Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Design Project ACTIVE DAMPING OF TALL BUILDING VIBRATIONS Franz Hover, 2 November 2007 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 charac-terize 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. • Lab 7: 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. • Lab 8: 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. • Lab 9: 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! 1Supplemental Notes 1. The Step Resp onse vs. the Impulse Response The prototypical input we have used in this class is the step function, u(s) = 1/s or u(t) = 1, for all t > 0, and zero otherwise. You should be familiar with the step of multiplying in the LaPlace domain the step function with the transfer function, to obtain the LaPlace transform of the step response: G(s) y(s) = G(s)u(s) = s where G(s) is the transfer function, and y(s) is the output. The step response is meaningful for many controlled systems, for example corresponding to a step change in torque in the open loop, or a step change in reference voltage in the closed-loop. In the tower labs, we will excite the physical system with an impulse function. The impulse, or delta function δ(t) is many things: (a) the derivative of the step function, (b) an infinitely tall and infinitesimally narrow spike, whose area is exactly one. (c) a strong but instantaneous hit to the system, like a hammer or a lightning strike! Using the differentiation rule in the LaPlace domain, we have L{δ(t)} = 1, leading to the observation that a transfer function is its own impulse response! 2. Suggested State Variables in Your Model Your 2.003 and 2.004 experience tells you that the lab structure is a coupled pair of masses. The mass of the building itself, assumed to be concentrated at the top, is connected to the ground through an effective spring and damper; the ”counterweight” is connected to the building mass through a spring and damper. The control is an additional force exerted between the two masses, in parallel with the counterweight spring and damper. Because the physical system consists of two masses, connected by springs and dampers, it is of fourth-order - the position and velocity of each mass is needed to completely define the state of the system. Using “F = ma”, develop the equations this way: �q = ⎧⎪⎫⎪⎪q⎨1 x q2⎬1 ⎪⎪⎪v ⎪ ⎪= ⎧⎪1⎪⎫⎪⎪⎨⎪⎬⎪, and ⎪⎩⎪q3 ⎪⎪⎩⎪⎪⎪x2 ⎭ ⎪⎪⎪⎭ q4 v2 d�q(t) = A�q(t) + Ba(t) + Gw(t), where dt 2where the derivative of the vector is the same as the vector of derivatives, a(t) is the control action (i.e., force from the voice coil), and w(t) is the external input. It is important to distinguish these two input gain vectors B and G because the two inputs enter the system differently: the actuator force works to separate the two masses, whereas the external input pushes on the building from outside. 3. Obtaining Model Parameters (a) The voice coil has similar properties to an electric motor, namely force = k × current back emf = k × velocity For each of the voice coils here, k = 7.1N/A = 7.1V /(m/s). Don’t forget these numbers (or the gain of the current amplifier) when you implement your design in hardware! (b) We weighed the entire upper assembly of the building without the sliding mass (m1), and then the sliding mass alone (m1) and found m1 = 5.11 kg m2 = 0.87 + 0.075n kg, where n is the number of additional steel collets placed on the shaft. (c) Then we attached the upper assembly without the sliding mass to the aluminum struts - this comprises “the building.” An accelerometer was placed to record lateral accelerations, and we observed the vibrations of the structure when re-leased from a non-equilibrium condition. These voltages with the time vector are recorded in the file tower_base_1.m which you can run within Matlab in the usual way. Your task is to estimate the natural frequency and the damping ratio of this structure, and thereby compute k1 and b1. For your reference, the accelerometer has the approximate calibration 0.0453V/(m/s2). (d) Finally, we fixed the building and studied the behavior of the sliding mass, when released from a non-equilibrium initial position. We used the voice coil as a velocity sensor in this case. The data is found in the file actuator_A_0.m This particular file has the spring clamp at its furthest extension, and no addi-tional masses affixed to the shaft. As above, your task is to estimate the natural frequency and damping ratio of the sliding mass, with the rest of the structure fixed. This will give you k2 and b2. 30 5 10 15 20 25 30−0.15−0.1−0.0500.050.10.15secondsVoltsImpulse Response of Building
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