Review Vibrations of SDOF Mechanical Systems
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Control System Design Methods
Christiansen-Sec.19.qxd 06:08:2004 6:43 PM Page 19.1 The Electronics Engineers' Handbook, 5th Edition McGraw-Hill, Section 19, pp. 19.1-19.30, 2005. SECTION 19 CONTROL SYSTEMS Control is used to modify the behavior of a system so it behaves in a specific desirable way over time. For example, we may want the speed of a car on the highway to remain as close as possible to 60 miles per hour in spite of possible hills or adverse wind; or we may want an aircraft to follow a desired altitude, heading, and velocity profile independent of wind gusts; or we may want the temperature and pressure in a reactor vessel in a chemical process plant to be maintained at desired levels. All these are being accomplished today by control methods and the above are examples of what automatic control systems are designed to do, without human intervention. Control is used whenever quantities such as speed, altitude, temperature, or voltage must be made to behave in some desirable way over time. This section provides an introduction to control system design methods. P.A., Z.G. In This Section: CHAPTER 19.1 CONTROL SYSTEM DESIGN 19.3 INTRODUCTION 19.3 Proportional-Integral-Derivative Control 19.3 The Role of Control Theory 19.4 MATHEMATICAL DESCRIPTIONS 19.4 Linear Differential Equations 19.4 State Variable Descriptions 19.5 Transfer Functions 19.7 Frequency Response 19.9 ANALYSIS OF DYNAMICAL BEHAVIOR 19.10 System Response, Modes and Stability 19.10 Response of First and Second Order Systems 19.11 Transient Response Performance Specifications for a Second Order -
C.4 Transient and Steady State Response Analysis
Introduction to Control Theory Including Optimal Control Nguyen Tan Tien - 2002.3 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ C.4 Transient and Steady State Response Analysis 4.1 Introduction Notice that when t = a , then y(t) = y(a) = 1− e−1 = 0.63 . The Many applications of control theory are to servomechanisms response is in two-parts, the transient part e−t / a , which which are systems using the feedback principle designed so approaches zero as t → ∞ and the steady-state part 1, which is that the output will follow the input. Hence there is a need for the output when t → ∞ . studying the time response of the system. If the derivative of the input are involved in the differential The time response of a system may be considered in two parts: equation of the system, that is, if a y& + y = b u& + u , then its • Transient response: this part reduces to zero as t → ∞ transfer function is • Steady-state response: response of the system as t → ∞ bs + 1 s + z 4.2 Response of the first order systems Y(s) = U (s) = K U (s) (4.4) as + 1 s + p Consider the output of a linear system in the form where K = b / a Y (s) = G(s)U (s) (4.1) z =1/ b : the zero of the system where p =1/ a : the pole of the system Y(s) : Laplace transform of the output G(s) : transfer function of the system When U (s) =1/ s , Eq. (4.4) can be written as U (s) : Laplace transform of the input K K z z − p Y(s) = 1 − 2 , where K = K and K = K s s + p 1 p 2 p Consider the first order system of the form a y& + y = u , its transfer function is Hence, − pt 1 y(t) = K1 − K 2e (4.5) Y (s) = U (s) { 14243 as +1 steady−state part transient part For a transient response analysis it is customary to use a With the assumption that z > p > 0 , this response is shown in reference unit step function u(t) for which Fig. -
Time Response 4
Time Response 4 Chapter Learning Outcomes After completing this chapter the student will be able to: • Use poles and zeros of transfer functions to determine the time response of a control system (Sections 4.1 –4.2) • Describe quantitatively the transient response of first-order systems (Section 4.3) • Write the general response of second-order systems given the pole location (Section 4.4) • Find the damping ratio and natural frequency of a second-order system (Section 4.5) • Find the settling time, peak time, percent overshoot, and rise time for an underdamped second-order system (Section 4.6) • Approximate higher-order systems and systems with zeros as first- or second- order systems (Sections 4.7 –4.8) • Describe the effects of nonlinearities on the system time response (Section 4.9) • Find the time response from the state-space representation (Sections 4.10 –4.11) Case Study Learning Outcomes You will be able to demonstrate your knowledge of the chapter objectives with case studies as follows: • Given the antenna azimuth position control system shown on the front endpapers, you will be able to (1) predict, by inspection, the form of the open-loop angular velocity response of the load to a step voltage input to 157 158 Chapter 4 Time Response the power ampli fier; (2) describe quantitatively the transient response of the open-loop system; (3) derive the expression for the open-loop angular velocity output for a step voltage input; (4) obtain the open-loop state-space representation; (5) plot the open-loop velocity step response using a computer simulation. -
Time Response of First Order Systems
Time Response of First Order Systems Consider a general first order transfer function (strictly proper) G(s) b G(s) = = 0 R(s) s + a0 It is common also to write G(s) as K b G(s) = = 0 τs + 1 s + a0 i.e., 1 K a = ; b = 0 τ 0 τ Example: 3 1:5 G(s) = = s + 2 0:5s + 1 a0 = 2 ; b0 = 3 τ = 0:5 ;K = 1:5 The pole of the system is at s = −a0 or s = −1/τ. τ is called the time constant. K is called the DC-gain or steady state gain. What is the differential equation corresponding to the input/output system K c(s) = R(s) τs + 1 becomes (s + 1/τ)C(s) = K/τR(s) which is equivalent to 1 c_(t) + c(t) = K/τr(t) τ Let us consider the effect of both the input r(t) and the initial condition c(0). Taking Laplace Transform of the differentiation equation | this time including the initial condition yields 1 K sC(s) − c(0) + c(s) = R(s) τ τ 1 or c(0) K/τ c(s) = + R(s): s + 1/τ s + 1/τ Note that the initial condition could be represented in the differential equation by an input c(0)δ(t) where δ is the unit impulse function 1 K c_(t) + c(t) = r(t) + c(0)δ? τ τ In block diagram from we have By superposition the response of the system is the sum of the response due to the initial condition alone (the free response) and the response due to the input R(s) (the forced response). -
Control Systems Engineering
WEBC04 10/28/2014 16:58:7 Page 157 Time Response 4 Chapter Learning Outcomes After completing this chapter the student will be able to: • Use poles and zeros of transfer functions to determine the time response of a control system (Sections 4.1–4.2) • Describe quantitatively the transient response of first-order systems (Section 4.3) • Write the general response of second-order systems given the pole location (Section 4.4) • Find the damping ratio and natural frequency of a second-order system (Section 4.5) • Find the settling time, peak time, percent overshoot, and rise time for an underdamped second-order system (Section 4.6) • Approximate higher-order systems and systems with zeros as first- or second- order systems (Sections 4.7–4.8) • Describe the effects of nonlinearities on the system time response (Section 4.9) • Find the time response from the state-space representation (Sections 4.10–4.11) Case Study Learning Outcomes You will be able to demonstrate your knowledge of the chapter objectives with case studies as follows: • Given the antenna azimuth position control system shown on the front endpapers, you will be able to (1) predict, by inspection, the form of the open-loop angular velocity response of the load to a step voltage input to 157 WEBC04 10/28/2014 16:58:7 Page 158 158 Chapter 4 Time Response the power amplifier; (2) describe quantitatively the transient response of the open-loop system; (3) derive the expression for the open-loop angular velocity output for a step voltage input; (4) obtain the open-loop state-space representation; (5) plot the open-loop velocity step response using a computer simulation. -
Second Order System and Higher Order
Second Order and Higher Order Systems 1. Second Order System In this section, we shall obtain the response of a typical second-order control system to a step input. In terms of damping ratio and natural frequency , the system shown in figure 1 , and the closed loop transfer function / given by the equation 1 1 2 This form is called the standard form of the second-order system. The dynamic behavior of the second-order system can then be description in terms of two parameters and . We shall now solve for the response of the system shown in figure 1, to a unit-step input. We shall consider three different cases: the underdamped 01 , critically damped 1, and overdamped 1 1) Underdamped Case : In this case, the closed-loop poles are complex conjugates and lie in the left-half s plane. The / can be written as 2 Where 1 , the frequency is called damped natural frequency. For a unit step- input, can be written 3 2 By apply the partial fraction expansion and the inverse Laplace transform for equation 3, the response can give by 1 | Page Control Laboratory 4 1 1 sin tan 1 If the damping ratio is equal to zero, the response becomes undamped and oscillations continue indefinitely. The response for the zero damping case may be obtained by substituting 0 in Equation 4, yielding 1cos 5 2) Critically Damped Case If the two poles of / are equal, the system is said to be a critically damped one. For a unit-step input, 1/ and / can be written 6 By apply the partial fraction expansion and the inverse Laplace transform for equation 6, the response can give by 1 1 7 3) Overdamped Case : In this case, the two poles of / are negative real and unequal. -
Chapter Six Transient and Steady State Responses
Chapter Six Transient and Steady State Responses In control system analysis and design it is important to consider the complete system response and to design controllers such that a satisfactory response is ¦¤ obtained for all time instants ¢¡£ ¥¤ , where stands for the initial time. It is known that the system response has two components: transient response and steady state response, that is §¨ §¥¨ §©¨ (6.1) The transient response is present in the short period of time immediately after the system is turned on. If the system is asymptotically stable, the transient response disappears, which theoretically can be recorded as "! § ¨ (6.2) However, if the system is unstable, the transient response will increase very quickly (exponentially) in time, and in the most cases the system will be practically unusable or even destroyed during the unstable transient response (as can occur, for example, in some electrical networks). Even if the system is asymptotically stable, the transient response should be carefully monitored since some undesired phenomena like high-frequency oscillations (e.g. in aircraft during landing and takeoff), rapid changes, and high magnitudes of the output may occur. Assuming that the system is asymptotically stable, then the system response in the long run is determined by its steady state component only. For control 261 262 TRANSIENT AND STEADY STATE RESPONSES systems it is important that steady state response values are as close as possible to desired ones (specified ones) so that we have to study the corresponding errors, which represent the difference between the actual and desired system outputs at steady state, and examine conditions under which these errors can be reduced or even eliminated.