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- Question : 1Q - Figure 2.23 shows a so-called inverted pendulum. If the driving force on the cart, u, is the input to the system and the cart position p, the output, a model of the system can be derived using Newton
- Question : 2Q - a. Find the transfer function, G(s)
- Question : 3Q - a. Linearize the inverted pendulum model from Problem 2.1 a. by using the method from Sect. 2.4 and assume the following stationary state, x0
- Question : 4Q - A simplified car suspension system is shown on Fig. 2.24. The components in the figure represent a quarter of a car (or half a motorcycle).Mis the body mass, misthe so-called unsprung mass (wheel, tire, brake etc.), kt and ks are the spring constants of the tire and the spring respectively and b is the dampingcoefficient of the shock absorber which is assumed to be a linear damper (the damping force is proportional to the velocity difference between the twomembers of the damper). u is the position of the road. Only motion in the vertical direction will be considered. A model is to be set up which is validaround a stationary state where the springs are in tension and shortened to their stationary length under the influence of gravity. a. Derive a model of the system using Newton
- Question : 5Q - The electrical circuits on Fig. 2.25 are called bridged-T-filters. a. Use Ohm
- Question : 6Q - Figure 2.26 shows a hydraulic servo cylinder. The piston is positioned symmetrically in the cylinder and the oil volume of the two cylinder chambers aretherefore of equal size V. The ring shaped area of the piston is Ac. The mass of piston and piston rod is M and the rod is loaded by an external force f. The servo valvedelivering the volume flows q1 and q2 is symmetric and dynamically ideal (massless and frictionless) and the relation between the driving input voltage, u,and the flows is assumed to be q1
- Question : 7Q - The following discrete time transfer function is given: H
- Question : 8Q - A nonlinear system is given by the equations, p_
- Question : 9Q - A system with 3 interconnected tanks is shown on Fig. 2.27. The tanks have the cross sectional areas A1, A2 and A3. The input flows are proportional to the input voltages, qa
- Question : 10Q - A high temperature oven with a product to be heat treated is shown on Figure 2.28. The outer surface temperature Ts of the oven is the same as the temperature in the inner oven space. The ambient temperature is Ta and the product temperature is Tg. All the temperatures are assumed to be uniform. The variables denoted q are the heat powers exchanged between the different parts of the system. The oven surface temperature is so high that the heat loss to the surroundings is not only due to convection but also to radiation. Assume that the convection power is proportional to the temperature difference, i. e., qc

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