? ?10 0 –5 ... (i – 1) cosφφ φφ i (i – 1) sinφφ φφ i (i – 1) cos2φφ φφ i (i – 1) sin2φφ φφ i 10 10 10 0 0 0 0 2900 1 ? ?10 0 1 ? ?10 3 18 0 ? ?10 10 –20 20 4 270 0 ? ?1 ? ?10 0 –3 –30 5 360 1 0 1 0 4 0 4 0 6 450 0 1 ? ?10 0 ... 16 .1. 3, Figures 16 .2 .1, 16 .2.2, and 16 .2.3 depict other common types of cams. Figures 16 .2.4 and 16 .2.5 depict various common FIGURE 16 .1. 1 A simple cam–follower pair. FIGURE 16 .1. 2 A translation

Ngày tải lên: 10/08/2014, 20:20

... Dynamics of Mechanical Systems 11 10 9 12 8 15 6 7 3 14 13 16 1 4 2 5 R FIGURE 18 .3.3... to multibody systems As the name implies, multibody systems contain many bodies Multibody systems ... substituting from Eqs. (17 .6.8) and (17 .6 .10 ) into (17 .6.7), we have: (17 .6 .11 ) Solving for ᐉ 2 , we obtain: (17 .6 .12 ) Similarly, for gear 1 we have: (17 .6 .13 ) where ᐉ 1 is the length of the contact ... Mechanical Systems or (17 .6 .16 ) From Eqs. (17 .5.2) and (17 .5.5), we then also have: (17 .6 .17 ) and (17 .6 .18 ) Hence, from Eqs. (17 .6 .12 ) and (17 .6 .13 ) the contact path segment lengths are found to be: (17 .6 .19 )

Ngày tải lên: 10/08/2014, 20:20

... 8 7 9 3 10 9 11 10 12 1 13 12 14 13 15 1 16 15 17 16 0593_C20_fm Page 704 Tuesday, May 7, 2002 9 :14 AM 704 Dynamics of Mechanical Systems 8 4 3 9 5 2 10 6 1 11 7 12 13 16 14 ... typical ... (19 .17 .12 ) and (19 .17 .13 ), we see that: (19 .17 .14 ) thus establishing Eq. (19 .17 .4). In view of Eq. (19 .17 .14 ) the governing equations, Eqs. (19 .17 .1) and (19 .17 .3), may be written as: (19 .17 .15 ) ... development of many of these methods As before, we seek to find for each body of the... Equations of Motion P19 .15 . 1: Using Eqs (19 .13 .11 ), (19 .14 .8), and (19 .15 . 1) , verify Eqs (19 .15 . 2)

Ngày tải lên: 10/08/2014, 20:20

... O 1 2 3 O O n 1 1 2 2 3 3 O1 O2 θ θ θ FF MM 12 21 12 21 =− =−and IIIII III kk k k k kkk 11 33 3 22 2 12 13 23 0 == = === () () , 0593_C20_fm Page 733 Tuesday, May 7, 2002 9 :14 ... May 7, 2002 9: 21 AM 750 Dynamics of Mechanical Systems III Solids, bodies 1 Parallelepiped, block: I 11 = m(a2 + c2) /12 I22 = m(b2 + c2) /12 I33 = m(a2 + b2) /12 2 Cylinder: I 11 = mr2/2 I22 ... integral... shell: I 11 = mr2/2 I22 = m(r2/2 + h2/9)/2 12 Half cone, half-conical shell: I 11 = mr2 (1 – 8/9Π2)/2 I22 = (m/36) (9r2 + 2h2) I33 = mh2 /18 + mr2 (1 – 16 / 9Π2)/4 I12 = –mrh/9Π 749 0593

Ngày tải lên: 10/08/2014, 20:20

... small of order c. The motion equations then become 6 NGUYEN VAN DAO (11 ) (12 ) M x + cx = e[ — A(x — t>) + hịX — /13 Ì 3], Ú = V, mi) = \(x — v). The solution of the system of equations (11 ) will ... the mass m of Lanchester damper is small quantity of order e is investigated. In this case the differential equations of motion (2 1 ) become miXi + (cI + c12)z 1 - c12x2 = sFit + c12( i 2 - ... vibrations of mechanical systems with finite degrees .of freedom and also system s w ith distributed param eters have been published [1- 20]. In paper [19 ] certain characteristics of the behaviour of

Ngày tải lên: 08/04/2015, 15:29

... Proc of IC N O -X I, 693-698, B u d a p e st 19 87 Nguyen Van D a o , D y n a m ic absorber for self-excited sy s te m with limit energy resource, J M ec n ic s, H3IÌOĨ, 14 11 12 13 14 No (19 89) 15 ... / / 31^ 1 èị = u /1, - -H 32vịaị)ì è2 = W, J a7 = (29) - - / / 32^ '? « ? - Ị ^ 3 ^ q 2)» where / / /12 = /11 I - Y j ( x ) d x - J A,[*l2 - ya(x,)]3f u •'*» / = /13 Ị Y?(x)YỈ(x)ji, ' (30) 1 # 33 ... V o lu m e : H A N O I 19 91 Proceedings of the National C entre f o r Scientific Research of Vietnam, Vol (19 91) (3 -18 ) Mechanics M U L T I- D Y N A M IC A D S O R B

Ngày tải lên: 08/04/2015, 15:29

... signal. 49 40 30 20 10 0 -10 0 2 4 6 8 10 12 0 2 4 6 8 10 12 20 15 10 5 0 -5 Figure 4 .11 The optimal control system output with both the actual c =1 and the estimated C =1. 1. The solid ... ε=0.2; λ =1. 0; Kd=0.3; K=0.5; r=0 .1; η=0 .1; Figure 4 .14 cmin=0.8; cmax =1. 2 λ=0 .1; Kv=0 .1; Kb=0 .1; Kz1=5; Kz2=2; Kz3=5; N/A K=0.0 01; Zm =10 ; S and T are initialized to be unit matrices. Number of hidden ... the closed-loop control system is H * (s) = = (C1 − C2 )G (1 + ∆G ) N 1 + C1G (1 + ∆G ) N C *G (1 + ∆G ) PN (3.5) 1 + (1 + ∆G ) PN + C *G (1 + ∆G ) PN where C1 = PG ? ?1 + C * P . C* could be designed by

Ngày tải lên: 03/10/2015, 21:57

... a K-linear differential automorphism of the field of fractions of K[S, T, d 11 ]/( f1 ) The group of the differential automorphism induce a group of permutations of the solutions of f1 (ii) We ... Proc Cambridge Philos Soc., 68 (19 70) 10 512 3 [F] C Faber Chow rings of moduli spaces of curves I: Yhe Chow ring of M Ann of Math 13 2 (19 90), 3 31- 419 [K] E.R Kolchin Differential algebra and algebraic ... of rational functions on a curve Proc Cambridge Philos Soc., 68 (19 70) 10 512 3 [F] C Faber Chow rings of moduli spaces of curves I: Yhe Chow ring of M 3 Ann of Math 13 2 (19 90), 3 31- 419 ...

Ngày tải lên: 26/10/2015, 14:01

... [1 1 11 2222 Az (9 3333 |4 4 4 4 [0 10 o đọ A= |1 00 0 4) 0 01 +1 |0 01 1 [1 -10 0 1 1090 of? ? e) A= 6) 0 O 11 -1 L0 01 1 [1- 11 0 Ð A=|l 19 1 @) 0 01- 1 |0 01 1 1. 8 Jordan Forms 39 4 The “Putzer Algorithm” ... the following matrices [11 0 (2) A=|0 10 |0 01 [11 0 (Œ) A=|0 11 |0 0: [10 0 () A=]0 0 -1 0 1 0 1 1 0 (d)A=l0 1 0 10 0 -1 10 0 () A=l0 1 1 |0 0 -1 fl oo (f)A=]0 0 1} 19 10 (a) List the five upper ... 2 0 |1 2 3 [ 10 0 (b) A=|-L 2 0 | 10 2 [1 1 2 () A=|0 2 1 |0 0 2 [2 1 2 (d) A=|0 2 1 |o 0 2 1 000 12 00 wmlies 1 2 4 4 r1 000 12 00 oles l1 10 2 [2 14 0 0 210 @MA =19 920 lo 00 2 [2 1 4 0 : 0 21 -1 Œ)

Ngày tải lên: 11/05/2018, 16:12

... Page 12 Monday, May 6, 2002 1: 43 PM 12 Dynamics of Mechanical Systems 1. 5 1. 6 1. 7 1. 8 1. 9 1. 10 1. 11 1 .12 1. 13 1. 14 1. 15 1. 16 1. 17 1. 18 1. 19 1. 20 1. 21 1.22 1. 23 1. 24 1. 25 Hsu, H P., Vector Analysis, ... Graphs .663 0593_ FM_fm Page 17 Tuesday, May 14 , 2002 10 :52 AM 19 .4 19 .5 19 .6 19 .7 19 .8 19 .9 19 .10 19 .11 19 .12 19 .13 19 .14 19 .15 19 .16 19 .17 19 .18 Angular Velocity of Robot Links 665 ... 333 × 10 –2 6 21 × 10 1 745 × 10 0 818 × 10 0 623 × 10 1 182 × 10 –3 818 × 10 0 303 × 10 –5 697 × 10 –2 999 × 10 2 6 21 × 10 1 373 × 10 1 022 × 10 –3 000* × 10 2 000* × 10 4 757 × 10 3 000* × 10 2 026 × 10 1 444...

Ngày tải lên: 10/08/2014, 20:20

... 0 .1 −0 .1 −0.2 8.5 9.5 10 10 .5 11 11 .5 12 12 .5 Imaginary part (m/N ) ω (rad/s) −0.05 −0 .1 −0 .15 −0.2 −0.25 −0.3 −0.35 8.5 9.5 10 10 .5 11 11 .5 12 12 .5 ω (rad/s) Figure Real and imaginary parts of ... Computations Applied to FEM 12 Real part (m/N ) 0.2 0 .1 −0 .1 −0.2 8.5 9.5 10 10 .5 11 11 .5 12 12 .5 Imaginary part (m/N ) ω (rad/s) −0 .1 −0.2 −0.3 −0.4 8.5 9.5 10 10 .5 11 11 .5 12 12 .5 ω (rad/s) Figure ... 0 (1 + i 1 )k1 + (1 + iη2 )k2 − (1 + iη2 )k2 (1 + i 1 )k1 0 0 + e1 0 − (1 + iη2 )k2 (1 + iη2 )k2 + (1 + iη3 )k3 +e2 1 0 (1 + iη2 )k2 − (1 + iη2 )k2 m1 + e3 −ω 1 m2 (1 + iη3 )k3 − (1 + iη2 )k2 (1 +...

Ngày tải lên: 12/01/2014, 21:45

... equations 17 2 10 .4 Local behavior near fixed points 17 4 Chapter 11 Periodic solutions 17 7 11 .1 Stability of periodic solutions 17 7 11 .2 The Poincar´ map e 17 8 11 .3 Stable and unstable manifolds 18 0 ... integral equations 11 5 11 5 11 8 12 3 12 7 Chapter §8 .1 §8.2 §8.3 Planar dynamical systems The Poincar´–Bendixson theorem e Examples from ecology Examples from electrical engineering 12 9 12 9 13 3 13 7 Chapter ... 1. 1 Newton’s equations 1. 2 Classification of differential equations 1. 3 First order autonomous equations 1. 4 Finding explicit solutions 11 1. 5 Qualitative analysis of first order equations 16 ...

Ngày tải lên: 31/03/2014, 15:56

... Underdamped (0 ≤ ξ < or ≤ c eq < cc ) ix 92 94 95 10 1 10 6 10 6 10 7 10 9 11 6 12 1 12 3 12 6 12 8 13 2 13 8 13 8 13 9 14 2 14 3 14 5 14 7 15 1 15 4 15 5 15 6 15 8 15 8 15 9 16 0 16 5 16 6 x Contents Case II: Critically Damped ... 1. 3.5 Viscous Dampers in Series and Parallel vii 5 6 8 10 10 11 12 13 14 16 16 17 19 22 viii Contents Dampers in Series Dampers in Parallel 1. 4 Free Vibration of an Undamped SDOF System 1. 4 .1 ... Poles and Zeros of Transfer Function Frequency Response Function Exercise Problems 16 6 16 6 16 8 16 9 17 0 17 3 17 5 17 5 17 6 17 9 Vibration of Two-Degree -of- Freedom -Systems 4 .1 Mass, Stiffness,...

Ngày tải lên: 08/04/2014, 11:49

... Xing and Fu Advances in Difference Equations 2 011 , 2 011 :14 http://www.advancesindifferenceequations.com/content/2 011 /1/ 14 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 He, Z, He, X: Monotone ... Equations 2 011 , 2 011 :14 http://www.advancesindifferenceequations.com/content/2 011 /1/ 14 Page 15 of 17 Proof Note that ||A-1B|| ≤ implies ||x(0)|| = ||A 1 Bx (1) || ≤ ||A 1 B|| · ||x (1) || ≤ ||x (1) || ... x (1) Xing and Fu Advances in Difference Equations 2 011 , 2 011 :14 http://www.advancesindifferenceequations.com/content/2 011 /1/ 14 Page 11 of 17 2.5 1. 5 0.5 −0.5 1 0.5 1. 5 2.5 Figure Figure of...

Ngày tải lên: 21/06/2014, 02:20

... http://www.advancesindifferenceequations.com/content/2 011 /1/ 10 Page of 13 is u(t) = G1 (t, s)h1 (s) ds, where ⎧ α 1 α 1 − (t − s)α 1 ⎪ t (1 − s) ⎪ , ≤ s ≤ t ≤ 1, ⎨ (α) G1 (t, s) = α 1 (1 − s)α 1 ⎪t ⎪ ⎩ , ≤ t ... (0, 1) ; (A2) ta - 1( 1 - t)s (1 - s)a - ≤ Γ(a)G1 (t, s) ≤ (a - 1) s (1 - s)a - 1, for t, s Î (0, 1) ; (A3) G1 (t, s) > 0, for t, s Î (0, 1) ; (A4) ta - 1( 1 - t)s (1 - s)a - ≤ Γ(a)G1 (t, s) ≤ (a - 1) (1 ... ||u|| (α − 1) (β) 1 G1 (l, s)q2 (s) q1 (r)k2 (r)dr ds (3:9) Zhao et al Advances in Difference Equations 2 011 , 2 011 :10 http://www.advancesindifferenceequations.com/content/2 011 /1/ 10 Page of 13 Hence,...

Ngày tải lên: 21/06/2014, 03:20

... have u t1 − u t− 1 t ut t1 t − s q 1 σ s ds Γ q t − t1 t1 t1 t1 − s q 1 σ s ds Γ q t1 − s q−2 σ s ds Γ q 1 J u t− b1 , 2 .11 u t1 −u t− β −t α g1 u α I1 u t− , Δu t1 I1 u t − t ∈ t1 , t2 2 .12 Advances ... M, k 1, 2, , p 3.4 Let H1 β α H2 Mp K1 β α β α Mp 1 Mp Mp pM, − s q 1 p s ds Γ q β α 1 1−s q 1 0

Ngày tải lên: 21/06/2014, 07:20

... Journal of Inequalities and Applications Nonoscillation Behaviors of Solutions of 1. 1 In this section, we obtain sufficient conditions for the nonoscillation of solutions of 1. 1 Theorem 2 .1 Let ... bounded solutions of 1. 1 are Proof Let y t be a bounded solution of 1. 1 on Ty , ∞ , Ty ≥ 0, such that |y t | ≤ M for ∞, there exists a t1 > t0 such that g t ≥ Ty for t ≥ t1 In view of t ≥ Ty Since ... 2, pp 13 8 14 8, 19 94 12 N Parhi and P Das, “Oscillation criteria for a class of nonlinear differential equations of third order,” Annales Polonici Mathematici, vol 57, no 3, pp 219 –229, 19 92 13 N...

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