Linearized System Equations

Consider the following second-order nonlinear autonomous system  

Define a 2n-vector of system states as y = [x (3.1) can be rewritten in state-space form as 

Vahid-Araghi and F. Golnaraghi, Friction-Induced Vibration in Lead Screw Drives, DOI 10.1007/978-l-4419-1752-2 3, Springer Science+Business Media, LLC 2011 

In many dynamical systems the equation of motion of the system takes the form 

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Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Let us assume that for a linear system, equation (10) has solution for all values of the parameters in the neighborhood 5 namely, there exist solutions 5 (/x), 5 (/x) to the equations... 

Some different forms of the Kramers-Kronig relations are presented in Table 22.1. Equations (22.55) and (22.56), called the Plemelj formulas, are obtained directly from consideration of causality in a linear system. Equations (22.57) and (22.58) are mathematically equivalent to equations (22.59) and (22.60), respectively, because... 

As shown in Sect. 6.9.2 in small-signal range the operators Ts and Fa can be approximated by the linear system equations... 

For the sake of completeness, we consider here the linearized system equation for such cases. Assume that the onset of the Painleve s paradox is at 0 = 0cr (i.e., A(0cr) = 0 or det [M(0cr)] = 0). As the system parameters are varied such that 0 crosses the surface 0 = 0cr an eigenvalue goes to infinity and becomes positive as shown in Fig. 3.3. Beyond the critical value of the parameters, the solution of the linear differential equation (3.8) diverges. 

In the absence of the velocity-dependent forces, i.e., Av = 0, the linearized system equation (3.8) reduces to... 

Note that the nonlinear force vector given by (7.7) is nonzero only when a trajectory reaches (or crosses) the stick-slip boundary (i.e., sgn(jj - - (2) sgn(Q)) or when the contact force changes sign (i.e., sgn(A ) sgn(i )). The linearized system equation in matrix form is given by... 

Minimizing the square of the gradient vector under the condition c/ = I yields the following linear system of equations... 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

Equations (213) are a system of 3N simultaneous linear differential equations in the 3N unknowns qj. It can be transformed to a... 

Example The differential equation My" + Ay + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A < 2 VkM. the roots of the characteristic equation... 

Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system -2 + yx i + yx = ( )- Assume that the homogeneous solution has been found by some technique and write yY = -I- Assume that a particular solution yl = andD ... 

Variable Coejftcients The method of variation of parameters apphes equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. 

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

Like thermal systems, it is eonvenient to eonsider fluid systems as being analogous to eleetrieal systems. There is one important differenee however, and this is that the relationship between pressure and flow-rate for a liquid under turbulent flow eondi-tions is nonlinear. In order to represent sueh systems using linear differential equations it beeomes neeessary to linearize the system equations. 

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

Deterministic Randomness. On the one hand, equation 4.8 is a trivial linear difference equation possessing an equally as trivial solution for each initial point Xq Xn = 2"a o (mod 1). Once an initial point is chosen, the future iterates are determined uniquely. As such, this simple system is an intrinsically deterministic one. On the other hand, look again at the binary decimal expansion of a randomly selected a o- This expansion can also be thought of as a particular semi-infinite sequence of coin tosses. 

Nonanalytic Nonlinearities.—A somewhat different kind of nonlinearity has been recognized in recent years, as the result of observations on the behavior of control systems. It was observed long ago that control systems that appear to be reasonably linear, if considered from the point of view of their differential equations, often exhibit self-excited oscillations, a fact that is at variance with the classical theory asserting that in linear systems self-excited oscillations are impossible. Thus, for instance, in the van der Pol equation... 

On the other hand, it is also well known that a system of two linear differential equations that replace each other at the instant when changes its sign, namely ... 

The change of n, with time was calculated according to first-order kinetics. It is given by a system of r linear differential equations and 0 r(r - 1) variables ... 

It should be immediately apparent that this formulation is in complete accord with all of the rate measurements for these systems. Equation (37) demands that k increase linearly with increasing catalyst concentration, and this is, in fact, what is observed. 

Equations (6 b) and (6 c), together with Eqs, (1) and (12) form a non-linear system with the following unknowns ... 

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