| WEEK # | LAB TOPICS | Pre-Lab Materials | LAB MATERIALS | SUPPORTING Materials |
|---|---|---|---|---|
| 2 | Lab 1: 1st-order Spring-damper System | Prelab 1 (PDF) | Lab 1 (PDF) |
Cameras (PDF) Lab 1 (ZIP) (The ZIP file contains: 11 .m files.) |
| 3 | Lab 2: Rotational 1st-order Inertia/Damper System | Prelab 2 (PDF) | Lab 2 (PDF) | |
| 6 | Lab 3: Translational 2nd-order Spring/Mass/Damper System; Natural Response; Fitting Models | Prelab 3 (PDF) | Lab 3 (PDF) | Lab 3 (ZIP) (The ZIP file contains: 14 .m files.) |
| 7 | Lab 4: Driven Response of Translational 2nd-order Spring/Mass/Damper System; Step Response | Prelab 4 (PDF) | GetData (PDF) | |
| 8 | Lab 5: 1st-order RC Circuits and 2nd-order LRC Circuit | Prelab 5 (PDF) | Lab 5 (PDF) | Lab 5 (ZIP) (The ZIP file contains: 4 .m files.) |
| 9 | Lab 6: Rotational 1st-Order Inertia/Damper System Driven through a Gear Train | Prelab 6 (PDF) | Lab 6 (PDF) |
Labs
Pre-Lab (PDF)
Lab 1 Description (PDF)
In this lab, the time response of a first-order system is demonstrated. This system consists of a spring and a damper, respectively represented by a cantilever and an air dashpot (Figure 1). The cantilever is made of spring-steel and can be modeled as a linear spring, i.e. the force at the tip of the cantilever is linearly dependent on its displacement. The dashpot can be modeled as a pure damper where the damping force is proportional to velocity as long as the volume of air behind the piston is not too large. The mass of the cantilever can be neglected, as long as the damping of the air dashpot is not too small. Students will observe that the system departs significantly from these idealizations in some circumstances. This nonideal behavior can be the motivation for postulating more complex models.
Figure 1. Idealization: spring and damper. (Image by Prof. Trumper.)
A schematic representation of the system is shown in Figure 2.
Figure 2. Idealization: spring and damper. (Image by Prof. Trumper.)
The glass dashpot is covered with rubber to protect the user if it breaks. In figure 3, the dashpot without the rubber coating is displayed, showing the plunger and the adjustment wheel. This wheel adjusts the damping by changing the opening through which air flows.
Figure 3. Air dashpot without rubber coating. (Image by Prof. Trumper.)
Figure 4 shows a simple ‘webcam’ that records the data as video. The recording software lets you configure the frame rate. By looking at the individual frames of the animation, a table of position and time can be constructed.
Figure 4. A simple camera records the data as an animation on the screen. (Image by Prof. Trumper.)
The data can then be processed into a plot that can be used to deduce the time constant tau (figure 5).
Figure 5. Response of a first order system to an initial displacement. (Image by Prof. Trumper.)
Figure 6. Video Demonstration. (Image by Prof. Trumper.)
Materials and Notes: Practical Info and Comments
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Airpot Precision Air Dashpot We use an adjustable glass dashpot that is covered with rubber to protect the user if it breaks. Model numbers we use are (2K)S95-A200 or (2K)S160-A200-F275 (603/382-4179). Look at the Web site of Airpot.
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Angle plates One of the angle plates is a mount for the dashpot, the other a mount and clamp for the spring cantilever. The 3X3X.375 inch angle plates were ordered from Pierce Aluminum. The drawings for these angle plates are: plate 1 and plate 2 (.tif files, use right mouse button and ‘save as…’).
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Bolts and ruler The bolts should be mounted to serve as stops for the cantilever to prevent damage of the airpot. At first we used plastic rulers as a scale for the deflection of the cantilever, but the plastic rulers appeared to be less visible with the webcam than the pieces of paper.
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Cantilever spring use flat sheets of spring steel, hole for attachment of damper is hard to machine
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Optical baseplate OptoSigma, black anodized aluminum, from stock:
- 18x24 inches, 1/4-20 on 1 inch centers: product number 145-1160
- 457x610 mm, M6 on 25 mm centers: product number 145-1165Prices can be found at their Web site. Go to >‘Optical Bases’>‘Baseplates’.
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Stand Edmund Scientific, clamps for webcam mounting: VWR supply room
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Webcam Look at Winnov / Solutions for information about these cameras.
Pre-Lab (PDF)
Lab 2 Description (PDF)
The dynamics of a spinning shaft and the effects of adding inertia and viscous fluid damping are examined in this lab. Figure 1 shows the actual apparatus.
Figure 1. First-order rotary system with damping supplied by cup filled with honey, video image shown on screen. (Image by Prof. Trumper.)
The drawing in figure 2 depicts the most important parts. The shaft and flywheel are modeled as an equivalent rotary inertia.
Figure 2. Drawing of the first-order rotary system. (Image by Prof. Trumper.)
A brass ring can be added to increase the inertia, as shown in figure 3.
Figure 3. A Brass ring can be added to increase the inertia. (Image by Prof. Trumper.)
The axial position constraint is provided by a ball bearing. This ball bearing is attached to the end of the shaft and rotates on a fixed hardened flat (figure 4). Damping is created by filling with honey the annular space between the end of the shaft and a clear tube. The height of the honey can be changed via the syringe shown in figure 1. The fluid damping can be modeled as an equivalent rotary damper that is linearly proportional to the height L of the viscous fluid.
Figure 4. The end of the shaft is supported by a ball bearing riding on a hardened flat. Honey filling an anular space provides the damping. (Image by Prof. Trumper.)
Figure 5 shows the idealization of this first-order system. We have chosen honey as the viscous fluid to attain time constants on the order of 1 second with relatively loose geometric constraints. Furthermore, honey is environmentally safe and can be cleaned up with water.
Figure 5. Idealization: rotary damping and inertia. (Image by Prof. Trumper.)
Figure 6 shows how the animation is being read from the screen. The data can be read by sticking a transparent angle sheet on the screen.
Figure 6. Idealization: rotary damping and inertia. (Image by Prof. Trumper.)
By reading every frame and processing the data (with for example MATLAB®) a plot of the angle versus time can be constructed (figure 7).
Figure 7. Plot of angle vs. time. (Image by Prof. Trumper.)
Clearly, we could automate this data collection by using for example an optical encoder or a tachometer. The reason for using a camera is to make the data acquisition more intuitive for the students in the early labs. To see an example of such a video capture click on the video button:
Figure 8: Video Demonstration. (Image by Prof. Trumper.)
Materials and Notes: Practical Info and Comments
- Air bearings New Way®, Each bearing consists of two parts:
(1) S301901 - 3/4" I.D. air bushing
(2) S8019P01 - pillow block housing for the air bushings
Prices for the New Way® bearings can be found at: New Way® Air Bearings. If you are looking for other air bearing types, note that the alignment is a major concern. An advantage of this type of air bearings is that the o-rings allow for some flexibility. Be careful not to damage the o-rings when assembling. Use alcohol as a lubricant when assembling the air bearings. A collar on the shaft is used to disable lifting the shaft to much to avoid honey coming into the bearings. However, it is possible to thoroughly clean the honey out of the bearings with alcohol while they are attached to an air supply. The part of the shaft that goes into the air bearing should be precision ground and unscratched for optimal results. Do not move the shaft inside the bearings when the bearings are not hooked up to air supply to prevent unnecessary scratching of the bearings and shaft. - Air supply Quick connectors and tubing from Beswick
- Angular sheet This is a small version of the polar plot we stick to the screen printed on a transparent sheet. A larger version of this .gif file can be found here. This image can of course be printed with the same dimensions, but with higher resolution.
Small polar plot .gif file. (Image by Prof. Trumper.)
- Collar Whittet Higgins aluminum split-collar clamps: partnumber SC-12A (split collar-size, aluminum). We use 3/4" inner diameter to match the shaft and air bearings. Look at their Web site at Whittet-higgins company go to ‘Collars’>‘SC-A’, order from Kaman Industrial Tech (781/935-7590).
- Flywheel
- Frame
- Honey We have chosen honey as the viscous fluid to attain time constants on the order of 1 second with relatively loose geometric constraints. Furthermore, honey is environmentally safe and can be cleaned up with water. However, honey cristallizes after being exposed to open air for a few days. It is therefore recommended to change the honey when it becomes too thick. Be sure that the honey does not get into the air bearings. For instance use a collar on the shaft to make sure the shaft can not be lifted too far up. If the shaft is not concentric with the look-through tube that holds the honey, the honey will not be distributed evenly across its circumference. This nonideal behavior is hard to model.
- Optical breadboard See Lab 1.
- Shaft, ball bearing and hardened flat Thompson shaft, precision ground 3/4 inch stainless steel; 1/2 inch ball bearing and hardened flat McMaster-Carr
- Stand See Lab 1
- Syringe 20 ml with luer-lok, connection to tubing: luer-lok to tubing Cobert Assoc.
- Webcam See Lab 1
Pre-Lab (PDF)
Lab 3 Description (PDF)
In this lab, the dynamics of a second-order system composed of a spring, mass and damper are examined. As shown in figure 1, the system consists of a cylindrical shaft riding on air bearings. A voice coil is attached at the left side to add variable damping. The voice coil armature is wound on an aluminum cylinder. If the coil is open-circuited, some damping is still present due to eddy currents in the aluminum. If we short-circuit the voice coil the damping is increased significantly, because of resistive losses in the wire. In the middle between the two air bearings, an adjustable spring is attached to the shaft. By decreasing the length of the spring, the natural frequency of the system is increased and the damping ratio is decreased. By adjusting the length of the spring, one can demonstrate overdamped, critically damped and underdamped behavior. A pulse can be applied to the system by allowing a small brass ball hanging on a piece of string to impact the plate attached to the shaft at the right. To measure the motion, an LVDT (Linear Variable Differential Transducer) is fixed on the right side of the shaft. See Prelab3 page 5 for an explanation of LVDT operation.
Figure 1. The second order system with voice-coil, air bearings, adjustable spring, shaft mass, and LVDT sensor. (Image by Prof. Trumper.)
Figure 2 shows the drawing of this system.
Note the schematic detail in figure 2 that shows the off-center attachment of the spring to the collar on the shaft. The reason for this is to allow for axial displacements of the end of the spring by free rotation of the shaft. If the spring is bend, it shortens a distance d, as shown in figure 2 on the bottom right. If the spring was not attached this way, the actual stiffness would be much larger. More importantly, the linearity of the stiffness would be much less because of the unwanted axial pulling force on the spring rod.
Figure 2. Drawing of the first-order rotary system. (Image by Prof. Trumper.)
Figure 3 shows the idealization of its dynamics.
Figure 3. A Brass ring can be added to increase the inertia. (Image by Prof. Trumper.)
Figure 4 shows the Close-up spring collar attachment.
Figure 4. Close-up spring collar attachment. (Image by Prof. Trumper.)
The signal passes through a signal conditioner and is made visible on an oscilloscope in Figure 5.
Figure 5. Overdamped impulse response. (Image by Prof. Trumper.)
The signal passes through a signal conditioner and is made visible on an oscilloscope in Figure 6.
Figure 6. Underdamped impulse response. (Image by Prof. Trumper.)
Materials and Notes: Practical Info and Comment
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Adjustable spring clamp
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Air bearings from New Way, see Lab 2. They do not mount directly to optical breadboard, so we made special adaptor plates from Pelrin. Another simpler option to attach the bearings to the baseplate would be to use toeclamps.
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Angle plate 3X3X3 inch ground steel (flat surfaces) from Suburban Tool, inc.
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Collars see Lab 2.
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Optical breadboard
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LVDT (Linear Variable Differential Transducer) Schaevitz HR-1000 ($450 each) was the best choice, because of the large clearance through the hole in the body of the sensor. LVDT’s with smaller clearance caused problems with alignment. This sensor has a hole all the way through. It is unguided, i.e. does not use bearing surfaces, to avoid friction.
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Separate signal conditioner ($167) is needed for this LVDT.
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Shaft Thompson shafting, necessary threading can be specified, precision ground stainless steel, diameter 3/4 inch to match the air bearings.
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Spring
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Stand
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Voice coil LA24-33 BEI-Kimco seems to be the only manufacturer for this type of part. Prices are much lower when ordering in large quantities ($315 each when ordering 30).
Pre-Lab (PDF)
In Lab 4, we will use the same system that we studied in Lab 3. The objective of this lab is to observe the driven step response of a second order system and to tune the system’s parameters to obtain a desired behavior. This can be done by adjusting the length of the spring rod and thereby changing the stiffness of the system. The voice coil that was short circuited in Lab 3 is used to supply the input force. A function generator and power amplifier provide a step in voltage to the voice coil, which results in a step in force on the shaft. The damping is fixed at the short-circuit value of Lab 3.
Figure 1. Idealization: spring, mass, damper with step force input. (Image by Prof. Trumper.)
Figure 2: The power amplifier is an Apex PA21 power op-amp in their EK21 evaluation kit.
Figure 2. The power amplifier is an Apex PA21 power op-amp in their EK21 evaluation kit. (Image by Prof. Trumper.)
Figure 3 shows a circuit diagram of this lab.
Figure 3. Circuit diagram of this lab. (Image by Prof. Trumper.)
Materials and Notes: Practical Info and Comments
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As in Lab 3:
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Adjustable spring clamp
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Air bearings
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Angle plate
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Optical breadboard
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LVDT (Linear Variable Differential Transducer)
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Shaft and collars
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Spring
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Stand
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Voice coil
New
Pre-Lab (PDF)
Lab 5 Description (PDF)
First Part
The experience students obtained from the mechanical systems in the previous four labs can be applied to study electrical systems. Systems with masses, springs and dampers have their electrical equivalents with respectively inductors, capacitors and resistors. A major difference between electrical and mechanical systems is that with electrical systems you can achieve nearly ideal behavior far more easily. In this lab, the transfer functions of a few first order electrical systems, shown in figure 1, are measured. The students measure the circuit step response and frequency response functions.
Figure 1. Examples of simple first order electrical circuits. (Image by Prof. Trumper.)
The hardware we need for this lab consists of a signal generator, an oscilloscope and a breadboard to easily put together the electrical components as shown in figure 2.
Figure 2. The power amplifier is an Apex PA21 power op-amp in their EK21 evaluation kit. (Image by Prof. Trumper.)
Materials and Notes: Practical Info and Comments
- Electronic kits Electronic School Supplies, kits with electronic components (about $20 each)
Second Part
This lab focuses on second-order electrical circuits, which contain inductors, resistors and capacitors. By measuring the step and frequency responses, the transfer functions can be determined. In figure 1, the idealization of this lab’s circuit is shown.
Figure 1. Idealization of the RLC circuit used in this lab. (Image by Prof. Trumper.)
The input signal of the system is provided by a function generator. Because the function generator and inductor have internal resistances, a more accurate model looks like the circuit in figure 2.
Figure 2. RLC circuit with internal resistances of the function generator and inductor. (Image by Prof. Trumper.)
The internal resistance of the function generator makes the output voltage vg differ from the desired input vi. Therefore an op-amp is used to work as a buffer, as shown in figure 3. Because the input impedance of the op-amp is very high and its output impedance very low, it forces vb to be almost equal to vi as long as the output current does not exceed the limits of the op-amp.
Figure 3. An op-amp buffer forces vb to be nearly equal to vi. (Image by Prof. Trumper.)
The damping ratio can now be adjusted by varying the resistance R1. The resistor be seen as an electrical equivalent of a mechanical damper. Predicted responses of this circuit can be compared with measured ones. Figure 4 shows how magnitude and phase shift can be displayed on an oscilloscope using a sine wave from the function generator. By varying the input frequency, the circuit Bode plot can be generated.
Figure 4. A sine input shows the magnitude and phase shift at a certain frequency. (Image by Prof. Trumper.)
Course Info
Instructors
Departments
As Taught In
Spring
2005
Level
Topics
Learning Resource Types
theaters
Demonstration Videos
grading
Exams
assignment
Problem Sets
