Assume the displacement angles of the pendulums are small enough to ensure that the spring is always horizontal. As we assumed the angles are small, we can approximate using sin 1 1, and cos 2 1. Finally the linearized equations 1 ; sin 2 2 , cos 1 of motion becomes,. Write the equations of motion of a pendulum consisting of a thin, 2-kg stick of length l suspended from a pivot. How long should the rod be in order for the period to be exactly 1 sec? The inertia I of a thin stick about an endpoint is 13 ml2.
Grandfather clocks have a period of 2 sec, i. This pendulum is shorter because the period is faster. But if the period had been 2 sec, the pendulum length would have been 1.
In general, Eq. However, we also can use the formular with a reference point other than mass center when the point of reference is …xed or not accelerating, as was the case here for point O. For the car suspension discussed in Example 2.
Find the value of b that you would prefer if you were a passenger in the car. Solution: The transfer function of the suspension was given in the example in Eq. This transfer function can be put directly into Matlab along with the numerical values as shown below. Note that b is not the damping ratio, but damping. What passengers feel is the position of the car. Some general requirements for the smooth ride will be, slow response with small overshoot and oscillation.
There is too much overshoot for lower values, and the system gets too fast and harsh for larger values. Write the equations of motion for a body of mass M suspended from a …xed point by a spring with a constant k. Some care needs to be taken when the spring is suspended vertically in the presence of the gravity. The static situation in b results from a balance between the gravity force and the spring. Automobile manufacturers are contemplating building active suspension systems.
The simplest change is to make shock absorbers with a changeable damping, b u1 : It is also possible to make a device to be placed in parallel with the springs that has the ability to supply an equal force, u2; in opposite directions on the wheel axle and the car body. Is this a good idea? But b is not constant so the system is non-linear with respect to u1 because the control essentially multiplies a state element. So if we add controllable damping, the system becomes non-linear.
However, it would take very high forces and thus a lot of power and is therefore not done. These features are now available on some cars An example is shown in Figure 2.
Figure 2. Modify the equation of motion for the cruise control in Example 2. Put the equations in the standard state-variable form with vr as the input and v as the state. We can see that the larger the K is, the better the performance, with no objectionable behaviour for any of the cases. The fact that increasing K also results in the need for higher acceleration is less obvious from the plot but it will limit how fast K can be in the real situation because the engine has only so much poop.
Note also that the error with this scheme gets quite large with the lower values of K. Determine the dynamic equations for lateral motion of the robot in Fig. Assume it has 3 wheels with the front a single, steerable wheel where you have direct control of the rate of change of the steering angle, Usteer , with geometry as shown in Fig. Assume the robot is going in approximately a straight line and its angular deviation from that straight line is very small Also assume that the robot is traveling at a constant speed, Vo.
The dynamic equations relating the lateral velocity of the center of the robot as a result of commands in Usteer is desired. Solution: This is primarily a problem in kinematics. Note that no dynamics come into play here. It was assumed that the velocity is constant and the front wheel angle time rate of change is directly commanded. Therefore, there was no need to invoke Eqs 2. As you will see in future chapters, feedback control of such a system with a triple integration is tricky and needs signi…cant damping in the feedback path to achieve stability.
Problems and Solutions for Section 2. A …rst step toward a realistic model of an op amp is given by the equations below and shown in Fig. Find the transfer function of the simple ampli…cation circuit shown using this model. Solution: As i. Show that the op amp connection shown in Fig. Give the transfer function if the op amp has the non-ideal transfer function of Problem 2. Solution: Ideal case:.
A common connection for a motor power ampli…er is shown in Fig. The idea is to have the motor current follow the input voltage and the connection is called a current ampli…er.
This expression shows that, in the steady state when s! If fact, the current ampli…er normally has no feedback from the output voltage, in which case Rf! An op amp connection with feedback to both the negative and the positive terminals is shown in Fig 2. From Eq. We can see that the pole is at the left side of the zero, which means a lead compensator.
We can see that the pole is at the right side of the zero, which means a lag compensator. There are a couple of methods to …nd the transfer function from Vin to Vout with set of equations but for this problem, we will directly solve for the values we want along with the Laplace Transform. From the …rst three equations, slove for V1; V2. Vin R1. Find the equations and transfer function for the biquad circuit of Fig.
The torque constant of a motor is the ratio of torque to current and is often given in ounce-inches per ampere. The electric constant of a motor is the ratio of back emf to speed and is often given in volts per rpm.
In consistent units the two constants are the same for a given motor. What is its torque constant in ounce-inches per ampere?
Some remarks on non SI units. We're sorry! We don't recognize your username or password. Please try again. The work is protected by local and international copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. You have successfully signed out and will be required to sign back in should you need to download more resources.
Feedback Control of Dynamic Systems, 8th Edition. Gene F. Franklin J. Davis Powell Abbas F. Feedback control fundamentals with context, case studies, and a focus on design Feedback Control of Dynamic Systems, 8th Edition, covers the material that every engineer needs to know about feedback control—including concepts like stability, tracking, and robustness.
Preface Preface is available for download in PDF format. An emphasis on design , beginning in Chapter 4, builds confidence in solving design problems from the start. Examples compare and contrast the design techniques afforded by the different design methods and complex real-world design problems are attacked using all the methods in a unified way.
Relationships used in design and throughout the book are collected inside the back cover for easy reference. Updated - New examples, updates, and additions keep the material relevant and up-to-date. New - Over 60 of the problems in this edition are either new or revised from the previous edition. New to This Edition. New examples, updates, and additions keep the material relevant and up-to-date.
Table of Contents 1. Laplace Transforms Appendix B. Solutions to the Review Questions Appendix C. Matlab Commands. Share a link to All Resources. Instructor Resources. Previous editions. Feedback Control of Dynamic Systems, 7th Edition.
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