Free Vibration of Cantilever Beam with Lumped Mass at Free End

2.4 Experimental setup

Fig. 2.4: An experimental setup for the free vibration of a cantilever beam


The experimental setup is consists of a cantilever beam, transducers (strain gauge, accelerometer, laser vibrometer), a data-acquisition system and a computer with signal display and processing software (Fig. 2.4). Different types of beam materials and its properties are listed in Table 2.1. Different combinations of beam geometries for each of the beam material are summarized in Table 2.2.

Accelerometer is a sensing element (transducer) to measure the vibration response (i.e., acceleration, velocity and displacement). Data acquisition system takes vibration signal from the accelerometer and encode it in digital form. Computer acts as a data storage and analysis system. It takes encoded data from data acquisition system and after processing (e.g., FFT), it displays on the computer screen by using analysis software.


Table 2.1  Material properties of various beams



Density (kg/m 3 )

Young’s Modulus (N/m 2 )



2.1×10 11



1.2×10 11



  0.69×10 11


Table 2.2  Different geometries of the beam












Example 2.1: Obtain the undamped natural frequency of a steel beam with l = 0.45 m, d = 0.003 m, and b = 0.02 m. The mass of transducer at the free end = 18.2 gm.






2.5 Photos of experimental setup

Fig. 2.5: Experimental setup of a cantilever beam


Fig. 2.5 shows an experimental setup of the cantilever beam. It includes a beam specimen of a particular geometry with a fixed end and at the free end an accelerometer is mounted to measure the free vibration response. The fixed end of beam is gripped with the help of clamp. For getting precise free vibration cantilever beam data, it is very important to ensure that clamp is tightened properly, otherwise it may not give fixed end conditions in the free vibration data.

Fig. 2.6: A close view of the fixed end of the cantilever beam


It is the most common contacting type sensor for the vibration (i.e., acceleration, velocity or displacement) measurement. It is available with connecting cable as-well-as wireless type. It is pasted onto the surface by either using magnetic base, or by using adhesive glue, or by threaded screw (Fig. 2.7).


Fig. 2.7: A close view of an accelerometer mounted on the free end of the beam

The basic principle of the measurement by an accelerometer is that it measures the force exerted by a body as a result of a change in the velocity of the body (i.e. which leads to acceleration). A moving body possesses an inertia which tends to resist change in velocity. The force caused by vibration or a change in motion causes the mass to "squeeze" the piezoelectric material which produces an electrical charge that is proportional to the force exerted upon it. Since the charge is proportional to the force, and the mass is a constant, hence the change is proportional to the acceleration.


Laser Doppler Vibrometer (LDV):
It is an instrument (Fig. 2.8) that is used to make non-contact vibration measurements of a surface. The laser beam from the LDV is directed at the surface of interest, and the vibration amplitude and frequency are extracted from the Doppler shift of the laser beam frequency due to the motion of the surface. The output of an LDV is generally a continuous analog voltage that is directly proportional to the target velocity component along the direction of the laser beam (give a line diagram of basic principle of the LDV).

Fig. 2.8: A laser vibrometer system


Fig. 2.9: A laser projector of a laser vibrometer fitted on a stand.


Fig. 2.10: The controller for the laser vibrometer


Fig. 2.11: The laser generator of a laser vibrometer


Fig. 2.8 shows a rotational laser vibrometer. The complete setup consists of a projector (Fig. 2.9), controller (Fig. 2.10), and laser generator (Fig. 2.11). To generate laser beam by the laser generator, all the settings related to the measurement are done in the controller. The laser goes to the reflector by an optical fiber cable. The beam is projected to the measurement surface and measurement signal is taken into the computer through a data-acquisition system.


Rotational Laser Vibrometer (RLV):
The optical measurement principle for the rotational vibrometer is based on laser interferometry. Use of the RLV is not limited to cylindrical parts. By using a special differential measurement process with two laser beams, independently of the shape of the object under investigation, only the rotational movement component is acquired and translational vibrations are predominantly suppressed. A schematic layout of the signal paths is shown in Fig. 2.12.


Fig. 2.12: Principle of measurement of rotational laser vibrometer


Dynamic acquisition of rotational vibrations is possible in a frequency range from 0 Hz to 10 kHz. It also cover challenging measurement tasks e.g. in the order analysis in rotors. The interferometric process works continuously, i.e. in principle there is no limit to the angular resolution as for example, this limitation exist when using optical encoders with a finite number of divisions.


Data acquisition system:
Data acquisition system typically involves the conversion of analog signals and waveforms into digital values, and processing the values to obtain desired information. Data acquisition systems, as the name implies, are products and/or processes used to collect information to document or analyze some phenomenon. The components of measurement and data acquisition systems include (see Fig. 2.13) (i) Sensors that convert physical parameters to electrical signals, (ii) Signal conditioning circuitry to coerce sensor signals into a form that can be converted to digital values, and (iii) Analog-to-digital converters, which convert conditioned sensor signals to digital values.


Fig. 2.13(a): An overall measurement system


Fig. 2.13(b): Data acquisition system

Data acquisition system receives voltage signal from sensors (e.g., accelerometer) and calibrate the data into equivalent physical quantity (e.g., acceleration) and send it to computer where by using a vibration measurement software these data can be analyzed in time history (e.g., acceleration-time, velocity-time or displacement-time) and in frequency domain (i.e., using FFT) Fig. 2.14.


Fig. 2.14: A typical response with time and the corresponding FFT plot


When the voltage signal from the accelerometer is sent to the data-acquisition system, it converts the signal to a mechanical vibration data (acceleration) and stores it to the computer. A typical screen-shot of a captured vibration signal by using vibration measurement software is plotted as shown in Fig. 2.14 and can be used for further analysis.