Abhishek Sharma - Research

Vibrations of axially moving continua in presence of obstacle at the supports

Axially travelling continuous systems have many applications in industries. These can be in the form of a string, beam, etc. Its most widespread uses include belt-pulley drives for power transmission, ropeways, elevators, conveyers, paper holding equipment, thread lines, magnetic tapes, etc.
Axially traveling continuous systems have many applications in industries, such as threadlines, high-speed magnetic and paper tapes, band saw blades, power transmission using belt pulley drive and chains, as well as pipes that contain flowing fluid. However, the biggest challenge in all these systems is instability, which is responsible for undesired large amplitude vibrations. These undesirable oscillations make it difficult for the entire system to operate smoothly and can reduce its performance and functionality. It is known that there exists a critical speed above which continuum structures might lose stability. This instability, caused by axial speed, might lead to self-excited vibrations. Various mathematical models have been developed to study such types of vibrations, but most of them have ignored the finite size of the obstacle/pulley over which these structures move. Therefore, it is necessary to develop an extensive mathematical model that considers the complexity of the moving boundary and associated wrapping nonlinearity. Vortex induced vibrations of such systems might be an interesting point of investigation.

Problem : Vibrations of string with curved obstacle at both the ends.

Figure 2: Convergence of relative amplitudes .

Figure 3: Modes converging to state of equipartition of energy.

Some important observations-

•Harmonic nature of the frequencies and their dependence on the static-wrapped length of the string.

•Time-varying modeshapes.

•Amplitude modulations.

•Reduction in the modulation frequency when the relative size of two curved obstacles approach each other.

•Existence of mode-locked periodic solution with equipartition of energy among the various vibrational modes.

Problem - Axially traveling string interacting with curved obstacles at both ends

Several engineering systems like band saw blades, threadlines, high-speed magnetic tapes, and pipes transporting fluids involve the transverse vibration of an axially moving continua. More specifically, our system is pertinent to applications like ropeways and conveyor belts, which have big pulleys at the ends. Due to their technological importance, various researchers have explored the dynamics of axially moving systems. String and beam are the most frequently used approximations to model these types of structures. Accordingly, in this problem, the string is used to model the ropeways and conveyor belt-like structures. There are curved obstacles at both ends, so our system could be relevant for studying the vibrations of axially traveling belts or ropeways with pulleys at both ends. It is worth noting that in this problem, end supports are curved, which is why there is a wrapping and unwrapping phenomenon, which makes the problem more complicated and engaging from the research perspective. In addition, a particularly complex issue evolves when the string's velocity exhibits harmonic fluctuations around a mean value. This harmonic fluctuation makes the system parametrically excited, resulting in intricate and diverse vibrational behaviours that are significantly different from those seen in stationary or uniformly moving strings. A significant number of studies exist in the literature addressing the vibrations of axially traveling strings or beams with pin supports at both ends. However, there is an apparent deficit in research that integrates the complexities of moving boundaries and parametric excitation, which could enhance the analysis for practical engineering applications. This study has effectively addressed the identified gap.

Figure: Stability diagrams for different mean velocities for the case of nonuniform tension

Some important observations-

Natural frequencies decrease with increasing velocity, leading to divergence instability at critical velocity.
Slow convergence of the stationary modes as compared to AMM.
Traveling modes give the most accurate results.
An increase in the velocity significantly affects the nonlinear modal interactions.
An increase in the mean velocity leads to the emergence of new stability tongues and widens the instability region.
In the lower excitation frequency range ($\Omega=0.1-2.2$), the case of longitudinal variation of the tension overestimates the stability.