First-Order Averaging

The eigenvalue analysis of the previous section does not reveal any information regarding the behavior of the nonlinear system once instability occurs. The existence of periodic solutions (limit cycles), region of attraction of the stable trivial solution, and the effects of system parameters on these features as well as the size of the limit cycles (amplitude of steady-state vibrations) are important issues that are addressed in this section. The method of averaging is used here to study the 

Before performing averaging, (6.5) must be transformed to the standard form [56]. To that end, some simplifications are necessary. In the following sections, first the equation of motion is simplified and then converted into a nondimensionalized form. Next, a small parameter, e, is introduced and the new dimensionless parameters are ordered to reach an approximate weakly nonlinear equation of motion accurate up to 0(e). 

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Since the zero-order average Hamiltonian of each RF field j]m(m Q) and the first-order average Hamiltonians between any two RF fields (m, n O)... 

Figure 3.3-2 shows typical results. The simple first-order averaging filter, panel (c) in Fig. 3.3-2, is most effective in reducing the noise, but also introduces the largest distortion, visible even on the broadest peaks. This is always a trade-off noise reduction is gained at the cost of distortion. The same can be seen especially with the narrower peaks, where the higher-order filters distort less, but also filter out less noise. In section 10.9 we will describe a more sophisticated filter, due to Barak, which for each point determines the optimal polynomial order to be used, and thereby achieves a better compromise between noise reduction and distortion. 

The tiny difference between hard pulses and their delta-function approximation can be exploited to control coherence. Variants on the magic echo that work despite a large spread in resonance offsets are demonstrated using the zeroth- and first-order average Hamiltonian terms, for C-13 NMR in Cgo- The Si-29 NMR linewidth of silicon has been reduced by a factor of about 70 000 using this approach, which also has potential applications in MR microscopy and imaging of solids. 

The method of first-order averaging is introduced in Sect. 3.6. This method is utilized in Sect. 4.1 and Chap. 6 to expand the results of the eigenvalue analysis in the study of negative damping instability mechanism. 

Theorem 3.1. (First-Order Averaging) Consider the following system in standard form... 

The method of first-order averaging applies to weakly nonlinear systems. This requirement is fulfilled here since friction and damping coefficients are small quantities. The first step is to convert (4.7) to a nondimensional form. Let... 

The first-order averaged equations are obtained by averaging the right-hand side of (4.15) and (4.16) over a period (i.e., T = 2n) while keeping a and 4> constant ... 

Thus, when the origin is unstable, trajectories are attracted by a stable limit cycle. There is, however, one more step needed before accepting the above nontrivial pure-slip solution The condition v>0 (i.e., pure-slip motion) must be checked. In terms of the nondimensional system parameters, this condition is satisfied when Lfflmax(M ) < Vb. The maximum velocity of the mass according to the first-order averaged solution is max(i< ) = a, thus from (4.19) we must have... 

This smoothing modification to the coefficient of friction function enables us to apply the method of first-order averaging to the cases where relative velocity becomes zero or even changes sign. There are, however, side effects that must be treated with caution ... 

Remark 4.3. The parameter r can be very large thus diminishing the boundary layer effects. This parameter, however, cannot be larger than <9( 1) in order to insure 0(e) accuracy of the first-order averaging results. ... 

Note that the first three terms of (4.25) are identical to (4.12). Substituting (4.25) into (4.17) and carrying out the integration, the first-order averaged amplitude equation is found as... 

Remark 4.4. For a>l/p, the simplifying assumption (i.e., u < v )lcc>L or y< Vb) leading to (4.25) is violated. This is the case where the moving mass overtakes the conveyor for a portion of a period. To obtain the first-order averaged equation, the... 

Fig. 6.7 First-order averaging results, c = 2 x 10 rey. truncated equation of motion black. amplitude of vibration from first-order averaging...

In this part, three examples are presented that correspond to the Cases 3 and 4 above. In these examples, the first-order averaged amplitude results are used to study the variations of the amplitude of the steady-state periodic solutions as the... 

The accuracy of the results obtained in Sect. 6.3 depends heavily on the size of the system parameters. In practical applications, such as the experimental example of Chap. 9, the 0(e) error of the first-order averaging may not be sufficiently accurate for the entire range of parameters in the domain of interest. A possible way to improve the accuracy of the amplitude and frequency estimates is to extend the averaging to higher orders. 

Appendix C First-Order Averaging Applied to the 2-DOF Lead Screw Model... 

Fig. B.l First-, second-, and third-order averaging results, (a) Numerical averaging results gray solid nonlinear system equation dotted black first-order averaging dashed-dot second-order averaging solid black third-order averaging, (b) black measurements gray, simulation results...

In this appendix, the method of first-order averaging is used to analyze the 2-DOF model of Sect. 7.1.1. The equations of motion are given by (7.4). Neglecting Fq and To for simplicity and dividing the second equation of motion by tan 2, yield... 

Notice that (C.48) is exactly the same as (6.35) if oj is replaced by a. The first-order averaged equations for fl], (C.48), and 02, (C.47), are decoupled. Furthermore, (C.47) shows that, to this order of approximations, the vibration component with the frequency (02 dies out exponentially, independent of aj. 

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