Parametric Frequency Analysis of Mathieu–Duffing Equation

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

Characteristics Analysis of Hydraulic System for Screw Distributor of Asphalt Paver

The hydraulic system of screw distributor plays an important role in asphalt paver. It belongs to the speed-regulating system of pump controlled motor. The motor speed and direction of rotation can be controlled by variable delivery pump. Thus the transferring speed of bituminous mixture can be adjusted. The mathematical model of the system is presented, and dynamical characteristics are analyzed. Simulation results indicate that the system is suitable to screw distributor and has high operating efficiency. By changing the angle of swashplate, the rotating speed of motor can be adjusted proportionally. The undamped natural frequency of the system is lower for two reasons. One reason is only one line controlled by the trapped oil spring rate. The other reason is larger volume of pump.

Quality Factors of Rotors With Hydrodynamic Bearings

The quality factor of a system is a measure of the maximum amplitude of vibration that occurs at resonance when the frequency of excitation is equal to the undamped natural frequency. This factor can be easily determined for a given mode of vibration, given its equivalent viscous damping ratio, as Q = 1/2ξ. Such a definition becomes complicated for a rotor mounted on hydrodynamic bearings. This note discusses some factors involved in estimating the quality factor of a rotor.

Determination of frequency response from step response: application to fluid-filled catheters

The performance of a fluid-filled catheter can be described by reporting its undamped natural frequency and damping ratio. These parameters can be measured by subjecting the catheter to sinusoidally varying pressures at a wide variety of frequencies to obtain the frequency response. They can also be computed from the response to a step change in pressure, which is often easier to produce. This paper derives the required equations and includes a graph which permits one to look up the undamped natural frequency after measuring the period and decay rate of the oscillation following a step change in pressure.

Estimating the Roots of the Characteristic Determinant for Multicoupled Systems

A procedure is developed, based on Couchy’s residue theorem, for bounding the undamped natural frequency, damping ratio, and real part of the roots of the characteristic determinant associated with a multicoupled system with several inputs and outputs. The method can lead to a locus of the roots as one or more parameters are varied. The underlying theory is developed and a numerical illustrative example is included.