Forced Vibration of a Cantilever Beam with a Lumped Mass at Free End



Experimental Procedure

1.  Choose a beam of a particular material (steel or aluminum), dimensions ( L , w , d ) and transducer (e.g., accelerometer or laser vibrometer).

2.  Clamp one end of the beam as shown in Fig. 3.7.

3.  Place an accelerometer (with magnetic base) at the free end of the cantilever beam to measure the forced vibration response (acceleration).

4.  Place the exciter stinger on the beam slightly offset from the middle of the beam (Fig.3.5). Properly check the connection of the exciter with the controller, and the level of input power.

5.  Connect another accelerometer very close to the exciter stinger location to measure the forced vibration response (acceleration).

6.  Make a proper connection of both the accelerometers with data acquisition system and with computer to capture the vibration data.

7.  Make the settings correctly to generate swept-sine signal by the exciter.

8.  During setting of the swept-sine parameter, make sure that, in the vibration measurement software, the time duration should be less than the total time of excitation.

9.  Start the experiment by giving force signal continuously to the exciter and allow the beam to vibrate.

10.  Record the variation of the vibration response with time obtained from the chosen transducer.

11.  Repeat the experiment for five to ten times to check the repeatability of the experimentation (i.e., vibration data).

12.  Repeat the whole experiment for different material, dimensions, and measuring devices.

13.  Record the whole set of data in a data base for further processing and analysis.

 

3.7 Experimental Identification of Resonance Conditions

It is relatively easy to obtain the resonance condition by observing the behavior of the system at varying frequency (e.g., swept-sine). During the experiment, the frequency of excitation force is swept between a chosen range of frequency with a fixed time-rate. The frequency range is selected in such a way that the natural frequency of system lies within the frequency range and the swept rate is selected by several trials of experiments (i.e., increment of 2 Hz to 3 Hz per sec).  As we know that in case of forced vibration, the system vibrates with the same frequency as that of exciter. When the frequency of excitation coincides with the natural frequency of the beam resonance occurs, which characterizes by a sudden increase in the response. But in our case it is linear because of property of exciter to generate force signal in such a manner to get linearly increasing displacement.

(a)

(b)

(c)

Fig. 3.11 (a): Displacement against frequency

(b): Magnification factor against frequency ratio.

(c): Phase change between 0 and 180 degree at resonance

 

At the resonance the phase difference between the force and response changes by 180 degree, which also implies the indication of resonance and corresponding frequency is the natural frequency of the system.

3.8 Calculations of Damping Ratio

The damping ratio can be calculated with the help of phase difference between the force and the response at any excitation frequency especially which is close to the natural frequency of the system (Fig. 3.11(c)).  The damping ratio can be calculated as follows. We have

                                  

                                                                (3.15)

 

Where, φ is phase difference between the force and the response, and

                                           

                                                            (3.16)

 

The above equation can be rewritten as,

                                      

                                                            (3.17)

 

Damping ratio can be calculated from above formula directly once we know the natural frequency, and phase difference at a particular excitation frequency.

 

Example 3.2: As seen from fig 3.11 (c), the natural frequency is 13.12 Hz, let ω = 8 Hz, and suppose at that point phase difference between force and response is 10 degree, then

The frequency ratio is

                                

Hence, the damping ratio is

                                  

3.9 Virtual experimentation

Virtual experimentation provides the interface, which gives facility to perform experiments virtually. It provides different options for material selection, instruments, and specimen dimensions. After making desired selection and running the program, it gives the result from a storage database for a particular configuration selected by the user. Fig. 3.12 shows an overall flowchart of a virtual laboratory in which several experiments can be performed through the internet by users remotely with the help of already stored measured data.

Fig. 3.12: Overview of measurement based virtual experiments

 

3.10 Explanation of virtual experiment

The virtual experiment program of forced vibration of single DOF cantilever beam is divided in many sections. The step by step description of program is as follows which also include the guideline to proceed in the virtual experiment…

 

1. Title Page - This is the first page of virtual experiment. It includes title of experiment and a photo of experimental setup with accelerometer. User starts the program by clicking the RUN button given at the top of the page.

 

 

2. Introduction - This section contains aim of experiment, some important definitions related to the experiment. This page also includes the instructions and guidelines to perform the experiment with effectively and efficiently.

 

 

User has to follow all the instruction strictly for successful completion of the virtual experiment. Also please don’t try to go back to previous pages in between the program.

3. Entrance test - This page offers an entry test to the user. This test is offered to ensure that user has adequate basic knowledge about the experiment. Total five objective types of questions are given; user has to select the right option among the given options. User can select only one option.

 

After submitting the test, if user passes, program automatically moves to next page otherwise it will stuck in the same page. User is suggested to read the manual carefully before starting the experiment.

4. Input section - This section contains various input options for the experiment i.e. beam material, beam dimensions, transducer for measurement, frequency sweep range and sweep rate. User has to select proper input to proceed for the experiment.

 

 

5. Questions - In between the virtual program, experiment offers some questions to the user to make the experiment more interactive. User should try to think about the answers of these questions with proper explanation. These questions are very basic and related to the experiment.

 

 

6. Experiment and data collection - In this page, the program takes related files from the database based on the input selection. Here these data sets are plotted which correspond to the real experiment. In this experiment two graphs are plotted, one is force against time and other is response at the free end against time. The force is provided by exciter and always measured by accelerometer, but for measuring response user has two options i.e. accelerometer or laser vibrometer.

In case of real experiment, user has to take care about the forcing amplitude as high amplitude may cause damage to the system. Also put the measuring sensor in the same side of the beam otherwise they will have initial 90 degree phase difference.

 

 

7. Answers -   This page contains the answers of the question asked before the experiment.

 

 

8. Variation of Force and response with frequency - This page basically converts the time domain graph into frequency domain plot. It shows the variation of force and response with respect to the frequency. As shown in the figure, first force decreases to minimum which occurs at resonance, after this again it increases. The displacement is increasing with constant rate.

 

 


9. Variation of (response/force) and phase difference with frequency -
This page shows two plots; one is the variation of magnification factor with frequency, and the other is the variation of phase difference with frequency.

 

 

10. Phase difference - This page shows the variation in the phase difference before and after the resonance. Before the resonance the phase difference between the force and response is around 0 degree but after the resonance it changes to 180 degrees.

 

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11. Theoretical calculations - The theoretical calculations are done based on the input configuration chosen by the user. Program decodes the input parameter and generates the related parameter like Young modulus, density and dimensions. By using these values, the theoretical calculation is done with the help of formula node. Refer the experimental manual for detailed analysis of formulation and calculation procedure.

 

 

12. Experimental calculations - The experimental calculations are done by using the data taken from the files. A waveform peak detector is used to get the peak values and its locations. The damped natural frequencies can be obtained by observing the peak of (Force/response) graph against the frequency. The location of peak is itself shows the resonance frequency. The resonance frequency can also be observed by lowest force or where the phase difference changes from 0 to 180 degree.

The damping ratio can be calculated by using the given formula at any instant before the resonance. Refer the manual for details.

 

 

13. Results - The result is the comparison of theoretical and experimental results. The result is basically comparison between the theoretical result and experimental result. Here, user can perform the experiment again by same or different input configurations.

 

 

14. Evaluation test -  After successful completion of the experiment, program offers an evaluation test to judge the knowledge gained by the user. Here five optional questions are given, user has to select right option and click the submit button.

 

 

15. Exercise - The exercise problems are given to the user to explore the knowledge about the basics of the experiment. User has to find out the answer of these questions.