MIT 2 004 - Modeling the System (10 pages)

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Modeling the System



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Modeling the System

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Pages:
10
School:
Massachusetts Institute of Technology
Course:
2 004 - Dynamics and Control II
Dynamics and Control II Documents
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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 highaspect ratio structure the skyscraper is exible You might not notice this until a strong wind storm sets up large scale vibrations in the rst 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 oor that damp out vibrations caused by wind In these nal three lab sessions you will study in detail a physical structure with a similar dynamic response creating a linear model from rst 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 sta 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 sti nes 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 gure 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 owing 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 1 April 10 2008 1 Figure 1 The experimental system Figure 2 Detailed view of the left side of the building 2 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 eld 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 sti ness 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 w in d fo r c e K F 1 K w m B 1 B 1 v 2 2 m F a c t 1 v 2 2 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 sti ness of the building structure m2 is the mass of the sliding element shaft and voice coil formers B2 is the viscous ction coe cient describing the eddy current losses in the voice coils K2 is the sti ness of the wire spring Two transfer functions for this system V1 s Fw s and 3 V1 s Fact s are 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 simpli ed 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 ywheel 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 rst 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 sta 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 Volts 0 05 0 0 05 0 1 0 15 0 5 10 15 seconds 4 20 25 30 The response is contained in the MATLAB le 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 collet for attaching the wire spring were weighed and found to have a weight of 8 53 N 4 Make a model of the sliding components m2 B2 K2 assuming that the base on which they sit is xed ie vm1 0 5 The response of the sliding components was measured and was found to be Impulse response of moving …


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