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1.
The parametrically excited lateral vibration of a mass-loaded string is investigated in this paper. Supposing that the mass at the lower end of the string is subjected to a vertical harmonic excitation and neglecting the higher-order vibration modes, the equation of motion for the mass-loaded string can be represented by a Mathieu's equation with cubic nonlinearity. Based on the stability criterion for Mathieu's equation, the critical conditions inducing parametric resonance are clarified. Theoretical analysis shows that when the natural frequency f
s of the string lateral vibration and the vertical excitation frequency f satisfy f
s= ( n/2) f, n= 1, 2, 3, ..., parametric resonance occurs in the case of no damping. For a damped system, parametric resonance most likely occurs when f is close to 2 f
s, and depends on the damping of the system and the vertical excitation. The critical excitation has been derived at different frequencies. If the natural frequency of the mass vertical vibration happens to be twice that of the string lateral vibration, the parametric resonance may occur due to a small disturbance. Numerical simulations show that the lateral vibration of the string does not increase infinitely at parametric resonance because the parametric excitation is self-tuned due to the coupling between the vertical and lateral vibrations. Finally, the theoretical results are supported by some experimental work. 相似文献
2.
The (2 M:1)-librational and ( M:1)-rotational resonances are discovered in the stochastic layer of a parametrically excited pendulum. The analytical conditions for the onset of a resonance in the stochastic layer are derived. Numerical predictions of the appearance of resonance in thestochastic layer are also completed. Illustrations of the stochasticlayer in the parametrically excited pendulums are given through thePoincaré mapping sections. This methodology can be used for resonantlayers in nonlinear Hamiltonian systems. However, the analyticalapproaches need to be improved for the better predictions of theresonant characteristics in the stochastic layer. 相似文献
3.
Experiments have been carried out to determine the dependence of the detonation velocity in porous media, on mixture sensitivity
and pore size. A detonation is established at the top end of a vertical tube and allowed to propagate to the bottom section
housing the porous bed, comprised of alumina spheres of equal diameter (1–32 mm). Several of the common detonable fuels were
tested at atmospheric initial pressure. Results indicate the existence of a continuous range of velocities with change in
Φ, spanning the lean and the rich propagation limits. For all fuels in a given porous bed, the velocity decreases from a maximum
value at the most sensitive mixture near Φ≈1 (minimum induction length), to V/V
CJ≈0.3 at the limits. A decrease in pore size brings about a reduction in V/V
CJ and a narrowing of the detonability range for each fuel. For porous media comprised of spherical particles, it was possible
to correlate the velocity data corresponding to a variety of different mixtures and for a broad range of particle sizes, using
the following empirical expression: V/V
CJ=[1–0.35 log( d
c
/ d
p)]±0.1. The critical tube diameter d
c
is used as a measure of mixture sensitivity and d
p denotes the pore diameter. An examination of the phenomenon at the composition limits, suggests that wave failure is controlled
by a turbulent quenching mechanism. 相似文献
4.
In this paper, we study a Targeted Energy Transfer (TET) problem between a p degrees-of-freedom (dof) linear master structure and several coupled parallel slave Nonlinear Energy Sink (NES) systems.
In detail, each lth dof l=1,2,…, p contains n
l
parallel NES; so the linear structure has ( n
1+ n
2+⋅⋅⋅+ n
l
+⋅⋅⋅+ n
p
) NES. We are interested to study analytically the TET phenomenon during the first mode of the compound system. To this end,
complexification, averaging, and multiple scales methods are used. 相似文献
5.
Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented
in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small
excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced
to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis
of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical
simulations and an essential difference between linear and nonlinear modes is pointed out. 相似文献
6.
To realize large scanning angles, torsional microscanners are normally excited at their natural frequencies. Usually, a bias DC voltage is also applied to scan around a desired nonzero tilt angle. As a result, a deep understanding of the mirror’s response to a DC-shifted primary resonance excitation is imperative. Along these lines, we use the method of multiple scales to obtain a second-order nonlinear approximate analytical solution of the mirror steady-state response. We show that the response of the mirror exhibits a softening-type behavior that increases as the magnitude of the DC component increases. For a given mirror, we can also identify a DC voltage range wherein the mirror exhibits a two-to-one internal resonance between the first two modes; that is, ω
2≈2ω
1. To analyze the mirror behavior within that range, we first treat the case where the excitation frequency is near the first-mode frequency; that is, Ω≈ω
1. Then we treat the case where the excitation frequency is near the second-mode frequency; that is, Ω≈ω
2. We analyze the stability of the response and compare the analytical results to numerical solutions obtained via long-time integration of the equations of motion. We show that, due to the internal resonance, the mirror exhibits complex dynamic behavior characterized by aperiodic responses to primary resonance excitations. This behavior results in undesirable oscillations that are detrimental to the mirror performance, namely bringing the target point in and out of focus and resulting in distorted images. 相似文献
7.
Discrete dipoles located near the crack tip play an important role in nonlinear electric field induced fracture of piezoelectric
ceramics. A physico-mathematical model of dipole is constructed of two generalized concentrated piezoelectric forces with
equal density and opposite sign. The interaction between crack and electric dipole in piezoelectricity is analyzed. The closed
form solutions, including those for stress and electric displacement, crack opening displacement and electric potential, are
obtained. The function of piezoelectric anisotropic direction, p
a
(θ)=cosθ+ p
a
sinθ, can be used to express the influence of a dipole's direction. In the case that a dipole locates near crack tip, the
piezoelectric stress intensity factor is a power function with −3/2 index of the distance between dipole and crack tip.
Supported by National Natural Science Foundation of China(No. 10072033) 相似文献
8.
This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative spring–mass–pendulum
system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities.
Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then
applied to the spring–mass–pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic
perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate
the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear
frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation
techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple
time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The
numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse
into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order–chaos–order sequence.
This phenomenon is also presented and discussed. 相似文献
9.
The single and double phase macroscopic permeabilities of bimodal reconstructed porous media have been studied. The structure of these bimodal media is characterized by the micro and macroporosities (vug system) and by the micro and macrocorrelation lengths l
p and l
v. For a single phase, if the vugular system does not percolate, it is shown that the absolute permeability K mainly depends on l
p and very little on the other parameters. However, when the vugs percolate, K is also influenced by the density of vugs. For double phase calculations (in strong wettability conditions), it is shown that a vuggy percolating system affects mainly the nonwetting phase permeability. Moreover, the relative permeabilities for a nonpercolating vuggy system are only slightly influenced by the porosity distribution. These predictions are in good agreement with some experimental data obtained with limestones. 相似文献
10.
When a plane detonation propagating through an explosive comes into contact with a bounding explosive, different types of diffraction patterns, which may result in the transmission of a detonation into the bounding mixture, are observed. The nature of these diffraction patterns and the mode of detonation transmission depend on the properties of the primary and bounding explosives. An experimental and analytical study of such diffractions, which are fundamental to many explosive applications, has been conducted in a two channel shock tube, using H 2-O 2 mixtures of different equivalence ratios as the primary and bounding or secondary explosive. The combination of mixtures was varied from rich primary / lean secondary to lean primary / rich secondary since the nature of the diffraction was found to depend on whether the Chapman-Jouguet velocity of the primary mixture, D
p, was greater than or less than that of the secondary mixture, D
s. Schlieren framing photographs of the different diffraction patterns were obtained and used to measure shock and oblique detonation wave angles and velocities for the different diffraction patterns, and these were compared with the results of a steady-state shock-polar solution of the diffraction problem. Two basic types of diffraction and modes of detonation reinitiation were observed. When D
p> D
s, an oblique shock connecting the primary detonation to an oblique detonation in the secondary mixture was observed. With D
p< D
s, two modes of reinitiation were observed. In some cases, ignition occurs behind the Mach reflection of the shock wave, which is transmitted into the secondary mixture when the primary detonation first comes into contact with it, from the walls of the shock tube. In other cases, a detonation is initiated in the secondary mixture when the reflected shock crosses the contact surface behind the incident detonation. These observed modes of Mach stem and contact surface ignition have also been observed in numerical simulations of layered detonation interactions, as has the combined oblique-shock oblique-detonation configuration when D
p> D
s. When D
p> D
s, the primary wave acts like a wedge moving into the secondary mixture with velocity D
p after steady state has been reached, a configuration which also arises in oblique-detonation ramjets and hypervelocity drivers. 相似文献
11.
The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due
to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations,
where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed
transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium,
found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled
this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ, scales according to an inverse square-root power law, τ∼( μ− μ
c
) −1/2, as the bifurcation parameter μ, is driven further away from its critical value, μ
c
. In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional
maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species
ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising
due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement
with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling
laws of one-dimensional discrete dynamical systems with saddle-node bifurcations. 相似文献
12.
The phenomenon of low amplitude self-sustained pitch oscillations in the transitional Reynolds number regime is studied numerically through unsteady, two-dimensional aeroelastic simulations. Based on the experimental data, simulations have been limited in the Reynolds number range 5.0×10 4<Re c<1.5×10 5. Both laminar and URANS calculations (using the SST k– ω model with a low-Reynolds-number correction) have been performed and found to produce reasonably accurate limit cycle pitching oscillations (LCO). This investigation confirms that the laminar separation of the boundary layer near the trailing edge plays a critical role in initiating and sustaining the pitching oscillations. For this reason, the phenomenon is being labelled as laminar separation flutter. As a corollary, it is also shown that turbulence tends to inhibit their existence. Furthermore, two regimes of LCO are observed, one where the flow is laminar and separated without re-attachment, and the second for which transition has occurred followed by turbulent re-attachment. Finally, it is established that the high-frequency, shear instabilities present in the flow which lead to von Kármán vortex shedding are not crucial, nor necessary, to the maintaining mechanism of the self-sustained oscillations. 相似文献
13.
The present paper envisages laminar mixing of a two‐dimensional jet of particulate suspension in an incompressible carrier fluid with a free stream in direction of the jet axis. Finite difference technique has been employed for finding out solution of governing equations. It is found that the diffusion parameter ε, the ratio of particle diffusion coefficient and kinematic viscosity of the carrier fluid, have significant influence on the concentration of particles. A large value of ε has the effect in increasing the perturbation velocity up and perturbation density ρp. It is observed that the volume fraction φ, has no significant effect on perturbation velocity u and up but has profound effect on perturbation velocity v and vp. It is also found that the particle phase as well as the carrier fluid velocity attain free stream value for the large ξ, the modified x‐co‐ordinate. Further the magnitude of the perturbation quantities u, up, v, vp decreases as ξ increases i.e. at far away from the nozzle exit. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
14.
The method of multiple scales is used to analyze the non-linear forced response of circular cylindrical shells in the presence of a two-to-one internal (autoparametric) resonance to a harmonic excitation having the frequency Ω. If ω r and a r denote the frequency and amplitude of a flexural mode and ω b and a b denote the frequency and amplitude of the breathing mode, the steady-state response exhibits a saturation phenomenon when ωb ≈ 2 ωr, if the excitation frequency Ω is near ω b. As the amplitude ƒ of the excitation increases from zero, a b increases linearly whereas a r remains zero until a threshold is reached. This threshold is a function of the damping coefficients and ωb−2 ωr. Beyond this threshold a b remains constant (i.e. the breathing mode saturates) and the extra energy spills over into the flexural mode. In other words, although the breathing mode is directly excited by the load, it absorbs a small amount of the input energy (responds with a small amplitude) and passes the rest of the input energy into the flexural mode (responds with a large amplitude). For small damping coefficients and depending on the detunings of the internal resonance and the excitation, the response exhibits a Hopf bifurcation and consequently there are no steadystate periodic responses. Instead, the responses are amplitude- and phase-modulated motions. When Ω ≈ ωr, there is no saturation phenomenon and at close to perfect resonance, the response exhibits a Hopf bifurcation, leading again to amplitude- and phase-modulated or chaotic motions. 相似文献
15.
Wakes, and their interaction behind two parallel cylinders lying in a plane perpendicular to the flow, have been investigated experimentally in the sub-critical Reynolds number regime. The experiments were performed in a water channel using laser Doppler velocimetry. The gap between the two cylinders was less than the cylinder diameter, a geometry referred to as strong interaction configuration. In this case the blockage is strong and a gap-jet appears between the cylinders. Two flow regimes of the near wake region have been identified: one below a critical Reynolds number Re
c ]1000;1700[, where the gap jet is stably deflected to one side and the double near-wake becomes asymmetric; the other, above Re
c, where the gap-jet deflection is unstable and a random flopping phenomenon takes place. When Re< Re
c, two different Strouhal numbers are identified, related to the Kármán vortex shedding behind each cylinder. When Re> Re
c, a third frequency appears in the near wake, related to the development of Kelvin-Helmholtz vortices in the separated shear layer of the cylinders [Prasad A, Williamson CHK (1997) J Fluid Mech 333:375]. The observed flopping behavior is attributed to the birth of these Kelvin-Helmholtz instabilities and their intermittent nature. Further downstream, beyond about five cylinder diameters, the random flopping flow phenomena disappear while a slightly asymmetric single wake persists. It is characterized by a Strouhal number St=0.13, a value that one would normally measure behind a single cylinder of twice its diameter. 相似文献
16.
In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and NavierStokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if ( u, p) is a solution to the NavierStokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and | u( x)| = o (| x| -(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as | x| ? 0 |x| \to 0 or u ? L2n/(n - 1)( BR) u \in L^{2n/(n - 1)}(B_R) , then ( u, p) is a distribution solution and if in addition, u ? Lb( BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR. 相似文献
17.
Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting
an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies
of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems,
associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact
periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of
a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to
reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of
two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong
resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of
period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact
systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T
in, T
on and T
out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact
motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the
vibro-impact systems exhibit very complicated quasi-periodic impact motions.
The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of
Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang. 相似文献
18.
Let D ⊂ R
N be either all of R
n or else a cone in R
N whose vertex we may take to be at the origin, without loss of generality. Let p
i, q j, i = 1, 2, be nonnegative with 0< p
1+ q
1≦ p
2+ q
2. We consider the long-time behavior of nonnegative solutions of the system
相似文献
19.
The mechanical behaviors of deep crack-weakened rock masses are different from those of shallow crack-weakened rock masses. The surrounding rock in shallow crack-weakened rock mass engineering is classified into loose zone, plastic zone and elastic zone, while the surrounding rock in deep crack-weakened rock mass engineering is classified into fractured zone and non-fractured zone, which occur alternatively. It is assumed that the deep rock masses contain one joint set, in which the probability density function describing the distribution of sizes is assumed to follow the Rayleigh distribution, and the probability density function describing the distribution of spacing is assumed to follow the Weibull distribution. On the basis of strength criterion of deep rock mass, the near-field stress redistribution around circular opening induced by excavation is determined. The strong interaction among cracks is investigated by using the dislocation model. The nucleation, growth, interaction and coalescence of cracks were analyzed based on the strain energy density factor theory. When cracks coalesce, failure of deep crack-weakened rock masses occurs, fractured zone is formed. Then, size and quantity of fractured zone and non-fractured zone are given out. The size and quantity of fractured zone increase with decreasing strength of rock mass. The size and quantity of fractured zone increase with increasing in situ stress. Zonal fracturing phenomenon occurs once value of in situ stress is larger than the unaxial compressive strength of rock masses. The size and quantity of fractured zone decrease with increasing λ when p2 > p1. The size and quantity of fractured zone increase with increasing λ when p2 < p1. 相似文献
20.
The adhesion versus vapor pressure ( p/ p
s) trend between two elastically hard rough surfaces is modeled and compared with experimental results. The experimental samples
were hydrophilic surface-micromachined cantilevers, in which the nanometer-scale surface roughness is on the order of the
Kelvin radius. The experimental results indicated that adhesion increases exponentially from p/ p
s=0.3 to 0.95, with values from 1 mJ/m 2 to 50 mJ/m 2. Using the Kelvin equation to determine the force-displacement curves, the mechanics of a wetted rough interface are treated
in two ways. First, the characteristics of a surface with rigid asperities of uniform height are derived. At low p/ p
s, menisci surrounding individual asperities do not interact. Beyond a transition value, [ p/ p
s] tr, a given meniscus grows beyond the asperity it is associated with, and liquid fills the interface. Capillary adhesion in
each realm is found according to the integrated work of adhesion. Second, a more general approach allowing an arbitrary height
distribution of Hertzian asperities subject to capillary forces is justified and developed. To compare with experimental results,
a Gaussian height distribution is first assumed but significantly underestimates the measured adhesion. This is because equilibrium
is found far into the Gaussian tail, where asperities likely do not exist. It is shown that by bounding the tail to more likely
limits, the measured adhesion trend is more closely followed but is still not satisfactorily matched by the model. The uniform
summit height model fits the data very well with a single free parameter. These results can be rationalized if the upper and
lower surfaces are geometrically correlated. 相似文献
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