MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring 2008 Laboratory Project: Active Damping of Tall Building Vibrations1 Week 1: Modeling the System Overview The high-rise building is a modern miracle - miles of steel beams and welds, thousands of fasteners allowing graceful structures of one hundred or more stories in height. Like any high-aspect ratio structure, the skyscraper is flexible. You might not notice this until a strong wind-storm sets up large-scale vibrations in the first bending mode. Then the motions will make you ill, or at a minimum cause fatigue. The motions certainly cause damage to the building, notably in the loss of windows which can crack or fall, and in long-term fatigue life reduction. Among potential remedies for building sway, the most common today is the passive or active mass concept. In fact, our own Hancock Tower in Boston has two 300-ton masses near the top floor, that damp out vibrations caused by wind. In these final three lab sessions you will study in detail a physical structure with a similar dynamic response, creating a linear model from first principles and using Simulink to characterize it, synthesizing an active control system design based on the model, and testing your controller on the actual device. The teaching staff will act as consultants. The Experimental Plant Figure 1 shows the experimental system that you will work with. The “building” consists of a block of metal atop a pair of steel side-plates with low lateral stiffnes so that the mass can sway from side-to-side. The passive damping elements, along with the actuator and motion sensors are mounted on the steel plate. The moving mass is a steel shaft, mounted in “frictionless” air bearings. A wire spring couples the moving mass to the building. At each end of the sliding shaft is a voice-coil actuator/sensor. Figure 2 shows a more detailed view of one side of the plant. One of the voice-coils is at the left side with one of the air-bearings. The wire-spring is at the right of the figure. • The air-bearings are connected to a high pressure air supply. The shaft is suspended on a cushion of air, providing an almost frictionless suspension. • The voice-coils are Lorentz force actuators, similar to loudspeakers, and produce a force proportional to the current flowing. At the same time they produce a back-emf voltage that is proportional to the velocity of the coil. They are energy conserving transducers (as we discussed in class) and so F = Kvc × i 1April 10, 2008 1Figure 1: The experimental system Figure 2: Detailed view of the left-side of the building. 2K 1 B 1 F a c t F vb = Kc × v where F is the force produced, i is the current, vb is the back-emf, v is the velocity, and Kvc is a constant. The value of Kvc =7.1 N/amp (or V-s/m). Thus the voice coil may be used as an actuator (by supplying current), or as a velocity sensor, by monitoring the voltage vb. In this case we use one voice-coil as an actuator and a second one to monitor the relative velocity between the building and the sliding shaft. The voice-coil consists of a copper coil, wound on an aluminum formed that slides in a strong magnetic field. As it moves, eddy-currents are set up in the former (similar to what happens in the lab rotating plant) leading to an inherent viscous drag as the coil moves. • The wire spring provides a restoring force proportional to the displacement. Its length can be adjusted to provide a variable stiffness. • In addition an accelerometer is attached to the building to sense its motion. The gain of the accelerometer is Ka =0.453 v-s2/m. m mK B 21 22w( w i n d f o r c e ) v 1 v2 Figure 3: Lumped parameter model of the experimental plant. Figure 3 shows a simple lumped parameter model of the system. In this model m1 is the lumped mass of the building. B1 represents the energy dissipation as the building moves. K1 is the lateral stiffness of the building structure. m2 is the mass of the sliding element (shaft and voice-coil formers). B2 is the viscous fiction coefficient describing the eddy-current losses in the voice coils. K2 is the stiffness of the wire spring. Two transfer functions for this system V1(s) V1(s)and Fw(s) Fact(s) 3are provided in the Appendix. You may need to generate additional transfer functions during the course of the project. Project Schedule This is a three week project: Week 1: Create a simplified model of the open-loop system, using the attached notes and data on the physical properties of the plant. You will write this model in state-space form, and use Matlab to (numerically) convert it to a transfer function. Week 2: Employing MATLAB, design an active damping system built on the PID controller you have used for the flywheel plant. You will model your system in Simulink, and test out your controller in simulation. Week 3: Test your controller on the real plant; prepare and turn in a report detailing your model, the controller design, and your results. Be sure to document the process and rationale for your controller design! This Week’s Goals: In this first week you will: • Be introduced to the experimental model building and the actuators and sensors that you will have available. • Use experimental results to estimate the basic system parameters. • Use MATLAB to determine some basic properties and responses of your model, and your model resulta against experimental measurements. Steps: 1. Make a model of the “building”, consisting of m1, B1 and K1. 2. The course staff removed the sliding components (m2, B2, K2) from the building and did some experiments and found: (i) The building weighs 50.13 N. (ii) The building was displaced a distance of 1 cm and released. The response was measured by an accelerometer and found to be highly oscillatory: Impulse Response of Building via Accelerometer, Trial 1 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Volts 0 5 10 15 20 25 30 seconds 4The response is contained in the MATLAB file BuildingResponse.m in the course locker. Download and use this data to estimate B1 and K1. 3. The sliding shaft with the two voice coil formers and the
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