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Unformatted text preview: TREATING MULTIDIMENSIONAL SYSTEMS 10 TREATING MULTI DIMENSIONAL SYSTEMS Chapter 1 through Chapter 9 developed basic principles of dynamics and control of single degreeoffreedom systems. The interacting parameters were composed of physical parameters, dynamic performance parameters (both natural and controlled), and control parameters. Since we were concerned with single degreeoffreedom systems, the relationships that were developed were temporal in nature, that is, associated with how quantities change in time. In this chapter, we’ll develop basic principles of dynamics and control of two degreeoffreedom systems. While doing this we’ll develop new relationships that are spatial in nature. We will be able to answers questions like: Where should we place the actuators? How many actuators should we use? Perhaps the first thing to realize about multi dimensional systems is that the additional degrees of freedom make these systems more complex than single degreeoffreedom systems and that a way of handling this added complexity is necessary. This added complexity is handled through another separation principle. This separation principle MAE 461: DYNAMICS AND CONTROLS TREATING MULTIDIMENSIONAL SYSTEMS enables us to treat the multidimensional system as separate single degreeoffreedom systems. Each of the single degreeoffreedom systems is a “mode” of the system. In Chapter 11 we’ll see under certain circumstances that a multi dimensional system can be controlled by controlling its individual modes. 1. Equations Consider the undamped two degreeoffreedom system shown in Fig. 10  1. Applying Newton’s second law of motion to each mass, we get the two equations that govern the dynamics of the system (10 – 1) 2 2 3 1 2 2 2 2 1 1 1 1 2 2 1 1 ) ( ) ( f x k x x k x m f x k x x k x m + − − − = + − − = & & & & Figure 10 – 1 Equation (10 – 1) can be rewritten in the matrix vector form (10 – 2) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − − + + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 2 1 2 1 3 2 2 2 2 1 2 1 2 1 f f x x k k k k k k x x m m & & & & or more compactly as MAE 461: DYNAMICS AND CONTROLS TREATING MULTIDIMENSIONAL SYSTEMS (10 – 3) F Kx x M = + & & where x is called the position vector, F is called the force vector, M is called the mass matrix, and K is called the stiffness matrix. Of course, systems with more than two degreesoffreedom can be written in this form, too. 2. The Eigenvalue Problem Let’s study freely vibrating two degreeoffreedom systems. We let F = 0 in Eq. (10 – 3) and try solutions in the form (10 – 4) st e φ = x Substitute Eq. (10 – 4) into Eq. (10 – 3) and divide the result by to get st e (10 – 5) K M = + φ ] [ 2 s Equation (10 – 5) admits a nontrivial solution only if the matrix in brackets is singular, in which case (10 – 6) ] det[ 2 = + K M s Equation (10 – 6) is called the characteristic equation of the system. It yields the values of of the system....
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 Fall '08
 Silverberg
 Force, Controls, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, Normal mode, MULTIDIMENSIONAL SYSTEMS

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