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Vibration Control of a Two Story Building using Active Mass Dampers
Elizabeth A. Magliula, Hani Sallum
Abstract – The objective of this research is to design a control system to dampen the vibrations in a building-like structure using a MIMO approach. Contributory objectives include testing the validity of a simplified mathematical model versus the experimental results using the Quanser Consulting Inc. hardware model, as well as investigating the effectiveness of the controller in regulating the motion of the structure by comparing it to the uncontrolled response.
This advanced active mass damping experiment is useful to study the benefits of employing individual AMD's on each floor, as opposed to the classical approach of using a single AMD on top of a building. The tall building-like structure consisting of two floors is instrumented with an accelerometer on each floor to measure their acceleration. The structure is flexible along its façade and two carts, each driven via a rack and pinion mechanism, are mounted on each floor.
The goal is to design a feedback controller that measures the cart positions and the accelerations of the floors to dampen the vibrations of the structure effectively. The system is supplied with a state feedback controller, though we have opted to design and employ our own. The results of this research indicate that the theoretical mathematical model closely modeled the experimental data, and prove that the controlled system was successful in reducing the stresses on the sidewalls of the structure by regulating the motion of the cart present on each floor of the structure.
I. INTRODUCTION
E
arthquakes and wind are two important external loads that must be taken into account when designing a structure, as they can greatly affect stability. For years, extensive research has been done to reduce the structure response of a building subject to lateral loads. In the United States, passive control devices have been implemented to address this area of research. Control devices such as Tuned Mass Dampers have been used in real structures. Japan has also implemented similar devices to improve building performance. The most widely used controller for use in tall buildings in the Linear Quadratic Regulator (LQR) (Soong, 1990; Housner et al., 1997). Doyle et al. (1989) introduced other control methods, such as H2 and H, to be discussed in the next section.
This report discusses the design of a two floor active mass damper system. The controller was designed for the purposes of minimizing the vibration present in a building like structure by applying a state feedback control law to control the motion of two masses, one located on the roof of the first floor, and one located on the roof of the second floor. The design of the controller was completed by implementing a state observer system, which uses the measurable state variables to estimate state variables that are not directly measurable.The ability of the controller to model the predicted responses of the state variables was tested using a MATLAB simulation, and finally, the ability of the control system to actively reduce vibrations in the structure was verified by comparing it with results obtained when the system was uncontrolled.
A. Prior Work
Various groups have proposed different control algorithms for use in tall buildings and towers. The most commonly used is the Linear Quadratic Regulator (LQR) (Soong, 1990; Housner et al., 1997), though other methods have been successfully employed. Sushardjo et al., (1990) used the H2 and H control methods in their frequency domain design, which primarily focused on the frequency domain characteristics of the structural response (Suhardjo et al. 1992; Spencer et al., 1994; Suhardjo and Kareem, 1997). Spencer et al. (1994) investigated the controllability of a three story-building model subjected to various introduced inputs, and Battaini et al. (1998) proved that bench-scale models could be used to investigate various full-scale structure issues that arise in real life scenarios. They utilized a two story active mass damper system, as we have in our investigations.
Finally, a group of students from a previous AM501 course also investigated the possibility of minimizing the stresses using a single floor of the same Quanser hardware setup that we employed to perform our own two story investigation. This group (Selha and Favaretto), determined that their experimental data closely modeled the theoretical simulation, that the same gain values could be used for both simple and complex input disturbances, and finally that their control system was successful in reducing the stresses in the sidewalls of the structure.
II. APPROACH
A. System
In order to understand how the simplified mathematical model of the Quanser Consulting Inc. hardware setup was devised, a brief summary of the setup in necessary. The structure consists of two vertical side supports which function as the frame, Plexiglas which acts as the first and second floors of the structure, and two DC motor driven carts, one on each floor, which run on rack and pinion gear systems. The shaker table located at the base of the structure consists of two metal plates with a DC motor mounted between. When an input signal is applied to the motor, the shaker moves via a rack and pinion drive system. The motor driven carts act as the active mass dampers, which move back and forth along a geared rack. An accelerometer supplied and manufactured by Quanser Consulting Inc. is mounted to the Plexiglas casing on each of the two floors. A MultiQ I/O board, also supplied by Quanser Consulting Inc., is used to implement the digital controller, in conjunction with WinCon real time controller software. The controller for the system was developed in SIMULINK.
B. Mathematical Model
A simplified model for a two story active mass damper was determined, as seen in Figure 2.
The moving elements in the system are the carts, the first and second floors, and the flexing beams. The kinetic and potential energies are derived as follows:
where
Mc1 = mass of cart 1 in kg = 0.9 kg
Mc2 = mass of cart 2 in kg = 0.9 kg
Mf1 = mass of first floor = 1.850 kg
Mf2 = mass of second floor = 1.777 kg
K1 = spring constant, first floor = 6EI/l3
K2 = spring constant, second floor = 6EI/l3
l = 0.4763 m
w = 0.10795 m
h = 0.001588 m
E = 2x1011
I = 3.599x10-12
The work in the system was also determined, and is expressed in the following equation:
where
Km = motor torque constant = 0.00767 Nm/A
Ke = back emf = 0.105 V-sec/rad
Kg = amplifier gain = 3.7
Ra = armature resistance = 2.6
rw = pinion radius = 0.00635m
After the total kinetic and potential energies were determined, the Lagrangian was taken, given by:
L = T – V
We then proceeded to develop equations of motion using the generalized Lagrange formulations. The four equations of motion are given as follows:
Equations 7 and 8 represent the accelerations of floors 1 and 2 respectively, and equations 8 and 9 represent the acceleration of carts 1 and 2, respectively. Substituting system parameters and converting to voltage input, we obtain the A and B matrices that are used in the simplified model simulation. The output components are q1 and q2. The states of the system are defined as:
where
q1 = displacement of floor 1
q2 = displacement of floor 2
x1 = position of cart 1
x2 = position of cart 2
= velocity of floor 1
= velocity of floor 2
= velocity of cart 1
= velocity of cart 2
C. Linear Quadratic Regulator
The weighting strategy employed for the LQR cost function consists of applying weights to individual state and input values separately (i.e. all non-diagonal Q and R matrix elements were zero).
Primary importance is placed on maintaining small values for q1 and q2, the floor displacements, and thus their weighting factors are chosen to be large. Additionally, because each cart has a very limited range of motion, similar importance is placed on maintaining small values for x1 and x2, and their weighting factors are chosen to be large as well. Weighting factors for the floor and cart velocities are not considered as important, and therefore their weighting factors are smaller.
Because the base displacement input x0 is not intended for use as part of the control law, a large weighting factor is placed on this term to ensure that the LQR calculation would assign negligible (near zero) gains for x0 in the control law. Alternately, the voltage inputs to the carts are the sole methods of control for the system, thus small weighting factors are placed on these terms.
The final weighting values used for the cost function are as follows:
Q11 = 10(weighting on q1)
Q22 = 10(weighting on q2)
Q33 = 1(weighting on x1)
Q44 = 1(weighting on x2)
Q55 = 1(weighting on q1dot)
Q66 = 1(weighting on q2dot)
Q77 = 0.1(weighting on x1dot)
Q88 = 0.1(weighting on x2dot)
R11 = 0.05(weighting on Vc1)
R22 = 0.05(weighting on Vc2)
R33 = 1020(weighting on x0)
D. State Observer System
A state observer system makes it possible to estimate the state variables of a system using the quantities that can be measured. The inputs of the state observer system are the inputs and outputs of the original system, and the outputs of the state observer are the estimated state variables.
The inputs into the observer are q1, q2, s, Vc1, Vc2. The outputs of the observer are the estimated values of the displacements of floors one and two.
The state equations for the observer system are given by:
Where G is the observer gain matrix. The matrix G can be found by defining the error in the estimated state variables, versus the actual state variables. The error in the estimated state variables is defined as:
Taking the derivative with respect to time of equation 13 and making the appropriate substitutions derives equation 14. The result is as follows:
The appropriate choice of G must be selected to assure the stability of the matrix quantity of equation 14, which is possible by assuring that the original system is controllable. The appropriate G matrix may be obtained using Matlab function lqr, using A=AT, B=CT, and the appropriate weighting factors R and Q.
The need for a state observer system arose from difficulties in reliably determining displacement and velocity values for each floor, which are critical for implementing the proposed control law. The only state data available for each floor was an accelerometer signal, which when integrated twice to yield displacement proved to contain excessive amounts of cumulative error.
Notch filters are implemented in an attempt to reduce the amount of sensor noise and signal offset variation. These filters, however, did not reduce the error to an acceptable degree for reliable closed-loop state control. Therefore, it was determined that a state observer, based on the already developed mathematical state model of the system, would need to be implemented.
III. ACTUAL MODEL versus SIMULATION
A state feedback controller was used to regulate the motion of the carts on each floor of the building. The controller was designed using SIMULINK.
After summing the individual state-gain contributions to the input signals, saturation blocks are placed on each signal for the control voltages to each cart. This is done to limit the voltage range, which the controller can impart to the carts in order to avoid electrical damage to the motors or mechanical damage to the rack-and-pinion mechanisms.
To validate our mathematical model we used Matlab, utilizing the matrices developed from our equations of motion. We held the displacements of the carts at zero and modeled the system as a simple 2DOF oscillator. We compared the results to those found using the actual hardware model (also holding the displacements of the carts at zero).
Comparisons between the acceleration responses of both floors using the simulation and the actual model were made. Plots of acceleration responses may be viewed in Figure’s 4 and 5. In order to assess whether or not the acceleration responses using our simulation model were comparable to the ones determined using the hardware model, we plotted the first and second floor responses using each method. The results for the first floor are displayed in blue, and the results for the second floor are displayed in magenta.
When the blue plots are compared, it is seen that both methods produce a similar response. The same can be seen for the magenta plots, though not as strongly after 4 seconds.
Overall, as can be seen in Figure’s 4 and 5, the responses using the simulation are similar to those found using the actual model, and they appear to be in phase.
Plots of the Fourier transform of the displacement data using both the actual and simulation models were also compared in order to determine if both systems had resonance peaks at approximately the same frequencies. This was done for both floors, and these plots are labeled Figure 6 and 7.
After comparing the two plots, it can be seen that both have resonance frequencies at approximately the same values; the first is near 1HZ and the second is near 3 HZ. These results are also validated by the results determined by Quanser Consulting Inc. in the manual accompanying the 2 Floor AMD setup. The Quanser sample model estimates resonant frequencies to be 1.16 Hz and 3.18 Hz. These values are close to those determined by our simulation model, 1.2 Hz & 3.1 Hz respectively.
In order to verify our mathematical model in a third way, we utilized the accelerations experienced by the floors using our mathematical simulation model. The Fourier Transform of the acceleration data was taken to see if the data once again yielded resonance peaks at approximately the same frequencies. The results may be seen in Figure 8.
It can be seen from Figure 8 that the resonance frequencies correspond to the ones determined using the other verification methods, though the amplitudes differ.
IV.VIBRATION REDUCTION
After the controller was implemented, we evaluated the response of the system subject to two different inputs. The first input was an impulse delivered to the floors themselves, and the second was an input from the shaker at the base of the structure, delivered every 2 seconds.
The response of the structure when driven by an impulse to the floors themselves can be seen in Figures 9 and 10. It can be seen from the plots that their exists a significant reduction in the vibration experienced by the structure while the controller is implemented, especially for large displacements. In the future we hope to incorporate the friction of the bearings into the model in order to obtain better small displacement control.
We also sought to investigate the response of the structure when driven by the shaker located at the base of the two story building structure. The response of the system when subjected to an impulse every 2 seconds may be seen in Figures 11 and 12.
It can be seen that there continues to be a significant reduction in vibration experienced by the structure. The control objective of designing a control system to dampen the vibration in the building like structure using a MIMO approach is met when an impulse is applied to the base of the structure via the shaker, as well as to impulses delivered to the floors themselves.
V. CONCLUSIONS
A mathematical model was derived, which succeeded in determining resonant frequencies and acceleration responses comparable to those determined by the hardware model and the Quanser sample mathematical model. A nonlinear controller, including a state observer system was successfully developed and implemented using SIMULINK, and LQR Control was successfully implemented using the appropriate weighting.
The objective of dampening the vibrations in the building-like structure was met. As can be seen from the previous figures, the state feedback controller was successful in regulating the motion of the carts on both floors, and hence that of the structure, when driven by the shaker using an impulse response, and by impulses delivered directly to the floors themselves. Future work to be done on this project includes
incorporatingiincorporating the friction of the bearings into the model in order to obtain better small displacement control, developing a more rigorous LQR weighting strategy (though there were no optimal specifications which needed to be met, we wish to investigate the effects of various weighting alterations to obtain even better control), and investigating the systems response when subject to other inputs.
REFERENCES
[1] Battaini, M., et al., "Bench-scale Experiment for Structural Control," Dept.. of Structural Mechanics, University of Pavia, Italy, 1998.
[2]Doyle, T.C., et al., “State Space Solutions to Standard H2 and H Control Problems.” IEEE, Trans. Aut. Contr., AC34(8), 831-847, 1989.
[3]Duke, S.J., et al., "Acceleration Feedback Control of MDOF Systems," ASCE., J. Engrg. Mech.,, vol. 122, pp. 907-917, 1996.
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[7]Suhardjo, J., and Kareem, A.., "Structural Control of Off-Shore Platforms," Proceedings of the 7th International Off-Shore and Polar Engineering Conference IOSPE-7, Honolulu, 1997.
[8]Suhardjo, J.,Spencer, Jr. B. F., and Kareem, A., “Frequency Domain Optimal Control of Wind Excited Buildings.” ASCE, J. Engrg. Mech., Vol.118, 2463-2481, 1992.