Differential Equation Models of Dynamic Systems Q1. A spring-mass-dashpot system is mounted on a massless'cart as...

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Differential Equation Models of Dynamic Systems Q1. A spring-mass-dashpot system is mounted on a massless'cart as shown in Figure Q1. In this system, u(t) is the displacement of the cart and is the input to the system. The displacement y() of the mass is the output. m is the mass, k is the spring stiffness and b is the damping constant of the dashpot. The sum of the forces, F, acting on the mass, m, results in acceleration of the mass F=my (1.1) The forces acting on m are the force due to the dashpot, F, and the force due to the spring Fr F=-by-u) (1.2) Massless cart ww m F =k(yu) (1.3) Figure Q1 (a) Express the system as a differential equation with all the input terms on one side and the output terms on the other side. (b) Derive expressions for the natural frequency and the damping ratio of the system. Differential Equation Models of Dynamic Systems Q1. A spring-mass-dashpot system is mounted on a massless'cart as shown in Figure Q1. In this system, u(t) is the displacement of the cart and is the input to the system. The displacement y() of the mass is the output. m is the mass, k is the spring stiffness and b is the damping constant of the dashpot. The sum of the forces, F, acting on the mass, m, results in acceleration of the mass F=my (1.1) The forces acting on m are the force due to the dashpot, F, and the force due to the spring Fr F=-by-u) (1.2) Massless cart ww m F =k(yu) (1.3) Figure Q1 (a) Express the system as a differential equation with all the input terms on one side and the output terms on the other side. (b) Derive expressions for the natural frequency and the damping ratio of the system.