HW 1

docx

School

New York University *

*We aren’t endorsed by this school

Course

4163

Subject

Mechanical Engineering

Date

Jan 9, 2024

Type

docx

Pages

Uploaded by MagistrateProton12881

Homework # 1 1. Reading assignment Chapter 1 – Principles of Feedback Control Systems. Sec 1.1 – 1.9. With Review questions answered by hand. Exercise 1.9.1 a) Name three applications for feedback control systems. An elevator, rover robot and a radar antenna. Exercise 1.9.2 (a) Name three reasons for using feedback control systems and at least one reason for not using them. Reasons for using them:  With control systems we can move large equipment with precision that would otherwise be impossible.  Control systems can also be used to provide convenience by changing the form of the input. For example, in a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output.  The ability to compensate for disturbances. Reason for not using them:  They are complex. Exercise 1.9.3 (a) Give three examples of open‐loop systems.

Toasters, mechanical systems consisting of a mass, spring, and damper with a constant force positioning the mass. Exercise 1.9.4 (a) Functionally, how do closed‐loop systems differ from open‐loop systems? Closed‐loop systems, then, have the obvious advantage of greater accuracy than open‐loop systems. They are less sensitive to noise, disturbances, and changes in the environment. Transient response and steady‐state error can be controlled more conveniently and with greater flexibility in closed‐loop systems, often by a simple adjustment of gain (amplification) in the loop and sometimes by redesigning the controller. Exercise 1.9.5 (a) State one condition under which the error signal of a feedback control system would not be the difference between the input and the output. The closed‐loop system compensates for disturbances by measuring the output response, feeding that measurement back through a feedback path, and comparing that response to the input at the summing junction. If there is no difference, the system does not drive the plant, since the plant's response is already the desired response. Exercise 1.9.6 (a) If the error signal is not the difference between input and output, by what general name can we describe the error signal? Actuating signal. Exercise 1.9.7 (a) Name two advantages of having a computer in the loop.  The advantage of using a computer is that many loops can be controlled or compensated by the same computer.  Any adjustments of the compensator parameters required to yield a desired response can be made by changes in software rather than hardware.  The computer can also perform supervisory functions, such as scheduling many required applications .

Exercise 1.9.8 (a) Name the three major design criteria for control systems. Transient response, steady-state response and stability. Exercise 1.9.9 (a) Name the two parts of a system's response. Natural response and forced response. Exercise 1.9.10 (a) Physically, what happens to a system that is unstable? It leads to self-destruction of the physical device if limits stops are not part of the design. Exercise 1.9.11 (a) Instability is attributable to what part of the total response? When the natural response is so much greater than the forced response that the system is no longer controlled. Exercise 1.9.12 (a) Describe a typical control system analysis task. Evaluate a system performance’s transient response and steady‐state error to determine if they meet the desired specifications. Exercise 1.9.13 (a) Describe a typical control system design task. For example, if a system's transient response and steady‐state error are analyzed and found not to meet the specifications, then we change parameters or add additional components to meet the specifications.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Exercise 1.9.14 (a) Adjustments of the forward path gain can cause changes in the transient response. True or false? True. Exercise 1.9.15 (a) Name three approaches to the mathematical modeling of control systems.  Kirchhoff's laws for electrical networks and Newton's law  Laplace transform  State-space representation Exercise 1.9.16 (a) Briefly describe each of your answers to the previous question.  Kirchhoff's and Newton's laws lead to mathematical models that describe the relationship between the input and output of dynamic systems.  Laplace transform: Although the transfer function can be used only for linear systems, it yields more intuitive information than the differential equation. We will be able to change system parameters and rapidly sense the effect of these changes on the system response. The transfer function is also useful in modeling the interconnection of subsystems by forming a block diagram similar to the one above with a mathematical function inside each block.  State space representation: ne advantage of state-space methods is that they can also be used for systems that cannot be described by linear differential equations. Further, state-space methods are used to model systems for simulation on the digital computer. Basically, this representation turns an nth‐order differential equation into n simultaneous first‐order differential equations. Let this description suffice for now; we describe this approach in more detail in subsequent chapters. 2. Write one paragraph by hand for each one of the following mathematicians. a. Leonhard Euler (April 15, 1707- September 18, 1783) Swiss mathematician and physicist, developed methods for solving problems in observational astronomy and demonstrated useful applications of mathematics in technology and public affairs. He was responsible for treating trigonometric functions through the so-called Euler

identity (e i θ = cos θ + i sin θ), with complex numbers (for example: 3 + 2 √ −1 ). He discovered the imaginary logarithms of negative numbers and showed that each complex number has an infinite number of logarithms. He introduced many current notations:  Σ for the sum  the symbol ‘ e’ for the base of natural logarithms  a , b and c for the sides of a triangle  A, B, and C for the opposite angle  the letter ‘ f’ and parentheses for a function  i for square root of √ −1 He also popularized the use of the symbol π (devised by British mathematician William Jones) for the ratio of circumference to diameter in a circle. b. Pierre-Simon Laplace (March 23, 1749—March 5, 1827, Paris), French mathematician, astronomer, and physicist who was best known for his investigations into the stability of the solar system. Laplace successfully accounted for all the observed deviations of the planets from their theoretical orbits by applying Sir Isaac Newton’s theory of gravitation to the solar system, and he developed a conceptual view of evolutionary change in the structure of the solar system. He formulated Laplace's equation, which is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as: c. Joseph Fourier (March 21, 1768- May 16, 1830), French mathematician, known also as an Egyptologist and administrator, who exerted strong influence on mathematical physics through his Théorie analytique de la chaleur (1822; The Analytical Theory of Heat ). He showed how the conduction of heat in solid bodies may be analyzed in terms of infinite mathematical series now called by his name, the Fourier series . Far transcending the particular subject of heat conduction , his work stimulated research in mathematical physics, which has since been often identified with the solution of boundary- value problems, encompassing many natural occurrences such as sunspots , tides , and the weather . His work enabled him to express the conduction of heat in two-dimensional objects (i.e., very thin sheets of material) in terms of the differential equation:

in which u is the temperature at any time t at a point ( x , y ) of the plane and k is a constant of proportionality called the diffusivity of the material. For the solution of problems in one dimension, Fourier introduced series with sines and cosines as terms: Source for all three paragraphs: https://www.britannica.com/

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version