Dynamics of Mechanical Systems 2009 Part 10 pot

432 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, Eq. (12.3.3), we obtain the governing equations: (12.4.20) (12.4.21) (12.4.22) If we let the joint moments (M 1 , M 2 , and M 3 ) be zero and if we let the point mass M also be zero in Eqs. (12.4.20), (12.4.21), and (12.4.22), we see that the equations are identical ∂∂= () −− () +− () [] + () −− () +− () [] Km M θθθθθθθθθ θθ θθ θθ θθ 2 2 12 2 1 2 3 3 2 2 12 2 1 23 3 2 12 3 12 2 2 l l ˙˙ sin ˙˙ sin ˙ sin ˙ sin ∂∂= () −− () +− () [] + () −− () +− () [] Km M θθθθθθθθθ θθ θ θ θθ θ θ 3 2 23 3 2 1 3 2 3 2 23 3 2 31 1 3 12 12 2 2 l l ˙˙ sin ˙˙ sin ˙ sin ˙˙ sin ∂∂= () ( ) +− () +− () [] + () +− () +− () [] Km M ˙˙˙ cos ˙ cos ˙˙ cos ˙ cos θθθθθθθθ θθ θθ θ θθ 1 2 12 213 13 2 12 21 3 13 12 143 3 12 2 2 2 l l ∂∂= () () +− () +− () [] + () +− () +− () [] Km M ˙˙˙ cos ˙ cos ˙˙ cos ˙ cos θθθθθθθθ θθ θθθ θθ 2 2 21 213 32 2 21 21 3 32 12 83 3 12 2 2 2 l l ∂∂= () () +− () +− () [] + () +− () +− () [] Km M ˙˙˙ cos ˙ cos ˙˙ cos ˙ cos θθθθθθθθ θθ θθθ θθ 3 2 32 32 1 13 2 32 32 1 13 12 23 12 2 2 2 l l 73 32 12 32 12 1212 313 2 2 12 3 2 13 12 12 3 13 2 () + () − () + () − () + () − () + () − () + () +− () [ +− () + ˙˙ ˙˙ cos ˙˙ cos ˙ sin ˙ sin ˙˙ ˙˙ cos ˙˙ cos ˙ θ θ θθ θ θθ θ θθ θθθ θθθθθθθ θ Mm 22 12 3 2 13 2 2 3 52 0sin ˙ sin sinθθ θ θθ θ− () +− () ] + ()() −+=gMMl 43 32 12 32 12 2121 323 2 1 21 3 2 23 21 21 3 23 1 () + () − () + () − () + () − () + () − () + () +− () [ +− () + ˙˙ ˙˙ cos ˙˙ cos ˙ sin ˙ sin ˙˙ ˙˙ cos ˙˙ cos ˙ θ θ θθ θ θθ θ θθ θθθ θθθθθθθ θ Mm 22 21 3 2 23 2 2 3 32 0sin ˙ sin sinθθ θ θθ θ− () +− () ] + ()() −+=gMMl 13 12 12 12 12 3131 232 1 2 31 2 2 32 31 31 2 32 1 () + () − () + () − () + () − () + () − () + () +− () [ +− () + ˙˙ ˙˙ cos ˙˙ cos ˙ sin ˙ sin ˙˙ ˙˙ cos ˙˙ cos ˙ θ θ θθ θ θθ θ θθ θθθ θθθθθθθ θ Mm 22 31 2 2 32 3 3 12 0sin ˙ sin sinθθ θ θθ θ− () +− () ] + ()( ) −=gMl 0593_C12_fm Page 432 Monday, May 6, 2002 3:11 PM Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 433 to Eqs. (8.11.1), (8.11.2), and (8.11.3) reportedly developed using d’Alembert’s principle. Although the details of that development were not presented, the use of Lagrange’s equations has the clear advantage of providing the simpler analysis. As noted before, the simplicity and efficiency of the Lagrangian analysis stem from the avoidance of the evaluation of accelerations and from the automatic elimination of nonworking constraint forces. In the following section, we outline the extension of this example to include N rods. 12.5 The N-Rod Pendulum Consider a pendulum system composed of N identical pin-connected rods with a point mass Q at the end as in Figure 12.5.1 (we considered this system [without the end mass] in Section 8.11). As before, let each rod have mass m and length ᐉ. Also, let us restrict our analysis to motion in the vertical plane. The system then has N degrees of freedom represented by the angles θ i (i = 1,…, n) as shown in Figure 12.5.1. This system is useful for modeling the dynamic behavior of chains and cables. We can study this system by generalizing our analysis for the triple-rod pendulum. Indeed, by examining Eqs. (12.4.1) through (12.4.10), we see patterns that can readily be generalized. To this end, consider a typical rod B i as in Figure 12.5.2. Then, from Eq. (12.4.1), the angular velocity of B i in the fixed inertia frame may be expressed as: (12.5.1) where n 3 is a unit vector normal to the plane of motion as in Figure 12.5.2. Next, from Eq. (12.4.2), the velocity of the mass center G i may be expressed as: (12.5.2) where as before, and as in Figure 12.5.2, the n iθ are unit vectors normal to the rods. Similarly, the velocity of the point mass Q is: (12.5.3) FIGURE 12.5.1 N-rod pendulum with endpoint mass. FIGURE 12.5.2 A typical rod of the pendulum. ωω B i 1 3 = ˙ θ n vnn n n G i i ii i =++…+ + () − − () ll l l ˙˙ ˙ ˙ θθ θ θ θθ θ θ11 22 1 1 2 vnn n Q NN =++…+ll l ˙˙ ˙ θθ θ θθ θ11 22 θ 1 2 3 n-1 n Q(M) θ θ θ θ θ G B n n n iθ ir 3 i i i 0593_C12_fm Page 433 Monday, May 6, 2002 3:11 PM 434 Dynamics of Mechanical Systems From Eq. (12.4.4) we see that the partial angular velocities of B i are: (12.5.4) Similarly, from Eq. (12.4.5), we see that the partial velocities of G i are: (12.5.5) From an examination of Eqs. (12.4.8), (12.4.9), and (12.4.10), the generalized forces are: (12.5.6) where M is the mass of Q. From a generalization of Eq. (12.4.11) the kinetic energy K of the N-rod system is: (12.5.7) where as before I is the central moment of inertia of a rod about an axis parallel to n 3 and is given by: (12.5.8) Then, by substituting into Eq. (12.5.7) from Eqs. (12.5.1), (12.5.2), and (12.5.3), K becomes: (12.5.9) where the coefficients m ij are: (12.5.10) and (12.5.11) ωω ˙ θ j i B ij ij = ≠ =    0 3 n ωω ˙ θ θ θ j i G j j ji ji ji = < () = >      l l n n2 0 FNi mg Mg i iiθ θθ=− − + () [] −12 llsin sin Km m m II I M GG G Q N = () () + () () +…+ () () + () () + () () +…+ () () + () () 12 12 12 12 12 12 12 12 22 2 22 2 2 vv v v BB B 12 N ωωωωωω Im= () 112 2 l Km m ij j N i N ij = () == ∑∑ 12 2 11 l ˙˙ θθ mNpMm ij p i j ij j i = () +− () + () [] − () ≠ 1 2 1 2 2 cos θθ and is the larger of and mNi Mm ii =−+ () + () [] 13 0593_C12_fm Page 434 Monday, May 6, 2002 3:11 PM Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 435 Finally, substituting from Eqs. (12.5.6) and (12.5.9) into Lagrange’s equations in the 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: (12.5.13) (12.5.14) (12.5.15) (12.5.16) (12.5.17) (12.5.18) 12.6 Closure The computational and analytical advantages of Kane’s equations and Lagrange’s equa- tions are illustrated by the examples. In each case, the effort required to obtain the governing dynamical equations is significantly less than that with d’Alembert’s principle or Newton’s laws. As noted earlier, the reason for the reduction in effort is that non- working constraint forces are automatically eliminated from the analysis with Kane’s and Lagrange’s equations; hence, an analyst can ignore such forces at the onset. Also, with Kane’s and Lagrange’s equations, the exact same number of governing equations are obtained as the degrees of freedom. Finally, Lagrange’s equations offer the additional advantage of using energy functions, which makes the computation of vector accelera- tion unnecessary. The disadvantages of Lagrange’s equations are that they are not applicable with nonholonomic systems, and the differentiation of the energy functions may be tedious and even unwieldy for large systems. In the following chapters we will consider applications of these principles in vibra- tions, stability, balancing, and in the study of mechanical components such as gears and cams. mn gk rj j rj j rj j N ˙˙ ˙ θθ++ () [] = = ∑ 2 1 0l mNpMm rj p i j rj j r = () +− () + () [] − () ≠ 1 2 1 2 2 cos θθ and is the larger of and mNr Mm rr =−+ () + () [] 13 nNpMm rj p i j rj j r =− () +− () + () [] − () ≠ 1 2 1 2 2 sin θθ and is the larger of and n rr = 0 krj rj =≠0 kNr Mm rr r =−+ () + () [] 1 2 sinθ 0593_C12_fm Page 435 Monday, May 6, 2002 3:11 PM 436 Dynamics of Mechanical Systems References 12.1. Kane, T. R., Dynamics of nonholonomic systems, J. Appl. Mech., 28, 574, 1961. 12.2. Kane, T. R., Dynamics, Holt, Rinehart & Winston, New York, 1968, p. 177. 12.3. Huston, R. L., and Passerello, C. E., On Lagrange’s form of d’Alembert’s principle, Matrix Tensor Q, 23, 109, 1973. 12.4. Papastavridis, J. G., On the nonlinear Appell’s equations and the determination of generalized reaction forces, Int. J. Eng. Sci., 26(6), 609, 1988. 12.5. Huston, R. L., Multibody dynamics: modeling and analysis methods [feature article], Appl. Mech. Rev., 44(3), 109, 1991. 12.6. Huston, R. L., Multibody dynamics formulations via Kane’s equations, in Mechanics and Control of Large Flexible Structures, J. L. Jenkins, Ed., Vol. 129 of Progress in Aeronautics and Astronautics, American Institute of Aeronautics and Astronautics (AIAA), 1990, p. 71. 12.7. Huston, R. L., and Passerello, C. E., Another look at nonholonomic systems, J. Appl. Mech., 40, 101, 1973. 12.8. Kane, T. R., and Levinson, D. A., Dynamics, Theory, and Applications, McGraw-Hill, New York, 1985, p. 100. Problems Section 12.2 Kane’s Equations P12.2.1: Consider the rotating tube T, with a smooth interior surface, containing a particle P with mass m, and rotating about a vertical diameter as in Problems P11.6.6 and P11.9.6 and as shown again in Figure P12.2.1. As before, let the radius of T be r, let the angular speed of T be Ω, and let P be located by the angle θ as shown in Figure P12.2.1. This system has one degree of freedom, which may be represented by θ. Use Kane’s equations, Eq. (12.2.1), to determine the governing dynamical equation. P12.2.2: Consider the pendulum consisting of a rod with length ᐉ and mass m attached to a circular disk with radius r and mass M and supported by a frictionless pin as in Problems P11.9.1 and P11.12.1 and as shown again in Figure P12.2.2. This system has one degree of freedom represented by the angle θ. Use Kane’s equations to determine the governing dynamical equation. FIGURE P12.2.1 A particle moving inside a smooth surfaced tube. FIGURE P12.2.2 A rod/disk pendulum. T r θ P(m) Ω ᐉ θ 0593_C12_fm Page 436 Monday, May 6, 2002 3:11 PM Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 437 P12.2.3: Consider the rod pinned to the vertically rotating shaft as in Problems P11.9.2 and as shown again in Figure P12.2.3. If the shaft S has a specified angular speed Ω, the system has only one degree of freedom: the angle θ between the rod B and S. Use Kane’s equations to determine the governing dynamical equation where B has mass m and length ᐉ. Assume the radius of S is small. P12.2.4: Repeat Problem P12.2.3 by including the effect of the radius r of the shaft S. Let the mass of S be M. P12.2.5: See Problems P11.9.4 and P12.2.3. Suppose the rotation of S is not specified but instead is free, or arbitrary, and defined by the angle φ as in Problem P11.9.4 and as represented in Figure P12.2.5. This system now has two degrees of freedom represented by the angles θ and φ. Use Kane’s equations to deter- mine the governing dynamical equations, assuming the shaft radius r is small. P12.2.6: Repeat Problem P12.2.5 by including the effect of the shaft radius r and the shaft mass M. P12.2.7: Consider a generalization of the double-rod pendulum where the rods have unequal lengths and unequal masses as in Figure P12.2.7. Let the rod lengths be ᐉ 1 and ᐉ 2 , and let their masses be m 1 and m 2 . Let the rod orientations be defined by the angles θ 1 and θ 2 , as shown. Assuming frictionless pins, determine the equations of motion by using Kane’s equations. P12.2.8: See Problem P12.2.7. Suppose an actuator (or motor) is exerting a moment M 1 at support O on the upper bar and suppose further that an actuator at the pin connection between the rods is exerting a moment M 2 on the lower rod by the upper rod (and hence a moment –M 2 on the upper rod by the lower rod). Finally, let there be a concentrated mass M at the lower end Q of the second rod, as represented in Figure P12.2.8. Use Kane’s equations to determine the equations of motion of this system. P12.2.9: Repeat Problems P12.2.7 and P12.2.8 using the relative orientation angles β 1 and β 2 , as shown in Figure P12.2.9, to define the orientations of the rods. P12.2.10: See Problems P11.6.8 and P11.9.8. Consider again the heavy rotating disk D supported by a light yoke Y which in turn can rotate relative to a light horizontal shaft S which is supported by frictionless bearings as depicted in Figure P12.2.10. Let D have mass m and radius r. Let angular speed Ω of D in Y be constant. Let the rotation of Y relative to S be FIGURE P12.2.3 A rod B pinned to a rotating shaft S FIGURE P12.2.5 A rod B pinned to a rotating shaft S. FIGURE P12.2.7 A double-rod pendulum with unequal rod lengths and masses. FIGURE P12.2.8 A double-rod pendulum with unequal rods, joint moments, and a concentrated end mass. S Ω θ ᐉ B S θ B φ O θ 1 1 2 2 ᐉ ᐉ θ θ 1 1 2 2 ᐉ Q(M) M M O 1 2 ᐉ θ 0593_C12_fm Page 437 Monday, May 6, 2002 3:11 PM 438 Dynamics of Mechanical Systems measured by the angle β and the rotation of S in its bearings be measured by the angle α as shown. Recall that this system has two degrees of freedom, which may be represented by the angles α and β. By following the procedures outlined in Problems P11.6.8 and P11.9.8, use Kane’s equations to determine the governing equations of motion. P12.2.11: See Problems P11.6.7 and P11.9.7. Consider again the right circular cone C with altitude h and half-central angle rolling on an inclined plane as in Figure P12.2.11. As before let the incline angle be β and let the position of C be determined by the angle φ between the contacting element of C and the plane and a line fixed in the plane as shown. Let ᐉ be the element length of C, and let r be the base radius of C. Let O be the apex of C, and let G be the mass center of C. Finally, let ψ measure the roll of C, as shown in Figure P12.2.11. This system has one 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 using Lagrange’s equations to obtain the equations of motion. Compare the analysis effort with that of using Kane’s equations. FIGURE P12.2.9 Double, unequal-rod pendulum with relative orientation angles. FIGURE P12.2.10 A disk spinning in a free-turning yoke and supported by a shaft S. FIGURE P12.2.11 A cone rolling on an inclined plane. O 1 1 2 2 ᐉ β β ᐉ Ω D S α β Y O r ψ G φ ᐉ α β C 0593_C12_fm Page 438 Monday, May 6, 2002 3:11 PM 439 13 Introduction to Vibrations 13.1 Introduction Vibration is sometimes defined as periodic (repeating) oscillatory movement. It occurs in virtually all mechanical systems. Vibration produces noise, unwanted wear, and even catastrophic failure. On the other hand, for many systems vibration is essential for the proper functioning of the systems. Therefore, analysis and control of vibration are principal problems of mechanical design. In this chapter we will develop a brief and elementary introduction to mechanical vibration. It is only an introduction and is not intended to replace a course or a more intense study. The reader is referred to the references, which provide a partial listing of the many books devoted to the subject. We begin with a brief review of solutions to second-order ordinary differential equations. We then consider single and multiple degree of freedom systems. We conclude with a brief discussion of nonlinear vibrations. 13.2 Solutions of Second-Order Differential Equations Vibration phenomena are often modeled by second-order ordinary differential equations. Solutions of these equations provide a representation of the movement of vibrating sys- tems; therefore, to begin our study, it is helpful to review the solution procedures of second- order ordinary differential equations. The reader is encouraged to also independently review these procedures. References 13.1 to 13.7 provide a sampling of the many texts available on the subject. We will consider first the so-called linear oscillator equation: (13.2.1) where, as before, the overdot represents differentiation with respect to time, and ω is a constant. In Eq. (13.2.1) the time t is the independent variable and x is the dependent variable to be determined. It is readily verified that the solution of Eq. (13.2.1) may be expressed in the form: (13.2.2) ˙˙ xx+=ω 2 0 xA tB t=+cos sinωω 0593_C13_fm Page 439 Monday, May 6, 2002 3:21 PM 440 Dynamics of Mechanical Systems where A and B are constants that may be evaluated by auxiliary conditions ( initial condi- tions or boundary conditions ). (We can verify that the expression of Eq. (13.2.2) is indeed a solution of Eq. (13.2.1) by direct substitution. It is in fact a general solution, as cos ω t and sin ω t are independent functions and there are two arbitrary constants, A and B .) Through use of trigonometric identities, we can express Eq. (13.2.2) in the form: (13.2.3) where  and φ are constants. To develop this, recall the identity: (13.2.4) Then, by thus expanding the expression of Eq. (13.2.3) we have: (13.2.5) By comparing this with Eq. (13.2.2), we have: (13.2.6) In the expression  cos( ω t + φ ) of Eq. (13.2.3),  is the amplitude , ω is the circular frequency , and φ is the phase . Recall the form of the cosine function as depicted in Figure 13.2.1. Observe that the function is periodic with period 2 π . By comparing Eq. (13.2.3) with Figure 13.2.1, we see that the displacement of the linear oscillator changes periodically between the extremes of  and –Â. Also, the period T of the oscillation is determined by: (13.2.7) The frequency is the rate at which the oscillation occurs; that is, the frequency f is the reciprocal of the period: (13.2.8) FIGURE 13.2.1 Cosine function. xA t=+ () ˆ cos ωφ cos cos cos sin sinαβ α β α β+ () ≡− ˆ cos ˆ cos cos ˆ sin sinAt At Atωφ ω φ ω φ+ () =− AA BA AAB BA == =+ =− ˆ cos ˆ sin ˆ tan φφ φ 222 ωπ πωTT==22 or fT f= 1 = or = 2ωπ ω π2 1.0 0 1.0 - y cos y π/2 π 3π/2 2π 0593_C13_fm Page 440 Monday, May 6, 2002 3:21 PM Introduction to Vibrations 441 As noted earlier, the constants in Eqs. (13.2.2) and (13.2.3) are to be evaluated by auxiliary conditions. These auxiliary conditions are generally initial conditions or boundary condi- tions. Initial conditions (where t = 0) might be expressed as: (13.2.9) Then, by requiring the solution of Eq. (13.2.2) to meet these conditions, we have: (13.2.10) and thus, (13.2.11) Boundary conditions might be expressed as: (13.2.12) Then, by requiring the solution of Eq. (13.2.2) to meet these conditions, we see that the constants A and B must satisfy: (13.2.13) The second expression is satisfied by either: (13.2.14) If B = 0, we have the trivial solution x = 0. Alternatively, if sin ω ᐉ = 0, ω ᐉ must be an integer multiple of π . That is, (13.2.15) Thus, there is a nontrivial solution only for selected values of ω — that is, for ω = n π / ᐉ . The solution for the displacement then takes the multiple forms: (13.2.16) Then, by superposition, we have: (13.2.17) where the constants B n are to be determined from other conditions of the specific system being studied. (Eq. (13.2.17) is a Fourier series representation of the solution.) xx xx00 00 () = () = and ˙˙ Ax Bx= 00 and = ˙ ω xx t x t=+ () 00 cos ˙ sinωωω xx00 0 () = () = and l AB==00 and sinωl B ==00 or sinωl ωπ ωωπll===nn n or xx B nt n nn == = …sin , ,π l 12 xBnt n n = = ∞ ∑ 1 sin π l 0593_C13_fm Page 441 Monday, May 6, 2002 3:21 PM [...]... (13 .10. 8) sin θ ≈ θ − θ 3 6 (13 .10. 9) ˙˙ θ − ( g 6 l) θ 3 + ( g l) θ = 0 (13 .10. 10) Suppose we approximate sinθ as: Then, Eq (13 .10. 8) becomes: Hence, by comparison with Eq (13 .10. 1), we can identify ⑀, f, and p as: ⑀ = g 6l (13 .10. 11) ( ) (13 .10. 12) ˙ f θ, θ = −θ 3 0593_C13_fm Page 465 Monday, May 6, 2002 3:21 PM Introduction to Vibrations 465 and p= g l (13 .10. 13) F( A, ψ ) = − A 3 sin 3 ψ (13 .10. 14)... g l A0 t + φ0 (13 .10. 18) where φ0 is a constant Therefore, from Eq (13 .10. 6), the approximate solution to Eq (13 .10. 8) is: θ = A0 sin { [ ] g l 1 − ( A0 16) t − φ0 } (13 .10. 19) Comparing Eq (13 .10. 19) with the linear equation solution, A0sinωt, we see that ω is: [ ] ω = 2 π T = g l 1 − ( A0 16) (13 .10. 20) 0593_C13_fm Page 466 Monday, May 6, 2002 3:21 PM 466 Dynamics of Mechanical Systems where as before... 448 Monday, May 6, 2002 3:21 PM 448 Dynamics of Mechanical Systems x t FIGURE 13.5.2 Underdamped oscillation By considering a free-body diagram of B and by using the principles of dynamics (for example, Newton’s laws or d’Alembert’s principle), we readily see that the governing equation of motion is: ˙˙ ˙ mx + cx + kx = 0 (13.5.2) where as before, m is the mass of B and k is the linear spring modulus... Page 458 Monday, May 6, 2002 3:21 PM 458 Dynamics of Mechanical Systems ˙ ξ 3 = − A3ω 3 sin ω 3t + B3 ω 3 cos ω 3t (13.8.19) Next, suppose that at time t = 0, the values of x1, x2, and x3 and their derivatives are x10, ˙ ˙ ˙ x20, x30, x10, x20, x30, respectively Then, from Eqs (13.8.4), (13.8.5), and (13.8.6), the initial ˙ ˙ ˙ values of the ξ1, ξ2, and ξ3 and of the ξ1, ξ2, and ξ3 are determined This... terms of Eqs (13.9.16) and (13.9.17) we have: k cos z ( dz = (1 2) cos (θ 2) = (1 2) 1 − k 2 sin 2 z dθ ) 1/2 (13.9.18) 0593_C13_fm Page 462 Monday, May 6, 2002 3:21 PM 462 Dynamics of Mechanical Systems and then ( dz = (1 2) 1 − k 2 sin 2 z dθ ) 1/2 (13.9.19) k cos z or dθ = 2 k cos z dz (1 − k 2 sin 2 z (13.9.20) ) 1/2 Regarding the limits of the integral of Eq (13.9.9) we have from Eq (13.9 .10) and... proceeds as follows: For equations that may be put in the form of Eq (13 .10. 1), let a function F(A,ψ) be introduced such that F( A, ψ ) = f ( A sin ψ , A p cos ψ ) (13 .10. 2) 0593_C13_fm Page 464 Monday, May 6, 2002 3:21 PM 464 Dynamics of Mechanical Systems where A is an amplitude to be determined and where ψ is defined as: D ψ = pt + φ (13 .10. 3) where φ is a phase angle to be determined Next, let dA/dt... Coordinates of particles in the fixed horizontal tube x2 n1 x1 P1 n2 P2 P3 T 0593_C13_fm Page 455 Monday, May 6, 2002 3:21 PM Introduction to Vibrations 455 FIGURE 13.7.4 Particle movement at lowest frequency ω1 FIGURE 13.7.5 Particle movement at intermediate frequency ω2 FIGURE 13.7.6 Particle movement at highest frequency ω3 13.8 Analysis and Discussion of Three-Particle Movement: Modes of Vibration... Then, Eqs (13 .10. 4) and (13 .10. 5) become: 2π   dA = −( g 6l)  1 2 π g l − A 3 sin 3 ψ cos ψ d ψ    dt   0 ∫ = (13 .10. 15) g l 3 sin 4 ψ 2 π |=0 A 0 12 π 4 and 2π dφ = −( g 6l)(1 2 π g l A) A 3 sin 4 ψ d ψ dt ∫ 0 =− g lA 3 sin 2 ψ sin 4 ψ  π +  ψ− | 12 π  8 4 32  0 2 (13 .10. 16) = − g l A 2 16 Upon integration of Eqs (13 .10. 15) and (13 .10. 16) we obtain: A = A0 (a constant) (13 .10. 17) and 2... we considered in the previous sections, but here the movement of the mass B is restricted by a “damper” in the form of a dashpot For simplicity of illustration, we will assume viscous damping, where the force exerted by the dashpot on B is proportional to the speed of B and directed opposite to the motion of B with c being the constant of proportionality That is, the damping force FD on B is: ˙ FD... ˙˙ mx1 + 2 kx1 − kx2 = 0 (13.8.1) ˙˙ mx2 − kx1 + 2 kx2 − kx3 = 0 (13.8.2) ˙˙ mx3 − kx2 + 2 kx3 = 0 (13.8.3) 0593_C13_fm Page 456 Monday, May 6, 2002 3:21 PM 456 Dynamics of Mechanical Systems TABLE 13.8.1 Modes of Vibration of Spring-Supported Particles Mode Frequency Normalized Amplitudes 1 [(2 − 2 ) (k m)] A1 = 1 2, A2 = 2 / 2 , A3 = 1 2 2 [ 2 k m] A1 = 2 / 2, A2 = 0, A3 = − 2 / 2 3 [(2 + 2 ) (k m)] . PM 434 Dynamics of Mechanical Systems From Eq. (12.4.4) we see that the partial angular velocities of B i are: (12.5.4) Similarly, from Eq. (12.4.5), we see that the partial velocities of G i . May 6, 2002 3:11 PM 436 Dynamics of Mechanical Systems References 12.1. Kane, T. R., Dynamics of nonholonomic systems, J. Appl. Mech., 28, 574, 1961. 12.2. Kane, T. R., Dynamics, Holt, Rinehart. 0593_C13_fm Page 447 Monday, May 6, 2002 3:21 PM 448 Dynamics of Mechanical Systems By considering a free-body diagram of B and by using the principles of dynamics (for example, Newton’s laws or d’Alembert’s