Analysis of predictor equations for determining the blast-induced vibration in rock blasting

The paper proposes a new empirical correlation designed to complement the‘‘site laws"currently used to evaluate the attenuation in the rock masses of vibrations induced by rock blasting.The formula contains a deformed exponential known as the K-exponential,which seems to well represent a large number of both natural and artificial phenomena ranging from astrophysics to quantum mechanics,with some extension in the field of economics and finance.Experimental validation of the formula was performed via the analysis of vibration data covering a number of case studies,which differed in terms of both operation and rock type.A total of 12 experimental cases were analysed and the proposed formulation exhibited a good performance in 11 of them.In particular,the proposed law,which was built using blast test data,produced very good approximations of the points representing the vibration measurements and would thus be useful in organising production blasts.However,the developed formula was found to work less well when a correlation obtained for a given site was applied to another presenting similar types of rocks and operations,and thus should not be employed in the absence of measurements from test data.

State-vector equation with damping and vibration analysis of laminates

Based on the modified mixed Hellinger-Reissner（H-R） variational principle for elastic bodies with damping, the state-vector equation with parameters is directionally derived from the principle. A new solution for the harmonic vibration of simply supported rectangular laminates with damping is proposed by using the precise integration method and Muller method. The general solutions for the free vibration of underdamping, critical damp and overdamping of composite laminates are given simply in terms of the linear damping vibration theory. The effect of viscous damping force on the vibration of composite laminates is investigated through numerical examples. The state-vector equation theory and its application areas are extended.

SOLUTIONS TO EQUATIONS OF VIBRATIONS OF SPHERICAL AND CYLINDRICAL SHELLS

The governing equations of the free vibrations of spherical and cylindrical shellswith a regular singularity. are solved by Frobenius Series Method in the form ofmatrix. Considering the relationship of the roots of the indicial equation, we get somevarious expressions of solutions according to different cases. This work lays afoundation of solving certain elastic problems by analytical method.

SOLVING VIBRATION PROBLEM OF THIN PLATESUSING INTEGRAL EQUATION METHOD

This paper deals with reducing differential equations of vibration problem of plates with concentrated masses, elastic supports and elastically mounted masses into eigenvalue problem of integral equations. By applying the general function theory and the integral equation theory. the frequency equation is derived in terms of standard eigenvalue problem of a matrix with infinite order. So that, the natural frequencies and mode shapes can be determined. Convergence of this method has also been, discussed at the end of this paper.

Existence of Solutions for Boundary Value Problems of Vibration Equation with Fractional Derivative

In this paper, we investigate the solvability of boundary value problems for a class of vibration differential equation describing the fractional order damped system with signal stimulus. By presenting kernel function through the Laplace transform, and using the eigenvalue and the improved Leray-Schauder degree, the existence of solutions for boundary value problems is established.

A Compact Difference Method for Viscoelastic Plate Vibration Equation

In this paper, a fourth-order viscoelastic plate vibration equation is transformed into a set of two second-order differential equations by introducing an intermediate variable. A three-layer compact difference scheme for the initial-boundary value problem of the viscoelastic plate vibration equation is established. Then the stability and convergence of the difference scheme are analyzed by the energy method, and the convergence order is <img src="Edit_0a250b60-7c3c-4caf-8013-5e302d6477ab.png" alt="" />. Finally, some numerical examples are given of which results verify the accuracy and validity of the scheme.

Mixed Finite Volume Element Method for Vibration Equations of Beam with Structural Damping

In this paper, for the initial and boundary value problem of beams with structural damping, by introducing intermediate variables, the original fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed. Numerical examples are provided to confirm the theoretical results. In the end, we test the value of δ to observe its influence on the model.

A case study of blasting vibration attenuation based on wave component characteristics

A typical blasting vibration wave is a composite wave,and its attenuation law is affected by the type of dominant wave component.The purpose of the present study is to establish an attenuation equation of the peak particle velocity(PPV),taking into account the attenuation characteristics of P-,S-and R-waves in the blasting vibration wave.Field blasting tests were carried out as a case to specifically apply the proposed equation.In view of the fact that the discrete properties of rock mass will inevitably cause the uncertainty of blasting vibration,we also carried out a probability analysis of PPV uncertainty,and introduced the concept of reliability to evaluate blasting vibration.The results showed that the established attenuation equation had a higher prediction accuracy,and can be considered as a promising equation implemented on more complex sites.The adopted uncertainty analysis method can comprehensively take account of the attenuation law of blasting vibration measured on site and discrete properties of rock masses.The obtained distribution of the PPV uncertainty factor can quantitatively evaluate the reliability of blasting vibration,which is a powerful and necessary supplement to the PPV attenuation equation.

Propagation characteristics of vibration waves induced in surrounding rock by tunneling blasting

The effect of blasting vibration waves on surrounding rock and supporting structures is an important field in underground engineering. In this paper, the separation variable method is used to solve the displacement potential function for the propagation of the blasting vibration waves. In the axis coordinate system, the particle motion and stress change with axial distance, radial distance and time is obtained in surrounding rock. The peak particle velocity law in surrounding rock under different blast loads and surrounding rock parameters is discussed.In addition, the particle vibration characteristics in the surrounding rock are studied using numerical simulations method. The results shows that the peak particle velocity in surrounding rock appears negative exponent attenuation with the increase of axial distance, but it appears positive and negative fluctuations in radial direction. This phenomenon is a new discovery and it has been rarely investigated before. Moreover, the peak particle velocity attenuates more quickly and intensely in the near blasting field,which means that the supporting structure in a shorter distance away from the heading face is vulnerable to the impact of blasting vibration. Theattenuation of blasting vibration velocity is closely related to charge length, blasting load amplitude,attenuation index and rock elastic modulus. The numerical simulation accomplishes the same results and then demonstrates the validity of theoretical results.

Comparison of dominant frequency attenuation of blasting vibration for different charge structures

Dominant frequency attenuation is a significant concern for frequency-based criteria of blasting vibration control.It is necessary to develop a concise and practical prediction equation describing dominant frequency attenuation.In this paper,a prediction equation of dominant frequency that accounts for primary parameters influencing the dominant frequency was proposed based on theoretical and dimensional analyses.Three blasting experiments were carried out in the Chiwan parking lot for collecting blasting vibration data used to conduct regression analysis of the proposed prediction equation.The fitting equations were further adopted to compare the reliability of three different types of dominant frequencies in the proposed equation and to explore the effects of different charge structures on the dominant frequency attenuation.The apparent frequency proved to be more reliable to express the attenuation law of the dominant frequency.The reliability and superiority of the proposed equation employing the apparent frequency were verified by comparison with the other five prediction equations.The smaller blasthole diameter or decoupling ratio leads to the higher initial value and corresponding faster attenuation of the dominant frequency.The blasthole diameter has a greater influence on the dominant frequency attenuation than the decoupling ratio does.Among the charge structures applied in the experiments,the charge structure with decoupling ratio of 1.5 and blasthole diameter of 48 mm results in the greatest initial value and corresponding fastest attenuation of the dominant frequency.

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube(FGNT)composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model.The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle,and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel.Several nominal variables are introduced to simplify the differential governing equation,integral constitutive equation,and boundary conditions.Rather than transforming the constitutive equation from integral to differential forms,the Laplace transformation is used directly to solve the integro-differential equations.The explicit expression for nominal torsional rotation and torque contains four unknown constants,which can be determined with the help of two boundary conditions and two extra constraints from the integral constitutive relation.A few benchmarked examples are solved to illustrate the nonlocal influence on the static torsion of a clamped-clamped(CC)FGNT under torsional constraints and a clamped-free(CF)FGNT under concentrated and uniformly distributed torques as well as the torsional free vibration of an FGNT under different boundary conditions.

Exact solution of free vibration of adjacent buildings interconnected by visco-elastic dampers

With consideration of a high-rise coupled building system,a flexible beams-based analytical model is setup to characterize the dynamic behavior of the system.The general motion equation for the two beams interconnected by multiple viscous/visco-elastic dampers is rewritten into a non-dimensional form to identify the minimal set of parameters governing the dynamic characteristics.The corresponding exact solution suitable for arbitrary boundary conditions is presented.Furthermore,the methodology for computing the coefficients of the modal shape function is proposed.As an example,the explicit expression of the modal shape function is derived,provided only one damper is adopted to connect the adjacent buildings.Finally,to validate the proposed methodologies,three case studies are performed,in which the existence of the overdamping and the optimal damping coefficient are revealed.In the case of using one damper in connecting two similar buildings,the estimating equations for the first modal damping ratio are formulated.

Evaluation and prediction of blast induced ground vibration using support vector machine

We present the application of Support Vector Machine (SVM) for the prediction of blast induced ground vibration by taking into consideration of maximum charge per delay and distance between blast face to monitoring point. To investigate the suitability of this approach, the predictions by SVM have been compared with conventional predictor equations. Blast vibration study was carried out at Magnesite mine of Pithoragarh, India. Total 170 blast vibrations data sets were recorded at different strate-gic and vulnerable locations in and around to mine. Out of 170 data sets, 150 were used for the training of the SVM network as well as to determine site constants of different conventional predictor equations, whereas, 20 new randomly selected data sets were used to compare the prediction capability of SVM network with conventional predictor equations. Results were compared based on Co-efficient of Determination (CoD) and Mean Absolute Error (MAE) between monitored and predicted values of Peak Particle Veloc-ity (PPV). It was found that SVM gives closer values of predicted PPV as compared to conventional predictor equations. The coef-ficient of determination between measured and predicted PPV by SVM was 0.955, whereas it was 0.262, 0.163, 0.337 and 0.232 by USBM, Langefors-Kihlstrom, Ambraseys-Hendron and Bureau of Indian Standard equations, respectively. The MAE for PPV was 11.13 by SVM, whereas it was 0.973, 1.088, 0.939 and 1.292 by USBM, Langefors-Kihlstrom, Ambraseys-Hendron and Bureau of Indian Standard equations respectively.

Vibrational Stabilization by Reshaping Arnold Tongues: A Numerical Approach

This paper presents two contributions to the stability analysis of periodic systems modeled by a Hill equation: The first is a new method for the computation of the Arnold Tongues associated to a given Hill equation which is based on the discretization of the latter. Using the proposed method, a vibrational stabilization is performed by a change in the periodic function which guarantees stability, given that the original equation has unbounded solutions. The results are illustrated by some examples.

Vibration Measurement of Pedestrian Bridge Using Double Magnetic Suspension Vibrator Based on Wavelet Analysis

Aiming at the problem of pedestrian bridge vibration measurement,a vibration measurement system of pedestrian bridge with dual magnetic suspension vibrator structure was designed according to absolute vibration measurement principle. The relationship between the magnetic repulsion force of vibrator and its displacement was obtained by the experimental method and the least square fitting method. The vibration equations of two magnetic suspension vibrators were deduced respectively,and the measurement sensitivity of the system was deduced. The amplitude-frequency characteristic of the system was studied. A simulation model of vibrator measurement system with double magnetic suspension vibrator was established. The analysis shows that the sensitivity of the vibration measurement system with double magnetic suspension vibrator is higher than that with single magnetic suspension vibrator. The four vibration waveforms were measured,that is,no one passes through a pedestrian bridge,there are cars running under the pedestrian bridge,single pedestrian passes through the pedestrian bridge and multiple pedestrians pass through the pedestrian bridge. The multi-scale one-dimensional wavelet decomposition function was used to analyze the vibration signals. The vibration characteristics were obtained using one dimension wavelet decomposition function under four different conditions. Finally,the vibration waveforms of four cases were reconstructed. The measured results show that the vibration measurement system of pedestrian bridge with double magnetic suspension vibrator structure has high measurement sensitivity. The design has a certain value to monitor a pedestrian bridge.

Distribution of Vibrational Energy Levels of Protein Molecular Chains

The distributions of the quantum vibrational energy levels of the protein molecular chain are found by the discretely nonlinear Schrodinger equation appropriate to protein obtained from the Davydov theory. The results calculated by this method are basically consistent with the experimental values. Furthermore, the energy spectra at high excited states have also been obtained for the molecular chain which is helpful in researching the properties of infrared absorption and Raman scattering of the protein molecules.

Vibrational shear flow of anisotropic viscoelastic fluid with small amplitudes

Using the constitutive equation of co-rotational derivative type for anisotropic viscoelastic fluid-liquid crystalline(LC),polymer liquids was developed.Two relaxation times are introduced in the equation:λn represents relaxation of the normal-symmetric stress components;λs represents relaxation of the shear-unsymmetric stress components.A vibrational rotating flow in gap between cylinders with small amplitudes is studied for the anisotropic viscoelastic fluid-liquid crystalline polymer.The time-dependent constitutive equation are linearized with respect to parameter of small amplitude.For the normal-symmetric part of stress tensor analytical expression of the shear stress is obtained by the constitutive equation.The complex viscosity,complex shear modulus,dynamic and imaginary viscosities,storage modulus and loss modulus are obtained for the normal-symmetric stress case which are defined by the common shear rate.For the shear-unsymmetric stress part,two shear stresses are obtained thus two complex viscosities and two complex shear modulus(i.e.first and second one) are given by the constitutive equation which are defined by rotating shear rate introduced by author.The dynamic and imaginary viscosities,storage modulus and loss modulus are given for each complex viscosities and complex shear modulus.Using the constituive equation the rotating flow with small amplitudes in gap between two coaxial cylinders is studied.

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.

Vertical vibrations of elastic foundation resting on saturated half-space

This paper is mainly concerned with the dynamic response of an elastic foundation of finite height bounded to the surface of a saturated half-space. The foundation is subjected to time-harmonic vertical loadings. First, the transform solutions for the governing equations of the saturated media are obtained. Then, based on the assumption that the contact between the foundation and the half-space is fully relaxed and the halfspace is completely pervious or impervious, this dynamic mixed boundary-value problem can lead to dual integral equations, which can be further reduced to the Predhohn integral equations of the second kind and solved by numerical procedures. In the numerical extortples, the dynamic colnpliances, displacements and pore pressure are developed for a wide range of frequencies and material/geometrical properties of the saturated soil-foundation system. In most of the cases, the dynamic behavior of an elastic foundation resting on the saturated media significantly differs from that of a rigid disc on the saturated half-space. The solutions obtained can be used to study a variety of wave propagation problems and dynamic soil-structure interactions.