The first few seconds of a launch is the most critical period for a rocket and...

 

 

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The first few seconds of a launch is the most critical period for a rocket and its payload since they are subjected to maximum • thrust and • acoustic pressure during that period. Experience shows that excessive thrust and acoustic pressure cause serious damage to sensitive electronics in the rocket and in the payload. As the rocket moves away from the earth's surface, the decreasing air density reduces the acoustic pressure, and the decreasing rocket mass and gravitational pull reduce the thrust requirement. Thus one of the key concerns in planning a launch mission is to ensure that the systems remain undamaged and functional after the first few seconds of a launch. In this problem, we'll focus on achieving the required acceleration for the rocket while minimizing the peak thrust. The rocket is to be moved from the ground level to a height of 5000 feet in 5 seconds. Let y(t) denote the height from the ground at time t and let u(t) denote the upward thrust at time t. Let's assume that the rocket has a nearly constant mass of m= 20, 000 lbs in the first five seconds. The equation of motion is given by d 2 y dt2 + mg = u(t) where g, the gravitational deceleration, is 32 ft-sec-2. (a) Formulate the problem of minimizing the peak thrust during the first five seconds. (b) What type of optimization problem is it? (c) Using Excel determine the optimal control strategy. Graph the thrust, height and velocity as functions of time. The first few seconds of a launch is the most critical period for a rocket and its payload since they are subjected to maximum • thrust and • acoustic pressure during that period. Experience shows that excessive thrust and acoustic pressure cause serious damage to sensitive electronics in the rocket and in the payload. As the rocket moves away from the earth's surface, the decreasing air density reduces the acoustic pressure, and the decreasing rocket mass and gravitational pull reduce the thrust requirement. Thus one of the key concerns in planning a launch mission is to ensure that the systems remain undamaged and functional after the first few seconds of a launch. In this problem, we'll focus on achieving the required acceleration for the rocket while minimizing the peak thrust. The rocket is to be moved from the ground level to a height of 5000 feet in 5 seconds. Let y(t) denote the height from the ground at time t and let u(t) denote the upward thrust at time t. Let's assume that the rocket has a nearly constant mass of m= 20, 000 lbs in the first five seconds. The equation of motion is given by d 2 y dt2 + mg = u(t) where g, the gravitational deceleration, is 32 ft-sec-2. (a) Formulate the problem of minimizing the peak thrust during the first five seconds. (b) What type of optimization problem is it? (c) Using Excel determine the optimal control strategy. Graph the thrust, height and velocity as functions of time.

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The required acceleration can be achieved by setting the peak thrust to u... View the full answer