Cansmart 2009

International Workshop SMART MATERIALS AND STRUCTURES 22 - 23 October 2009, Montreal, Quebec, Canada

2009 Cansmart Workshop

ADAPTIVE PENDULUM MASS DAMPER FOR THE CONTROL OF

STRUCTURAL VIBRATIONS

R. Lourenco1, A. J. Roffel2, S. Narasimhan3

1Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Ontario, Canada

rlourenco@uwaterloo.ca2,3Department of Civil and Environmental Engineering, University of Waterloo, Waterloo

Ontario, Canada 2ajroffel@uwaterloo.ca, 3snarasim@uwaterloo.ca

ABSTRACT Incorporating adjustment mechanisms to compensate for de-tuning in traditional tuned mass dampers is essential for the advancement of this technology. In this paper, experimental and simulation results on the control of structural vibrations using an adaptive pendulum mass damper are presented. The adaptive pendulum mass damper is a spherical pendulum augmented with a tuning frame to adjust its length and two adjustable air dampers connected to the mass for achieving the damping adjustment. The mechanical adjustments are implemented using three independent stepper motors, one micro-controller and three drives. The mass damper is used to control the vibration responses of a bench-scale two-story model structure with sufficiently long fundamental period representative of flexible structures such as towers. The basic architecture of the system proposed consists of two components; identification and control, one followed by the other in that order. The identification is carried out using traditional Fourier methods and a second-order blind identification method in the time-domain. The control phase consists of position control based on the identified frequency from the identification phase. The paper focuses on the hardware and software aspects of the real-time implementation of this adaptive mass damper. The identification methods used in this study rely only on acceleration measurements collected from high accuracy-low frequency accelerometers mounted on the structure, and do not utilize the excitation information. This study is primarily intended to demonstrate the feasibility of employing adaptive pendulum mass damper designs to enhance the robustness of passive tuned mass dampers to structural, environmental and design changes.

Keywords: Adaptive Pendulum Mass Dampers, TMDs, Structural Control.

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INTRODUCTION

Tuned mass dampers (TMDs) have the potential to reduce structural vibrations by adding additional damping to flexible structures. The literature on this topic is extensive [1-8] and a comprehensive review of TMDs is not attempted here. A large number of TMDs are of pendulum type, called pendulum-TMDs (PTMDs), which essentially consist of a suspended mass tuned to a fraction of the fundamental period of the structure, and a viscous damper which acts as an energy sink transferring the mechanical motion of the TMD to heat that is eventually dissipated to the environment. A major drawback of TMDs is their sensitivity to changes in their operating environment such as structural deterioration, which results in de-tuning. Failure to account for de-tuning may cause the performance of TMDs to degrade over time. In this paper, an adaptive pendulum TMD, called adaptive pendulum mass damper (APMD) henceforth, is presented to compensate for the de-tuning and other environmental effects. The main goal of this paper is to illustrate a relatively simple method of harnessing the advantages of semi-active control using pendulum TMDs, while focusing on the algorithmic and design aspects involved in achieving the optimal tuning in the proposed APMD configuration. Semi-actively controlled TMDs have been studied by various researchers and several effective designs have been presented. For example, variable stiffness TMDs consisting of adjustable stiffness elements have been studied for seismic and wind applications [3, 4]. Pendulum mass dampers with nonlinear dampers have also been studied by researchers for structural vibration reduction and a comprehensive review of the literature on this subject has been published recently [9]. In this paper, a pendulum tuned mass damper (PTMD) is enhanced with adaptation mechanisms to adjust its length and damping through feedback. A pendulum TMD is chosen as the underlying passive system because a large number of TMDs in operation today are of this type. In this paper, the effective length is related to the identified natural frequency of the structure through simulation studies incorporating the three-dimensional behaviour of the pendulum mass. Once the effective length is known as a function of the fundamental period of the structure, the frequency tuning is implemented in two stages: first, the identification phase is carried out while the structure and TMD are in operation and in a pre-assigned configuration, followed by a control phase where a tuning frame is hoisted to the desired position to achieve the fundamental natural frequency. The level of damping is adjusted by closing or opening a valve at one end of the air-damper using a stepper motor. The paper is organized as follows. The equations of motion for the PTMD attached to a flexible structure are presented first. Then, an experimental set-up of a new APTMD is presented followed by brief experimental results using this set-up. Finally, important conclusions of this paper are presented.

EQUATIONS OF MOTION FOR A PTMD Fig. 1 shows the position of the auxiliary mass, . The origin of the system is set up to coincide with the suspension point of the auxiliary mass. Critical parameters of the geometry include

am

θ , the angle of the swing of the auxiliary mass away from the vertical, ϕ , the angle 2009 Cansmart Workshop 18

of the auxiliary mass rotating about the vertical line, and , the length of the pendulum. The vectors u, v, and w are the displacements of the suspension point in the x-, y-, and z-directions, respectively. All of the aforementioned parameters vary with time with the exception of the pendulum length, which has been kept constant for this derivation. Consider also a linear spring and damper in both horizontal directions connected to the pendulum length at a distance xh and away from the suspension point. yh

Fig. 1: Formulation of the equations of motion for the PTMD system. The position, relative to the origin, of the auxiliary mass is [ ] [ ] [ ]sin cos sin sin cosa a a ar u L v L w Lθ ϕ θ θ= + ⋅ + + + − ⋅k

L ⎤⎦

ϕ ⋅i j (1)where i, j, and k are unit vectors along the x-, y-, and z-directions, respectively. The velocity vector of the auxiliary mass is cos sin sin cos sin sin cos

sin

a a a a a

a

v L cos L

L

u v L

w

θ ϕ θ ϕϕ θ ϕθ θ ϕϕ

θ

θ

θ

⎡ ⎤ ⎡= + − ⋅ + + + ⋅⎣ ⎦ ⎣⎡ ⎤+ + ⋅⎣ ⎦

&&& & &&

&&

i j

k(2)

The linear spring constant is xk in the x-direction and in the y-direction. Similarly, the damping coefficient is

yk

xc in the x-direction and in the y-direction. Ignoring vertical movement of the spring/damper attachment point, the position of the attachment point, relative to a vertical line passing through the moving suspension point, in the x- and y-direction is

yc

[ ], sin cosp x xr h θ ϕ= ⋅ i (3a)

, sin sinp y yr h θ ϕ⎡ ⎤= ⋅⎣ ⎦ j (3b)The velocity at the attachment point is

, cos cos sin sinp x x xv h hθ ϕθ θ ϕϕ⎡= −⎣& & i⎤ ⋅⎦ (4a)

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⎤ ⋅⎦

, cos sin sin cosp y y yv h hθ ϕθ θ ϕϕ⎡= +⎣& & j (4b)

The kinetic energy of the auxiliary mass is (

)

2 2 2 2 22221 1 sin 2 cos cos2 2

2 sin sin 2 cos sin 2 sin cos 2 sin

a a a a a a a a

a a a a

u u

u

T m v v m v w

v vL w

L L L

L L L

θ ϕ θθ θ

θ ϕ θ

ϕ

θ ϕϕ θ ϕ θ ϕ θ

= ⋅ = + + + + +

+ +

−

+

& &&& &

& &&& && &

& &

&

(5)

The potential energy of the auxiliary mass is ( )

( )

2 2 2 2 2, ,

2 2 2

1 1 1cos sin cos2 2 2

1 sin sin cos2

a x p x y p y a a x x

y y a a

U k r k r m g w L k h

k h m g w L

θ θ ϕ

θ ϕ θ

= + + − =

+ −

+ (6)

where g is the acceleration due to gravity. The Raleigh dissipation function for the auxiliary mass is (

) ()

2 2 2 2 2 2 2, ,

2 2 2 2 2 2 2 2 2

2 2 2 2

1 1 1 cos cos 2 cos cos sin sin2 2 2

1sin sin cos sin 2 cos cos sin sin2

sin cos

a x p x y y p x x x

x y y y

y

F c v c v c h h

h c h h

h

θ ϕϕ θ ϕθ θ ϕϕ

θ ϕϕ θ ϕθ θ ϕθ θ ϕϕ

θ ϕϕ

= + = −

+ + +

+

&& &

& && &

&

(7)

The kinetic energy of the main mass is

{ } [ ]{ }{ }

[ ][ ][ ]

[ ] [ ] [ ] [ ] { }

1 12 2

uu uv uw ur

vu vv vw vr

wu w

T

v ww

T

wr

ru rv r

m

rw r

m m m Mm m m Mm m m MM M M M

u uv v

T M w w

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪= Δ Δ = ⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪Δ Δ⎭ ⎦⎩ ⎩

& &

& && &

& &

& &⎭

(8)

where [ ]M is the mass matrix for the entire main structure. The first three rows and columns correspond to the three degrees of freedom (DOF) of the main structure at the suspension point. The potential (strain) energy of the main mass is

{ } [ ]{ }

{ }

[ ][ ][ ]

[ ] [ ] [ ] [ ] { }

1 12 2

uu uv uw ur

vu vv vw vr

wu wv ww wr

ru rv r

T

r

m

w r

T k k k Kk k k Kk

u uv v

Uk

KK

K K Kw wk

K

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪= Δ Δ = ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪Δ Δ⎩ ⎭ ⎩ ⎭

(9)

where [ ]K is the stiffness matrix of the main structure. The Raleigh dissipation factor is given by

{ } [ ]{ }{ }

[ ][ ][ ]

[ ] [ ] [ ] [ ] { }

1 12 2

uu uv uw ur

vu vv vw vr

wu w

T

v ww

T

wr

ru rv r

m

rw r

c c c Cc c c Cm c c CC C C C

u uv v

F C w w

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪= Δ Δ = ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪Δ Δ⎭ ⎦⎩ ⎩

& &

& && &

& &

& &⎭

(10)

where [ is the damping matrix of the main structure. ]C

20

2009 Cansmart Workshop

m The total kinetic energy is aT T T= + , the total strain energy is , and the dissipation function is . The Lagrangian of the system is

aU U U= + m

maF F F= + L T U= − . The Lagrange’s equation [10] is

0r r r

d L L Fdt q q q

∂ ∂ ∂− + =

∂ ∂ ∂& & (11)

where and are the general velocities and coordinates of the system. For the PTMD system, the general coordinates are u, v, w,

rq& rq{ }Δ , θ , and ϕ and the general velocities are u ,

, ,

&

v& w& { }Δ& , θ& , and ϕ& . Substituting equations (7) through (10) into (11) produces the following system of equations for the MDOF system with a PTMD suspended from the top floor, corresponding to the first three DOF in the mass, damping, and stiffness matrices:

[ ]

[ ][ ][ ]

[ ] [ ] [ ] [ ] { }[ ]

{ }[ ]

{ }2 2cos cos sin cos 2cos sin sin s

0 0 00 0 00

in sin coscos s

0 00 0 0

i

0

n

a

a

aa a

u u uv v v

C K m L

mm

w w wM

m

θ ϕ θ ϕθ θ θ ϕ ϕ θ ϕ θ ϕϕθ

θθϕ

ϕ

⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪+ + = ×⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ΔΔ Δ ⎩ ⎭⎩ ⎭ ⎩ ⎭

− + + + +−

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

+

⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

&& &

&& &

&& &

&& &

&& & & &&

&&

& &

[ ]

2 2

2

sin sin 2cos cos sin cos sin sin

sin cos

0a

gL

θ ϕ θ ϕ ϕ θ ϕ θ ϕ

θθ θ

θ θ

θ

ϕ⎧ ⎫⎪ ⎪− − +⎪ ⎪⎪ ⎪⎨ ⎬− − −⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

& & &&& &

&& &

ϕ

(12)

(

) ( )

2

22 22 2 2

22 2

sin cos cos cos cos sin sin sin

sin cos cos sin cos sin cos cos

cos cos sin sin cos sin cos cos sin sin 0

a a

y yx x x x

a a a a a a

y y

a a

L L u v w g

k hk h c hm L m L m L

c hm L

θ θ θϕ θ ϕ θ ϕ θ θ

θ θ ϕ θ θ ϕ θ ϕ

θ ϕ θ ϕϕ θ ϕθ ϕ ϕ

θ

θ θ ϕ

− + + + +

+ + +

− + −

&& & && && &&

&& &

&2

=

(13)

( )

( )

( )

2 2

22

22

sin 2 cos sin cos sin sin cos

cos cos sin sin sin

cos cos sin sin cos 0

x x y ya a

a a

x x

a a

y y

a a

k h k hL L u v

m L

c hm L

c hm L

θϕ θθϕ ϕ ϕ θ ϕ ϕ

θ ϕ ϕθ θ ϕϕ

θ ϕ ϕθ θ ϕϕ

−+ − + −

+ − +

+ + =

&&& & && &&

& &

& &

(14)

The equations of motion were cast in state-space form and implemented in MATLAB. A simple 5DOF (u, v, w, θ , and ϕ ) was developed for simulating the response of the structure with the PTMD. The frequency response, for the controlled and uncontrolled cases using 25% of that excitation in the y-direction as in the x-direction is shown in Fig. 2. The magnitude is normalized with respect to the maximum uncontrolled magnitude in the x-direction and the frequency is normalized with respect to the natural frequency of the system.

21

Fig. 2: Frequency response in x- and y-directions for controlled and uncontrolled

EXPERIMENTAL SET-UP

The experimental set-up of the APMD is shown in Fig. 3. The basic mechanism consists of a 1.5 kg suspended mass, three stepper motors; one for hoisting a tuning frame, and the other two to adjust the valves in the two air dampers, in two orthogonal directions (only one motor and one damper is shown here for clarity). The motors, pendulum and the dampers are mounted on an aluminum frame. The structural model shown in Fig. 3 consists of two steel plates, weighing approximately 120 N each, that serve the function of floor masses, and four aluminum angles that provide the story flexibility. This structure has the first fundamental frequency in the range of 2.1-2.3 Hz with diagonal bracing on the upper floor. The fundamental frequency of the APMD is adjusted by hoisting or lowering a tuning frame whose ends are connected to a sliding aluminum rail through a smooth Teflon bearing assembly. It is worth mentioning here that if one were to hoist the suspended mass instead, torque requirements increase significantly. Hence, this idea was abandoned by the authors during the early design stages. The damping is controlled using two air-dampers with valve adjustments that are capable of generating peak (adjustable) forces between 0-88.0 N.s/m. The valves of the dampers are mated through a customized connector to stepper motors in order to adjust the opening and closing of the valves. The APMD is controlled using a DSP controller and data acquisition board (dSPACE 1104) with 8 parallel output and 4 parallel and 4 mixed analog inputs. The board also contains extensive digital I/O capabilities that are necessary to control the motors. In order to minimize the overall weight of the adaptive components, the stepper-motor controller was custom-built and consists of an Atmel-Atmega8 microcontroller and three L293 drivers capable of driving three stepper motors at upwards of 2A and 12V each. All three stepper motors are geared with bi-polar windings operating at 2.75W per phase and running torques of 0.21 N-m and 0.85 N-m at 240 PPS. A draw wire potentiometer is used to determine the vertical displacement of the pendulum mass relative to the moving pivot point on the tuning frame. Two laser sensors (Wenglor) are mounted orthogonally in the horizontal plane to determine the location of the pendulum mass. Each measurement sensor outputs an analog voltage that is sent to the analog inputs of the dSPACE controller and data acquisition board.

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Damper

Tuning Frame

Fig. 3: Experimental set-up of the APMD.

IDENTIFICATION ALGORITHMS FOR TUNING THE ATMD

The effective length of the pendulum is available from the optimization study (from the simulation model) as a function of the imposed excitation and the fundamental natural frequency of the structure. An issue that is critical to the application of adaptive TMDs is the fact that the effective length should be based on as-built conditions and while the structure is in operation. Hence, the identification has to be carried out while the structure is in operation. This is the approach taken in this paper. The motions of the TMD mass is limited by lowering the tuning frame so that it rests flush with the surface of the pendulum mass. Once the motion of the mass is limited in this fashion, the identification phase is carried out. Once the effective length is computed using the identification algorithm, the tuning frame is hoisted to its final location in the control phase. Two algorithms, one in the frequency domain and the other in the time-domain, are implemented, both using the SIMULINK environment provided by dSPACE. The first method is based on the power spectral density estimate of the data in a moving window. The signal is buffered and windowed at each time step and the buffer length is set at 1024. The process of determining the fundamental frequency is achieved by a peak-picking process and averaging the frequency estimates for multiple windows of the samples with appropriate threshold conditions. The algorithm involves several trigger and state-flow features that cannot be presented here due to space limitations. The second method is carried out in the time-domain and is based on second-order blind identification method, SOBI [11]. A brief summary is provided below.

SOBI is carried out by diagonalizing one or more covariance matrices of measurements. The basic problem statement is cast in the form of a linear static mixtures problem as: ( ) ( )

( ) ( )k k

k k

⎫= ⎪⎬

= ⎪⎭

x As

y Wx (15)

where is the instantaneous mixing matrix and ij n na

×⎡ ⎤= ⎣ ⎦A n n×W is the un-mixing matrix, to

be determined. The sources are then given by the vector ( )ky , which provides an estimate of

23

the sources. SOBI seeks to determine the un-mixing matrix W using the information contained in only. The first step in this method is the simultaneous diagonalization of

two covariance matrices

( )kx

( )ˆ 0xR and ( )ˆx pR evaluated at zero time-lag and non-zero time lag

p, defined as ( ) ( ) ( ){ } ( )

(

2009 Cansmart Workshop

) ( ) ( ){ } ( )

0 0T Tx s

T Tx s

E k k

p E k k p

⎫= = ⎪⎬

= = ⎪⎭

R x x AR

R x x AR

A

A (16)

where, ( ) ( ) ( ){ }Ts p E k k p=R s s − . The simultaneous diagonalization is performed in three

basic steps: whitening, orthogonalization and unitary transformation. Whitening is a linear transformation in which ( ) ( ) ( ) ( )( )1

ˆ 10 Nx k

Tk kN == ∑ x xR is first diagonalized using

singular value decomposition ( )ˆ 0 Tx x x x=R V VΛ where xV are the eigenvectors of the co-

variance matrix of x. Then, the standard whitening is realized by a linear transformation expressed as ( ) ( ) ( )1

2 Tx xk k −= =x Qx V xΛ k (17)

Because of whitening, ( )ˆx pR becomes ( )x̂ pR which is given by the equation

( ) ( ) ( ) ( )( ) ( )ˆ 11 N T

x xkTp pk kN =

= =∑ QRxR Qx (18)

The second step, called orthogonalization, is applied to diagonalize the matrix ( )ˆx pR

whose eigen-value decomposition is of the form ( )ˆ 0 Tx x x x=R V Λ V . Using equations (16) and

(18), ( ) ( )ˆ T T

x sp p=R QAR A Q (19)If the diagonal matrix xΛ has distinct eigen-values, then the mixing matrix can be estimated uniquely by the equation 1

21ˆ ˆ

ˆx x x x

−= =H Q V V VΛ (20)It is easy to see that the product QA is a unitary matrix (since the sources are assumed to be uncorrelated and scaled to have a unit variance), and the problem now becomes one of unitary diagonalization [12] of the correlation matrix ( )ˆ

x pR at one or several non-zero time

lags. Equation (19) is a key result, which states that the whitened matrix ( )ˆx pR at any non-

zero time lag p, is diagonalized by the unitary matrix, QA . Since ( )s pR is a diagonal matrix (since the sources are assumed to be mutually uncorrelated), the problem now becomes one of diagonalizing the matrix ( )ˆ

x pR resulting in the unitary matrix, QA . Both the frequency domain and the time-domain identification algorithms were implemented using MATLAB scripts and functions embedded into SIMULINK and executed aboard the DSP chip on the dSPACE1104 controller board. Since the identification phase occurs prior to the control phase, time-delay issues are not critical in the current configuration, and a time-step of 0.01 seconds did not cause implementation or stability issues.

24

BRIEF SUMMARY OF THE EXPERIMENTAL RESULTS

In order to test the experimental set-up proposed earlier, two tests were conducted. First, a forced excitation test was conducted using a harmonic forcing function of 2.2 Hz and the identification carried out using the algorithms described earlier. The valve of the air damper (in one direction) was nearly open. The time-step in the experiment was set to 0.01 sec. For the de-tuned case, the frame was lowered to about 5 mm from the top of the suspended mass, and for the tuned case the frame was hoisted using a motor to the correct height from the center of the suspended mass once the identification phase was completed. The results are shown in Fig. 4. The results of identification from both the identification methods described earlier were identical, and hence only one set of results are presented.

0 2 4 6 8 10 12 14 16 18 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sec

Nor

mal

ized

Top

Flo

or A

ccel

erat

ion

Tuned

De-tuned

Fig. 4: Forced excitation at 2.2 Hz

The second test was carried out using an impact excitation from an electro-dynamic shaker. The impact was in the form of a half-sinusoid with a total duration of approximately 0.25 sec. The optimum suspended length was adjusted based on the identified fundamental frequency and the results from the optimal length from simulations performed under similar conditions. The results of this test are shown in Fig. 5. Both these results appear to be promising. These results are of preliminary nature, and efforts are currently underway to conduct a comprehensive set of tests to quantify the behaviour of the APMD systems under various de-tuning conditions, excitations and damping. Having said that, these preliminary tests do demonstrate conclusively that the proposed APMD system, both hardware and software designs, is feasible for practical implementation.

CONCLUSIONS

The issue of de-tuning in TMDs is an important problem that needs effective and practical compensation designs. In this paper, an adaptive pendulum tuned mass damper has been proposed that is capable of tuning its properties to compensate for the de-tuning caused due to varying operating conditions, environment or imposed loading. The hardware and software design used to identify and control the APMD system has been presented. Preliminary experimental results are promising and demonstrate that the proposed APMD system is both

2009 Cansmart Workshop 25

feasible and effective for the control of structural vibrations. More comprehensive experimental studies on this device are currently underway at the University of Waterloo.

1 2 3 4 5 6-1

-0 .8

-0 .6

-0 .4

-0 .2

0

0 .2

0 .4

sec

Nom

aliz

ed T

op S

tore

y A

ccee

ratin

De-tunedTuned

Fig. 5: Impact test

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source separation technique using second-order statistics.” IEEE Transactions on signal processing, IEEE,45(2), 434–444.

12. Golub, G. H. and Loan, C. (1989). Matrix Computations. 2009 Cansmart Workshop 26