Dynamic Stresses in a Driven Pile During Installation-Classical Wave Equation Model Solution Using Partial Differential Equations

2014 ◽  
Author(s):  
Syed Muhammad Mohsin Jafri ◽  
Phayak Takkabutr
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Closed Form Solution ◽  
Equation Of Motion ◽  
Form Solution ◽  
Design Parameters ◽  
Partial Differential ◽  
Equation Analysis ◽  
Driven Pile

This paper derives and solves the governing dynamic wave equation of motion of a driven pile during the installation phase, when the driven pile is subjected to hammer blows. The pile is assumed as an elastic solid body. The equation of motion is a partial differential equation in space (axial coordinate) and time. The governing partial differential equation of motion is solved for installation boundary conditions, and simplified soil resistance models. The solution of the governing equation yields important design parameters, such as stress variation at any cross-section along the pile length with respect to time, and propagating wave speed. The resulting closed-form solution can be easily implemented using a standard spreadsheet or an engineering calculation program. This approach is compared with conventional wave equation analysis (WEAP) used in industry practice. The conventional wave equation analysis is based on discretization of the pile into mass-spring-damper elements (lumped parameter approach), rather than continuous modeling. The models and solutions from these two approaches are compared.

scholarly journals Behaviour under Moving Distributed Masses of Simply Supported Orthotropic Rectangular Plate Resting on a Constant Elastic Bi-Parametric Foundation

2020 ◽  
pp. 64-92
Author(s):  
T. O. Awodola ◽  
S. Adeoye
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourth Order ◽  
Closed Form Solution ◽  
Form Solution ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Simply Supported ◽  
Orthotropic Rectangular Plate

This work investigates the behavior under Moving distributed masses of orthotropic rectangular plates resting on bi-parametric elastic foundation. The governing equation is a fourth order partial differential equation with variable and singular co-efficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al[1]. This is then simplified using modified asymptotic method of Struble. The closed form solution is analyzed, resonance conditions are obtained and the results are presented in plotted curves for both cases of moving distributed mass and moving distributed force.

The fundamental solution for the axially symmetric wave equation II

1980 ◽  
Vol 87 (3) ◽  
pp. 515-521
Author(s):  
Albert E. Heins
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Fundamental Solution ◽  
Free Space ◽  
Axially Symmetric ◽  
Partial Differential ◽  
Symmetric Wave ◽  
Alternate Forms ◽  
Image Position

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equationcould be written asif b > rorif r > b.

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

On the integrals of a partial differential equation

1947 ◽  
Vol 43 (3) ◽  
pp. 348-359 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Partial Differential ◽  
One Dimensional ◽  
First Order ◽  
Progressive Waves ◽  
Image Position ◽  
Straight Lines ◽  
Dimensional Wave

The ordinary one-dimensional wave equationhas special integrals of the formwhich satisfy the first-order equationsrespectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight linesin an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

Casimir Effect on Superharmonic Resonance of Bio-NEMS Circular Plate Resonator Sensors

2018 ◽  
Author(s):  
Martin Botello ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Circular Plate ◽  
Casimir Effect ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Amplitude Response ◽  
Partial Differential ◽  
Superharmonic Resonance ◽  
Mems Resonator

Casimir effect on superharmonic resonance of electrostatically actuated bio-nano-electro-mechanical system (Bio-NEMS) circular plate resonator sensor is investigated. The plate sensor resonator is clamped at the outer end and suspended over a parallel ground plate. The sensor can be used for detecting human viruses. Superharmonic resonance of the second order, frequency near one-fourth the natural frequency of the resonator, is induced using Alternating Current (AC) voltage. The magnitude of the AC voltage is also large enough to be consider hard excitation acting on the resonator. Beside Casimir effect, other external forces (i.e. electrostatic force and viscous air damping) acting on the MEMS resonator create a nonlinear behaviors such as bifurcation and pull-in instability. Hence, numerical models, such as Method of Multiple Scales (MMS) and Reduced Order Model (ROM), are used to predict the frequency-amplitude response for MEMS resonator. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. While, ROM, based on the Galerkin procedure which uses the mode shapes of vibration of the resonator as a basis of functions, transforms the nonlinear partial differential equation of motion into a system of ordinary differential equation with respect to dimensionless time. The frequency-amplitude response allows one to observe the behavior of the system for a range of frequencies near the superharmonic resonance. The effects of parameters such as Casimir effect, voltage, and damping on the frequency-amplitude response are reported.

Numerical Analysis of the Exact Solution of the Wave Equation of the Longitudinal Seismic Oscillations of Soil and the Construction as a Point Insertion

2018 ◽  
Vol 931 ◽  
pp. 152-157 ◽  
Author(s):  
Kamil D. Yaxubayev ◽  
Dinara D. Kochergina
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Wave Equation ◽  
Numerical Analysis ◽  
Differential Equations ◽  
Partial Differential ◽  
Seismic Oscillations

The numerical analysis of the exact solution of the system of the differential equations which includes the partial differential equation of the longitudinal seismic oscillations of the soil and the ordinary differential equation of oscillations of the construction in the form of a point rigid insertion.

scholarly journals Three ways to solve for bond prices in the Vasicek model

2004 ◽  
Vol 8 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Rogemar S. Mamon
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytic Solution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Bond Pricing ◽  
Rate Process ◽  
Bond Price ◽  
Martingale Approach ◽  
Forward Measure

Three approaches in obtaining the closed-form solution of the Vasicek bond pricing problem are discussed in this exposition. A derivation based solely on the distribution of the short rate process is reviewed. Solving the bond price partial differential equation (PDE) is another method. In this paper, this PDE is derived via a martingale approach and the bond price is determined by integrating ordinary differential equations. The bond pricing problem is further considered within the Heath-Jarrow-Morton (HJM) framework in which the analytic solution follows directly from the short rate dynamics under the forward measure.

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