Investigation of dynamic responses using rectilinear and torsional apparatuses ECP-210/205

1. Introduction

Mechanical systems are generally classified as open-loop and closed-loop systems according to their architecture. The open-loop systems are driven by externally applied forces and moments with preprogrammed dependence on time. The closed-loop systems apply the feedback control principle based on sensing system motion parameters. Mechanical or robotic system architecture typically includes simple mechanisms executing either rectilinear or rotational motion.

The main objective of this experimental work and assignment is to understand behavior and to learn basic dynamical properties of a simple mass-spring-damper system (ECP-210) and a rotating disk (ECP-205) under action of

externally applied force and torque, respectively.

The external driving force or torque can be formed as a predefined function of time allowing you to understand different aspects of their dynamics and compare the observations with related mathematical models. You will be asked to draw conclusions from your experimental study and confirm them using theoretical analysis.

This experiment laboratory sheet includes two parts. You only need to collect data and submit a report from the rectilinear machine. The torsional experiment is optional. Please read the guideline carefully and submit your report in due time. Any questions regarding the lab report should be raised

through the discussion board.

2. Description of Experimental Apparatus

2.1 Rectilinear dynamic plant ECP M210

Experimental apparatus ECP M210 shown in Figure 1(a) includes two or three trolleys connected by elastic coil springs and one dashpot connected to one of the trolleys. Through a rack and pinion drive the first trolley is connected to an electric motor generating force F(t), which can be appropriately programmed. Each trolley is connected to an optical encoder measuring its linear displacement x (see Figure 21 (b)).

There is a special hardware control block and software with user-friendly interface, which supports a range of controller specifications, control input trajectory generation, data acquisition and plotting features. The ECP M210 apparatus can be transformed into a number of dynamic configurations, which can represent different classes of “real-life” mechanical systems, for

example, to study flexibility in linear drives, gearing and belts, the vibration of coupled rigid bodies controlled by a single actuator and a number of sensors.

To start the experiment:

a) turn on the PC with the ECP Executive program installed in it,

b) turn on the power to Control Box (press on the black switch),

c) open the ECP Executive program

2.2. One Degree of Freedom Model (1 DOF)

A schematic representation of one degree of freedom mechanical system is shown in Figure 2, where a trolley of mass m experiences elastic force produced by attached coil spring with stiffness coefficient k and linear viscous friction produced by attached dashpot or damper with coefficient c. The following equation of motion follows from the Newton’s second law:

Applying the Laplace transform to equation (1) and assuming zero valued initial conditions equation

(1) may be equivalently represented by the transfer function:

2.3. Experimental study (open loop)

This section describes ECP M210 configuration and experimental investigations. It gives instructions for selection and execution of different control inputs (trajectories), acquire and plot experimental data. First experiment deals with the open-loop 1 DOF mass-spring-damper system (see Figure 3) and identification of its parameters, such as damping factor z and natural frequency

Set up the ECP M210 system. Assemble ECP M210 system with only one trolley carrying two 500g mass blocks being connected to the left constraint by a spring with stiffness coefficient k1=800 N/m. Dashpot is not connected to the trolley in this experiment to minimize dissipative forces in the system.

Download the configuration file, default.cfg from the FileLoad Settings menu and the control personality file, M210-v38.pmc from the UtilityDownload Controller Personality File menu.

Ensure that Control Loop Status indicates “Open Loop”. Otherwise, press the “Abort Control” button at the down-right corner of the main screen.

• Execute a Step Input Control. Enter the Command menu and select Trajectory.

• Select Selections: Step

• Then push Setup.

• In the “Configure Step Trajectory” window ensure that “Open Loop” is checked (see box on the right).

• Set control effort = 1 V, dwell time =4,000 ms and no. of repetitions = 1.

• Exit this window by pressing OK.

• Go to the Command menu. This time select Execute and with the Normal Sample Data box checked.

• Press Run button to execute selected control input (trajectory) and wait for the data to be uploaded from the real-time Controller. You should notice a step displacement of the trolley for duration of 4 seconds and return back to initial position.

Plot Data. Now enter the Plotting menu and choose Setup Plot. In the new window:

o Ensure that Encoder 1 and Commanded Position are appeared in the left window for plotting (in the left axis)

o Press Plot Data button. You can now see Plot window with Open Loop Step response of the system. An example of the plot is shown in Error! Reference source not found. (a).

• Export Raw Data.

o Enter Data menu and choose Export Raw Data.

o Choose a path and save the text file.

o The file contains acquired data in a format suitable for reviewing, editing, or exporting to other engineering/scientific packages such as MATLAB. You can recreate, analyse and process the system response there with following information. 1 revolution = 16,000 encoder counts = 7.06 cm = 2.780 in, 11625.45 Khw =. Figure 4

(b) describe the motion after editing in MATLAB.

2.4. Trolley-spring-dashpot system

Air-dashpot is a typical mechanical damper generating resistive force proportional to velocity and acting opposite to motion. Connect the dashpot to the trolley by two screws at it is shown in Fig. 5. After running again experiments with this trolley applying step signal. Turn the dashpot “knob” to reduce overshoot in system response and finally make this response critically damped (zero overshoot). Save data for a few representative dynamic processes.

2.5. Two degrees of freedom system (2 DOF)

Dynamics of two degrees-of freedom system is more complicated in comparison with 1DOF system. Two-trolley system shown in Fig. 6 has two natural frequencies and therefore its response to periodical force excitation will show two resonance peaks while frequency will gradually increase covering a broad range of frequencies.

Connect a second trolley via a spring as it is shown in Fig. 6. Use a sine sweep vibration test (i.e. constant amplitude periodical force with linearly increasing frequency). The force amplitude is 0.5v, and the frequency changes from 0 to 7Hz in 30 seconds. Analyse system response to this type of excitation. Safe recorded processes for further detailed analysis.

2.6. Identification of the 1DOF system parameters (open loop)

In your report using open-loop system transient response plot or raw data (see Section 2.3), determine the damping factor z, natural frequency n , effective mass meff and the friction coefficient 𝑐𝑒𝑓𝑓.

Use the following properties of unit step transient response for the standard two degrees of freedom system (see Figure 5).

3. Torsional dynamic plant ECP M205

The ECP M205 system schematically shown in 6 consists of an electromechanical plant and a set of control hardware controlled via user-friendly software interface. The user interface to the system supports a broad range of controller specification, trajectory generation, data acquisition, and plotting features. The electromechanical apparatus may be transformed into a variety of dynamic configurations, which represent important classes of “real-life” systems.

The Model 205 torsion apparatus represents many such physical plants including a number of rigid bodies with elastic connections. It allows study of flexibility in linear drives, gearing and belts and investigation of

mechanical vibrations and their active control in torsional multi-dimensional system with elastic linkages similar to robotic arms, transmission lines, etc.

3.1. One Degree of Freedom (1DOF) configuration torsion plant

The most general form of one degree of freedom torsional plant is shown in Figure 7, where friction is idealized as being viscous. Applying Newton’s second law to attached rotating disk using free body diagram method the following differential equation can be derived:

Applying the Laplace transform to differential equation (1) and assuming zero valued initial conditions one can derive the following transfer function for the plant:

3.2. Experimental study: analysis of the open-loop system

Experimental study of the open-loop plant includes investigation of the rotating disk response to an externally applied torque having rectangular pulse. Provided below instructions allow one to determine the moment of inertia, dry and viscose friction effect, and physically simulate the open-loop system behavior by acquiring data for transient processes.

Dynamic properties of the open-loop 1 DOF torsional system (see Figure 8) is characterized by system dynamic response to a unit step input for Khw =14.928.

You need to identify the moment of inertia J of the attached disk with two cylinders and evaluate the viscose friction coefficient c .

3.3. Experimental study: analysis of the open-loop system Identification of the 1DoF system parameters The plot in Figure 10 represents the disk velocity obtained by numerical differentiation of the angular position data from the open loop experiment.

The identification procedure starts by plotting tangent lines at point 1, point 2, and point 3 and evaluating respective tangents, 𝛼1, 𝛼2, velocity 𝜔2, and finally the tangent 𝛼3 which represent angular accelerations of the disk at these three points.

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