weight reduction, vibration suppression and … · • optimum design for harmonic base excitation...

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WEIGHT REDUCTION, VIBRATION SUPPRESSION

AND ENERGY HARVESTING FOR TUNED MASS-

DAMPER – INERTER (TMDI) EQUIPPED SUPPORT-

EXCITED STRUCTURES

Laurentiu Marian* PhD Candidate

School of Engineering and Mathematical Sciences

e-mail: [email protected]

Dr. Agathoklis Giaralis* Senior Lecturer in Structural Engineering



______________________________________________________________

______________________________________________________________

________________________________________

______________________________________________________________

*Structural Dynamics Research Group

Department of Civil Engineering

City University London

Introduction

• Passive vibration control of civil structures - Tuned Mass Damper (TMD).

• Tuned Mass Dampers Inerter (TMDI) passive control solution.

• Mass amplification devices – Inerter.

Proposed TMDI for single-degree-of freedom primary structures

• Equations of motion.

• Optimum design for harmonic base excitation and white noise base excitation.

• TMDI weight reduction effect

Proposed TMDI for multi-degree-of freedom primary structures

• Equations of motion.

• Optimum design for stochastic seismic excitation.

• TMDI performance assessment. TMDI as a lighter passive control solution.

Energy harvesting enabled TMDI

• Model description and characterization.

• Quantification of energy harvesting potential

Concluding remarks

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WEIGHT REDUCTION, VIBRATION SUPPRESSION AND ENERGY HARVESTING FOR TUNED MASS-DAMPER – INERTER (TMDI) EQUIPPED SUPPORT-EXCITED STRUCTURES 2/38

PRESENTATION OUTLINE

Seismic Design applications for buildings • Conventional design philosophy: allows controlled inelastic deformations (a)

• Passive Vibration Control of Civil Structures: the incorporation of various

devices to passively control the vibratory motion of structures.

______________________________________________________________________

• Passive response control systems: (b) seismic isolation, (c) energy

dissipation devices, (d) Tuned Mass Dampers (Simple design; Linearity)

(b) (c) (d) (a)

INTRODUCTION

WEIGHT REDUCTION, VIBRATION SUPPRESSION AND ENERGY HARVESTING FOR TUNED MASS-DAMPER – INERTER (TMDI) EQUIPPED SUPPORT-EXCITED STRUCTURES 3/38

______________________________________________________________________

Tuned Mass Damper (TMD) for Passive Vibration Control

− The oldest device historically used for passive vibration control (e.g. Frahm, 1911)

− Comprises a mass (mTMD) attached to the primary structure via a (linear)

spring (kTMD) and a dashpot (cTMD).

4/38

INTRODUCTION

SDOF

MDOF

WEIGHT REDUCTION, VIBRATION SUPPRESSION AND ENERGY HARVESTING FOR TUNED MASS-DAMPER – INERTER (TMDI) EQUIPPED SUPPORT-EXCITED STRUCTURES

______________________________________________________________________


– The larger the attached mass considered, the more effective an optimally

designed TMD becomes to suppress excessive primary structure vibrations.

Wind and traffic-induced vibration : mTMD = 1% - 10% of the total

mass of the structure

5/38

– Examples from TMD applications in civil engineering structures:

Earthquake-induced vibrations: mTMD = 25% - 100%

INTRODUCTION


______________________________________________________________________

(Ni/Zuo/Kareem 2011)

(Zuo/Tang 2013)

INTRODUCTION


6/38 WEIGHT REDUCTION, VIBRATION SUPPRESSION AND ENERGY HARVESTING FOR TUNED MASS-DAMPER – INERTER (TMDI) EQUIPPED SUPPORT-EXCITED STRUCTURES

______________________________________________________________________

Tension leg platforms (TLPs)

(Taflanidis/Angelides/Scruggs 2009)

WEIGHT REDUCTION, VIBRATION SUPPRESSION, AND ENERGY HARVESTING FOR TUNED MASS-DAMPER –INERTER (TMDI) EQUIPPED SUPPORT-EXCITED STRUCTURES

INTRODUCTION


7/38

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Off-shore wind turbines Suspension and cable-stayed (foot-) bridges

WEIGHT REDUCTION, VIBRATION SUPPRESSION, AND ENERGY HARVESTING FOR TUNED MASS-DAMPER –INERTER (TMDI) EQUIPPED SUPPORT-EXCITED STRUCTURES

INTRODUCTION


8/38

______________________________________________________________________

- Recently proposed passive vibration control configuration - considers the

classical tuned mass-damper (TMD) in conjunction with a two terminal flywheel

device – inerter for vibration suppression of SDOF and MDOF system.

- The TMDI – a generalization of classical TMD systems and thus, optimum

design (“tuning”) of the classical TMD are readily applicable to achieve

“optimal” performance for the new TMDI configuration.

- Taking advantage of the “mass amplification effect” of the inerter, the TMDI

improves the effectiveness of the classical TMD

9/38

The Tuned Mass Damper Inerter (TMDI) (Marian & Giaralis, 2013, 2014)

INTRODUCTION


______________________________________________________________________

- The Inerter has two terminals free to move independently and develops an

internal (resisting) force proportional to the relative acceleration of its

terminals. (M.C. Smith, 2002)

1 2( - )F b u u

1 1 1 1 11 1)( gc x k x mb axm

- Single-degree of freedom oscillator connected to the ground via an inerter:

- The inerter increases the m1 mass:

m1 m1+b

“mass amplification effect”

- Constant of proportionality b fully characterises its linear dynamical behaviour.

10/38

The concept of the Inerter and its mass amplification effect

____________

Note:

1 2( - )F k u u

1 2( - )F c u u

INTRODUCTION


______________________________________________________________________

1 2( - ),F b u u

2 2

2 21

( )n

f if

if i

rb m

p pr

- Assume a mechanical realisation of the inerter comprising a plunger that

drives a rotating flywheel through a rack, pinion and gearing system:

- The constant of proportionality b (inertance) - mass units (>> physical mass

of device)

- Inerter can have an inertance ("apparent" mass) much grater then its

physical mass .

11/38

Inerter – two terminal flywheel-based device (M.C. Smith, 2002)

INTRODUCTION


______________________________________________________________________

Equations of motion

1 1 1 1 1 1 1

0

0

TMDI TMDI TMDI TMDI TMDI TMDI TMDI TMDI TMDI

g

TMDI TMD TMDI TMDI

m b x c c x k k x ma

m x c c c x k k k x m

Time domain:

Frequency domain:

2 2

21

1 1 22 2 2 2

2 2

1 1

1 2 1( )

1 2 2

TMDI TMDI TMDI

g

TMDI TMDI TMDI TMDI TMDI TMDI

ixG

ai i

1

TMDIm

m

TMDI

TMDI

TMDI

k

m b

2( )

TMDI

TMDI

TMDI TMDI

c

m b

1

TMDI

TMDI

1

b

m

12/38

TMDI FOR SDOF BASE EXCITATION


______________________________________________________________________

Optimum design

- Design problem: given an undamped primary system with specific dynamic

characteristics (m1,k1,c1=0), determine the TMDI parameters (kTMDI, cTMDI)

which minimise the response of the primary system, given mTMDI and b.

- Solution - ‘Equal points’ design method: there exist two fixed points where

the FRF curves intersect (noted P1 and P2), independent of cTMDI (ζTMDI).

- Optimum response is obtained if and only if there exists two local maxima and

both have the same amplitude (Den Hartog,1956)

-mass ratio μ=0.1,

-inertance ratio β=0.1,

-frequency ratio υTMDI=0.5.

13/38

TMDI FOR SDOF HARMONIC

BASE EXCITATION


______________________________________________________________________

− Minimum response <=> (if and only if) |G1(ω)| has two local maxima with

equal amplitudes at the stationary points P1 and P2.

− Enforce:

Optimum design

2 2

1 10

lim ( ) lim ( )TMDI TMDI

G G

1) |G1(ω)| is independent of ζTMDI :

2) Equal amplitude at points P1

and P2 for the limit ζTMDI → ∞:

1 1 1 2lim ( ) lim ( )TMDI TMDI

P PG G

1 (1 )(2 )

1 2(1 )TMDI

- Optimum frequency ratio υTMDI :

1 2

1 1( ) ( )0

P P

G G

3) |G1(ω)| is maximized locally

at points P1 and P2: 2 26 (1 ) (1 )(6 7 )

8(1 )(1 )[2 (1 )]TMDI

- Optimum TMDI damping ratio ζTMDI :

14/38


BASE EXCITATION


______________________________________________________________________

Optimum design parameters

- setting b=β=0, the optimal tuning formulae for the classical TMD reported in

the literature are retrieved (TMDI is a generalisation of classical TMD!)

- υTMDI decreases as β increases for all

values of μ considered.

- υTMDI decreases as mTMDI mass

increases

______________________________

- ζTMDI increases as β increases for all μ

values considered

- ζTMDI increases as mTMDI mass

increases

Optimum frequency ratio υTMDI Optimum TMDI damping ratio ζTMDI

15/38


BASE EXCITATION


______________________________________________________________________

Optimally designed TMDI equipped undamped SDOF primary

structure

For the same mTMDI mass, the inclusion of the inerter (TMDI vs. TMD):

- reduces the peak response of the primary structure (at ω1, ωP1 and ωP2),

- All FRF curves attain two local maxima of equal height at the frequencies

ωP1 and ωP2 whose location depend on the ratio β.

- increases robustness to “detuning effects” - Large β values - wider range of

frequencies peak response reduction compared to the classical TMD.

16/38

- Practically, the inerter furnishes all the positive effects of increasing the

attached mass without the negative effect of the added weight.


BASE EXCITATION


P1 P2

______________________________________________________________________


structure

- TMDI reduction saturates as the ratio β increases

- Peak response amplitude for optimally designed TMDI normalized by the

peak response amplitude of the optimally designed classical TMD (β =b=0).

- Incorporation of the inerter to the classical TMD system is more effective for

vibration suppression for smaller attached masses mTMDI.

17/38


BASE EXCITATION


______________________________________________________________________

Optimally designed TMDI as a lighter passive control solution

- Required additional oscillating mTMDI mass values for achieving prescribed

levels of structural response in the H∞ sense:

- |G1(ω)| = 2.9 if:

- The TMDI configuration represents a much lighter passive control solution

compared to the case of classical TMD.

OR 2)TMDI solution - mass ratio μ=0.34

(for β=0.1)

1) TMD solution: mass ratio μ=0.63

Example:

18/38


BASE EXCITATION


______________________________________________________________________

Optimum design

22

1 1( ) ( )G S d

• Design problem: given an undamped primary system with specific

dynamic characteristics (m1,k1,c1=0), determine the TMDI parameters

(kTMDI, cTMDI) which minimise variance of the relative displacement x1 :

S(ω)=S0 (ideal white noise)

19/38

TMDI FOR SDOF WHITE NOISE

BASE EXCITATION

2 2

1 10 and 0TMD

[ ( 1) (2 )(1 )]1

1 2(1 )TMDI

( ) (3 ) (4 )(1 )

2 2(1 )[ (1 ) (2 )(1 )]TMDI

Impose:

- Optimum frequency ratio υTMDI :

- Optimum TMDI damping ratio ζTMDI :


______________________________________________________________________

Optimum design parameters

20/38

Optimum frequency ratio υTMDI Optimum TMDI damping ratio ζTMDI

- setting b=β=0, the optimal tuning formulae for the classical TMD reported in

the literature are retrieved (TMDI is a generalisation of classical TMD!)

- υTMDI decreases as β increases for all

values of μ considered.

- υTMDI decreases as mTMDI mass

increases

- ζTMDI increases as β increases for all μ

values considered

- ζTMDI increases as mTMDI mass

increases

______________________________


BASE EXCITATION


______________________________________________________________________

• Displacement response variance for white noise base excitation for the

proposed TMDI configuration (β>0) and classical TMD (β=0)

21/38


structure

- TMDI reduction saturates as the ratio β increases




BASE EXCITATION


______________________________________________________________________

Optimally designed TMDI as a lighter passive control solution


levels of structural response in the H2 sense:

- The TMDI configuration represents a much lighter passive control solution

compared to the case of classical TMD.

22/38


BASE EXCITATION


______________________________________________________________________

The Tuned Mass Damper Inerter (TMDI) for MDOF

23/38

- Consider the TMDI as an inter-story connective device placed in a ‘diagonal’

configuration.

- The primary structure is assumed to behave linearly in alignment with

current trends in performance based requirements for minimally damaged

structures protected by passive control devices.

TMDI FOR MDOF PRIMARY STRUCTURES


______________________________________________________________________

Equations of motion

( )gM X C X K X M a t

- Displacement vector

- Mass matrix - Damping matrix - Stiffness matrix

1 2( ) ( ) ( ) ( )T

TMD nX x t x t x t x t

1

2

3

0 0 0

0 0 0

0 0

0 0 0

0

0 0

TMDI

n

m b b

m

b m bM

m

m

1 1

1 1 2 2

2

1

1 1

0 0

0

0

0 0 0

0 0

TMDI TMDI

TMDI TMDI

n

n n n

c c

c c c c

c c c cC

c

c

c c c

1 1

1 1 2 2

2

1

1 1

0 0

0

0

0 0 0

0 0

TMDI TMDI

TMDI TMDI

n

n n n

k k

k k k k

k k k kK

k

k

k k k

24/38

Time domain:

Frequency domain:

1

2( 1)

( )( ) ( )

( )O n

Y sG s C sI A B

A s

1 1

1 1,

n nO IA

M K M C

1 1

1 1

n n

n n

O IB

I I



______________________________________________________________________

Optimum design for stochastic seismic excitation.

• Aim: Given a TMDI equipped primary system with specific dynamic

characteristics, determine the TMDI parameters (CTMDI , KTMDI) which

minimise the mean square displacements of the top floor.

|G1(ω)|2 - squared modulus of the frequency response function

S(ω,tmax) – evolutionary power spectral density function of the seismic

excitation modelled by a non stationary stochastic process .

• The following performance index is considered:

0/ ,TMDIPI J J Where: 2

1 max0

( ) ( , ) ,TMDIJ G S t d

J0 - the variance of the top floor displacement for the uncontrolled primary

(linear) structure

25/38

tmax - the instant in time at which the proposed non-stationary power spectrum

is maximized



______________________________________________________________________


Optimisation method:

• MATLAB® built-in “min-max” constraint optimization algorithm employing a

sequential programming method (Salvi & Rizzi , 2011).

Initial estimates:

• Optimum TMDI parameters for an undamped linear primary structure under

white noise base excitation (Marian & Giaralis, 2014).

Constraints:

• Appropriate constraints are imposed to the sought design parameters

relying on physical considerations.

0.5 1.10 0 1.00TMDI TMDIand

26/38



______________________________________________________________________


Structure Story Mass (kg) Stiffness

(N/m)

3DOF

1 (top) 30 x 10^3 10 x 10^5

2 30 x 10^3 30 x 10^5

3 30 x 10^3 30 x 10^5

Structure Mode Period

(s) Frequency

(rad/s)

3DOF

1st 0.50 12.56

2nd 0.23 27.79

3rd 0.12 52.31

Modeling of the seismic excitation (C-P EPS compatible with EC8 spectrum)

Characteristics of primary

structures considered

2 4

2

2 22 2 2 2

2 2

1 4

( , ) exp( )2

1 4 1 4

g

g f

g g

g g f f

btS t C t

____

____

____

____

____

____

____

____

_

C

(cm/sec2.5)

b

(1/sec) ζg

ωg

(rad/sec) ζf

ωf

(rad/sec)

17.76 0.58 0.78 10.73 0.90 2.33

27/38

Parameters for the definition of C-P evolutionary power

spectrum compatible with EC8 spectra

(Giaralis A, Spanos PD,2012)



______________________________________________________________________

Optimum TMDI parameters for stochastic seismic excitation.

- 3DOF primary structure

28/38


Optimum stiffness kTMDI Optimum TMDI damping cTMDI _

__

__

__

__

__

__

__

__

__

__

__

__

__

__

_

- kTMDI increases as β increases

- kTMDI decreases as mTMDI mass

increases

- cTMDI increases as β increases

- cTMDI increases as mTMDI mass

increases


______________________________________________________________________

TMDI performance assessment for stochastic seismic excitation.

29/38


- The proposed TMDI configuration reduces the value of the Performance

Index as the value of b increases




TMDI as a lighter passive control solution

______________________________________________________________________

30/38



levels of structural response (Performance Index) for different b values.

Example:

- PI = 0.136 if:

1) TMD solution:

additional mTMDI =31000Kg

OR 2) TMDI solution

additional mTMDI =6000Kg (for b=48000Kg)


Performance assessment for field recorded EC8 compatible

accelerograms

31/38



______________________________________________________________________

Ground acceleration no.

#1 #2 #3 #4 #5 #6 #7 Average

3 DOF structure

(T1=0.5s)

Primary structure

alone

14.2

9

10.3

6

11.2

8

10.0

2

11.5

4

12.4

3 14.45 12.05

TMD

(m=9000 Kg,

b=0 Kg)

5.58 5.90 6.79 6.93 5.30 6.46 5.54 6.07

(50.4%)

TMDI

(m=9000 Kg,

b=68000 Kg)

3.98 3.73 4.86 4.09 4.15 4.28 5.05 4.31

(35.7%)

Maximum top floor displacements (cm)

______________________________________________________________________

- The potential of energy harvesting from structures equipped with TMDs has

been recently recognized in the literature and attracted the interest of various

researchers (eg. Buelga et al, 2014).

A typical energy harvesting enabled TMD configuration

- Objective: channel part of the kinetic energy of the attached mass to a

device which can transform part of the kinetic into electric energy

- Functionality: the harvester device “resists” the relative motion (x1-xTMD) by

developing an additional electromechanical “damping” force FEM.

32/38

ENERGY HARVESTING ENABLED TMDI


______________________________________________________________________

Model description and characterization

- Electromagnetic device connected in parallel to the TMDI spring and damper

2

( )EM

C L

Jc

R R

- Energy harvesting enabled TMDI total damping : ,TMDI EM Mc c c

1( )TMDIV J x x - Moving magnet travels within a magnetic

field of constant flux density J generating voltage:

- Force “transmitted” to the “mechanical domain”: 1( )EM EM TMDIF c x x

RL - resistive load

RC - internal “parasitic” resistance

J - magnetic field flux density

33/38

- cEM - electromechanical damping coefficient:



______________________________________________________________________

Quantification of energy harvesting potential

- Energy harvesting enabled TMDI - optimally designed for vibration suppression!

The power P that can be harvested

2

2

2( )

( )RV L

C L

JP G R

R R

|GRV(ω)| - relative velocity between m1 and

mTMDI mass

______________________

______________________________________________________________________

- |GRV(ω)| decreases as β increases (for a fixed μ) - enhanced vibration suppression

- |GRV(ω)| reduction is not beneficial in terms of energy harvesting!

34/38



______________________________________________________________________

Energy harvesting capabilities Vibration suppression capabilities

__________________

______________________________________________________________________

- The increase of the ratio β has a negative effect in terms of the available

energy for harvesting at resonant frequency.

- The peak values of the magnitude of FRFs saturates for β>0.5, but the

range of frequencies that the FRFs take on non-negligible values increases

- It confirms that: “an optimal absorber is not an optimal harvester”.

- However, the TMDI is not confined by apriori fixed inertial properties of the

TMD.

35/38




SIMULTANEOUS ENERGY HARVESTING

AND VIBRATION SUPPRESSION

36/38

_____________________________________________________________________

- optimally designed TMDI for vibration suppression with mass ratio μ= 0.1 and

inertance ratio 𝛽=0.6


- by keeping constant the weight of the TMDI, changes in the inertance b allows for

controlling the trade-off between energy harvesting and vibration suppression


______________________________________________________________________

Bobby/Spence/Kareem 2014

37/38

SIMULTANEOUS ENERGY HARVESTING

AND VIBRATION SUPPRESSION


______________________________________________________________________

• The TMDI system, exploits the mass amplification effect of the inerter to:

- improve the classical TMD performance for a fixed oscillating

additional mass, or to

- “replace” part of the TMD vibrating mass by achieving an overall

lighter passive control solution

• The energy harvesting capabilities of TMDI depends on the inerter constant

b in a optimally designed TMDI for vibration suppression, which can vary by

means of a gearbox for the case of a flywheel based inerter

38/38

CONCLUDING REMARKS


______________________________________________________________________

39/38

• H. Frahm, “Device for Damping Vibrations of Bodies,” U.S. Patent 989958, 1911.

• J. Ormondroyd, J.P. Den Hartog, “The theory of the dynamic vibration absorber,” J. Appl. Mech., vol.

50, pp. 9–22, 1928.

• J.P. Den Hartog, Mechanical Vibrations, 4th ed., New York: McGraw-Hill, 1956.

• S. Krenk, “Frequency analysis of the tuned mass damper,” J. Appl. Mech. ASME pp. 936–942, 2005.

• A. Gonzalez-Buelga, L. R. Clare, A. Cammarano, S. A. Neild, S. G. Burrow and D. J. Inman, “An

optimised tuned mass damper/harvester device, Struct. Control Health Monit.,2014, DOI:

10.1002/stc.1639.

• X. Tang and L. Zuo, “Simultaneous energy harvesting and vibration control of structures with tuned

mass dampers,” J. Intelligent Material Sys. and Struct., vol. 23(18), pp. 2117–2127, 2012.

• S. Adhikari, F. Ali., “Energy Harvesting Dynamic Vibration Absorbers,” J. App.Mech., vol. 80, pp. 1-9,

2013.

• L. Marian, A. Giaralis, “Optimal design of inerter devices combined with TMDs for vibration control of

buildings exposed to stochastic seismic excitations,” In: Proceedings of the 11th ICOSSAR

International Conference on Structural Safety and Reliability for Integrating Structural Analysis, Risk

and Reliability 2013; New York, US) (eds: Deodatis G, Ellingwood BR and Frangopol DM), CRC Press.

• L. Marian, A. Giaralis, “Optimal design of a novel tuned mass-damper-inerter (TMDI) passive vibration

control configuration for stochastically support-excited structural systems,” Probab. Eng. Mech.

DOI:/10.1016/j.probengmech.2014.03.007.

• M.C. Smith, “Synthesis of mechanical networks: The Inerter,” IEEE Trans. Autom. Contr., vol. 47-10,

pp. 1648-1662, 2002.

SELECTED REFERENCES


THANK YOU!

______________________________________________________________________

Laurentiu Marian PhD Candidate School of Engineering and Mathematical Sciences


Dr. Agathoklis Giaralis Senior Lecturer in Structural Engineering



weight reduction, vibration suppression and … · • optimum design for harmonic base excitation...

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