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Topics covered:

1. Dynamic Behavior of a system of mass spring & damper

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Experiment details:

The experimental control system is comprised of the three subsystems shown in Figure 2. The first of these is the electromechanical plant which consists of the spring/mass mechanism, its actuator and sensors. The design features a brushless DC servo motor, high resolution encoders, adjustable masses, and reconfigurable plant type. 

Next is the real-time controller unit which contains the digital signal processor (DSP) based real- time controller1, servo/actuator interfaces, servo amplifier, and auxiliary power supplies. The DSP is capable of executing control laws at high sampling rates allowing the implementation to be modeled as being continuous or discrete in time. The controller also interprets trajectory commands and supports such functions as data acquisition, trajectory generation, and system health and safety checks. A logic gate array performs motor commutation and encoder pulse decoding. Two optional auxiliary digital-to-analog converters (DAC's) provide for real-time analog signal measurement. This controller is representative of modern industrial control implementation.

The third subsystem is the executive program which runs on a PC under the DOS or Windows™ operating system. This menu-driven program is the user's interface to the system and supports controller specification, trajectory definition, data acquisition, plotting, system execution commands, and more. Controllers may assume a broad range of selectable block diagram topologies and dynamic order. The interface supports an assortment of features which provide a friendly yet powerful experimental environment.

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The system is schematically shown in Figure 1. The apparatus consists of three mass carriages interconnected by bi-directional springs.  The mass carriage suspension is an anti-friction ball bearing type with approximately +/- 3 cm of available travel.  The linear drive is comprised of a gear rack suspended on an anti-friction carriage and pinion (pitch dia. 7.62 cm (3.00 in)) coupled to the brushless servo motor shaft.  Optical encoders measure the mass carriage positions – also via a rack and pinion with pinion pitch diameter 3.18 cm (1.25 in).

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Procedure:

First, the relevant parameters such as the mass, spring stiffness and damping coefficients will be determined in various configurations shown in Fig. 4.

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Before starting, identify the way the mass carriages are labeled: one, two and three from left to right in Fig. 4.

1. Clamp the second mass to put the mechanism in the configuration shown in Figure 4a above using a shim (e.g. 1/4 inch nut) between the stop tab and stop bumper. Connect the first and second mass carriages by a spring.

2. Secure four 500g masses on the first and second mass carriages. The 2^nd mass carriage is fixed not to move in this part of the experiment. 

3. Set up the data acquisition. With the controller powered up, enter the Control Algorithm box via the Set-up menu and set Ts = 0.00442 s. Enter the Command menu, go to Trajectory and select Step, Set-up.  Select Open Loop Step and input a step size of 0, a duration of 3000 ms and 1 repetition.  Exit to the background screen by consecutively selecting OK.  This puts the controller in a mode for acquiring 6 sec of data on command but without driving the actuator.  This procedure may be repeated and the duration adjusted to vary the data acquisition period.

4. Go to Set up Data Acquisition in the Data menu and select Encoder 1 as data to acquire and specify data sampling every 2 (two) servo cycles  (i.e. every 2 Ts's).  Select OK to exit.  Select Zero Position from the Utility menu to zero the encoder positions.

5. Select Execute from the Command menu.  Manually displace the first mass carriage approximately 2.5 cm in either direction. select Run from the Execute box and release the mass approximately 1 second later. The mass will oscillate and attenuate while encoder data is collected to record this response. Select OK after data is uploaded.

6. Select Set-up Plot from the Plotting menu and choose Encoder #1 Position then select Plot Data from the Plotting menu. You will see the first mass time response.

7. Choose several consecutive cycles (say ~5) in the amplitude range between 5500 and 1000 counts (This is representative of oscillation amplitudes during later closed loop control maneuvers. Much smaller amplitude responses become dominated by nonlinear friction effects and do not reflect the salient system dynamics). Divide the number of cycles by the time taken to complete them being sure to take beginning and end times from the same phase of the respective cycles.  Convert the resulting frequency from Hz to rad/sec.  This damped frequency, ω_d, approximates the natural frequency, ω_n, according to:

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where the "m₁₁" subscript denotes mass carriage #1, trial #1.

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Next, remove the four masses from the first mass carriage and repeat Steps 5 through 7 to obtain w_{n_m12} (natural frequency for mass carriage #1, trial #2) for the unloaded carriage. Shorten the test duration set in Step 3 if needed.

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10.    Repeat Steps 5 through 9 for the second mass carriage. Here in Step 6 you will need to remove Encoder #1 position and add Encoder #2 position to the plot set-up. Hence obtain ω_{n_m21} , ω_{n_m22} and ζ_m22. How does this damping ratio compare with that for the first mass?

11. Connect the mass carriage extension bracket and dashpot to the second mass as shown in Figure 6.1-2c. Open the damping (air flow) adjustment knob 2.0 turns from the fully closed position. Repeat Steps 5, 6, and 9 with four 500 g masses on the second carriage and using only amplitudes ≥ 500 counts in your damping ratio calculation. Hence obtain ζ_d where the "d" subscript denotes "dashpot".

12.  The mass of each brass weight is 0.5kg.  Let m_w be the total mass of the four weights combined. Use the following relationships to solve for the mass of the unloaded carriage, m_c2, and spring constant, k:                       

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Find the damping coefficient c_m2 by equating the first order terms in the following equation:

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where ζ is ζ_m22 , ω_n is ω_{n_m22}, and m is m_c2 in this case.  Repeat the above for the  first mass carriage, spring and damping m_c1, c_m1 and k respectively. Calculate the damping coefficient of the dashpot, c_d.

Report requirements:

The report for this part is expected to include: 

·        Abstract and Introduction

·        Procedures and Steps

·        Experiment Related (Plots & Calculations)

·        Discussion and Conclusion

Experiment Related (Plots & Calculations)

Plots used to determine the natural frequency of the system, along with titles, labels to clearly show which plot corresponds to which situation. Plots include: 

• Plot of Mass 1 loaded

• Plot of Mass 1 unloaded

• Plot of Mass 2 loaded

• Plot of Mass 2 unloaded

• Plot of Mass 2 loaded with dashpot

Calculations showing how you found the following values, along with units for every quantity.           

• Mass 1 loaded natural frequency, w_nm11

• Mass 1 unloaded natural frequency, w_nm12

• Mass 1 unloaded 2 damping ratio, z_m12 

• Mass 2 loaded natural frequency, w_nm21

• Mass 2 unloaded natural frequency, w_nm22

• Mass 2 unloaded damping ratio, z_m22

• Mass 2 damping ratio loaded with dashpot connected, z_d 

• Spring constant, k

• Mass of Carriage 1 plus driving unit, m_c1

• Mass of Carriage 2, m_c2

• Mass 1 & driving unit damping coefficient, c_m1

• Mass 2 damping coefficient, c_m2

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