lecture_13 - Lecture 13 1 State space(phase space view of 1...

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Lecture 13 1 State space (phase space) view of 1 DOF sys- tems So far we have concentrated mostly on understanding of time depen- dence of the response - position, velocity or acceleration - of a vibration system to excitation. The excitation was either in non-zero initial veloc- ity, non-equilibrium displacement, or some time-dependent oscillatory, or non-periodic forcing. In this section we will expand the point of view that we indicated once earlier, in section 3, that we called the state- space representation. You have already used it quite a bit in this class, any time we asked you to simulate the equations of motion in MAT- LAB. In particular, to use any of the numerical integration routines, it was necessary to represent the second order ordinary differential equa- tion, say m ¨ x + kx = 0 , (1) as two first order ordinary differential equations that described the change in time of position and velocity Recall: since ˙ x = y, ¨ x = ˙ y ˙ x = y, ˙ y = - k m x. In multi-degree of freedom vibrations case, that we will study next, we will make good use of representing our equations in the matrix form. To introduce and review essential matrix manipulations, and their use in vibrations, note that the set of equations (2) can be written as ˙ x ˙ y = 0 1 - k/m 0 x y = A x y (2) where A = 0 1 - k/m 0 The eigenvalues of the matrix A play the crucial role in the solution of that system of equations. They are obtained by solving for λ in det( A - λI ) = 0; 1
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where I is the identity matrix. This gives det - λ 1 - k/m - λ = λ 2 + k m = 0 , implying that λ 1 , 2 = ± r - k m = ± i r k m = ± n . where ω n = p k/m is the natural frequency of the system in rad/s . The eigenvalues of the matrix A are imaginary numbers whose position in the complex plane is shown in figure The solution of the system (3) is Figure 1: Eigenvalues of the mass-spring system in the complex plane. obtained by superposition of exponentials of the form e λt for the two possible values of λ . For example, the position reads x ( t ) = Ae i ( k/m ) t + A c e - i ( k/m ) t Where A is the constant to be determined from initial conditions and A c its complex conjugate. The phase portrait (state space representation) of solutions is shown in figure 2. It is obtained by plotting y ( t ) vs. x ( t ) (i.e. velocity vs. position) in the two-dimensional, ( x, y ) plane. The set of pairs ( x ( t ) , y ( t )), plotted for all time t , starting at t = 0 from the initial condition ( x (0) , y (0)) is called the trajectory of the system passing through ( x (0) , y (0)). This graph represents the evolution of trajectories of the system in position-velocity space, for an arbitrary initial condition in that space. Look at the arrow anchored by a red dot at the point x = 0 , y = y 0 . This indicates the initial position of 2
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x y y 0 Figure 2: State-space portrait (or phase portrait) of a mass-spring system.
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