Constant Force Spring System With a Spiral

2020 ◽  
Vol 12 (6) ◽  
Author(s):  
Richard B. Hetnarski
Keyword(s):  
Differential Equation ◽  
Initial Position ◽  
The Other ◽  
Helical Spring ◽  
Constant Force ◽  
Spiral Groove ◽  
Logarithmic Spirals ◽  
Spring System

Abstract The paper is devoted to the theory of a mechanism called constant force spring system. The system consists of a linear helical spring, a spiral drum, a take-up pulley, and two cords. The spiral drum and the take-up pulley are attached rigidly to each other. One of the cords connects the spring with the spiral drum, and at the initial position of the system fills the entire spiral groove on the drum. During the operation of the system, the spiral drum may rotate about its center, and the cord may gradually unwind from the spiral and wind again. The process of winding/unwinding causes the spring to change its length and change the force it exerts on the spiral drum. Due to the shape of the spiral, the distance of the cord from the center of the drum changes, so that the force in the other cord which is wound on the take-up pulley remains constant. Creation of that constant force is the goal of the system. The heart of the system is a specially designed spiral. The solution to the associated differential equation is provided. The system may allow to eliminate weight towers in exercise machines; eliminate counterweights in elevators, as well as in windows that open by moving upwards. The landing path of fighter planes’ landing on aircraft carriers may be reduced. The spiral of the system exhibits an important property which may interest mathematicians; its behavior is compared with that of the Archimedes’ and logarithmic spirals. Because of this property, the spiral may find other applications.

On the theory of the perturbations of the planets

1837 ◽  
Vol 3 ◽  
pp. 98-98
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
The Other ◽  
The Sun ◽  
Constant Force ◽  
First Case ◽  
Complete Determination ◽  
Analytical Formulae ◽  
Sole Action

The methods hitherto employed by mathematicians for determining the variations which the elements of the orbit of a planet undergo in consequence of perturbation, and for expressing these variations analytically in the manner best adapted for computation, are found to depend upon a theory in mechanics, of considerable intricacy, known by the name of the Variation of the Arbitrary Constants . In seeking the means for abridging the severe labour of the calculations, we must separate the general principles on which they are founded from the analytical processes by which they are carried into effect; and in some important problems great advantage is obtained by adapting the investigation to the particular circumstance of the case, and attending solely to the principles of the method in deducing the solution. The author suggests the possibility of simplifying physical astronomy by calling in the aid of only the usual principles of Dynamics, and by setting aside every formula or equation not absolutely necessary for arriving at the final results. The present paper contains a complete determination of the variable elements of the elliptic orbit of a disturbed planet, deduced from three differential equations, that follow readily from the mechanical conditions of the problem. In applying these equations the author observes, the procedure is the same whether a planet is urged by the sole action of the constant force of the sun, or is besides disturbed by the attraction of other bodies revolving round the luminary; the only difference being that, in the first case, the elements of the orbit are all constant, whereas in the other case they are all variable. The success of the method followed by the author is derived from a new differential equation between the time and the area described by the planet in its momentary plane, which greatly shortens the investigation by rendering it unnecessary to consider the projection of the orbit. But the solution given in the present paper, although it makes no reference to the analytical formulæ of the theory of the Variations of the Arbitrary Constants , is no less an application of that method and an example of its utility, and of the necessity of employing it in very complicated problems.

Extensional flows with viscous heating

2007 ◽  
Vol 571 ◽  
pp. 359-370 ◽  
Author(s):  
JONATHAN J. WYLIE ◽  
HUAXIONG HUANG
Keyword(s):  
Critical Value ◽  
The Other ◽  
Viscous Heating ◽  
Constant Force ◽  
Temperature Dependent ◽  
Downstream Location ◽  
Temperature Dependent Viscosity ◽  
Pulling Forces ◽  
Extensional Flows ◽  
Dependent Viscosity

In this paper we investigate the role played by viscous heating in extensional flows of viscous threads with temperature-dependent viscosity. We show that there exists an interesting interplay between the effects of viscous heating, which accelerates thinning, and inertia, which prevents pinch-off. We first consider steady drawing of a thread that is fed through a fixed aperture at given speed and pulled with a constant force at a fixed downstream location. For pulling forces above a critical value, we show that inertialess solutions cannot exist and inertia is crucial in controlling the dynamics. We also consider the unsteady stretching of a thread that is fixed at one end and pulled with a constant force at the other end. In contrast to the case in which inertia is neglected, the thread will always pinch at the end where the force is applied. Our results show that viscous heating can have a profound effect on the final shape and total extension at pinching.

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

An Inverse Force Analysis of a Planar Two-Spring System

1994 ◽  
Author(s):  
Thomas M. Pigoski ◽  
Joseph Duffy
Keyword(s):  
The Other ◽  
Spring Stiffness ◽  
Force Analysis ◽  
Degree Polynomial ◽  
Real Solutions ◽  
Variable Elimination ◽  
Spring System ◽  
The Common ◽  
Resultant Length ◽  
Inverse Force

Abstract A closed-form inverse force analysis was performed on a planar two-spring system. The two springs were grounded to pivots at one end and attached to a common pivot at the other. A known force was applied to the common pivot of the system, and it was required to determine all of the assembly configurations. By variable elimination, a sixth degree polynomial in the resultant length of one spring was derived, and from this, six real solutions of the point of application of force were obtained. Following this, the applied force was incremented along a line and the six paths of the moving pivot were tracked starting from the zero-load configurations. An analysis of these results showed stability phenomena indicating the workspace of this system contained regions of negative spring stiffness and points of catastrophe.

The Optimum Groove Geometry for Spiral Groove Viscous Pumps

1990 ◽  
Vol 112 (2) ◽  
pp. 409-414 ◽  
Author(s):  
Yuichi Sato ◽  
Kyosuke Ono ◽  
Akihiko Iwama
Keyword(s):  
Differential Equation ◽  
Flow Rate ◽  
Groove Width ◽  
Design Parameters ◽  
Width Ratio ◽  
Maximum Pressure ◽  
Spiral Groove ◽  
Clearance Ratio ◽  
Viscous Pump ◽  
Outflow Rate

The optimum geometries of disk and cylindrical sprial groove viscous pumps to provide the maximum pressure or flow rate are investigated theoretically. The geometrical design parameters, such as the groove angle, groove to ridge clearance ratio, groove width ratio and ridge clearance ratio, are considered as functions of meridional coordinate. Results are obtained from the solution of a differential equation for the smoothed overall pressure distribution of a spiral groove viscous pump. It is found that outflow rate increases with the increase of groove to ridge clearance ratio λ, and that for each value of λ there exist “optimum” values of groove angle and groove width ratio, which give a maximum outflow rate. However, the increase of λ decreases the ridge clearance.

scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

scholarly journals Klein–Gordon particle near RN black hole, generalized sl(2) algebra and harmonic oscillator energy

2019 ◽  
Vol 34 (31) ◽  
pp. 1950196
Author(s):  
J. Sadeghi ◽  
M. R. Alipour
Keyword(s):  
Differential Equation ◽  
Information Theory ◽  
Black Hole ◽  
Energy Spectrum ◽  
Harmonic Oscillator ◽  
Three Dimensional ◽  
The Other ◽  
Thermodynamical Properties ◽  
Heun Functions ◽  
Klein Gordon

In this paper, we consider Klein–Gordon particle near Reissner–Nordström black hole. The symmetry of such a background led us to compare the corresponding Laplace equation with the generalized Heun functions. Such relations help us achieve the generalized [Formula: see text] algebra and some suitable results for describing the above-mentioned symmetry. On the other hand, in case of [Formula: see text], which is near the proximity black hole, we obtain the energy spectrum. When we compare the equation of RN background with Laguerre differential equation, we show that the obtained energy spectrum is same as the three-dimensional harmonic oscillator. So, finally we take advantage of harmonic oscillator energy and make suitable partition function. Such function help us to obtain all thermodynamical properties of black hole. Also, the structure of obtained entropy lead us to have some bit and information theory in the RN black hole.

scholarly journals Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 110 ◽  
Author(s):  
Sooie-Hoe Loke ◽  
Enrique Thomann
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Ruin Probability ◽  
Risk Model ◽  
Constant Force ◽  
Time Of Ruin ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
Force Of Interest ◽  
The Time Of Ruin

In this paper, a dual risk model under constant force of interest is considered. The ruin probability in this model is shown to satisfy an integro-differential equation, which can then be written as an integral equation. Using the collocation method, the ruin probability can be well approximated for any gain distributions. Examples involving exponential, uniform, Pareto and discrete gains are considered. Finally, the same numerical method is applied to the Laplace transform of the time of ruin.

Design and Research of Constant Spring Hangers

2012 ◽  
Vol 482-484 ◽  
pp. 457-460 ◽  
Author(s):  
Li Hua Chen ◽  
Hao Zou ◽  
Chang Qing Sun
Keyword(s):  
Differential Equation ◽  
Engineering Design ◽  
Coordinate Transformation ◽  
Mechanical Model ◽  
Constant Force ◽  
Working Principle ◽  
Curve Equation

Compared with link constant hangers, main-compensating constant hangers are structural symmetric, having high theoretical constant force. Swing cams are important component of the device, therefore the design of cam curve is the key of constant hangers. Based on the analysis of working principle of constant hangers, a mechanical model of constant hanger is established, and the differential equation of cam curve is derived by means of coordinate transformation .Meanwhile, spring parameters are determined. Curve equation is resolved and the cam curves are plotted. Then, the influence of spring parameters on cam curves is discussed. Finally, force values acting on the cam and the central rod are calculated according to cam curve. The simulations in this paper can be used to instruct the actual engineering design.

THE CENTER PROBLEM FOR DISCONTINUOUS LIÉNARD DIFFERENTIAL EQUATION

1999 ◽  
Vol 09 (09) ◽  
pp. 1751-1761 ◽  
Author(s):  
B. COLL ◽  
R. PROHENS ◽  
A. GASULL
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Differential Equations ◽  
General Expression ◽  
The Other ◽  
Center Problem ◽  
Homogeneous Polynomials ◽  
Lyapunov Constants

We prove that the Lyapunov constants for differential equations given by a vector field with a line of discontinuities are quasi-homogeneous polynomials. This property is strongly used to derive the general expression of the Lyapunov constants for two families of discontinuous Liénard differential equations, modulus some unknown coefficients. In one of the families these coefficients are determined and the center problem is solved. In the other family the center problem is just solved by assuming that the coefficients which appear in these expressions are nonzero. This assumption on the coefficients is supported by their computation (analytical and numerical) for several cases.

Sign in / Sign up

Export Citation Format

Share Document