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lecture_15 - Lecture 15 Forced vibrations 2-DOF systems 0.1...

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Lecture 15: Forced vibrations, 2-DOF systems 0.1 Matrix methods for 2-DOF system with forcing We have seen how matrix methods simplify analysis for free vibration of 2-DOF. Now we use those to enable analysis of forced systems. We will see how the use of diagonalization discussed in the previous lecture enables us to solve such, relatively complex problems. Consider the vector x = ( x 1 , x 2 ) of positions for a linear vibrat- ing system with 2DOF. Let the force on the system be expressed as Bu ( t ) where B is a 2 × 2 matrix and u ( t ) the forcing vector u ( t ) = ( u 1 ( t ) , u 2 ( t )) T . Equations of motion can be written in matrix form as: M ¨ x = - Kx + Bu ( t ) . We premultiply by M - 1 and rewrite this as ¨ x = - M - 1 Kx + M - 1 Bu ( t ) . From the previous lecture, we know that we can simplify this system by using the change of voordinates y = V - 1 x where V is the eigenvector matrix. We obtain ¨ y = Dy + V - 1 M - 1 Bu ( t ) . The expression V - 1 M - 1 Bu ( t ) = v ( t ) is a column vector that we can write in components as v ( t ) = ( v 1 ( t ) , v 2 ( t )). Thus the equations for y 1 , y 2
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