Method of Determining Damping and Elastic Properties of Rubber Used in Tires

1968 ◽  
Vol 41 (4) ◽  
pp. 943-952
Author(s):  
D. F. Moore ◽  
D. B. Larson
Keyword(s):  
Differential Equation ◽  
Elastic Properties ◽  
Nonlinear Differential Equation ◽  
Viscoelastic Behavior ◽  
Least Square ◽  
History Of ◽  
Dynamic Penetration ◽  
Voigt Model ◽  
Two Parameters ◽  
Tread Rubber

Abstract Dynamic penetration of tread rubber by asperities of a roadbed for a rolling tire was simulated by impact of a rubber specimen on different arrangements of small spheres. The penetration history of the draping rubber was then analyzed to determine its damping and elastic properties. The method of analysis essentially varied two parameters in a nonlinear differential equation that described the system, so as to minimize the error between the integrated equation and test data. Least square of minimization was used. The results indicated that the viscoelastic behavior of the rubber can be represented by a simple Voigt model. The importance of jointly considering both draping and traction zones in the contact patch of a rolling tire is discussed.

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

Efficacy of galcanezumab in patients with episodic cluster headaches and a history of preventive treatment failure

2021 ◽  
Vol 4 ◽  
pp. 251581632110156
Author(s):  
Brian Plato ◽  
J Scott Andrews ◽  
Mallikarjuna Rettiganti ◽  
Antje Tockhorn-Heidenreich ◽  
Jennifer Bardos ◽  
...  
Keyword(s):  
Cluster Headache ◽  
Treatment Failure ◽  
Preventive Treatment ◽  
Mean Difference ◽  
Least Square ◽  
Double Blind Study ◽  
Double Blind ◽  
Cluster Headache Attack ◽  
Episodic Cluster Headache ◽  
History Of

Objective: The efficacy of galcanezumab was evaluated in patients with episodic cluster headache and history of preventive treatment failure. Methods: In the randomized, 8-week, double-blind study (CGAL), patients with episodic cluster headache received once-monthly subcutaneous injections of galcanezumab 300 mg or placebo. Patients who completed CGAL and enrolled in an open-label study were queried for preventive treatment history. In a subset of patients with a known history of failure of verapamil or any other prior preventive treatment, a post hoc analysis of least square mean change from baseline in weekly cluster headache attack frequency across Weeks 1 to 3 was assessed. Results: Fifteen patients provided data for known history of prior preventive treatment failure (6 placebo, 9 galcanezumab), of whom 11 failed verapamil. The mean reduction in the weekly frequency of cluster headache attacks was greater with galcanezumab compared to placebo among patients with prior preventive treatment failure (8.2 versus 2.4); mean difference 5.8 (95% confidence interval [CI] 2.0, 13.6) and among patients with verapamil failure (10.1 versus 1.6); mean difference 8.5 (95% CI 0.4, 16.7). Conclusion: In this exploratory analysis of patients with a known history of prior preventive treatment failures, treatment with galcanezumab resulted in greater mean reductions in weekly cluster headache attacks compared with placebo. ClinicalTrials.gov: NCT02397473 (I5Q-MC-CGAL) NCT02797951 (I5Q-MC-CGAR)

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

scholarly journals Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Cemil Tunç ◽  
Muzaffer Ateş
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Cauchy Formula ◽  
Boundedness Of Solutions ◽  
Constant Coefficients ◽  
Particular Solution

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

Magnetohydrodynamic Squeeze Film Characteristics Between Parallel Circular Plates Containing a Single Central Air Bubble in the Inertial Flow Regime

1999 ◽  
Vol 66 (4) ◽  
pp. 1021-1023 ◽  
Author(s):  
R. Usha ◽  
P. Vimala
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Bubble Radius ◽  
Fluid Film ◽  
Squeeze Film ◽  
Parallel Plates ◽  
Circular Plates ◽  
Air Bubble ◽  
Approximate Analytical Solutions

In this paper, the magnetic effects on the Newtonian squeeze film between two circular parallel plates, containing a single central air bubble of cylindrical shape are theoretically investigated. A uniform magnetic field is applied perpendicular to the circular plates, which are in sinusoidal relative motion, and fluid film inertia effects are included in the analysis. Assuming an ideal gas under isothermal condition for an air bubble, a nonlinear differential equation for the bubble radius is obtained by approximating the momentum equation governing the magnetohydrodynamic squeeze film by the mean value averaged across the film thickness. Approximate analytical solutions for the air bubble radius, pressure distribution, and squeeze film force are determined by a perturbation method for small amplitude of sinusoidal motion and are compared with the numerical solution obtained by solving the nonlinear differential equation. The combined effects of air bubble, fluid film inertia, and magnetic field on the squeeze film force are analyzed.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

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