Referring to the system in (a), obtain the nonlinear model in terms of two ODES for...

 

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Referring to the system in (a), obtain the nonlinear model in terms of two ODES for G and 0. Since the system is nonlinear, you cannot discard the weight. Let be the static deflection due to the weight. You can choose to define G as the absolute displacement, or as the displacement relative to . Linearize about 0 = 0, G = 0, and zero velocities and accelrations, and obtain the transfer matrix G(s) such that (SXC) = G(s) (80) SU where SXG and Se are small variations (Laplace transformed) of the kinematic variables, induced by the small variations (Laplace transformed) SU, U2 of the inputs 1, 2. Set L = 0.5 and L2 = 0.5. qui K b I 1U Ma IXG ib 1 u (a) (b) The system sketched in the figure refers to a lumped parameter model of of a terrestrial vehicle. The representation (a) represents the vehicle as a rigid body in plane motion, with kinematics described by the two degrees of freedom #G and 0, repsectively referring to the vertical translation of the center of mass with respect to an equilibrium defined by the gravity, and the rotation of the body. By defining with respect to the static euilibrium defined by the gravity, the weight does not need to be included in the linear model of the system. Details are delegated to the course in mechanical vibratuions. In (a), the coupling between the vehicle's body and the terrain is schematized by two series of visco-elatic elements (linear spring and damper), lumping the properties of the suspensions and the tires. The action of the terrain is described in terms of the vertical displacement inputs and 2, which are measured with respect to a reference profile. For example, the reference profile could be defined as a flat terrain, parallel to the body at the equilibrium with = 0 under the static effect of the gravity. The sketch in figure (b) shows a furter simplification, in which the rotational defree of freedom is neglected, and a single degree of freedom lumped parameter model with a scalar input is adopted. Referring to the system in (a), obtain the nonlinear model in terms of two ODES for G and 0. Since the system is nonlinear, you cannot discard the weight. Let be the static deflection due to the weight. You can choose to define G as the absolute displacement, or as the displacement relative to . Linearize about 0 = 0, G = 0, and zero velocities and accelrations, and obtain the transfer matrix G(s) such that (SXC) = G(s) (80) SU where SXG and Se are small variations (Laplace transformed) of the kinematic variables, induced by the small variations (Laplace transformed) SU, U2 of the inputs 1, 2. Set L = 0.5 and L2 = 0.5. qui K b I 1U Ma IXG ib 1 u (a) (b) The system sketched in the figure refers to a lumped parameter model of of a terrestrial vehicle. The representation (a) represents the vehicle as a rigid body in plane motion, with kinematics described by the two degrees of freedom #G and 0, repsectively referring to the vertical translation of the center of mass with respect to an equilibrium defined by the gravity, and the rotation of the body. By defining with respect to the static euilibrium defined by the gravity, the weight does not need to be included in the linear model of the system. Details are delegated to the course in mechanical vibratuions. In (a), the coupling between the vehicle's body and the terrain is schematized by two series of visco-elatic elements (linear spring and damper), lumping the properties of the suspensions and the tires. The action of the terrain is described in terms of the vertical displacement inputs and 2, which are measured with respect to a reference profile. For example, the reference profile could be defined as a flat terrain, parallel to the body at the equilibrium with = 0 under the static effect of the gravity. The sketch in figure (b) shows a furter simplification, in which the rotational defree of freedom is neglected, and a single degree of freedom lumped parameter model with a scalar input is adopted.