Proceedings of



The Journal of Rowan Engineering/

Mechanical

Fall 2001

ME-F01-06

Visual beams II

|JOSEPH PLITZ, ME |MICHAEL RESCINITI, ME |

|ADITYA CHAUBAL, ECE |Frank Brown, CEE |

Abstract

The goal of the Visual Beams II project was to design, build, and test an instrumented simply supported beam with a computer interface. By the end of the semester, two structurally sound supports were designed and constructed and a computer interface was designed and programmed to meet and surpass the original goal. The result was a visualization of the reactions of an infinite number of combinations of beam shapes, point load and support positions, and point load forces shown on a user-friendly computer program.

introduction

Visual Beams was an idea started in the spring semester of 2001 at Rowan University with the introduction of a cantilever beam designed with load and torque cells for data acquisition. The successful design made it possible for professors, students, and other interested users to see the different reaction forces and torques with an applied force or torque. Though the mechanical design of the cantilever beam was sound, the data acquisition had room for improvement. The Visual Beams II project kept the same principles of having an interactive visualization tool to aid students and others interested in beam bending, but took the visualization aspect farther.

The data acquisition of the Visual Beams I project plotted many different numbers on a single graph. These numbers represented the reaction forces of whatever weight or force the user put on the beam. Again, these are only numbers and not a visual tool. The beam used, however, was made of a flexible enough plastic so the user could actually see and get a feel for deflection, but no connection between the actual bending and theoretical bending was made.

The purpose of Visual Beams II was to have the user actually see the connection between what happens in reality and what happens on paper when a force is on a beam. As the user applies weight and sees some deflection in the beam, a computerized version of the beam would show that deflection magnified to any extent. In addition, the moment and shear stress diagrams for that load could be seen by clicking to a different window.

It is hard for some students to believe that equations have been made that can model a beam with a force on it. Similarly, some students have trouble visualizing what kind of deflection or stresses a force can create. With the aid the Visual Beams, these problems and many more will be solved.

In order to accomplish the goal of Visual Beams II, many tasks needed to be completed.

MECHANICAL DESIGN

Knowing the beam is simply supported, the actual mechanical design had to be as ideal as possible. In a simply supported beam one support must be a pivot and the other support must be a roller. The beam needed to be able to pivot and slide without friction. Below is a schematic of the engineering model.

A specialty hinge was designed to accomplish the task of a pivot and a support for the load cell. The hinge is a simple pin joint, but is designed to support both the slider, and the load cell. It was also designed to resist any forces applied normal to the side view.

Also, the beam needs to slide with no friction; therefore, a linear slider made of self-lubricating frelon-j was used to give this effect. The advantage of using this slider was its very low friction and it could be used to either lock the bar in place (pivot support) or allow the bar to slide freely (roller support). A disadvantage of using this slider is it can only adjust to adapt to a small range of bar widths. Also the length of the slider is 3.125 inches. This is a problem when the beam is bending. The actual deflection is not going to be exactly the same as the calculations, because the beam cannot bend inside of the rigid box.

[pic]

Figure 1: Frictionless Slider

When the DAQ system reads the load cell, the reaction forces must show only the vertical components. Many people assume that the forces would always be vertical, but if the beam is under heavy deflection, the reaction forces will also have a horizontal component. Therefore, steps were taken to ensure the load cells would read only the vertical load. To accomplish this, four clearance holes were drilled into the support block and the bottom hinge. Then four steel dowel pins were inserted into the holes. This allowed the hinge and slider assembly to slide vertically on steel dowel pins with minimal friction. To reduce the chance of the pins and the bottom hinge from binding up, the holes were drilled very deep and also reamed to size. This allows a longer pin to fit deeply inside causing a smaller angle to form between the bottom hinge and pins. The smaller the angle, the less force is on the top of the pin, therefore the less chance the pin will bind up.

[pic]

Figure 2: Short pin in hinge

[pic]

Figure 3: Long pin in hinge

Also, this design must be user friendly and must be able to handle a wide variety of applications. A track system with locking thumbscrews was installed to allow the user to move either support to the desired position with ease. Below is a solid works model of the final design.

[pic]

Figure 4: Final Design

To load the beam, a design was created that allowed the user to add standard gym weights to a point load without interfering with the supports system. This design also allows the user to remove the load or move it to any place on the bar with ease. Below is a solid works model of the weight hanger.

[pic]

Figure 5: Hanger

The support blocks were chose to be relatively small to prevent additional torque to the base and also to prevent high school students from breaking or knocking the visual beam over during usage. After building and testing the project, it was then realized that if the blocks had been taller, heavy weights could be more easily applied. As it stands, only standard 10 lb gym weights can be placed on the bar. It might have been more convenient to use heavier weights for loading, but the design still allow for maximum loading conditions.

[pic]

Figure 6: Complete Design

The load cells were chose for both positive and negative loading. This was necessary because of the semi-cantilever case. The range of the load cells was chosen because the beam material could not with stand too heavy of a load and also they were cost effectiveness. These load cells were also very simple to adapt to the assembly.

Beam material selection

Various materials and shapes had to be considered for the beam. A solid square bar, a hollow square tube, and an I-beam (cross sections are shown below, Figure 7) were selected to illustrate how the arrangement of mass in the beam can affect its strength and therefore performance. In addition, these shapes are some of the most commonly used in structures and can be easily be associated with. A solid beam with certain overall dimensions is very strong but uses much material. The same amount of material can be better arranged in an I-beam or hollow tube shape in order to reduce the stresses and therefore increase the overall strength. The main constraint of the shape of beams was that the overall dimension had to be no larger than 1.5 inches square in order to accommodate the slider that would mount them.

[pic]

Figure 7: Beams

Using the various shapes, the stresses for different materials were calculated. Bending and shear stresses were calculated as a measure for beam strength. These stresses were based on the worst case loading scenarios. The loading cases were created from the arrangement of the supports and the point load. The worst case-loading scenario included the arrangement of the two supports about an inch apart with the rest of the beam overhanging one support. The support toward the end of the beam acted as a pin, while the other support near the overhang performed as a roller. The load was applied at the end of the overhang which produced the maximum moment and the maximum shear force for a beam loaded with a given point load. These maximums were experienced in the overhanging section of the beam and therefore caused the maximum bending and shear stresses. Calculations included the use of various materials, such as aluminum, PVC, and acetal-copoly. Plastic materials were preferred due to their deflection characteristics and lightweight. It was decided to use acetal-C and PVC material early in the project and as a result calculations focused on the use of these materials. Microsoft Excel spreadsheets were used to make calculations concerning the use of acetal-c in different loading conditions and also used for some general strength properties. A comparison between charts show how stresses are kept below the maximum values for a given material. It is important to note that research shows that PVC and acetal-c have approximately the same strength.

Other considerations included the availability of shaped or raw material stock. Certain materials only come in certain shapes and machining them to other shapes was often difficult and labor intensive. For this reason, a square hollow tube and two square solid bars of acetal-c were ordered. One of the acetal bars would be used in its existing shape, while the other bar would be machined into an I-beam. The choice of these materials and shapes completed the beam material selection portion of the project.

bEAM Deflection mODELING

Although many geometries can be found for the simply supported beam with a single point load, each can be derived from two more general geometries as seen below.

Geometry 1

Geometry 2

Where ‘Ra’ and ‘Rb’ are the reaction forces of the two supports; ‘P’ is the point load; and ‘a’ and ‘d’ and ‘b’ and ‘c’ are the distances from the ends to the outside forces and the distance between the outside forces and the inside force respectively.

Deflection can be solved for both geometries with the following steps.

1) Newton’s 2nd Law: entire bar

The point load can be written in terms of the reaction forces by summing the forces and summing the moments at a certain point.

ΣF = -Ra + Rb - P = 0

P = Rb - Ra

Geometry 1

ΣMA = -P (b) + Rb (b+c) = 0

P = Rb (b+c)/(b)

Geometry 2

ΣMA = Rb (b) - P (b+c) = 0

P = Rb (b)/(b+c)

Since the reaction forces are known from the load cells, the magnitude and placement of the point load can be found with these equations.

2) Newton’s 2nd Law: sections of bar

By taking four “slices” of the bar and taking the sum of the forces and the sum of the moments at each slice, the shear forces and the moments can be found. Below are the slices for Geometry 1. Geometry 2 is solved in a similar fashion.

Section 1

ΣF = -V1 = 0

V1 = 0

ΣM = M1 = 0

M1 = 0

Section 2

ΣF = Ra - V2 = 0

V2 = Ra

ΣM = -Ra (x-a) + M2 = 0

M2 = Ra (x-a)

Section 3

ΣF = Ra - P - V3 = 0

V3 = Ra - P

ΣM = -Ra (x-a) + P (x-a-b) + M3 = 0

M3 = Ra (x-a) - P (x-a-b)

Section 4

ΣF = Ra - P + Rb - V4 = 0

V4 = Ra - P + Rb

ΣM = -Ra (x-a) + P (x-a-b) - Rb (x-a-b-c) + M4 = 0

M4 = Ra (x-a) - P (x-a-b) - Rb (x-a-b-c)

Where ‘x’ is the distance from the left end, V is the shear force, and M is the moment. Note that each equation only works for the section it is in.

3) Solve for deflection

Since M = E I d2y/dx2 where ‘E’ is the modulus of elasticity and ‘I’ is the moment of inertia, the deflection ‘y’ of each section can be attained by integrating.

M1 = 0 = E I d2y1/dx2

E I dy1/dx = C1

E I y1 = C1 x + C2

M2 = Ra (x - a) = E I d2y2/dx2

E I dy2/dx = Ra (x2/2 - ax) + C3

E I y2 = Ra (x3/6 - ax2/2) + C3 x + C4

M3 = Ra (x - a) - P (x - a - b) = E I d2y3/dx2

E I dy3/dx = Ra (x2/2 - ax) - P (x2/2 - ax - bx) + C5

E I y3 = Ra (x3/6 - ax2/2) - P (x3/6 - ax2/2 - bx2/2) + C5 x + C6

M4 = E I d2y4/dx2 = Ra (x - a) - P (x - a - b) - Rb (x - a - b - c)

E I dy4/dx = Ra (x2/2 - ax) - P (x2/2 - ax - bx)

- Rb (x2/2 - ax - bx - cx) + C7

E I y4 = Ra (x3/6 - ax2/2) - P (x3/6 - ax2/2 - bx2/2)

- Rb (x3/6 - ax2/2 - bx2/2 - cx2/2) + C7 x + C8

Plug boundary conditions into these equations to solve for the constants, ‘Cn’.

i. @ x = a : y1 = 0

ii. @ x = a : dy1/dx = dy2/dx

iii. @ x = a : y2 = 0

iv. @ x = a + b : y2 = y3

v. @ x = a + b : dy2/dx = dy3/dx

vi. @ x = a + b + c : y3 = 0

vii. @ x = a + b + c : y4 = 0

viii. @ x = a + b + c : dy3/dx = dy4/dx

Once the constants are known, the deflection equations are as follows for Geometry 1.

y1 = (a-x)(b3Ra+3b2cRa+3bc2Ra-c3Rb)/(6(b+c)E I)

y2 = -(a-x)(-b3Ra-3b2 cRa+a2(b+c)Ra+c3Rb-2a(b+c)Rax

+cRax2+bRa(-3c2+x2))/(6(b+c)E I)

y3 =(a+b+c-x)(a2(b+c)Rb+b3(Ra+Rb)-b2(3Ra+2Rb)x

+cRbx(c+x)+bx(-c(3Ra+Rb)+Rbx)

+a(b+c)(b(3Ra+2Rb)-Rb(c+2x)))/(6(b+c)E I)

y4 = -(a+b+c-x)(2b3Ra+6b2cRa+3bc2(Ra-Rb)-2c3Rb)

/(6(b+c)E I)

These equations are used to govern the data acquisition.

DATA ACQUISITION

In order to display the beam bending, shear and moment diagrams on a computer, it was necessary to acquire force readings from the beam. This was done by means of two load cells placed at the two supports of the beam, as seen in Figure 8. Both these load cells were acquired from the Omega Corporation. The specifications for the load cells are as follows:

0-100 lbs range for tension/compression

10VDC input voltage

200mV full-scale output voltage

[pic]

Figure 8: Setup of the Visual Beams apparatus

The output from the two read cells was read in using a data acquisition card that would convert the analog input voltage to a digital signal. There are two ways in which this could be implemented.

1. The analog signal can be converted to digital using an external ADC. The readings from the ADC could then be sent to the computer using communication standards such as RS-232 or Ethernet.

2. The ADC used for the conversion could be in the form of a PC ‘plug and play’ card, which would interface directly to the PCI or ISA bus inside a computer.

The advantage of using an external ADC is that they are significantly cheaper than their computer-based counterparts and also offer significantly more flexibility in features such as sampling rate. However they can only handle a few (1-4 channels) whereas the plug-and-play ADC cards can generally handle at least 16 channels simultaneously. Additionally, the computer-based ADCs would be more convenient for this particular application since the sole purpose of it is to send the readings to the computer. As such it was decided to use a PCI-based ADC card manufactured by National Instruments for the data acquisition portion of the project. The card being used was the National Instruments PCI-6023E DAQ card. This card had the following specifications:

12-bit resolution

200 kS/s sampling rate

16 analog input channels

One of the main reasons for acquiring the PCI-6023E was that it comes with a National Instruments driver that can be used with the LabVIEW™ software. This software was one of the two that were being considered for displaying the acquired data on the computer. As such, it was decided the PCI-6023E DAQ card for data acquisition.

SOFTWARE INTERFACE

Creating the software interface was divided into the following main sections.

Selecting the interface:

The data acquired from the DAQ card needs to be displayed on the host computer. This can be done by means of commercially available software. Two such programs were considered for use in the project.

1. LabVIEW

2. MATLAB

Each type of software had its advantages and disadvantages. MATLAB is very useful for solving complex differential equations and other forms of mathematical manipulation. However, its graphic capabilities are limited and it is not very user friendly. LabVIEW, on the other hand, is very good at handling graphics but it cannot solve complex equations. After some consultation, it was decided that the equations that needed to be solved were in fact very simple and as such it was decided to use LabVIEW. Later on, it also emerged that LabVIEW has the capability of importing MATLAB scripts for solving equations.

Upgrading Hardware:

The computer being used for the Visual Beams project was found to have limited memory as well as hard disk space. It had 64MB of RAM and about 2GB of disk space. The operating system for the Machine was MS Windows™ version 4.0, which needs about a significant amount of both, RAM and disk space. As such, it was decided to upgrade the computer’s hardware. The project team replaced the two DIM (Dual Inline Memory) RAM chips with two new ones giving a total of 128MB of RAM. A new hard drive having capacity 39.8GB was also installed.

Acquiring Data:

As mentioned before, the PCI-6023E DAQ card has drivers designed for use with LabVIEW. The card being used had been setup by the clinic team who had worked on this project in spring semester 2001. Documentation regarding the setup can be found in the Journal of Rowan Engineering/Mech, Spring 2001. The first six acquisition channels had been labeled as follows:

1) 0Top 2) 1Bottom 3) 2Left

4) 3Right 5) 4Vertical 6) 5Torsion

The current project only required the use of the first two channels and so it was decided not to re-label the channels. These channels correspond to the following pins on the NI-DAQ board:

Channel 1 – Pins 34 and 68

Channel 2 – Pins 33 and 66

In order to ensure that the setup was functioning properly, it was necessary to display the acquired data in LabVIEW. This was done using a standard LabVIEW VI (virtual instrument) called ‘Analog Input One Point.vi’. This VI acquires data from the ADC one point at a time from each channel. As input, it requires the device number of the card (0 in this case) and the channels to be read from (0Top, 1Bottom). It was verified that the data acquisition process was working properly by creating a VI named ‘Analog.vi’, which would display the channels on the screen.

The team also experimented with certain types of filters such as Butterworth and Chebychev to remove the noise in the channels but did not succeed. Instead, this Vi was used to find the average signal sent by the load cells when there was no load. This number was then subtracted from the signal to null out the effects of noise and weight of the beam itself.

Displaying the bending diagram:

The first part of the project was to display the bending diagram of the beam based on equations shown earlier in the report. A Virtual Instrument in LabVIEW is divided into two sections – the front panel and the wiring diagram. The VI that displays the bending diagram is called ‘Combined_Graph.vi’. The front panel for this VI consists of a ‘X-Y chart’ that displays the bending of the beam as shown in Figure 8. There are four values that the user needs to input for the VI to function properly:

1. Position of the supports A and B

2. Position of the weight

3. Scaling factor along the Y-axis

4. Number of points to be used in the display

[pic]

Figure 9: Bending diagram of beam

In the wiring diagram, the VI is divided into three sections that execute in sequence. The first section acquires all the data from the user and processes it to create the variables that are used in equations whichever. These variables are then passed to the second section. The second section initially creates an array of numbers going from 0 to 30 (length of the beam). This is the X input to the chart that plots the bending diagram. For each value of X, this section calculates the corresponding value of Y. This calculation is done by a VI called ‘Decision.vi’.

This VI takes as input, the position variables and the reactions at the two supports. It contains the equations for y1, y2, y3 and y4 shown in the ‘Beam Deflection Model’ section of the report. Based on these equations, it calculates the Y-value and sends it back the second section of Combined_Graph.vi. The X and Y values are then sent to the front panel to be displayed. The third section of the Combined_Graph.vi is simply a wait function that waits for 0.5 seconds after each screen update. This wait function was introduced only to give the operating system some time to clear the RAM after the calculations had been performed.

Displaying the shear and moment diagrams:

The VI for displaying the shear and moment diagrams is called ‘Diagrams.vi’. This VI functions in the same way as Combined_Graph.vi. The main difference is in the equations that are used to compute the diagrams. Since these equations are not very complicated, they are programmed in Diagrams.vi itself and no sub-VI is created. The front panel of this VI that has the shear and moment diagrams can be seen in Figure 10. There are only two user inputs since this VI assumes that the supports are fixed at the end of the beam. The inputs are:

1. Position of weight

2. Number of points to be used in the display

[pic]

Figure 10: Shear and Moment Diagrams

An implicit assumption made by this VI is that the weight is always between the supports. This is always the case since the VI assumes that the supports are always at the end of the beam. Some modifications will be required before this VI can handle more generic inputs.

TECHNICAL IMPACT STATEMENT

Economic Impact

The first question that this project raises is if the cost of the visual beam is feasible. Currently the cost for a load cell is 295 dollars each. This project only requires two load cells, therefore is already less expensive then Visual beam I. Beam material and other raw material can be mass-produced at a cheaper rate, which would drive down the price of the product. The final cost of this project would be low enough for the public institutions.

Environmental

This product is made with environmentally friendly parts, therefore having no ill effects on the environment.

Sustainability

The life span of this product will be extremely long. It is a very simple design with low wear surfaces.

Manufacturability

This product can be machined very easily. It is composed of aluminum and the beam material is made out of acetal. The slider, load cell, and the track were purchased separately. Everything else was machined at Rowan University’s manufacturing facility.

Health and Safety

The materials chosen are safe to handle since Visual Beams is a hands-on tool for learning. This product also complies with the safety and health standards put forth by the Rowan University faculty in regards to the operation in the machine shop.

SUMMARY/CONCLUSIONS

This project was completed with great success. It was designed, built and tested for durability and functionality. This semester, the simple support blocks were manufactured, beam material selected and shaped, calculations were derived, and a functioning LABVIEW interface was programmed.

Visual Beams is an ongoing project that will soon be implemented in many learning facilities that teach the intricate concepts of solid mechanics. This tool will bring the education of solid mechanics to another level for both students and professors.

FUTURE WORK AND IMPROVEMENTS

There is always room for improvement on any project. The design of the system could be redone to work out some minor discrepancies; for example, the beam could be higher to give more clearance for loading conditions.

To improve the quality of future work concerning this project the following recommendations have been made. In respect to the beam design and material selection, the cross-sectional area should be kept constant for all the chosen shapes, and allocations should be made for the use of different overall dimensioned beams. This would allow beam mass to be distributed differently for each shape demonstrating many solid mechanic principles. For example, this would more clearly illustrate the advantages of using a taller and narrower I-beam over a smaller height but wider solid beam. The overall dimension on our design was fixed the dimensions of the sliders.

Future work with this project would include writing an OpenGL program so that a real-time stress strain visual aid could be shown. Also further work would include the manufacturing of cantilever beams of different shapes and sizes. In addition to this tests must be done to insure the accuracy of the torque transducer readings.

Acknowledgments

Faculty:

• Dr. Jennifer Kadlowec

Dr. Clay Gabler

Dr. Douglas Cleary

Staff

• Chuck Linderman

NI Support Engineers:

Randy Hoskin

Najib Hassan

References

1. NI-DAQ information manual, National Instruments, , Accessed: December 18, 2001

2. Journal of Rowan Engineering/Mech, Spring 2001 Vol 1.

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