Dynamics of Mechanical Systems 2009 Part 5 pps

... Weights of Figures P6.8.7A and B i θθ θθ i (%) r i (in.) ᐉ i (in.) W i (lb) 1 45 4. 45 11.7 5. 0 2 15 6 .5 14 .5 3.0 3 30 2 .5 6.0 1. 15 C A D B O 3.7 in. 31 in. 17 in. OB = 13 in. E 10. 05 in. C ... P6.8.1 Masses and Coordinates of a Set of Particles P i m i x i y i z i P 1 6 –102 P 2 43–74 P 3 34 5 5 P 4 8807 P 5 1 –2 –3 –9 P 6 91–20 P 7 47–83 P 8 5 –34–2 P...

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... Analysis and Discussion of Three-Particle Movement: Modes of Vibration 455 059 3_ FM_fm Page 14 Tuesday, May 14, 2002 10:19 AM 6 Dynamics of Mechanical Systems Two kinds of vectors occur so ... coordinates (r, θ, z) of P. b. Find the spherical coordinates (ρ, φ, θ) of P. Vij k=++3 4 12 ft sec 059 3_C01_fm Page 12 Monday, May 6, 2002 1:43 PM 24 Dynamics of Mech...

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... ⋅=     = 1 2 2 dW dt dK dt ii = 059 3_C10_fm Page 330 Monday, May 6, 2002 2 :57 PM 328 Dynamics of Mechanical Systems If P is a particle with mass m, the kinetic energy K of P is defined as: (10 .5. 1) where v is the velocity of ... ⋅m 059 3_C10_fm Page 329 Monday, May 6, 2002 2 :57 PM 330 Dynamics of Mechanical Systems The left side of Eq. (10.6.2) may be recognize...

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... sum of products of the elements of the ith row of A with the corresponding elements of the jth column of B. Specifically, (2.10 .5) where n is the number of columns of A and the number of rows of ... − − =− − − 123 123 13 8 724 46023 059 3_C02_fm Page 33 Monday, May 6, 2002 1:46 PM 58 Dynamics of Mechanical Systems the derivative is calculated. Thus, instead...

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... the derivative of c in R is: (4 .5. 4) Next, consider the product R ω B × n 1 : (4 .5. 5) Recall that: (4 .5. 6) By differentiating these expressions we have: (4 .5. 7) Hence, Eq. (4 .5. 5) may be written as: (4 .5. 8) Because ... Figure 5. 3.4. From Eq. (5. 3.9), the velocity and acceleration of Q are: (5. 3.12) where Ω is . In terms of n x and n y , v Q and a Q are: (5. 3.13)...

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... × () ααωωωω 059 3_C 05_ fm Page 151 Monday, May 6, 2002 2: 15 PM 154 Dynamics of Mechanical Systems (5. 8.31) (5. 8.32) (5. 8.33) Observe that the velocity of O is zero, as expected, but the acceleration of O ... vnn GO 22 223221 22 / ˙ =× () = () ωω llθ 059 3_C 05_ fm Page 144 Monday, May 6, 2002 2: 15 PM 142 Dynamics of Mechanical Systems Because Q also moves in...

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... terms of the unit vectors n 1 , n 2 , and n 3 of Figure P7 .5. 1. P7 .5. 7: See Problems P7 .5. 5 and P7 .5. 6. Verify that (α, β = a, b, c) is given by: P7 .5. 8: See Problems P7 .5. 3 and P7 .5. 5. Verify ... ft 7 .5 ft 13. 75 ft θ v (a) (b) 10 lb 10 lb 16 cm 8 cm 5 kg 3 kg 059 3_C08_fm Page 274 Monday, May 6, 2002 2: 45 PM 266 Dynamics of Mechanical Systems where a...

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... 355 The number of degrees of freedom of a mechanical system is the number of coordinates of the system if it were unrestricted minus the number of constraint equations. For example, if a particle ... x n 059 3_C11_fm Page 370 Monday, May 6, 2002 2 :59 PM 364 Dynamics of Mechanical Systems Observe in Eq. (11 .5. 1) that the number of generalized forces is the same...

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... | F | = 5N P(2 kg) ᐉ = 0 .5 m n n y x θ θ 1 2 ᐉ P(M) ᐉ θ 1 2 β ᐉ P(M) ᐉ β 059 3_C11_fm Page 4 05 Monday, May 6, 2002 2 :59 PM 386 Dynamics of Mechanical Systems degrees of freedom. ... a set of particles P i (i = 1,…, N) as parts of a mechanical system S having n degrees of freedom. Then, by superposing (or adding together) equations as Eq. (11.12 .5)...

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... x )) + () [] = () −+ () xx km 23 1 12 22ξ 059 3_C13_fm Page 456 Monday, May 6, 2002 3:21 PM 436 Dynamics of Mechanical Systems References 12.1. Kane, T. R., Dynamics of nonholonomic systems, J. Appl. Mech., 28, 57 4, 1961. 12.2. ... form of Eq. (12.3.3), we obtain the governing dynamical equations of the system: (12 .5. 12) where the coefficients m rj , n rj , and k rj are:...

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