Linear Spring Constant Research Articles

Vibration and buckling analysis of rectangular plates with nonlinear elastic end restraints against rotation

The fundamental natural frequencies and buckling loads of a rectangular plate with nonlinearly rotational restraints are obtained by using the finite element technique. If the rotational springs are unsymmetric the iterative scheme must be employed to acquire the solutions of the nonlinear problem. The values which describe the free vibration and stability behaviour of the plate will increase when either the parameters of rotational spring or the initial rotational angles increase. Incidentally, it can be concluded that these results grown nonlinearly with respect to either the linear or nonlinear rotational spring constants. Finally, both the frequency and stability parameters are evaluated for several boundary conditions which are quite useful in engineering analysis and design.

Mechanical impedance of a cracked cantilever beam

In this paper the influence of a transverse surface crack on the mechanical impedance of a cantilever beam is investigated both analytically and experimentally. Modeling the flexibility induced by the crack by a linear spring constant, relations linking the changes of the mechanical impedance to the location and size of the crack are obtained for both longitudinal and flexural vibrations. The results show that the mechanical impedance changes substantially due to the presence of the crack in case of flexural vibrations. The changes follow definite trends depending upon the location and size of the crack and consequently the impedance can be used as an additional defect information carrier for crack appearance and possible estimation of crack location.

A method for obtaining a linear spring for a permanent magnet levitation system using electromagnetic control

Since the electromagnetic force is a function of the gap between magnets, the spring constant has large nonlinearities with respect to the displacement. Hence the spring constant varies significantly with the initial load. This implies that the linear control theory is limited to a range of very small displacement. To treat the system with variable initial loads and large displacements, this paper presents a procedure for obtaining a linear spring with variable spring constants by using a repulsive type actuator with permanent magnets and electromagnets. The analytical expression for obtaining a linear spring is presented. Experimental tests were carried out, and are compared with theoretical results. As an example, the results are applied to the problem of dynamic absorbers. In this actuator, the feedback loop for obtaining the linear spring constant has no dependence on the damping force generated by a velocity feedback loop. Hence the velocity feedback coefficient can be decided without consideration of its effect on the spring constant.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Identification of Single-Degree-of-Freedom Nonlinear Vibrating System by Piecewise Linear Approximation.

A new method for parameter identification of single-degree-of-freedom nonlinear vibrating systems is proposed in the time domain. The nonlinearity is approximated by piecewise linear characteristics. The nonlinear characteristics of restoring force and damping can be obtained from the linear spring constants and linear viscous damping coefficients identified on each piecewise linear section. Two experimental procedures, that is, the impact excitation method and the additional mass method without excitation, are performed. The response and excitation data for the impact excitation are required in the impact excitation method and the free vibration responses of the system with and without additional mass are required in the additional mass method. System parameters of the vibrating systems with various nonlinear characteristics, Duffing-type system and clearance system, can be accurately identified by both the procedures. It is shown that the additional mass method is valid for torsional nonlinear systems on which the measurement of excitation force is difficult.

Shock-Absorbing Mechanism of Human Leg.

The shock-absorbing mechanism to minimize the vertical acceleration when jumping down onto a hard surface was investigated. The dynamic model of a human leg was built with a link joint. The muscle model consisted of a contractile element, a elastic element and a viscous element. The equivalent linear spring constant and equivalent linear damping coefficient decreased upon lowering of the hip by varying the bending angles of the ankle and knee. The numerical results showed that shock could be absorbed passively by deep bending, and to a greater degree actively in the same bending position.

Discrete element modelling of a broken ice field — Part II: simulation of ice loads on a boom

A two-dimensional discrete element model which simulates the dynamics and interaction forces between individual ice floes in a broken ice field has been described in the paper “Discrete Element Modelling of a Broken Ice Field — Part I: Model Development”. The present paper elaborates on the application of this model using three examples. The first example presents a simulation of the forces exerted on a boom when it is pulled through a broken ice sheet. This example refers to small-scale tests, in which the kinematic behaviour and calculated forces are compared with, and calibrated against, tests with such a boom in the ice tank at the National Research Council Canada (NRC, Ottawa). The two other examples, utilizing the calibration from the small-scale testing, study the full-scale interaction between a broken ice field and the boom. Example 2 shows that for an ice thickness of 1.0 m and a boom speed of 0.2 m/s, the average forces on a 250 m wide boom were 50 kN and 119 kN for ice concentrations of 0.4 and 0.6, respectively. In Example 2 the floe diameters ranged from 20 m to 36 m. The values of the parameters in the visco-elastic-plastic rheology (linear spring constant, viscous damping coefficient and friction coefficient) appeared to have marginal influence on the force exerted on the boom. The average force increases with increasing ice concentration, ice thickness, boom speed and boom width. Keeping the ice concentration, ice thickness and boom speed fixed, the average force on the boom is proportional to the boom width.

Hope bifurcation in viscous, low-speed flows about an airfoil with structural coupling

The locations of Hopf bifurcation points associated with the viscous, incompressible flow about a NACA 0012 airfoil with structural coupling are computed for very low Re ( < 2000). A semi-implicit, first-order-accurate time-integration algorithm is employed to solve the streamfunction-vorticity form of the Navier-Stokes equations. The algorithm is modified to include simple structural elements affixed to the airfoil and the resulting airfoil motion. The equations describing the airfoil motion are integrated in time using a fourth-order Runge-Kutta algorithm. Numerical experiments are performed to determine if the structural model of the airfoil has an effect upon the location of the Hopf bifurcation point when compared with the fixed airfoil. Results are reported for a variety of structural characteristics, including variation of torsional and linear spring constants, inertial properties, structural coupling and structural damping.

A self-consistent model for multi-fiber crack bridging

A self-consistent model for determining the equivalent spring stiffness in fiber crack-bridging problems is proposed. The model is compared with the shear-lag model for ideal bonding of the fiber-matrix interface, in which case the springs have a linear spring constant. A comparison is also made, in the homogeneous limit when fiber and matrix elastic properties are identical, with results from particle and ligament crack-bridging analyses. The self-consistent model is seen to agree with these other models. A simplified parametric equation is given which approximates the results over the considered range of material properties.

Static Deflection of Continuous Beams with Nonlinear Supports

A method to calculate the static deflection, the bending moment, the reactions at supports, etc. is proposed for continuous beams which have nonlinear supports. The calculation method is a kind of relaxation method. We linearize the system by replacing nonlinear supports with linear springs, and carry out the calculation of static deflection for the line-arized system. By iterating the linear calculations in which the linear spring constants are determined from the previous calculation result, we can obtain the solution which also satisfies the characteristics of nonlinear supports. The characteristics of the following supports are described as the examples of nonlinear ones: (1) a section of stern tube bearings made of lignumvitae, (2) a rigid support with backlash, and (3) a spring of which the end is separable from the beam when unloaded. As an example of numerical calculation, a calculation result for a propeller shaft of a ship is illustrated.

DYNAMIC RESPONSE OF THE SHIP AND THE BERTHING FENDER SYSTEM AFTER IMPACT

This paper describes a mothod of analysis for evaluating the portion of ship kinetic energy and impact force transmitted to a berthing structure provided with fenders which have linear or nonlinear spring constants. In the analysis, presented herein, the dynamic responses of the ship, fender and berthing structure, after impact, are considered, and derived equations for the selection of different parameters needed for the solutions of the dynamic equations are included. These are comprised of the virtual mass of the ship, in both translational and rotational motion, in addition to the time interval required for the solutions of the motion equations by numerical integration methods.