| Citation: |
Guozhong Xiu, Jian Yuan, Bao Shi, Liying Wang. HEREDITARY EFFECTS OF EXPONENTIALLY DAMPED OSCILLATORS WITH PAST HISTORIES[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2212-2223. doi: 10.11948/20180344
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HEREDITARY EFFECTS OF EXPONENTIALLY DAMPED OSCILLATORS WITH PAST HISTORIES
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Abstract
This paper presents hereditary effects of exponentially damped oscillators with past histories. Unlike the classical viscously damped oscillators, the nonviscously damped ones involve damping forces which depend on time-histories of vibrating motions via convolution integrals. As a result, equations of motion of such systems are a set of coupled second-order Volterra integro-differential equations. In this work, initial value problems for the integro-differential equations are revisited. The initial conditions should contain time-histories of vibrating motions. Then, initialization response of exponentially damped oscillators is obtained. It is used to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved from the theoretical viewpoint and verified by numerical simulations. This reveals that the hereditary effects gradually recede with increasing of time. -
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References
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Guozhong Xiu, Jian Yuan, Bao Shi, Liying Wang. HEREDITARY EFFECTS OF EXPONENTIALLY DAMPED OSCILLATORS WITH PAST HISTORIES[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2212-2223. doi: 10.11948/20180344
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Figure 1. Single-degree-of-freedom exponentially damped oscillator, damping force
$ {{f}_{d}}\left(t \right) = c\int_{-\infty }^{t}{G\left(t-\tau \right)\dot{x}\left(\tau \right)d}\tau $ . - Figure 2. Initialization responses with different past histories of vibrating motion.