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Analysis of free and forced vibrational dynamics of the second order system

## Objective

1.Calibrate the second order system, consisting of a cantilever beam attached with the spring at its free end and calculate its natural frequency.
2.Analyse the effect of periodic forced vibration frequency on the phase and amplitude of the above system.
3.Analyse the effect of damping on the response of above system and cross-compare it with the undamped response.

 Equipment Number Manufacturers/Remarks Motor Speed Controller 1 HAC 110 (part of PA hilton kit) Tachometer 1 HAC 90 (part of PA hilton kit) LVDT Displacement Sensor 1 Ã¯Â¿Â½ Out of balance Motor 1 Ã¯Â¿Â½ Oleodynamic damper 1 Ã¯Â¿Â½ Cantilever beam 1 Ã¯Â¿Â½ Spring 1 Spring constant to be detemined Weight Holder 1 Ã¯Â¿Â½ Weights 5 5 Ã¯Â¿Â½N each Software kit 1 Picoscope 6

The experimental setup (without data acquisition system) is shown in Fig. 1. As it can be seen from the figure, it consists of a fixed reaction frame containing an horizontal metal beam attached by one side with an hinge and held on the other side by a spring (its stiffness is not known and has to be measured experimentally). Due to the hinge, the beam can not rotate at the point of attachment, thus the bending moment transmitted to the beam is zero. The spring holds the beam in a perfect horizontal position, balancing out its weight. An oleodynamic damper (its damping coefficient is not known and has to be calculated) can be attached to the beam to introduce a localized damping force into the system which otherwise is assumed to be frictionless.

To the left side of the damper, there is an out of balance mass represented by two not-homogeneous disks (one on each side of the beam). The disks act as an inertial load when rotated with the help of electric motor (will be discussed in detail in subsequent section). The frequency of vibration can be regulate by changing the angular velocity of motor with the help of speed-controller. The angular velocity of motor can be varied upto to 62 rad/sec and monitored in real time using tachometer.

The static deflection of the beam is measured with the help of Linear Variable Differential Transformer (LVDT) displacement sensor. Since, the maximum measurement range is Ã¯Â¿Â½ 15 mm, it is place near the hinge, where displacement is within its range and the displacement near spring is calculated (within reasonable assumption) with the help of trigonometry.

The results obtained from from LVDT displacement sensor and tachometer is monitored on computer screen using the software kit. The graphical user interface of the software is shown in Fig. 2. In the figure, the blue and red curve corresponds to tachometer and LVDT sensor respectively. The x-axis represents time (in ms) and y-axis represents voltage (in V). Since the y-axis values are displayed in the voltage, corresponding displacement (applicable only for red curve) can be obtained using formula given in section "Conversion formulae".

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