Vibration can be introduced in all mechanical fields in our life. Engineers try to avoid its negative effect leading in some cases to deformation in the machines. Many researches are dedicated to study the identification of damping especially in multi degree of freedom systems with particular attention to the source of energy dissipation. They focus on developing new tools or methods which may be used in real problems to obtain accurate results about the amount (or value) and the location of energy dissipation in the structure. The aim of this paper is to present an original procedure aims to experimentally determine the modal damping ratio of a mechanical structure. The proposed procedure consists of extracting the Frequency Response Function of the vibrating system using the video magnification method and then calculate the modal damping ratio using the 3-dB method. These experimental measurements are carried out by giving an external force on a cantilever beam, then the modal damping ratios are extracted using motion magnification. The obtained results show a relative error less than 4.2% between the experimental measurements and the analytical calculation for the Frequency Response Function (FRF) curves. The novelty of the paper is to combine the video magnification technique and the 3dB method in a procedure that aims to experimentally measure the modal damping of a mechanical structure. The proposed procedure in this paper represents the damping identification as a simple and easy engineering application.

Nowadays, damping problems have significant concerns in many industrial systems. Damping could be a significant factor in many different fields: the design of buildings in active seismic areas, bridges in windy regions, etc. Damping has many types such as viscous, hysteretic and coulomb friction. The mathematical approximation of viscous damping has been used in awareness limitation to approach as much as possible from the real event. However, the challenge is to represent and identify all the sources of damping which is very different in terms of nature, extent and distribution within the structure. The separation and the evaluation of all the sources of damping are practically impossible in a structure, still it can be estimated by many ways and methods, that have their own advantages and limitations.

Rayleigh [

Material damping, or the energy dissipated throughout the material of the structure.

Boundary damping, or the energy dissipated by the interface between the parts of the structure

Viscous damping, or the dissipation of energy associated with the contact between the structure and a fluid.

Despite that the variety of the micro structural mechanisms affects the material damping according to Bert [

The identification of damping ratio is very important in many fields, like structural dynamic and condition monitoring. Lancaster [

Okuma et al. [

Some research efforts have also been made to update the damping matrices. Phani et al. [

Recently, numerical FEM modelling are used for motion detection. Such numerical models need to be validated experimentally to predict the behavior of the structure. Contact sensors (accelerometers) are used for modal analysis with high accuracy [

In this paper, a new procedure of modal damping identification is proposed for a multiple degree of freedom systems based on video magnification technique. The main focus of this work is the adaptation of the video magnification method for experimental damping identification, then use it in a real problem to obtain valuable information about amplitude and energy dissipation in order to represent the damping behavior in a suitable manner for engineering problems. The research focuses on the experimental analysis of an external impulse that excites a cantilever beam. The existence of the damping will be proven and then the modal damping ratio will be measured. The novelty of this paper consists of proposing a new non-contact experimental procedure aims to extract the modal damping ratio of a mechanical structure using the video magnification technique.

In this paper, we intend to introduce the video magnification technique as an accurate experimental damping identification method. For this sake, a simple but widely used mechanical structure, a cantilever beam (fixed-free) is considered. The dimensions of the beam are presented in

The dimensions of the used cantilever beam and the material properties are as follows:

Elastic modulus:

Density:

Length:

Width:

Thickness:

First, an analytical model of the steel cantilever beam will be used in order to determine the modal behavior. Since the length to the width ratio L/b=14.54 and the length to the thickness ratio L/h=375 are both greater than 10, this beam can be treated as Euler Bernoulli beam. According to the analytical model discussed minutely in Inman [

where

Visual vibrating motion which occurs at different amplitude and over large frequency range hides important information such as natural frequencies, mode shapes and damping ratios. Video magnification is a methodology that aims to exaggerate the displacement in recorded video [

The proposed procedure in this paper starts by recording a suitable frame per second (fps) video of the vibrating structure. A free vibration system of a cantilever beam subjected to a hammer impact will be considered. Actually, an impact to a mechanical structure is a perfect impulse, which causes a constant amplitude in the frequency domain. This result shows the excitation of all vibration modes. The recorded video will be processed using the video magnification technique along the frequency interest range. This method analyses the recorded video using a phase-based motion magnification technique [

The FRF presents the response spectrum of a vibrating system in response to an excitation force. It consists of the magnitude and the phase of the vibration response as function of frequency, in comparison to the excitation. An FRF is a measure of how much displacement, velocity or acceleration response a structure has at an output point, per unit of excitation force at an input point. The peaks in the FRF curve will represent the natural frequencies of the structure, and the width of the resonant peak about the peak’s center frequency is proportional to the damping ratio.

Hallal et al. [

Damping is the energy dissipation properties of a material or system under cyclic stress. Damping in mechanical systems causes the system to gradually stop moving over time. The more damping presents, the shorter the time is to stop moving. Damping can be presented in different forms such as the loss factor, damping factor or modal damping.

The following approach aims to extract damping values based on phase angle response around the resonance. Actually, the response of an undamped vibrating mechanical system is either in or out of phase with the source of excitation. Due to damping, a phase relationship between the structure and the excitation source exists. The slope of the phase angle response at resonance is defined by the damping in the system. Modal damping ration

where

Another approach, is referred to as the half power approach which is also called 3 dB method. This method uses the bandwidth at resonance, obtained from the FRF, to estimate the damping:

where:

Whatever the adopted approach, the FRF of the vibrating structure needs to be extracted in advance. In this paper, we will use the second approach which consists of the 3-dB method.

The measurement procedure consists first of extracting the FRF curve of the cantilever beam using the video magnification technique. And then, calculate the damping ratio of the structure using the half bandwidth method.

A hammer impact is applied at the base of cantilever beam and the transverse response is observed at an arbitrary point as shown in

The applied impact to a cantilever beam is considered as a perfect impulse which causes constant amplitude in the frequency domain. A 240 fps at 1920*1080 resolution video of the transverse vibration (y direction) of the beam is recorded. The motion in this video is amplified using the motion magnification technique on a range up to 120 Hz. This maximum frequency is chosen based on the Nyquist limit. Actually, in order to avoid aliasing in the processed video, frequency range is restricted to the half of the fps rate of the recorded video. The normalized displacement of the chosen point is shown in

Analytical | Experimental | Relative error | |
---|---|---|---|

First natural frequency (Hz) | 5.11 | 5 | 2.2% |

Second natural frequency (Hz) | 32.01 | 33.3 | 4.03% |

Third natural frequency (Hz) | 89.6 | 93.3 | 4.13% |

The modal damping ratios for the first three modes are calculated using Eq. (

The vibration measurements are proven to be accurate based on the Euler Bernoulli beam analytical model. Therefore, the obtained modal damping ratios are considered to be accurate. The experimental measurement procedure results show that the modal damping ratio can be easily, however, accurately measured using the video magnification technique.

The experimentally measured FRF is proven accurate using an analytical model. The relative error is less than 4.2%. This will lead to believe that the proposed procedure in this paper is accurate, reliable, and low cost experimental modal damping measurement method. This procedure could be automated through smartphone application or simply a computer software. It will re-present the damping identification as simple and easy engineering application.

The FRF curve obtained using the video magnification technique is not continuous. It is broken into discrete data points at a frequency interval. As described before, this frequency interval is defined based on the number of fps of the recorded video. Actually, for high fps video, the FRF curve will fit more the real FRF curve and therefore, we expect to obtain more accurate damping ratio.

One should pay attention also if two -or more- successive natural frequencies are close to each other. In this case, the peak would appear wider and mislead the calculation. Thus, we recommend to apply the video magnification technique on a narrow band around each natural frequency. This can successfully separate the two modes influence on each other and lead to an accurate measurement of modal damping ratio.

Since the measurement is non-intrusive, and based on video recording, the proposed procedure in this paper can be applied to different mechanical structure regardless its dimension. We intend to verify this procedure on more complicated structure such as composite material in future works, like two single plates, two single plate with air void or other structures.