Ali Vakil

Vibration analysis of a Timoshenko beam on a moving base

In this paper free vibration of a Timoshenko beam with a tip payload, which is mounted on a cart (referred to as TBC) is studied. The cart (base) can only have lateral displacement and the tip payload has both mass and mass moment of inertia. The center of mass of the payload does not coincide with the point where the beam connects to the payload. Therefore, the tip of the beam is exposed to an extra bending moment due to the inertial force of the payload. By employing Hamilton’s principle, the governing equations of motion and the associated boundary conditions for the TBC are first derived and then transferred into dimensionless forms. By using these governing equations and their associated boundary conditions, the closed-form frequency equation (characteristic equation) of the TBC is derived. This closed-form frequency equation is validated both analytically and numerically. The closed-form expressions for the mode shapes of the TBC and their orthogonality are also presented. By using the closed-form characteristic equation, a sensitivity study is performed and the changes in the natural frequencies versus changes in the physical parameters are investigated. The results presented in this paper are valuable for precise dynamic modeling and model-based control of flexible mobile manipulators; a flexible mobile manipulator is a flexible link manipulator with a moving base.

Stagnation Pressure, the Bernoulli Equation, and the Steady-Flow Energy Equation

The Bernoulli equation is arguably the most commonly used equation in fluid mechanics. For the incompressible, inviscid flow along a streamline, the Bernoulli equation states that the total head of the fluid (the sum of the pressure head, velocity head, and elevation head) is constant. Neglecting elevation changes, the Bernoulli equation therefore limits the maximum pressure coefficient in a flow to 1, which occurs at stagnation points in the flow. Normally, the action of viscosity causes the total head in a fluid to decrease in the streamwise direction, which means the stagnation-point pressure coefficient is less than 1. However, at low to moderate Reynolds numbers, where viscous forces are most significant, stagnation-point pressures exceed 1. This counterintuitive result is explained by reference to the shear work term in the steady-flow energy equation.