“…In order to validate the two lowest frequencies and decaying rates of a cantilever micro-beam with end subtangential force in the absence of the material gradient index, material length scale parameter, and piezoelectric layers, Figure 3(a) and (b) is plotted for imaginary and real parts of eigenvalue versus the follower force, respectively, and compared with those given by Leipholz (2013). It is evident that for the range 0≤P≤20, the real part of eigenvalue vanishes, while the first mode of the imaginary parts of the lowest eigenvalues increases and the second mode decreases, and they join together as P tends toward Pcr=20 and imaginary part of eigenvalue reaches to Im(Ω)=11.03.…”
This investigation aims to explore the non-conservative instability of a functionally graded material micro-beam subjected to a subtangential force. The functionally graded material micro-beam is integrated with piezoelectric layers on the lower and upper surfaces. To take size effect into account, the mathematical derivations are expanded in terms of three length scale parameters using the modified strain gradient theory in conjunction with the Euler–Bernoulli beam model. However, the modified strain gradient theory includes modified couple stress theory and classical theory as special cases. Applying extended Hamilton’s principle and Galerkin method, the governing equation and corresponding boundary conditions are obtained and then solved numerically by the eigenvalue analysis, respectively. The results illustrated effects of non-conservative parameter, length scale parameter, different material gradient index, and various values of piezoelectric voltage on the natural frequencies, flutter and divergence instabilities of a cantilever functionally graded material micro-beam. It is found that both the material gradient index and applied piezoelectric voltage have significant influence on the vibrational behaviors, divergence and flutter instability regions. Furthermore, a comparison between the various micro-beam theories on the basis of modified couple stress theory, modified strain gradient theory, and classical theory are presented.
“…In order to validate the two lowest frequencies and decaying rates of a cantilever micro-beam with end subtangential force in the absence of the material gradient index, material length scale parameter, and piezoelectric layers, Figure 3(a) and (b) is plotted for imaginary and real parts of eigenvalue versus the follower force, respectively, and compared with those given by Leipholz (2013). It is evident that for the range 0≤P≤20, the real part of eigenvalue vanishes, while the first mode of the imaginary parts of the lowest eigenvalues increases and the second mode decreases, and they join together as P tends toward Pcr=20 and imaginary part of eigenvalue reaches to Im(Ω)=11.03.…”
This investigation aims to explore the non-conservative instability of a functionally graded material micro-beam subjected to a subtangential force. The functionally graded material micro-beam is integrated with piezoelectric layers on the lower and upper surfaces. To take size effect into account, the mathematical derivations are expanded in terms of three length scale parameters using the modified strain gradient theory in conjunction with the Euler–Bernoulli beam model. However, the modified strain gradient theory includes modified couple stress theory and classical theory as special cases. Applying extended Hamilton’s principle and Galerkin method, the governing equation and corresponding boundary conditions are obtained and then solved numerically by the eigenvalue analysis, respectively. The results illustrated effects of non-conservative parameter, length scale parameter, different material gradient index, and various values of piezoelectric voltage on the natural frequencies, flutter and divergence instabilities of a cantilever functionally graded material micro-beam. It is found that both the material gradient index and applied piezoelectric voltage have significant influence on the vibrational behaviors, divergence and flutter instability regions. Furthermore, a comparison between the various micro-beam theories on the basis of modified couple stress theory, modified strain gradient theory, and classical theory are presented.
“…Remarkably, linearization in [57] reduces to the classical Beck column [30] with an external damping [107], where the acceleration term is neglected. Ling et al [99] extended the nonlinear model by including a distributed follower load as in the classical Leipholz column [33,108] and allowing three-dimensional perturbations to the non-deformed filament. As a result, bifurcations to both planar flapping states and non-planar spinning motions have been found, reminiscent of the cone-like beating motion of cilia [99].…”
Section: Flutter In Biology At Different Scales (A) Motility and Prop...mentioning
confidence: 99%
“…Among the many, the following ones are recalled (figure 2): Reut’s double pendulum [29], differing from the ‘Ziegler’s double pendulum’ only in the load application. More specifically, the constant magnitude load P slides along a rigid bar to maintain its application line always coincident with the undeformed state;Beck’s column [30], an elastic cantilever rod differing from the ‘Ziegler’s double pendulum’ only for the continuous and uniform distribution of mass, bending stiffness and dissipation;Pflüger’s column [31,32], which enhances the Beck’s column by considering the presence of a lumped mass at the application point of the follower load;Leipholz’s column [33,34], an elastic cantilever rod subject to a uniform distribution of follower tangential load acting along its axis;Nicolai’s column [35,36], an elastic cantilever rod with a follower twist load applied at the free end, so that three-dimensional motion is realized, instead of a planar one.Interestingly, these structural models do not only display flutter instability but, similarly to the ‘Ziegler’s double pendulum’, also counterintuitive behaviours, often referred as ‘paradoxes’ [37–49]. Figure 2Discrete and continuous structural models displaying flutter instability when subject to non-conservative loads.…”
The phenomenon of oscillatory instability called ‘flutter’ was observed in aeroelasticity and rotor dynamics about a century ago. Driven by a series of applications involving non-conservative elasticity theory at different physical scales, ranging from nanomechanics to the mechanics of large space structures and including biomechanical problems of motility and growth, research on flutter is experiencing a new renaissance. A review is presented of the most notable applications and recent advances in fundamentals, both theoretical and experimental aspects, of flutter instability and Hopf bifurcation. Open problems, research gaps and new perspectives for investigations are indicated.
“…This system appears as a generalization of the n-DOF Ziegler column where the load parameter is itself distributed on the system and may vary on each joint. It also may be considered as a discretized Leipholz column [Leipholz 1987]. In this case, the geometric degree of nonconservativity increases with the number of bars and the degree of freedom.…”
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