A collar of mass m slides without friction on a horizontal rigid rod and is restrained by a pair of identical springs with spring constant k. A pendulum consisting of a uniform rigid bar of length L and mass Mis suspended from the collar by a frictionless pivot. L, M 2.1 Select a complete and independent set of generalized coordinates for this system. 2.2 Derive using Lagrange's principle, the equations of motion of the system in the form of a matrix differential equation. 2.3 Develop the state-space model. å 2.4 Derive an eigenvalue problem of the form [K]{a}=¹[M]{a}

3. Dynamics and control of two tank in series The following system is given: A water storage Tank 1 in which water is stored at atmospheric pressure and distributed by gravity flow in a downfeed in the same type water storage Tank 2. Where: h1, h2 is water level(m), S1, S2 is squared area of botom of the tank cilinders (m²),F1 in/F2in-water inflow, F1/F2out- water outflow(m³/h), k1, k: Valve constant (m²5/h), pipe length is 2.5 ommited. F1in h1 S1 Tank 1 k1 h2 F1 out F2 in S2 k2 F2 out Tank 2 3.1 Describe the dynamic behavior of a tank 1 in terms of ordinary differential equations. 3.2 Describe the dynamic behavior of the level h2/ tank 2/, due to the input flow Flin. in terms of ordinary differential equations. 3.3 Determine the transfer function of the level h2 as output, due to the input flow Flin. 3.4 Select the strategy to control the plant / level control system / with Flin as control action and output- level h2/ tank 2 / Design the proper instrumentation and Sketch the P&ID…

The natural exponential function can be expressed by . Determine e2by calculating the sum of the series for:(a) n = 5, (b) n = 15, (c) n = 25For each part create a vector n in which the first element is 0, the incrementis 1, and the last term is 5, 15, or 25. Then use element-by-element calculations to create a vector in which the elements are . Finally, use the MATLAB built-in function sum to add the terms of the series. Compare thevalues obtained in parts (a), (b), and (c) with the value of e2calculated byMATLAB.

Customers arrive at a ticket counter at a rate of 50 per hour and tickets are issued in the order of their arrival. The average time taken for issuing a ticket is 1 min. Assuming that customer arrivals form a Poisson process and service times are exponentially distributed, find the average waiting time in queue.