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Fourth-Order Ordinary Differential Equations
Published in Saad A. Ragab, Hassan E. Fayed, Introduction to Finite Element Analysis for Engineers, 2018
Saad A. Ragab, Hassan E. Fayed
This chapter presents a finite‐element model for fourth‐order ordinary differential equations. In solid mechanics, a typical equation is the Euler‐Bernoulli beam equation that governs deflection of beams. The beam equation is then combined with the bar equation of Chapter 2 to analyze structures such as plane frames and trusses. Principles of virtual displacements and minimum total potential energy are frequently used to derive finite‐element models in structural and solid mechanics. Starting with the equations of equilibrium of a continuum and boundary conditions, we derive the principle of virtual displacements, which is mathematically identical to the weak form of the problem. We then present the principle of minimum total potential energy and demonstrate its use for deriving finite‐element models for bars and beams. In fluid mechanics, a well‐known fourth‐order equation is the Orr‐Sommerfeld equation which is encountered in hydrodynamic stability analysis of shear flows. However, it is a differential eigenvalue problem and will be presented in Chapter 7.
Single Degree‐of‐Freedom Damped Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
Euler helped develop the Euler‐Bernoulli beam equation, used extensively in mechanics of materials. Euler‐Bernoulli beam theory is a simplification of the linear theory of elasticity and provides a means of calculating the load‐carrying and deflection characteristics of beams. He worked on it circa 1750.
A backstepping-based fault compensation scheme for a class of Euler–Bernoulli beam-ODE cascade systems
Published in International Journal of Control, 2021
Dong Zhao, Bin Jiang, Hao Yang, Gang Tao
It should be noticed that the Euler–Bernoulli beam system is connected with an ODE system through the boundary signals of the Euler–Bernoulli beam system. In fact, the model expressed by the Euler–Bernoulli beam equation has been widely used in practical engineering, such as helicopter rotor, space structure, space aircraft, flexible robotic manipulators and turbine blades (Nwokah & Hurmuzlu, 2002; Wu & Wang, 2014). When part represents the controlled plant, the PDE part denotes the actuator dynamics. Alternatively, the ODE part describes the sensor dynamics when the PDE part expresses the controlled plant. For a practical problem belonging to the first situation, the longitudinal control of flexible hypersonic vehicles is a special one which can be depicted by (1)–(4). While it inevitably leads to an approximate model for such a control problem under the assumption that a light beam describes the actuator dynamics.