Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

“…In the two timescale-expansion form of the method of multiple timescales [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45], a uniformly valid expansion is sought in the following form…”

Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

“…Equations (3a)-(3c) form a set of linear partial differential equations which can be solved via the method of separation of variables [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. As such, the general solution for the displacement field of each subsystem can be expressed as a series expansion of the following form…”

“…(5d), (5e) and Eqs. (14) and following the procedure explained in the paragraph after Eq. (7) gives −2k cos(β n ξ s ) sin(β n ξ s ) sinh β n sin β n + 2k sin(β n ξ s ) × cosh(β n ξ s ) sinh β n sin β n − sinh(β n ξ s )β n k sin(β n ξ s )Γ × cos 2 (β n ξ m ) cosh β n cos β n + 2k cosh(β n ξ s ) cos(β n ξ s ) × sinh β n cos β n − 2 sinh(β n ξ s )k sin(β n ξ s ) sinh β n cos β n +k sinh(β n ξ s )β n cosh(β n ξ s )Γ cosh β n sin β n cos 2 (β n ξ m ) +β n k cos 2 (β n ξ s ) cosh(β n ξ m )Γ cosh β n sinh(β n ξ m ) sin β n +βk cosh(βξ s ) sinh(βξ m ) cos(βξ s )Γ sinh β cosh(βξ m ) × cos β + 2k sin(β n ξ s ) cosh(β n ξ s ) cosh β n cos β n −2k sin(β n ξ s ) sinh(β n ξ s ) cosh β n sin β n + 2k cos(β n ξ s ) × sinh(β n ξ s ) sinh β n sin β n − cosh(β n ξ s )k sin(β n ξ s )β n Γ × cosh(β n ξ m ) sinh β n cos β n cos(β n ξ m ) + sinh(β n ξ s )β n kΓ × cosh(β n ξ m ) sin(β n ξ s ) sinh β n sinh(β n ξ m ) cos β n +β n k cos(β n ξ s ) sinh(β n ξ s )Γ sinh β n cos β n +β n k cosh(β n ξ s )Γ sin(β n ξ s ) sinh β n cos β n +β n cosh 2 (β n ξ s )k cosh 2 (β n ξ m )Γ cosh β n cos β n + sinh(β n ξ s )β n k cos(β n ξ s )Γ cosh β n sin β n −k sinh(β n ξ s )β n cosh(β n ξ s )Γ sinh β n cos β n −kβ n cosh 2 (β n ξ s )Γ sinh β n sin β n cos 2 (β n ξ m ) +β n k cosh(β n ξ s )Γ sin(β n ξ s ) cosh β n sin β n +β n k cos(β n ξ s ) cosh(β n ξ m ) sin(β n ξ s )Γ sin(β n ξ m ) +2k cos(β n ξ s )β cosh(β n ξ s ) sinh(β n ξ m )Γ sin(β n ξ m ) −2 cosh(β n ξ s )k sin(β n ξ s ) − k sinh(β n ξ s )β n cosh(β n ξ s )Γ × cosh β n sin β n + 4β 3 n cosh β n cos β n − β n k cos(β n ξ s )Γ × sin(β n ξ s ) sinh β n cos β n − 2β n k cosh(β n ξ s ) cos(β n ξ s )Γ × sinh β n sin β n − β n k cos 2 (β n ξ s )Γ sinh β n sin β n × cosh 2 (β n ξ m ) − 2k sinh(β n ξ s ) cosh(β n ξ s ) sinh β n sin β n −β n k cos 2 (β n ξ s )Γ cos 2 (β n ξ m ) cosh β n cos β n −β n k cos(β n ξ s )Γ sin(β n ξ s ) cosh β n sin β n −2β n k cosh(β n ξ s ) sin(β n ξ s )Γ cos(β n ξ m ) sinh(β n ξ m ) −2 sinh(β n ξ s )β n k cosh(β n ξ m ) cos(β n ξ s )Γ sin(β n ξ m ) +2 sinh(β n ξ s )k cos(β n ξ s ) + cosh(β n ξ s )k cosh(β n ξ m )β n × sinh(β n ξ s )Γ sin(β n ξ m ) + k sinh(β n ξ s )β n cosh(β n ξ s )Γ × cos(β n ξ m ) sinh(β n ξ m ) + β n k cos(β n ξ s ) sinh(β n ξ m ) × sin(β n ξ s )Γ cos(β n ξ m ) + 2 sinh(β n ξ s )β n k cosh(β n ξ m ) × sin(β n ξ s )Γ cos(β n ξ m ) − cosh(β n ξ s )k sin(β n ξ s )β n × sinh(β n ξ m )Γ sinh β n sin β n cos(β n ξ m ) − β n k cos(β n ξ s ) × sin(β n ξ s )Γ sin(β n ξ m ) cosh β n cos β n cos(β n ξ m ) −β n k cosh(β n ξ s ) cosh 2 (β n ξ m ) cos(β n ξ s )Γ cosh β n cos β n +β n k cosh(β n ξ s )Γ cos 2 (β n ξ m ) cos(β n ξ s ) cosh β n cos β n −β n sinh(β n ξ s )k cos(β n ξ s ) sinh(β n ξ m )Γ sinh β n × cos β n sin(β n ξ m ) − 2 sinh(β n ξ s )k cos(β n ξ s ) cosh β n cos β n −2k cos(β n ξ s ) cosh(β n ξ s ) cosh β n sin β n + β n k cos(β n ξ s )Γ × sin(β n ξ s ) sinh β n cos β n cosh 2 (β n ξ m ) − 2β 4 n cosh 2 (β n ξ m )Γ × cosh β n sin β n + 2β 4 n Γ cos 2 (β n ξ m ) sinh β n cos β n −k sin(β n ξ s )β n Γ cosh 2 (β n ξ m ) sinh(β n ξ s ) cosh β n cos β n − sinh(β n ξ s )β n k cos(β n ξ s )Γ cosh β n sin β n cos 2 (β n ξ m ) + sinh(β n ξ s )β n k cos(β n ξ s )Γ cosh β n sin β n cosh 2 (β n ξ m ) +k sinh(β n ξ s )β n cosh(β n ξ s )Γ sinh β n cos β n cosh 2 (β n ξ m ) −β n k cosh(β n ξ s )Γ sin(β n ξ s ) cosh β n sin β n cos 2 (β n ξ m ) −β n k cosh(β n ξ s )Γ sin(β n ξ s ) cosh β n sin β n cosh 2 (β n ξ m ) +β n k cosh(β n ξ s ) cos(β n ξ s )Γ sinh β n sin β n cosh 2 (β n ξ m ) +k sin(β n ξ s )β n cosh(β n ξ s )Γ cos 2 (β n ξ m ) sinh β n cos β n −β n k cos(β n ξ s ) sinh(β n ξ s )Γ sinh β n cos β n cos 2 (β n ξ m ) −β n k cos(β n ξ s ) sinh(β n ξ s )Γ sinh β n cos β n cosh 2 (β n ξ m ) −β n k cosh(β n ξ s )Γ sin(β n ξ s ) sinh β n cos β n cosh 2 (β n ξ m ) +β n k cosh(β n ξ s ) cos(β n ξ s )Γ sinh β n sin β n cos 2 (β n ξ m ) +k sin(β n ξ s )β n sinh(β n ξ s )Γ cos 2 (β n ξ m ) sinh β n sin β n −k sin(β n ξ s ...…”

Nonlinear Dynamical Behavior of Axially Accelerating Beams: Three-Dimensional Analysis

“…The nonlinear partial differential equations of motion are derived assuming that: (1) shear deformation and rotary inertia are neglected, i.e., the Euler-Bernoulli beam theory is employed [26][27][28][29]; (2) the nonlinear behavior is due to the stretching effect of the midplane of the beam; (3) the beam is under a constant pretension p in the axial direction; (4) the cross section of the beam is uniform along the entire length [30,31]; and (5) the rotation of beam cross section is assumed to be small, which neglects the nonlinearities in the curvature-displacement relation [32][33][34][35][36][37][38][39].…”