CIE 616 Fall 2010

Experimental Methods in Structural Engineering Prof. Andrei M Reinhorn

An Introduction to Hybrid Simulation – Displacement-Controlled Methods

Mehdi Ahmadizadeh, PhD Andrei M Reinhorn, PE, PhD Initially Prepared: Spring 2007

Presentation Outline

• Structural Test Methods and Hybrid Simulation • Displacement-Controlled Hybrid Simulation • Development Challenges • Hybrid Simulation System at SEESL • A Typical Hybrid Simulation • Simulation Models 2

Structural Seismic Test Methods

• Shake Table Tests – The most realistic experimentation of structural systems for seismic events.

3

Structural Seismic Test Methods

• Shake Table Tests – Limitations: • Limited capacity of shaking tables • Scaling requirements and resulting unrealistic gravitational loads  It is generally accepted that shake table tests provide an understanding of overall performance of structures subjected to seismic events.

4

Structural Seismic Test Methods

• Quasi-Static Tests – Generally used for evaluation of lateral resistance of structural elements.

5

Structural Seismic Test Methods

• Quasi-Static Tests – Limitations: • Unable to capture rate-dependent properties of structural components • Slow application of loads may result in stress relaxation and creep, even in rate-independent specimens  The results of quasi-static tests generally have limited dynamic interpretation. 6

Structural Seismic Test Methods

• Hybrid Simulation – Pseudo-Dynamic – A parallel numerical and experimental simulation.

Test Structure Numerical Model Experimental Substructure 7

Pseudo-Dynamic Testing (Shing, 2008)

Test Structure Numerical Model Experimental Substructure 8

Pseudo-Dynamic Testing (Shing, 2008)

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Displacement Controlled Hybrid Simulation

• Equation of Motion (SDF):

ma



cv



kd

 

mu g

• Numerical Solution: – A time-stepping method, such as Newmark’s Beta:

a n v n

 1  

mu m



v n

 1  

t



kd n



c v n

  1   

a n

 1  

a n d n



d n

 1  

t v n

 1  

t

2    1 2   

a n

 1  

a n

  – For solution in implicit form, tangential stiffness matrix is needed, or iterations should be used.

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Displacement Controlled Hybrid Simulation

• Equation of Motion (for Hybrid Simulation)

ma



cv



kd

• Numerical Solution: – Newmark’s Beta Method:

mu g a n v n

 1  

mu m



v n

 1  

t



kd n c v n

 1   

a n

 1  

a n



d n



d n

 1  

t v n

 1  

t

2    1 2   

a n

 1  

a n

  – Tangential stiffness matrix or iterations?

11

Displacement Controlled Hybrid Simulation

• Typical Block Diagram (Also Called Pseudo-Dynamic Test) Integrator / Simulation

Analysis

Signal Generation

Commands (Desired Values)

d c

Experiment

D/A PID Controller Hydraulic Supply A/D Specimen Transducers Servo-valve Actuator

d m

,

r m

Measurements (Achieved Values)

12

Pseudo-Dynamic Implementation (Pegon, 2008)

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Structural Seismic Test Methods

• Hybrid Simulation – Advantages: • Lower cost than shake table tests (construction, moving mass) • Less scaling and size requirements • Able to capture rate-dependent properties of experimental substructure • Provides better understanding of component behavior – Limitations • Inertia and rate-dependent terms are artificial • The number and quality of boundary conditions • Unrealistic gravitational loads 14

Development Challenges

• Error Sources – Analytical: • Discretization of structural system in time and space, and simplifications such as lumped-mass models • Errors of the utilized integration methods – Experimental • Measurement contaminations – For example, noise in measurements may lead to excitation of high-frequency modes; if not, it will certainly affect the accuracy • Actuator tracking errors – The most important error source in hybrid simulation – the achieved displacement almost never equals the desired displacement 15

Development Challenges

• Delay in servo-hydraulic actuators

Command Achieved Delay Time

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Development Challenges

• Delay in servo-hydraulic actuators – How delay affects the simulation:

Linear Specimen Without Delay Displacement

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Development Challenges

• Delay in servo-hydraulic actuators – How delay affects the simulation:

Linear Specimen With Delay Displacement

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Development Challenges

• Delay in servo-hydraulic actuators – How to compensate delay: • First, measure the delay amount (in order of a few milliseconds) • Extrapolate displacements: send a command ahead of desired displacement to the actuator • Or modify forces: extrapolate force measurements, or seek the desired displacements in the force and displacement measurements 19

Development Challenges

• In hybrid simulations experimental substructures are involved  Iterations should be avoided, as they may damage the experimental substructures,  A complete tangent stiffness matrix of the experimental substructure may be difficult to establish due to contaminated or insufficient measurements.

 As a result, most integration procedures are either explicit, or use initial stiffness matrix approximations, whose applications are limited.

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Development Challenges

d n c a c n v n c



d m n

 1  

t v c n

 1  1

m

 

mu



v c n

 1  

t c n

 1 2 

kd n m



r n m



c v n c

  1   

a n c

 1  

a n c

Displacement to actuator Estimated Acceleration for Next Computation Estimated Velocity for Next Computation  Apply displacement, measure restoring force, update acceleration and velocity vectors.

 Explicit methods are conditionally stable, and have stringent time step requirements for stiff systems and systems containing high-frequency modes.

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Development Challenges

• Or use initial linear stiffness matrix instead of its tangent stiffness,  Apply explicit displacement:

d n



d n

 1  

t v n

 1  1 2   

n

 1  Measure the restoring force and find velocity and acceleration, while updating displacement and measured force vectors:

d n



d n r n



n



n d n m

  This is only an approximation. The accuracy may not be sufficient for highly nonlinear systems.

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Development Challenges

• Or use an iterative scheme only in numerical substructure, • Or find a way for global iterations without damage to the experimental setup, • Or use “non-physical” iterations on the measurements, • Or develop a fast method for finding tangential stiffness matrix during the simulation.

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UB Real-Time Hybrid Simulation

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UB Real-Time Hybrid Simulation

• Essential Components of Displacement-Controlled Hybrid Simulation Host PC

TCP/IP

Simulator (Running MATLAB Simulink)

SCRAMNet

Controller

SCRAMNet

Transducers STS Controller Actuators Test Substructure 25

UB Real-Time Hybrid Simulation

• Available test setup 26

UB Real-Time Hybrid Simulation

• Test Setup Properties: – Degrees of Freedom: up to 2 – Actuators: ± 3.0 inches, ± 5.0 kips – Experimental stiffness matrix can be altered by using different number of coupons. With two pairs in the first story and one pair in the second story:

K

27.7

  8.5

 8.5

3.9

  kips/in – Experimental mass is very small:

M

50  0 0 lb  – The rate-dependency of specimens is negligible 27

UB Real-Time Hybrid Simulation

• Fundamental periods of 0.4 s and above have been tested to work fine with the available equipment; a fundamental period of 0.6 s and above is recommended to minimize the noise in the measurements.

• If time scaling is acceptable, virtually any natural period can be tested.

• Available procedures allow for linear numerical system and linear transformations only.

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A Typical Hybrid Simulation

• Test Structure: 29

A Typical Hybrid Simulation

• Required information: – Total number of degrees of freedom: 4 – Experimental degrees of freedom: 2 – Numerical stiffness and total mass matrices:

K

     30  12 0 0  12 20  8 0 0  8 12  4 0 0  4     kips/in 4

M

    8.75

  0 0 0 0 6.25

0 0 0 0 3.75

0 0 0 0 1.25

     kips/

g

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A Typical Hybrid Simulation

• Required information: – Inherent damping ratio: 5% – Numerical damping matrix (in addition to the inherent damping):

C

     0 0 0 0 0 0 0 0 0 0  0  0   0 0 0 0  – Influence vector:

Mι

    8.75

6.25

   3.75

   1.25

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A Typical Hybrid Simulation

• Required information: – Transformation matrix for displacement (from global to actuator local coordinate system):

T

 1 1   1 0 0 1 0 0   – Displacement factor in actuator coordinate system: 1 – Measured force factor: 1 – Ground motion: 1940 El Centro, 200% 32

A Typical Hybrid Simulation

• Additional requirements for model-based integration: x 2 r 2 s 2

K

l E

 

k k

11 21

k

12

k

22  

P

s

1  0 0

s

2   x 1 r 1 s 1 – Total number of essential stiffness parameters: 2 – Transformation matrix to parameter coordinate system:

T

p

1/

l

1   1/ 0 1/

l

2   33

Detailed Description of Simulation Models

• Simulation and control models are prepared in MATLAB Simulink environment on Host PC.

• The models are then ‘downloaded’ to real time computers running MATLAB xPC kernel.

• After simulation, the results are ‘uploaded’ to Host PC for observation and analysis.

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Simulink Diagrams

35

Simulink Diagrams

36

Simulink Diagrams

37

Input file for Matlab: .m file

% ***General Information**** NDOF=4; NACT=2; % number of degrees of freedom % number of actuators involved in the simulation NPAR=2; % ***NUMERICAL MODEL**** k1 = 5.543*2; k2 = 3.89; % number of important parameters for formation of stiffness matrix % DOF 1 STORY 1 (two pairs of coupons) % DOF 2 STORY 2 l1=43; l2=46; l=l1+l2; % ***NUMERICAL MODEL DATA*** MT = [7 0 0 0; 0 5 0 0; 0 0 3 0; 0 0 0 1]*1.25/g; % Total mass matrix ME=[0 0 0 0; 0 0.05 0 0; 0 0 0.025 0; 0 0 0 0]/g; % Experimental Mass Matrix K = [30 -12 0 0; -12 20 -8 0; 0 -8 12 -4; 0 0 -4 4]; % Global analytical stiffness KEP = [k1*l1^2 0; 0 k2*l2^2]; C=zeros(NDOF,NDOF); % Parameteric experimental stiffness in intrinsic coord. system % Analytical damping matrix dr=0.05; L=-MT*ones(NDOF,1); % Damping ratio forstifness proportional damping % Influence vector for base motion % COORDINATE SYSTEM TRANSFORMATIONS ***** TDGA=[-1 1 0 0; -1 0 1 0]; % Displacement from global to actuator cs **** FDGA=1; FFAG=1; % Displacement scale factor from global to actuator coordinates % Force scale factor from actuator to global coordinates TDAP=[1/l1 0; -l/l1/l2 1/l2]; % Actuator displacements to parameter cs *** % Simulated experimental model properties Parameters.K1 = k1; % one column Parameters.K2 = k2; % one column Parameters.Uy = 0.20; Parameters.Ep = 0.00; Parameters.Ga = 0.45; Parameters.Be = 0.55; Parameters.N = 1.5; massA=0.025; eyd=[Parameters.Uy; Parameters.Uy*3]; % Actuator weight (kips) % experimental substructure yield displacement 38

Sequence of Operations

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