Introduction to Experiment 3: Vibrations

1.0 Overview:

From jackhammers to buildings, automotive suspension systems to aircraft gas turbine engines, it is obvious to even the most casual observer that vibrations are extremely important. In this experiment you will examine the vibrations of the spring, mass, damper system. In the experimental investigation you will look at the effect of sinusoidal excitation on a cantilever system with small damping. The thrust will be to gain an understanding of the natural frequency of a system. You will vary the dimensions (length) of the beam, hence affecting its frequency response characteristics. You will also use a delta impulse function for each case to determine the system’s damping coefficient. In the simulation portion of this investigation you will verify the theory behind the experimental investigation by comparing the system response using a simulated single degree of freedom system. You will expand your database by repeating the simulation using various materials, end masses and damping coefficients to quantify the system’s phase and amplitude response characteristics. This part of the lab provides insight into how the engineer and scientist can use damping to minimize the negative effects of vibrations. Before you begin this lab you should review second order differential equations with constant coefficients. Keep in mind what quantities in the governing second order differential equation for the spring mass damper system that you are changing throughout the experiment. This should be discussed in some detail in the lab write-up.

1.1 Elementary Theory:

1.1.1 A Simple Spring-Mass System

Many oscillating systems can be modeled as a spring-mass system using the differential equation of motion. The displacement, y(t), of such systems can be found using

(1)

where m is the mass of the object, c is the coefficient of viscous damping, k is the spring constant, and F(t) is some forcing function. Each term in this expression is actually a time-dependent force: my,, is the inertial force, cy, is the frictional or damping force, and ky is the spring force. Modeling the damping of a system in this way assumes that the damping force is proportional to the velocity of the mass; this is called viscous damping.

It is convenient to express damping non-dimensionally by calculating the viscous damping factor, , using

(2)

Here, is the natural frequency of the system as defined below. For , can be estimated by

(3)

where Y₁ and Y₂ are the values of any two consecutive maximum displacements from the time response that are one cycle apart.

The natural frequency of a system is the frequency at which an undamped system will freely vibrate, and can be calculated by using

  (3)

  (4)

1.1.2 Modeling a Cantilever Beam

The cantilever beam is an example of a system, which can be modeled as a simple spring-mass system. In order to model the vibration of the cantilever beam, the end of the beam is chosen as a reference point at which the characteristics and response of the beam are measured (see Figure 1). An equivalent system is then built that has a response, y(t), that is identical to that of the actual system. The spring constant, k, of the equivalent system is identical to that of the cantilever beam, and can be calculated quite easily using beam deflection formulae. Calculation of an equivalent mass is necessary because all points along the beam’s length do not have the same response as the end of the beam. This means that the equivalent mass, m, cannot be determined simply by adding the masses m_beam and m_end, but must be found by equating the energies of the two systems as they vibrate. This type of analysis is called lumping. A discussion of this technique can be found in J. E. Shigley, Mechanical Engineering Design, McGraw Hill, 1977.

Figure 1: Modeling the cantilever as a simple spring / mass system

The energy, U, stored in the simple spring-mass system is

(5)

where w, is the radial frequency and Y is the Mass’s amplitude. To find the energy stored in the cantilever, the energies of both the beam and the mass at the end of the beam must be considered.

  (6)

  (7)

Performing the integration, the energy stored in the actual cantilever system is found to be

  (8)

Hence, by comparing equations (5) and (8), we see that the real cantilever system can be modeled as a simple spring-mass system with a mass that is given by

(9)

1.2 Required Equipment:

• Shaker (single degree of freedom)
• Power supply for shaker
• B & K function generator
• 2 accelerometers
• 2 BNC to SMB coaxial cables
• Carbon steel bar
• Charge Amplifier (power supply for the accelerometers)
• National Instruments PXI system with LabVIEW 6.1
• PXI-8171 series controller with 862 MHz Processor, 256MB of RAM, 18GB memory with Windows XP OS.
• 1 x 4472 A/D card with 8 channel simultaneous sample and hold acquisition (SMB)
• 24 bit resolution
• Analog low pass filters
• 100 kHz max sampling rate.
• Measuring tape or ruler

1.3 Experiment Apparatus:

The apparatus consist of a steel cantilever beam mounted on a single degree of freedom shaker. Two accelerometers are mounted to the beam, one at the base (input) and one at the free end (response) of the beam. The charge amplifiers supply excitation to the transducers. In addition, the base of the beam is fixed using a removable clamp that can be used to adjust the length of the beam (which will change the natural frequency of the system).

The LabVIEW software will be used to acquire the experimental data. The format for some of the files is found following the "Topics for Discussion".

2.0 Experiment

Procedure:

Measure the dimensions of the beam for use in calculation of the theoretical natural frequency. We will be performing this test for three different bar lengths. Be sure to include the uncertainty associated with the measuring device. Let’s concentrate on the first bar length for now. Also, note the material of the beam (most likely carbon steel). The mass of the accelerometer is small and has little effect on the system’s response. Be sure to measure only the portion of the beam that will be vibrating. Do not include the portion of the beam located in the clamp.

2.1 The Shaker Test

In this experiment, the bars forced frequency is investigated. When the forcing frequency matches the beam's natural frequency, resonance is observed.

We begin by sweeping through several input functions to see where the different modes of the system’s natural frequency exist. .

• Make sure that the bar is clamped securely for a given bar length.
• Turn on the accelerometer’s charge amplifier and wait approximately 5 minutes for the accelerometers to stabilize.
• Turn on the power supply to the shaker (2 clockwise clicks), and power on the input function generator, (B & K unit).
• Set the output function on the function generator to sweep mode. This will increase the input frequency of the shaker table from 2Hz to 60Hz and back to 2Hz at a constant rate.
• Record the resonance frequency (s) observed.

2.2 The Accelerometer Test

In this test, the natural frequency and damping of the bar’s free response is investigated.

• Ensure that the first accelerometer (at the beams free end) is connected to channel 1 of the charge amplifier’s input. Use the coaxial cable to connect the output from the power supply (channel 1) to channel 0 of the PXI system. Do the same for the second accelerometer located at the base of the cantilever beam. That is channel 2 of the charge amplifier is connected to channel 1 of the PXI system.
• Open the VI (located on the desktop) "VIBEXPnew.vi" and make an observation of the various tabs and controls on the screen. The top plot is the time series response of the two accelerometers, where as the bottom two plots are the frequency spectra of the accelerometers 1 and 2 respectively.
• We will be sampling at 512Hz, low pass filtering at approximately 206Hz, and acquiring 1024 samples per scan. The "Save Spectra & Time series Data?" and "Save Peak Data" can be activated and/or deactivated at anytime during the experiment.

2.2.1 Recording the Natural Frequency of the System

We will be running this experiment for three different beams lengths. Therefore, be sure to properly name the output file accordingly, as not to rewrite over previous data; i.e. the first beam length experiment could use the file name "……\Desktop\MAMELab\Vibrations\2006-2007\monday\415-1\peakCSL1.txt".

• From your observations made during the shaker test, determine the frequency’s incremental spacing that would allow you to best capture the systems natural frequency. (For example: if you observed large excitations at 15Hz from the shaker table test, then perhaps you should choose 11 total increments; that is 5 decrements of 1Hz each below the 15Hz observation, and 5 increments of 1Hz each above 15 Hz. Therefore the total range over which you are acquiring data is from 10Hz to 20Hz, with 1Hz increments.)
• Change the Input Function Generator’s output to Sine Wave and dial the input frequency to your first desired setting. Be aware that the beam is now vibrating and should not be touched!
• Activate the "Continue Acquisition" switch to "YES" and run the "VIBEXPnew.vi" by activating the arrow in the top left corner of the screen.
• Type in the excitation frequency that you selected on the B&K function generator device and press the "Press to Acquire Data" button.
• Wait for the PXI system to fully acquire the data before moving to the next incremented frequency, (this should only take a couple of seconds: 1024 samples / 500 samples per second = 2.048 seconds.)
• Be sure to activate the "Save Peak Data" button.
• When you have moved through all of the input frequencies, depress the "Continue Acquisition" button so that is says "NO" and acquire 2 more data points. These last two data points may be disregarded.

2.3 The Hammer Test.

Using the Allen wrench supplied, remove the beam from the clamp and place it in the similar style clamp mounted to the solid bar located above the shaker. Try at best to preserve the same length used in the previous study, as you will be trying to compare natural frequencies between these two investigations. In this part it is best that you practice the timing between the impulse hammer and the data acquisition system before saving any data to file as you have only a 2.048 second window to get things right!

• Turn off the "Save Peak Data" and the ‘Continue Acquisition" buttons on the vi interface and activate the "Save Spectra & Time Series Data" button.
• Acquire the input signal by pressing the "Press to Acquire Data" button, while immediately impacting the beam with the impact hammer.
• You will notice that the time series will appear at a frequency equal to that of the natural frequency of the system, with additional peaks (seen in the bottom spectra plot) located at higher frequencies. These higher frequency peaks are the other modes of the system’s natural frequency. For this lab we will only concern ourselves with the first mode of vibration.
• Repeat the entire procedure starting from the section titled "The Shaker Table Test" but with two different lengths of the beam.

2.4 Exercises

• Calculate the theoretical natural frequency of the beam for the different lengths used.
• Calculate the damping coefficients for each of the cantilever systems from the hammer test (delta function).
• Calculate the natural frequency of the different length beams using the data from the accelerometer test and compare.
• On the same graph, plot the theoretical natural frequency vs. mass curve for the cantilever beam system and all of the experimental data points.
• Be sure to include a full uncertainty analysis with your results!

2.5 Topics for Discussion

• Describe the system’s response (i.e. the amplitude and frequency) to different forcing frequencies.
• Compare the various measures of natural frequency.
• How does the geometry of the system effect the natural frequency of the system?
• Discuss the type of damping (if any) that is present in the cantilever.

3.0 Simulation: Experiment validation, Effects of Changing Damping and End Mass

Note: Every time you save a run two .txt files will be created. The first will have the system’s response in voltage vs. time and the second in voltage vs. frequency.

3.1 Experimental Validation.

In this simulation, the results obtained from the experimental part of the lab will be duplicated and compared. If one is performing this part of the lab first, it is advisable to first measure the dimensions and take note of the material types of the beam used in the experimental part of this lab.

• Launch the vi labeled "VIBSIMv8" from the desktop.
• Run the vi by pressing the arrow located in the top left corner of the screen.
• Choose the Simulation portion of the lab and press “Continue”.
• Turn off the "Save Data to File" button so that it reads "NO"
• Input the required fields as they apply to the experimental characteristics just performed.
1. Length: begin with 450mm and upon completing the simulation do 500mm and 600mm
2. Width: as measured
3. Thickness: as measured
4. End-weight = 0
5. Damping coefficient (c): assume 0.01
6. Material type: Carbon Steel
• If the required input fields are correct, press the "Acquire" button.
• Toggle between the "Magnitude Ratio" and "Phase Angle" button located above the bottom graph and take note of the system’s natural frequency and phase lag, respectively. This will have more relevance in the next section.
• To continue press the "Continue" and "Continue Acquisition" buttons. You will have to update the simulation by running through this routine twice before obtaining accurate results.
• Activate the "Save Data to File" button and re- "Acquire" the signal.
• Perform this for the other two lengths used in the experimental portion of this lab.

3.2 The Effects of Damping.

We will now quantify the sensitivity of the systems frequency response to varying damping coefficients.

• Set the length to 600mm, and the width and thickness to match the geometry of the experiment beam as to start from a baseline set of measurements. End-weight should be zero.
• Choose a damping coefficient of 1.00.
• If the required input fields are correct, press the "Acquire" button.
• Toggle between the "Magnitude Ratio" and "Phase Angle" button located above the bottom graph and take note of the system’s natural frequency and phase lag, respectively.
• If the response looks correct, change the "Save Data to File" to "YES".
• Repeat the above procedure for damping coefficient values of 3, 5, 7, and 9.

3.3 The Effects of End Mass and Material Type.

We will now quantify the sensitivity of the systems frequency response to varying weights applied to the end of the cantilever beam and to the beam’s material.

Effect of End-Mass

• Set the length to 600mm, and the width and thickness to match the geometry of the experiment beam as to start from a baseline set of measurements. Damping Coefficient should be zero.
• Choose an "End Weight" of 0.15 kg.
• If the required input fields are correct, press the "Acquire" button. (You will have to update the simulation by running through this routine twice before obtaining accurate results as was previously done in the "Experimental Validation" section of this lab.)
• Toggle between the "Magnitude Ratio" and "Phase Angle" button located above the bottom graph and take note of the system’s natural frequency and phase lag, respectively.
• If the response looks correct, change the "Save Data to File" to "YES".
• Repeat the above procedure for end masses of 0.25kg and 0.5kg.

Effect of Material Type

• Set the length to 600mm, and the width and thickness to match the geometry of the experiment beam as to start from a baseline set of measurements. Use a damping coefficient and an end-weight of zero.
• Choose the “Material” to be Carbon Steel.
• If the required input fields are correct, press the "Acquire" button. (You will have to update the simulation by running through this routine twice before obtaining accurate results as was previously done in the "Experimental Validation" section of this lab.)
• Toggle between the "Magnitude Ratio" and "Phase Angle" button located above the bottom graph and take note of the system’s natural frequency and phase lag, respectively.
• If the response looks correct, change the "Save Data to File" to "YES".
• Repeat the above procedure for Stainless steel and Aluminum.

3.4 Exercises

• Calculate the theoretical natural frequency of the beam for the different materials and end masses used, and compare to the simulation data.
• On the same graph, plot the system’s frequency response to different end masses and discuss.
• On the same graph, plot the system’s frequency response to different damping coefficients and discuss.
• On the same graph, plot the system’s phase lag / lead to different damping coefficients and discuss. What effect does the damping coefficient have on the amplitude of the system’s natural frequency? How may this play a role (critical) in designing engineering systems? Discuss.