Excerpt

## Contents

Introduction

Introduction

Motivation

Goals

1 System Description

1.1 Hardware

1.2 Software

2 Modeling

2.1 Modeling of the Shake Table

2.1.1 Continuous Transfer Function

2.1.2 Continuous State Space Representaion

2.1.3 Discrete Transfer Function

2.1.4 Discrete State Space Representaion

2.2 Modeling of a Frame

2.2.1 Time Continuous Transfer Function

2.2.2 Time Continuous State Space Representation

2.2.3 Time Discrete Transfer Function

2.2.4 Time Discrete State Space Representation

3 Control

3.1 Control of the Shake Table

3.1.1 State Controller

3.1.2 State Controller with preset Integrator

3.2 Shake Table Observer

3.2.1 Complete Observer

3.2.2 Reduced Observer

3.3 Control of the Shake Table with a Frame

3.3.1 State Controller

3.3.2 State Controller with preset Integrator

4 Fault Detection

4.1 Tolerance Band Method

4.1.1 Constant Band Method

4.1.2 Proportional Band Method

4.2 Residual Method

4.3 Anti Jitter Automatism

4.4 Application in the real System

4.5 Further Ideas of Fault Detection Methods

4.5.1 Finite State Machine analyzing Reference Velocity

4.5.2 Integration of the Table Acceleration

Conclusion

References

A Simulink Models

A.1 State controlled Shake Table

A.2 State controlled Shake Table with preset Integrator

A.3 Observers

A.4 Controlled Shake Table with Frame

A.5 Fault Detection

B Matlab Script Files

## List of Figures

1 The shake table(1) with the mountable frame(2), the control unit(3) and the power supply unit(4)

2 Shake table: table plate(1), lead screw with ball nut(2), ball bearing(3), limit-/center sensors(4), shaft(5), motor with encoder(6), accelerome- ter(7)

3 Frame with sheet metal stays(1), cover plate(2), bottom plate(3) and accelerometer(4)

4 PCI Data aquisition board(1) with terminal board(2)

5 WinCon user interface with system initializing button, graph launching button,

6 WinCon ControlPanel user interface for online modification of control parameters

7 Matlab/Simulink model for the shake table identification experiment with voltage step excitation, output to the motor amplifier, input from the motor encoder and Safety & Enable circuit with the in- and outputs of the Data Aquisition bord

8 Stepresponse of the uncontrolled shake table with integration gain KI and delay τ

9 Continuous state space representation of the shake table

10 Discrete state space representation of the shake table

11 Simplification of the Frame Structure with Point Mass, Damper and Sheet Stay

12 Matlab/Simulink model for the decay experiment with the analog ac- celerometer input and the necessary enable circuit

13 Acceleration of the frame cover plate in the decay process

14 Shake Table with state feedbacks k1 and k2

15 Stepresponse with overshoot Mp and settling time T95%

16 Relation between pole locations of a) s-plane and b) z-plane

17 Matlab/Simulink model of the real shake table with state feedbacks k1 and k2

18 Control behaviour of the model and state controlled shake table with feedback kexcitated by a rectangular reference signal

19 Shake Table with state feedbacks ki1, ki2 and preset integral controller with integral gain KI

20 Matlab/Simulink model of the real shake table with state feedbacks k1, k2 and KI

21 Control behaviour of the table and the table model with feedback kI and preset integral controller excitated by rectangular reference signal . .

22 Shake table with state controller and complete observer (Observer matrix H)

23 Matlab/Simulink model of the Shake Table with state controller and a parallel conected complete observer

24 States of the shake table and the complete observer at rectangular exci- tation signal for the controlled Shake Table

25 Matlab/Simulink model of the Shake Table with state controller and feedback of the states estimated by the complete observer

26 Shake table with state controller and reduced observer (observer param- eter hr )

27 Matlab/Simulink model of the Shake Table with state controller and a parallel conected reduced observer

28 Derivated platform position and estimated velocity of the reduced ob- server excitated by a rectangular reference signal for the Shake Table . .

29 Matlab/Simulink model of the Shake Table with state controller and feedback of the table velocity estimated by the reduced observer

30 Matlab/Simulink model for positioning the controlled real shake table without consideration of the mounted frame

31 Floor acceleration while moving table without considering the mounted frame

32 Dividing the 5th order system into two 2nd order systems and one 1st order system

33 Control structure with the state feedback ktf of the Shake table with frame

34 Simulation results of the shake table with Frame and state control . . .

35 Control structure of the shake table with frame with the state feedback ktfi and preset integral controller

36 Dividing the 6th order system into three 2nd order systems

37 Simulation results of the control behaviour and manipulating variable of the state controlled shake table with frame and preset integrator

38 Control behaviour and manipulating variable of the state controlled shake table with frame and preset integrator

39 Model signal with constant tolerance band of half bandwidth Bk

40 Matlab/Simulink realization of the constant band method with half band- width Bk

41 Matlab/Simulink model for the functionality test of the constant band method with an arbitrary signal and an additive fault signal

42 Verification of the functionality of the CB-method with additive faults affecting the sensor signal

43 Model signal with proportional tolerance band of half bandwidth Bp . .

44 Matlab/Simulink realization of the proportional band method with max- imum percental deviation p

45 Matlab/Simulink model for the functionality test of the proportional band method with an arbitrary signal and an multiplicative fault sig- nal

46 Verification of the functionality of the PB-method with multiplicative faults affecting the sensor signal

47 Blockdiagram of the residual generation with reference position xr (k), measured table position θ(k) with fault, residual e(k) and sensor state Tr

48 Blockdiagram of the functionality test of the residual method with the model of the controlled shake table , the residual generator and the ad- ditive fault

49 Simulation results of the residual method with rectangular reference sig- nal, model output with fault fΘ, residual e(k) and sensor state Tr

50 Blockdiagram of the Anti Jitter Automatism with inputs T, clock and output Switch

51 Simulink/Stateflow-Realization of the Anti Jitter automatism with states, transition conditions, input T and output Switch

52 Matlab/Simulink model for the functionality test of the Anti Jitter Au- tomatism with a binary testsignal and clock

53 State machine behaviour with input T and output Switch

54 Matlab/Simulink model of the shake table model with three fault detec- tion methods and Anti Jitter Automatism

55 Matlab/Simulink model of the real shake table with three fault detection methods and Anti Jitter Automatism

56 Table position Θ, fault fθ and residual e(k) of the real system

57 Matlab/Simulink realization of a further Fault Detection Method with a shake table model parallel conected to the real shake table, a finite state machine analyzing the reference velocity and the output of the constant band method T

58 Matlab/Simulink realization of a further Fault Detection Method with the real shake table, the table platform accelerometer and the constant band method

59 Shake table with state control and the subsystems Shake Table, State Controller and Enable & Safety

60 Subsystem Shake Table

61 Subsystem State Controller without preset integrator

62 Subsystem Enable & Safety

63 State controlled Shake table with preset integrator and the subsystems Shake Table, State Controller with preset integrator and Enable & Safety

64 Subsystem State Controller with preset Integrator

65 State controlled Shake Table with observers and the subsystems Shake Table, complete and reduced Observer, State Controller and Enable & Safety

66 Subsystem Complete Observer

67 Subsystem Reduced Observer

68 Controlled Shake Table with mounted Frame and the subsystems Con- trolled Shake Table, Frame Model, Outer Loop Control and Enable Safety

69 Subsystem Frame Model

70 Subsystem Outer Loop Control

71 State controlled Shake Table with Fault Detection and the subsystems Shake Table, Model of Shake Table with State Control, Constant Band-, Proportional Band- and Residual Fault Detection Method, Anti Jitter Automatism, Fault and Enable & Safety

72 Subsystem Model of Shake Table with State Control

73 Subsystem Constant Band Fault Detection

74 Subsystem Proportional Band Fault Detection

75 Subsystem Residual Fault Detection

76 Subsystem Anti Jitter Automatism

77 Subsystem Fault

## Intoduction

### Introduction

In this student research project a control and fault detection is designed for a shake table with a mounted structure. The focus is on the modeling and analysis of the shake table and the structure as well as on the the controller design and its technical implementation. Moreover, an approach was made to build an fault detection automatism which can detect sensor faults of the shake table. The goal of the control was to move the table with and without mounted structures according to desired reference positions. Equipped with this feature the table can be used to simulate for example real earthquakes in small scale.

In section one the hardware of the shake table and the involved equipment is described. The modeling of the shake table and buiding frame is done in chapter two. Chapter three is dedicated to the controller design of the shake table. The fault detection is subject of chapter four.

This student research project was realized during an Erasmus program inter- change at the Department of System- and Automation Engineering (DISA) which belongs to the Polytecnical University of Valencia/ Spain.

I want to express my gratitude to Professor Dr. Rolf Isermann and Profes- sor Dr. Pedro Albertos Pérez who made this project possible. Especially, I give my thanks to Professor Dr. Ángel Valera Fernández who supported me during the entire project with his experience, knowledge and sympathic per- sonality. I don´t want to miss to thank my colleages Alessandro and Miguel, who spent a plenty of humourus time together with me in the laboratory.

### Motivation

Today´s technical standard allows a high level of automation and offers po- tential for applications which were not realizable some decades ago due to missing technicals tools . Especially in the digital technology this develop- ment is very distincted. Due to powerful computer arquitectures it is possible to process programs in real time. This is very convenient for process conrol. Complex control algorithms can be developed, simulated on computers and afterwards implemented on processors or controllers. In order to design such a control, the knowledge of the process to be controlled is essential. By mod- eling, these processes are brought into a mathematical representation form. Based on these models, there can be designed digital control system. The increasing degree of automation brings with itself the necessity to supervise the fault behaviour and breakdowns of the components. This is necessary in order to keep away damage from material and environment of the auto- mated device. Modern fault detection and fault diagnosis methods allow to make automated system safer. In this student research project some of the previous mentioned aspects shall be realized in small scale with a shake table in order to get experience in the topics modeling, controller design and fault detection.

### Goals

The concrete goals of this student research project were to

- Obtain models of the shake table and the structure. These models should represent the systems in transfer function form and in state space form. Moreover the models have to be time continuous and time discrete.

- Analyze the shake table and the structure in order to determine the parameters of the before developed models.

- Implement the obtained models with the Matlab/Simulink Tool.

- Design different digital state controllers with Matlab. The control must enable the shake table to follow desired reference positions. If the structure is mounted onto the shake table, then the table and the structure shall be able to follow a desired reference position while the structure oscillations are to be kept minimal.

- Design observers in order to estimate the non measurable shake table velocity. The observer has to be implemented in Matlab/Simulink and has to be tested regarding the convergence of the observer.

- Implement the obtained controllers in Matlab/Simulink and simulate the closed loop behaviour. The controllers and observers are to be implemented in a real time executable program by the Matlab/RealTime Workshop Tool. The controllers togehter with the observers have to be tested in the real system.

- Develop different fault detection methods in order to detect faults affecting the position sensor.

## 1 System Description

### 1.1 Hardware

The total system consitst of a shake table with a mountable frame, the control unit and the power supply unit. The next picture (1) shows the total device.

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Figure 1: The shake table(1) with the mountable frame(2), the control unit(3) and the power supply unit(4)

The shake table contains a 12V DC servo motor, which drives a spherically seated lead screw. This lead screw moves a ball nut, that is coupled rigidly with the table platform. The table platform istmounted on four ball bearings and slides on 2 hardened steel shafts. The motor voltage is provided by an amplifier, which is integrated in a Universal Power Module. Moreover, an independent power supply is integrated. It supplies the sensors with energy. These sensors are two acceleration sensors, one encoder and three light barriers. One accelerometer is fixed on the table platform and the other one is mounted on the frame. The range of these accelerometers is ±5 g. The encoder is coupled to the motor shaft and its resolution corresponds to a horizontal platform movement of 3 µm. The three light barriers serve as limit switches and as an indicator of the center position. They are fixed on the base platform. The following picture (2) shows the shake table with the sensors.

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Figure 2: Shake table: table plate(1), lead screw with ball nut(2), ball bearing(3), limit/center sensors(4), shaft(5), motor with encoder(6), accelerometer(7)

The frame onsists of two vertical sheet metal stays which are connected by two horizontal arranged plastic plates at top and bottom. At the top plate there is fixed an accelerometer. The folowing picture (3) shows the frame.

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Figure 3: Frame with sheet metal stays(1), cover plate(2), bottom plate(3) and accelerometer(4)

For the control of the table, a computer with a pentium processor is used. It is equipped with a PCI data aquisition board, which communicates via a terminal board with the peripheral devices motor, encoder, power module and sensors. This PCI Board disposes of analog/digital- and digital/analog- converters with a resolution of 12 bits. The two boards are shown in the next picture (4).

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Figure 4: PCI Data aquisition board(1) with terminal board(2)

### 1.2 Software

The software that is used is the simulation tool Matlab/Simulink with the RealTime Workshop toolbox ([Mat]). It is supplemented by the realtime capable WinCon software. WinCon is implemented as a real time operating system for Windows ([WiCo]). With help of the RealTime Workshop the controller modeled in Matlab/Simulink can be converted automatically in a program of language C. This programm can be executed by WinCon in real time and communicates via the PCI cardwith the peripheral devices. The manufacturer provides a supplementing library for Matlab/Simulink. This library contains models for the PCI cards and and makes it possible to model the totatl system with Matlab/Simulink. The user interface of the WinCon environment is shown in the following figure (5).

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Figure 5: WinCon user interface with system initializing button, graph launching button,

Via this user interface the control system can be initialized and started, signal processing tools and online graphs of system variables can be launched. More- over, WinCon disposes of a graphical user interface ControlPanel ([WiCo]). It allows the online modification of control parameters. An example is shown in the next figure (6).

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Figure 6: WinCon ControlPanel user interface for online modification of control param- eters

## 2 Modeling

### 2.1 Modeling of the Shake Table

In this part, a model of the shake table is to be deduced. As the motor parameters (for example the armature resistance, torque constant) and the reaction of the table platform on the motor are unknown, the theoretical modeling method is unsuited. As an alternative, the experimental modeling method is used. This method determines the coefficients of an assumed model equation that represents approximately the transfer behaviour of the system.

#### 2.1.1 Continuous Transfer Function

In order to obtain a model for the transfer behaviour of the shake table, the motor of the shake table was excited by a voltage step signal and the resulting table position was measured. The Matlab/Simulink model for this experiment is shown in the following figure (7)

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Figure 7: Matlab/Simulink model for the shake table identification experiment with voltage step excitation, output to the motor amplifier, input from the motor encoder and Safety & Enable circuit with the in- and outputs of the Data Aquisition bord.

The model contains the voltage step block which generates the voltage signal given to the Digital/Analog converter of the Data Aquisition board. This output signal is the reference value for the amplifier driving the motor of the shake table. The position of the table is calculated by scaling the measured motor angle that is put in from the motor encoder to the encoder input of the Data Aquisition bord. The output voltage and the input position are saved to the Matlab Workspace for evaluation purpose. The safety of the experiment is given by the safety circuit. If the limit sensors or the encoder indicate a forbidden position of the table, then the experiment run is aborted automatically. The Power Amplifier is enabled by the enable circuit which was designed and provided by the manufacturer. This circuit sends a digital enable signal to the microcontroller of the amplifier.

The response of the shake table excitated by a voltage step input is shown in figure (8).

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Figure 8: Stepresponse of the uncontrolled shake table with integration gain KI and delay τ

The relation between the motor input voltage U(t) and the table position θ(t) is assumed to be represented by an integrator with a first order delay:

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KI is the integrational gain, τ the delay constant and θ0 the initial position of the table. Derivating this equation with respect to the time yields

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This relationship can be transformed by the Laplace transform ([Og2])

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into the frequency domain and equals to

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This can be written in the transfer function representation

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The above characteristic θ(t) in figure ([8]) was obtained in an experiment and can be used to determine the values KI and τ :

- The integral gain KI can be calculated with the gradient m of the characteristic θ(t) in the linear proceeding region and the amplitude U0 of the voltage input step to

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- The delay τ is obtained directly by the horizontal difference between θ(t) in the linear proceeding region and the step characteristic of an ideal integrator with the same integral gain as as the delayed integrator.

With m=0.574m/sec ,U0=1 V and Δt=0.058sec,we get the sought values

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This leads to the transfer function

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#### 2.1.2 Continuous State Space Representaion

In the following a state space representation of the motor shall be deduced. The general form of this representantion for systems of order N with m inputs ui and n outputs yi is given with

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where the designations and dimensions of each vector and matrix are given in the following table 1:

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Table 1: Designations and dimensions of vectors and matrices

To get the concrete values of these vectors and matrices, we again regard equation (2)

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The state variables x1 und x2 are introduced and assigned to the variables θ(t) und θ(t) as followed:

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So, the state variables x1 und x2 represent directly the physical quantities angle and angular frequency of the motor.

The output quantity of the shake table is the platform position and therefore

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These equations (2),(12),(13) and (14) can be combined to

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and written in the more clearly matrix form

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A block diagram of this equation is given in figure (9).

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Figure 9: Continuous state space representation of the shake table

Put in the values KI und τ (Equations 7, 8), we obtain

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This entire transformation into the state space representation is also implemented in Matlab and can be executed by the command ss([Mat]).

#### 2.1.3 Discrete Transfer Function

For the time discrete representation of transfer funcions, the Z -transformation is used. A detailled description can be found in ([Is1],[Og1]). The representation of discrete transfer functions resembles to the continuous Laplacerepresentation and has the general form

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The variable z is an abbreviation for the term [Abbildung in dieser Leseprobe nicht enthalten] , where T0 is the sam- pling time of the discrete system. Applicating the transformation of Laplace- transfer funtions to Z-transfer functions (details in [Is1]), we have to calculate the partial-fraction expansion of the expression [Abbildung in dieser Leseprobe nicht enthalten] which is obtained to

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This expression can be Z-transformed according to [Is[1]] to the expression

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and yields the Z-transfer function with zero-order hold

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With the values of the parameters KI , τ and

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this results in the time discrete Z-transfer function of the shake table

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This transformation from the time continuous to the time discrete domain is also implemented in Matlab and can be executed by the command c2d([Mat]).

#### 2.1.4 Discrete State Space Representaion

The time discrete state space representation of the shake table is to be de- duced. For this, the general form of representation resembles the one of continuous systems. Instead of the continuous time t the dicrete time is used. It is counted in multiples k·T0 of the sampling time T0. For discrete systems of order N with m inputs ui and n outputs yi the differential equation system is

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The designations and dimensions of each vector and matrix are completely the same as used for the time continuous representation in table 1. The transformation of models from the continuous to the discrete state space is described detailed in [Is1] . First the time discrete system matrix [Abbildung in dieser Leseprobe nicht enthalten] is to be calculated. It can be obtained by

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where

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The argument of the inverse Laplace transform is

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and can be transformed with the inverse Laplace transformation to

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With (8), (26) and (30) we get

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The time discrete input vector b is obtained by calculating

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With the parameter values (8) and (26) the input vector results in

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The output vector [Abbildung in dieser Leseprobe nicht enthalten] stays the same and so we can write the time discrete state space representation as

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A block diagram of this matrix differential equation is shown in figure (10).

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Figure 10: Discrete state space representation of the shake table

This transformation from the continuous to the discrete state space is also implemented in Matlab. It can be executed by the command c2d([Mat]).

### 2.2 Modeling of a Frame

In this section the modeling of the frame is described. We are looking for the dynamic transfer behaviour between the movements of the table and the cover plate. The frame to be indentified consists of two vertical sheet metal stays, which are connected by two horizontal arranged plastic plates at top and bottom. For the theoretical modeling of the frames the knowledge of the material-specific characteristics e.g. the modulus of elasticity and the density of the involved frame elements is necessary. As these properties are not indicated by the manufacturer and therefore are unknown, we would have to accomplish a complex identification of these properties. To avoid this procedure, we apply the experimental modeling method. In advance the following considerations are necessary:

- Since the two vertical stays have the same geometrical dimensions and consist of the same material, they theoretically can be reduced to one stay with other dimensions and material properties. For further sim- plification this single stay can be regarded as a single spring with a damper.

- The cover plate will be simplified to a concentrated point mass which is mounted on the end of the spring.

- The base plate is screwed directly on the shake table and thus does not affect to the dynamical behaviour of the frame. Therefore it can be neclected in this section.

A frame results with the applied simplifications to the structure shown in the following figure (11):

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Figure 11: Simplification of the Frame Structure with Point Mass, Damper and Sheet Stay

#### 2.2.1 Time Continuous Transfer Function

This system is a mass-spring-damper system and has in general the folowing simplified time continuous Laplace transfer function of second order

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with

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Here Xp and Xt are the positions of the cover plate and the shake table. The parameters K, D, ω0 and ωe have the following meaning:

- K: Gain ; proportional gain between input and output in steady state

- D: Damping ratio (also damping); indicates the decay of an oscillation

- ω0: natural frequency; describes the frequency of the system without damping

- ωe: damped natural frequency; describes the frequency of the system with damping

The determination of each parameter is done as follows:

An accelerometer is mounted on the cover plate of the frame. It measures the horizontal acceleration of the cover plate. The cover plate is declined while the table does not move. Then the cover plate is released and the frame is oscillating. The oscillations are recorded for further evaluation.

The Matlab/Simulink model for this experiment is shown in the following figure(12):

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Figure 12: Matlab/Simulink model for the decay experiment with the analog accelerometer input and the necessary enable circuit

**[...]**

- Quote paper
- Dipl-Ing. Thomas Heidenreich (Author), 2004, Modeling, Control and Fault Analysis in Electromechanical Systems applicated on a Shake Table, Munich, GRIN Verlag, https://www.grin.com/document/113105

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