cantilever theory and experiment

Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

Experiment: Measurement of Young's modulus

View a definition of Young's Modulus .

A cantilever beam is fixed at one end and free to move vertically at the other, as shown in the diagram below.

Geometry of the cantilever beam test.

For each of three strips of material (steel, aluminium and polycarbonate), the strip is clamped at one end so that it extends horizontally, with the plane of the strip parallel to the plane of the bench. A small weight is hung on the free end and the vertical displacement, δ , measured. The value of δ is related to the applied load, P , and the Young’s Modulus, E , by

where L is the length of the strip, and I the second moment of area (moment of inertia). View derivation of equation .

For a prismatic beam with a rectangular section (depth h and width w ), the value of I is given by

By hanging several different weights on the ends of the strips, and measuring the corresponding deflections, a graph can be can be plotted which allows the Young's modulus to be calculated. This is repeated for each of the three materials. The calculated values for the Young’s modulus may be compared with the values in this properties table .

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Experiment to determine Young's modulus

Measuring Young's modulus. (Click on image to view a larger version.)

View QuickTime video of experiment (1.2 MB) ... in separate window ... video alone .

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Sine Sweep of base excitation of cantilever

A. Beams and Cantilever

Beams are structural members that have smaller dimensions of cross sections compared to its length (its axis) and are subjected to loads perpendicular to its axis; i.e. they are subjected to transverse loads. The whole beam deforms in the plane containing the axis and the transverse loads. We say that the beam bends. The beams are usually supported at both ends and they are termed differently depending on the support conditions. When one end of a beam is fixed, and the other free, it is called a Cantilever beam, or simply a Cantilever. When both end-supports re simple, the beam is called a Simply Supported Beam. If both ends of a beam are fixed, it is a Fixed-Fixed Beam or simply a Fixed Beam.

B. Physical systems that can be modeled as cantilever:

The diving board on a swimming pool, the slab on a porch, wall mounted structures, overhanging booms of cranes, etc can be modeled as cantilever. These physical systems can be idealized with loss of some accuracy and generalization but ability and simplicity of analysis. The vibration characteristics of these systems can be very well understood by knowing the vibrations of its cantilever model. As explained in the general theory, the characteristics of natural vibration are extremely important in knowing the response of the systems to forced excitations.

In this experiment, we shall find out the natural frequencies of a cantilever from its response to harmonic support-excitation.

C. Natural Vibration of a Cantilever - Natural frequencies and mode shapes

A Cantilever is a continuous system-its mass and elasticity are distributed all over its volume. It can be considered to have infinite very small masses connected by infinite very small springs resist the banding of the Cantilevers. Hence there are infinite degrees of freedom and infinite natural frequencies. And also, corresponding to every natural frequency, it has a particular shape of vibration, called Mode Shape. The lowest natural frequency is called Fundamental natural frequency and corresponding mode, fundamental mode or simply the first mode. Here are animations for first three modes of vibration of a Cantilever. Click on the figures to see the modes.

A cantilever of rectangular cross section bxd; Area of cross-section, A = bxd; and length L is shown in the figure. Cross-sectional dimensions are small compared to its length.

Let us consider its natural vibration in vertical plane, perpendicular to its length L. Let I be the second moment of the area of cross section about neutral axis perpendicular to the plane of vibration;

Let E be the modulus of elasticity of the material from which the cantilever is made. For steel value E is taken as E = 210GPa (210x109 N/m 2 ) and for Aluminum, it is 70 GPa Let ρ be the density of the material; for steel, ρ = 7800 kg/m 3 ; for Aluminum, ρ = kg/m 3

D. Equation of Motion

Once disturbed from its position of equilibrium and left to its own, the cantilever will vibrate naturally; it will perform natural vibration. From theory, we know that the vibration of a cantilever is governed by the equation

and Shear force

For small amplitudes of vibration of the cantilever, the motion can be assumed to be harmonic and we can write this equation in terms of amplitude of vibration as a function of x alone. The equation is as follows:

There are infinite sets of V(x) and λ which together satisfy the above equation. Such problems are called Eigenvalue problems and the solutions are called eigenvalues λi, and eigenvectors V(x)i.

V(x) is function of x that shows shape of the cantilever (Amplitudes of vibration at different values of x) corresponding to the respective frequencies of natural vibration λi. The shape of cantilever vibrating with certain natural frequency is called mode shape of cantilever for that frequency. Three of them were shown in figures earlier.

E. Harmonic excitation, Resonance and Natural frequency

When the cantilever is subject to forced vibration, it will vibrate. This can be done by holding the cantilever in a fixture, firmly mounting the fixture on a table of vibration shaker, and vibrating the shaker harmonically up and down with a known frequency. If we gradually change the frequency of vibration of the table within a range, say from 20 Hz to 2000 Hz, and note the amplitude of vibration of the tip of cantilever, we can plot a graph of amplitude of vibration of the cantilever versus frequency of excitation. We know that when the frequency of excitation matches with the natural frequency, the amplitude of response is large. Thus the frequencies in the graph mentioned just now corresponding to high values of response must correspond to the natural frequencies of the cantilever in the selected range of excitation frequencies. Thus, we find first few natural frequencies of the cantilever under test by exciting the support harmonically and carrying out the "Sine-Sweep"

Vibratory systems around us

Here are some examples of physical systems where the vibrations are prominent and can be observed easily. In musical instruments the vibrations are intentional. The parts of musical instruments are designed so that they generate sounds that are pleasant to listen. In many cases the vibrations are unwanted and we try to minimize them.

A chandelier hanging from ceiling oscillates to and fro following an initial disturbance; maybe due to a breeze of air.

The oscillations of the chandelier at cathedral of Pisa, Italy, were studied by the famous scientist Galileo Galilee.

A load attached at end of a wire-rope of a crane oscillates to and fro due to initial disturbance; maybe due to sudden stopping of carriage of the crane while revolving about the vertical axis.

The pendulum used in clock of olden days used to oscillate to and fro once every second. i.e. it had a period of oscillation of one second.

String of a guitar, when plucked and left to its own, vibrates and makes a musical sound. It comes to rest after a while; the vibrations die out. Similarly, the diaphragm of a table vibrates when hit and left to its own. It also comes to rest after some time.

All these are examples of vibratory systems that are set into vibration following an initial disturbance. All these systems have three components: mass, due to which the system possesses inertia; elasticity, due to which potential energy can be stored; and components that dissipate energy causing the vibratory motion to be damped which bring them to rest after some time.

Vibration or vibratory systems are classified in number of ways. Some of the classifications are given below:

Free and forced vibration - A free vibration occurs due to initial displacement or velocity, or both, applied to the system only initially. There is no external force acting on the system when the system is vibrating. A forced vibration occurs when the system vibrates in response to external force applied continuously. When the force applied is periodic, i.e. it repeats itself after a fixed interval of time, the forced vibration is called periodic. If the periodic force and hence the resulting vibration varies sinusoidally with respect to time, the vibration is called harmonic. If the force is not periodic, the forced vibration is called aperiodic or random.

Damped and undamped vibration - When the vibratory system has elements that offer resistance to motion, energy is continuously dissipated and the free vibrations of such systems come to halt after some time. This is called damped vibration and such systems are called damped systems. Forced vibration of a damped system continues as long as the force acts but some of the work done by the external force is lost in overcoming the resistance offered by the damping elements. Systems without damping elements are called undamped systems and their vibrations are called undamped vibrations. All systems in nature have some or the other damping element and their natural vibrations are damped. Hence they come to rest after some time following free vibrations. Nevertheless, we study the vibration of undamped systems because the concepts developed in studying them are useful in analyzing and understanding the phenomena occurring in vibration of damped as well as complicated systems. When the force of resistance offered by a damping element is proportional to velocity of mass of the system, it is termed as viscous damping and the damping element is called a viscous damper. If the force of resistance has a constant value, it is termed as Coulomb damping. Damping due to dry friction shows this kind of behavior. Coulomb damping can occur when the system has components rubbing over each other. There are other types of damping also which shall be discussed later.

Degrees of freedom : The vibratory systems are classified as single-degree-of-freedom systems, Multi-degree-of-freedom-systems or continuous systems. The number of degrees of freedom corresponds to the number of independent co-ordinates required to completely describe the motion of the system. In fact, it is the sum of the possible ways each mass can move independently of other masses. The translation of a mass along the three axes, X, Y and Z, and the three rotations about each of these axes constitute possible ways of motion of a mass. Many times, many of these six motions of a mass are restricted and a mass can have one or two degrees of freedom, i.e. only translation or translation and rotation of a single mass about any one of the axes.

Linear and Non-linear Vibrations : Vibration is said to be linear if the damping force is proportional to velocity, inertia force is proportional to mass, and restoring force is proportional to displacement. If any of this proportionality is not satisfied, the system is said to be non-linear.

Solving engineering problems : Analytical methods are usually applied to models of actual systems. We carry out experiments on models if physical systems are not available for testing. While preparing such models, we exclude superfluous details of the system but include all essential and important features of the actual system. While doing so, we idealize and approximate important behaviour of the system without affecting much the accuracy in predicting the behaviour. The system model so developed provides ease of application of analytical and experimental techniques. Once a satisfactory model is developed, laws of Physics can be applied which give a set of mathematical equations relating the properties and variables of the system. Such a set of mathematical equations is called mathematical model of the system. Solving the set of equations (or a single mathematical equation) provides expression for the system variable in terms of location and time. We call this as 'solution' of the problem. As an illustration of the concepts described above, see the example given below.

Mathematical model

Using Newton's second law of motion, the equation of motion of the mass is written as

The first term is the inertia force which is equal to mass multiplied by acceleration and the second term is the spring force given by stiffness of the spring multiplied by its elongation or compression.

The differential equation is a mathematical model of the system.

General Solution

The solution to the above differential equation is given by

A and B are constants that depend the initial conditions, i.e. the displacement and velocity of the mass when we started measuring our time.

These are known as initial conditions.

Particular solution obtained from the initial conditions

Now the expression for x becomes

And we can obtain the value of x at any time t from this expression.

Thus we have obtained the expressions for natural frequency and position of the head at any given time 't' and the problem stated by the problem statement is solved.

AOE3054 - Experiment 2 - Static Deflection of a Beam

Experiment 2 - static response of a beam.

1. Introduction

Characterizing how an engineering structure will deform under a steady load is key to the design and development of a broad range of engineering hardware. Theoretical methods for predicting deformation, such as you have already met in your coursework, obviously have an important role to play. However, the structural elements of real vehicles are often complex - consider the wing of a large transport aircraft, the rudder assembly of an oil tanker or the robotic arm of a space probe. Theoretical methods, by themselves, are not completely reliable for such "real-world" structures. Predictions for these structures inevitably involve simplifying assumptions about the structure and material, and will be based on incomplete or uncertain information about the environment (i.e. the boundary conditions) in which the structure must operate. Experimental testing of structures is thus important. Testing is used not just to prove that the final design works as intended, but also to examine whether the theoretical models or boundary information used to predict the behavior of the structure are correct and complete. Testing is subject to error as well. Thus a good experiment characterizes not just the properties of the structure in question, but also investigates and quantifies the errors in the measurements themselves. Only then can one be sure enough to make decisions, based on the test, that influence the form of the vehicle, the success of the mission, or the lives and investments that they may involve.

The objectives of Experiment 2 are: 1. to introduce you to some basic techniques used to measure the static deformation of a structure, 2. to provide you with a set up where you can use those techniques to examine validity and consistency of a simple theoretical approach (beam theory) when applied to a simple "real-world" structure, 3. to provide you with a set up where you can investigate the validity and accuracy of the measurement techniques themselves.

You will find that this manual does not prescribe a specific set of measurements you should make or conditions you should test. These will be your choices to make as a group, based upon your understanding of the theory, the set up and measurement techniques, and the specific goals provided for you. It is therefore critical that you prepare for this lab by making sure you understand and can use the theory, are familiar with the descriptions of the measurement techniques and experimental set up as described below, and are organized as a group.

The apparatus to be used in this experiment consists of a simple beam (this will be your test structure), a frame in which that beam can be mounted and its deformation measured, and a loading fixture that allows weights to be attached to the beam.

The Beam The beam, shown in Figure 1 , is constructed from aluminum alloy and has a rectangular cross-section that is nominally 1/4 by 1 1/2-inches. The corners at one end of the beam are beveled. You should measure the actual cross-sectional dimensions with the dial caliper provided, and use those actual dimensions in any calculations you perform (note that the two beams may have slightly different dimensions).

Attached to the beam are two electrical resistance strain gages, one on each side. As you learned in the online lecture, the stretching of an electrical conductor increases its resistance. The gages are designed to take advantage of this effect to measure the strain. They consist of a single conducting strip deposited on a film that is then glued to the beam. At first sight the conducting strip appears to have the form of a simple rectangular region. On closer inspection, perhaps with a magnifying glass, you will see that in fact the conducting region consists of a thin conducting path that zigzags back and forth along the long axis of the beam. Confirm in your own mind that this will make the gage primarily sensitive to normal strains along that long axis. Be careful of the strain gages and their electrical connections, they are delicate and may be important for your test.

Loading Fixture and Weights The loading fixture and one of the weights are shown in Figure 2 . The two Allen bolts in the loading fixture can be loosened so that it can be slid onto the beam. Do not slide the loading fixture over the strain gage. Completely remove the fixture (by undoing the Allen bolts using the key provided) if you need to move it from one side of the strain gage to the other. The weights are machined from aluminum bronze rod stock of 5-inch nominal diameter. They are nominally 2, 3, 5, 10, and 15 pounds, and each has a hook threaded into a tapped hole on one side for hanging them from the eye screw of the load fixture. A digital weighing machine is provided for you to measure the exact values of these weights, the loading fixture and any other items you feel are relevant.

The Frame The loading frame, built according to a design developed by Durelli et al. (1965), is shown in Figure 3 with the beam, loading fixture and weight assembled in one possible configuration.

The frame includes two vertical beam supports of the same height. These can each be placed at any one of 7 positions along the bottom of the frame. The positions are at exactly 2.5 inch intervals. The beam can also be fixed using the slot cut into the left hand side of the frame, and the clamp, see Figure 3 . The slot holds the beam at the same height as the vertical supports. The clamp is tightened using Allen bolts on the outside of the frame, and a threaded knob that passes through the top of the frame. With the with one of the beam supports in its left-most position, the clamp and slot can be used in combination to approximate a cantilever support. Note that if used, the both the bolts and threaded knob must be tightened firmly to produce a repeatable support. A second threaded knob and other threaded holes for the these knobs are provided along the top of the frame, so that it is possible to pin the beam to vertical supports at other locations along the frame, if necessary.

The top of the frame contains a series of machined holes, spaced at two inch intervals along the frame. These holes are designed to accept dial indicator gages to measure beam deflection. Figure 3 shows an indicator mounted in one such hole. Note that, for this particular beam configuration, this indicator wouldn't tell you very much, as it is located at a position where the beam deflection is ensured to be zero by the vertical support.

Note that this device is very flexible and allows for many beam configurations. It is thus ideal for testing simple theoretical predictions or measuring material properties and assessing instrumentation limits and errors through beam theory. For example, Figure 4 shows an arrangement producing a cantilevered beam, Figure 5 shows a simply supported beam, and  Figure 6 shows a simply supported beam with overhang. Note that Figure 3 shows an indeterminate beam configuration.

You have a variety of instrumentation available with each set of apparatus. Minor items include a ruler which you can use to accurately position the beam and the loads, a dial caliper to check beam dimensions and a digital camera which you can use to record your various set ups, and photograph items that you want pictured in your logbook. Your lab TA will be able to explain operation of the digital camera to you. The remaining items of instrumentation are those used to measure the beam deformation, namely the dial indicators and strain gage system.

Dial Indicators A number of basic mechanical dial indicators (see Figure 3 ) and two electronic indicators (not pictured) are available to measure beam displacements. Either type may be inserted through the holes on the top of the loading frame so that the end of the plunger rests on the beam or, if it at the same location, the top of the loading fixture.

The mechanical dial indicators are manufactured by the Chicago Dial Indicator Company (Des Plaines Il), model number 2-C100 1000.  Each gage has a range of one inch with each graduation on the large dial representing one thousandth of an inch. One complete revolution of the pointer on the large dial represents a displacement of 0.100 inches. The number of revolutions are counted on the small dial. It is possible to set the large dial to read zero, for example when the beam is unloaded, by loosening the thumb screw on the top right of the dial and rotating the outer ring of the dial face. In principle it is possible to read thes indicators to four significant figures by visually estimating the location of the needle between the smallest divisions on the scale.

The electronic indicators, Mitutoyu Model 575-123, have a range of one inch and read in increments of 0.0005 inches, and are generally easier to use.  They have inch/metric conversion, zero setting to any position, +/- counting direction, and position memory (the dimension indicated always reflects the movement of the stem from the last set zero position, also known as absolute (ABS) positioning). Their accuracy is ±0.0008 inches. Note that this is greater than their resolution. When inserting the electronic indicators into the holes in the loading frame, thread the indicator spindle through one or two of the washers provided at the workbench. This will raise the body of the indicator a few mm and enable it to work when the beam is in the unloaded position, or higher.

Note that the above accuracies are under ideal conditions and you may find that other factors tend to overwhelm the level of accuracy, particularly for the mechanical indicators. One of these factors is that the spindle of the dial indicator may stick. You can usually avoid this by sliding the spindle in and out a few times to loosen the gears before the indicator is installed. A second factor that may influence the accuracy of your measurements is the force exerted by the spindle on the beam, and the variation of that force with extension of the spindle. You should be aware of this factor as you proceed with your test, so that you are prepared to quantitatively assess its influence (and perhaps correct for that influence) during your tests.

You can also use the dial indicators to check that the load is supplied symmetrically to the beam. This can be done by inserting two dial indicators into the loading frame such that their spindles contact the beam at two widthwise points on the top surface that have the same spanwise location ( Figure 8 ). By monitoring the readings of these two dial indicators to make sure they remain the same, you can establish that the beam does not twist under load, and thus that no torque is applied to the beam as it is loaded.

Strain Gage System

The electrical resistance strain gages bonded to each beam are manufactured by Micro-Measurements Division, Measurements Group, Inc., Raleigh NC. As described above they consist of a single conducting strip deposited on a film that zigzags back and forth along the long axis of the beam. The gages are therefore sensitive to strain in that direction. Technically, the gage designation number is EA-13-125BZ-350.  The "EA" of this designation indicates these are polyimide-backed constantan material foil gages, the "13" designates the self temperature compensation of the gage with respect to aluminum, the "125" designates that the active gage length is 0.125 inches, "BZ" denotes the grid and tab geometry (narrow, high resistance pattern with compact geometry), and the "350" means the resistance of the gage in ohms. The engineering data sheet for these gages is included with the experimental set-up. To set up the strain gages you will need the gage factor from this data sheet, of 2.145.

Since the strain gage is just a single conductor, there are only two connections to make, one at each end. You will notice there are three wires coming from each gage. The red wire connects to one side of the gage. The black and white wires both connect to the other side. The strain gage is operated using the Wheatstone bridge circuit that you were introduced to in class, in a quarter bridge configuration. This circuit, shown in Figure 9 is arranged in a diamond shape. Three sides of the diamond consist the simple resistors,  R2, R3 and R4. The strain gage itself is connected to the fourth side of the diamond. Voltage is applied to the top and bottom corners of the diamond, P+ and P-, and the output signal, which will be a voltage proportional to the resistance of the gage and thus the strain it experiences, is measured across the left and right corners of the diamond, S+ and S-. Note that the points S- and D350 on the diagram are actually connected together through the leads to the strain gage.

The Wheatstone bridge circuits for our strain gages are provided by a Measurements Group model SB10 Switch and Balance Unit,  Figure 10 . Strain gages are connected into its bridge circuits using the array of electrical terminals on the right hand side of the unit. Note that there are 10 sets of terminals, each labeled P+, P-, S+, S-, and D corresponding to the 5 points in the bridge circuit in Figure 9 . There are 10 rows of terminals because this unit actually contains 10 bridge circuits that could be used to operate 10 strain gages simultaneously. We, of course, will only need to use rows 1 and 2 for the two strain gages on the beam. Consistent with Figure 9 one side of the first gage (the red wire) is connected to the P+ terminal in row 1. The other side of the gage (black and white wires) are connected to the S- and D terminals of row 1. The second gage is connected in the same way to row 2.

To supply the voltage to the P+ and P- terminals of the bridge circuits, and to read the strain from the S+ and S- terminals we have a Measurements Group model P-3500 Strain Indicator, also shown in Figure 10 . Obviously this device has to be connected to the switch and balance unit containing the bridges. The connections are straightforward. Use the banana plug cables provided to connect P+ on the P3500 to P+ on the SB10, P- to P-, S+ to S+, S- to S- and D350 to D. You should also connect the two ground (GND) sockets. The black knob on the left hand side of the SB10 controls which of the bridge circuits, and thus strain gages the strain indicator is connected to.

For the switch and balance unit to accurately output the strain the gage factor (of 2.145) must be entered. This is done by pressing the 'Gage factor' button (3rd from left) and adjusting the knobs immediately above it so that the display reads '2.145'. It is likely that the gage factor will be set equal, or close, to this value already. To use the strain gage system to make a measurement, first select the gage you are interested in (black knob SB 10) then press the 'Run' button (4 from left on the P3500). The display shows the strain in microstrain (strain times 10 -6 ). That is, a reading of 1234 means a strain = change in length / length = 0.001234. Your measurement will be the difference between this displayed value when the load is applied, and when it is not. Note that for convenience you can usually set the offset of the display to zero when there is no load by turning the 'balance' knob on the SB-10 that corresponds to the strain gage you are looking at.

If you need further help setting up or understanding this system instruction booklets are provided with both the SB10 and P3500 units.

where b is the width ( x -direction) and h (y-direction) is the height.

where y = h /2 at the bottom of the beam and y = - h /2 at the top of the beam.

D. Example displacement and moment calculation Problem: Determine expressions for the midspan displacement and axial strain for the three-point bend test pictured in Figure 5 . Take the distance between the vertical supports as L and assume that the load is applied at midspan (midway between the supports).

Consequently, the midspan displacement is

A. Getting familiar with the apparatus and instrumentation. The following procedure is designed to help you get a feel for the apparatus and instrumentation and, most importantly, its limitations. This is a group activity, but one in which it is important that you get individual hands on experience and that everybody understand and appreciate the items and issues raised below.  Discussion is key - each student must know how to use the apparatus, what the problems are, what measurements (in addition to those made by the dial indicators and strain gages) will be critical. Ensure that all the important observations (measurement/error lists, error estimates etc.), both qualitative and quantitative are properly recorded, along with any explanations, in the electronic lab book.

• Try out the weighing machine. Measure some weights. Measure the weight of the loading fixture (don't put more than 17 pounds on the scales).
• Try assembling the beam, frame and loading fixture into one of the configurations pictured in Figures 3 through 6 . Be sure to position the beam so that the strain gages are at a location where they will have something to measure.
• Insert two dial indicators at the location of the loading fixture (plungers should rest on top of the loading fixture, away from the Allen head bolts) so as to monitor both the deflection of the beam here, and to warn of any twisting. Insert a third dial indicator elsewhere (touching the beam) to monitor deflection here. Try setting the ring of the indicators to indicate zero deflection in this no-load case.
• Add a load to the beam and look closely at the configuration you have assembled. Make sure you and your lab partners understand how you could do a beam theory calculation for this configuration to predict the deflection produced at each of the dial indicator locations and the strain at each of the two gages.  Compile a list of all the measurements you would need to make accurately before you could get an actual number from a beam theory calculation of your configuration (e.g. exact position of the load, supports, dimensions of the beam, locations of gages etc.).
• Compile a list of any factors that you think might introduce error (items that are not included in the theory but have an effect on the experiment) and think about how you might minimize those factors. You will probably want to add to the list as you try the items below.
• Try deflecting/twisting the beam with your hand to get an idea of how well the indicators work. Do the dial indicators show a deflection, a twist? Do you understand how to read the dials? How big is the deflection compared to the resolution of the dial? Do the dial indicators stick? (If so try moving the spindle in and out a few times to loosen it up.)
• Try tapping the beam at different points. Do the dial indicators return reliably to their starting positions? If not, how big is the change? How does the size of that change compare to the deflections you see under (hand) load? Try adding and removing a weight - do you see any change in the start position of the indicators? Note that repeatability in the indicators can be improved somewhat by making sure that the bolts on the loading fixture and clamp (if you are using it) and the threaded knob(s) (if you are using these) are tight. However, some inconsistency in the dial indicators is inevitable, try to quantify this error. The influence of this inconsistency on your results can be minimized, for example, by basing them on averages of measurements made with different loads, and in different configurations.
• Try removing the dial indicators one by one. When you remove an indicator do the remaining indicators show any significant change in deflection (indicating that the force exerted by the plunger may be significant)? If so, what is the magnitude of the change? Think about whether it will influence your results. Think about how you could estimate the spindle force using the beam apparatus.
• Wire up the two strain gages. Dial in the gage factors, balance the bridges (see the strain gage section for help here) with zero load. Add a load to the beam (with your hand if you like). Do the changes in the indicated strain make sense given what you understand about the stretching and compression of the fibers of the beam? Do their magnitudes make sense? What are the units? Remove the load. Do the indicators return to their starting readings exactly? If not, how unrepeatable are they? Try to quantify this error. Leave the strain gages and beam assembly alone for a couple of minutes. Do the readings drift? If so, estimate the rate of drift? How can you minimize the influence of this drift on your results?

Goal 1. Design, conduct and document a sequence of tests to measure, as reliably as possible, the Young's modulus of the beam material, and to to put a number on the reliability (i.e. the likely error, a.k.a. the uncertainty). Suggestions. Such a test will obviously require some application of beam theory.  The reliability of your result, and your ability to estimate that reliability, will be enhanced if you make multiple determinations of Young's modulus. For example, it is easy to make multiple measurements for a sequence of different loads. Even better are multiple measurements obtained in situations where the errors are unlikely to be the same, i.e. using independent measurement techniques and/or using different beam configurations. Keep careful documentation of what you do, why you do it, set up, characteristics, expected results, unexpected results, analysis, photos and plots in the electronic lab book as you proceed. Analysis should include uncertainty estimates for all results.

Goal 2. Design, conduct and document a sequence of tests to examine the validity (and extent of validity) of Maxwell's reciprocal theorem. Again, you should aim to gather whatever information you need to assess how good your result is. Suggestions. Do not feel constrained to a single 'reasonable' test. How unreasonable can the test be? How far apart can your two locations be? what may be between them? is there a limit to the deflection for the theorem to be valid? can you use the strain gages (you can at least use them to get additional data points on Young's modulus, for goal 1)? what about the sources of error you identified above? Keep careful documentation of what you do, why you do it, set up characteristics, expected results, unexpected results, analysis, photos and plots in the electronic lab book as you proceed. Analysis should include uncertainty estimates for all results.

Note that your grade does not depend upon how close your results agree with beam theory, Maxwell's theorem or any other pre-conception of what the answers should be . Instead it depends upon how open mindedly and objectively you assess your data, its accuracy, and what it shows, or appears to show when combined with the theory.

The group should leave few minutes at the end of the lab period for discussion and to check that everybody has everything they need. As a group go through the exit checklist .

6. Recommended Report Format

Before starting your report read carefully all the requirements in Appendix 1.

Title page As detailed in Appendix 1 .

Introduction State logical objectives that best fit how your particular investigation turned out and what you actually discovered and learned in this experiment (no points for recycling the lab manual objectives). Then summarize what was done to achieve them. Follow this with a background to the technical area of the test and/or the techniques and/or the theory (and its mathematical results). This material can be drawn from the manual (no copying), the online class or even better, other sources you have tracked down yourself.  Finish with a summary of the layout of the rest of the report.

Apparatus and Instrumentation Begin this section with a description of the beam, loading frame and related hardware, and connect these items to your goals.  For example, you might write "A frame of the type designed by Durelli et al. (1965) was used to load the beam in several configurations for the purpose of determining its Young's modulus." Labeled dimensioned diagrams of the beam(s) and strain gages are probably critical. Labeled photographs may suffice for other items.  Don't forget to objectively describe any deficiencies, irregularities, imperfections in the apparatus relevant to your goals, and given any primary uncertainties. Then describe the instrumentation used to monitor the beam, how it was used, and its accuracy and limitations. For example "The dial indicators were inserted through the top of the frame to measure beam displacement. These indicators allowed measurements to a resolution of... but their actual accuracy was reduced to about ... inches as a result of ... The indicator spindles exerted a force on the beam of about ??? which complicated measurements somewhat (see below), particularly when... Dial indicators were also used to check for twisting...". With your description of the strain gages include a circuit diagram, and properly reference the manufacturers and model numbers. If there were any uncontrolled variables in your experiment this is a good place to mention them and assess there likely impact, for example -  "For the three point bend test the location of the load was set visually at centerspan but, due to an oversight, the actual location was not measured or checked. The maximum error in the load location is estimated at 0.25 inches and the implications for this error for the results of this test are discussed with the results below" - or - "While two actual beams were used during the tests, the data were combined as though they referred to a single structure. While the differences in dimensions were small, it is not actually known that the beams were constructed from the same type of aluminum alloy, introducing a potential error into the results. The maximum size of this error is estimated through further analysis of the beam results in the following section". Such honest assessments greatly raise the quality of a report (and your grade)

Results and Discussion Before writing the results and discussion make sure all your results are analyzed and plotted, and any theoretical derivations have been completed and compared. Make sure your plots are formatted correctly - default Excel plotting format is not acceptable, see Appendix 1 . Plotting results in more than one way may make them clearer or serve more than one purpose. For example, you may have plotted beam displacement against load for each of a series of configurations for the purpose of illustrating the beam deformations used in estimating Young's modulus.  Once you have computed a value of Young's modulus for each measurement/configuration, you may then consider plotting the whole set of estimates together, say against displacement to see of the fact that the deformations are actually finite has influenced the estimates (the error would increase with load). Alternatively you might consider cross-plotting estimates obtained separately from the strain gages and dial indicators against load. That might reveal any errors associated with additional forces, from the loading fixture say. Such pictures make it much easier for you to explain and discuss the significance and implications of your results.

Since the beam configurations you used are probably key to your goals they may well be best shown and described in this section along with the results. You will really need clear dimensioned and labeled diagrams for these.

A good way of writing this section may be to tie each set of tests and results to one of your objectives stated in the introduction. (If you find it hard to do this, try changing your objectives!) For example, you might begin with "The beam was tested in two configurations for the purpose of  determining the Young's modulus of the aluminum from which it was constructed. The two configurations, a three-point bend test and a cantilever test, are illustrated in Figures 7 and 8. Loads of ??? were applied at the locations marked with the symbol 'P'. Figure 9 show the displacements measured at the dial indicator locations (marked 'A') plotted against nominal load. Figure 10 shows strains measured on the at locations 'B' on the upper and lower surfaces of the beam. Note that the results do not show exactly a straight line behavior....due to non-repeatability in the... To use these results to estimate Young's modulus, it is necessary to use technical theory to analyze the beam deformation. As a first step, we note that the moment distribution for these two beams can be written as: ... (note coordinate directions defined on figures 7 and 8)......The estimates of Young's modulus based on equation ?? and the dial indicator results are shown in...Estimates based on ... strain results..." Don't forget to describe your plots. If you have trouble finding deeper things to say about a plot, ask yourself why you are presenting it.

Make sure your results and discussion include (and justify) the conclusions you want to make. Also remember to include any uncertainty estimates in derived results. You should reference a table (copied out of your Excel file) or appendix containing the uncertainty calculation.

Conclusions Begin with a brief description of what was done. Then a sequence of single sentence numbered conclusions that express what was learned. Your conclusions should mesh with the objectives stated in the introduction  (if not, change the objectives) and should be already stated (although perhaps not as succinctly) in the Results and Discussion. 7. References

• Beer, F. P., and Johnston, E. R., Jr., 1992. Mechanics of Materials , McGraw-Hill Book Company, New York, pp. 608-610.
• Durelli, A. J., Parks, V. J., and DeMarco, M, 1965. "Multipurpose Loading Device," Final Report, Mechanics Division, The Catholic University of America, Washington D. C. 20017, September (Sponsored by the National Science Foundation, Grant No. NSF-G22968).
• Megson, T. H. G., 1990. Aircraft Structures for Engineering Students , Second Edition, Halsted Press, pp 98-102.

Introductory Material

Experiment manuals, instrumentation manuals, course organizer: aurelien borgoltz.

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An undergraduate experiment on the vibration of a cantilever and its application to the determination of Young’s modulus

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Keith Turvey; An undergraduate experiment on the vibration of a cantilever and its application to the determination of Young’s modulus. Am. J. Phys. 1 May 1990; 58 (5): 483–487. https://doi.org/10.1119/1.16480

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An experiment is described to test the theory for the frequencies of lateral vibrations of a uniform cantilever. Good agreement between experiment and theory is obtained. Young’s modulus for the cantilever material (steel) is determined from the observed frequencies of the fundamental mode for a series of different lengths, and the result is compared with that obtained from static deflection of the cantilever. The agreement between the dynamic and static determinations is satisfactory.

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