Dragi Radomirovic

Deflection and potential energy of linear and nonlinear springs: approximate expressions in terms of generalized coordinates

The method for finding approximate expressions for the deflection and the potential energy of linear and certain nonlinear springs is presented. These expressions are given in terms of the constant static deflection, which can be zero or non-zero, and the deflection considered in two mutually orthogonal directions and the corresponding deflections, which need to be truncated to a certain order of the generalized coordinate. An example is given to demonstrate their use and benefits. These approximations can be useful for teaching courses on vibration. In addition, it is expected that they can be utilized by students and researchers to define the potential energy of springs as well as to derive the equations of motion more easily.

An equivalent spring for nonlinear springs in series

This work is concerned with nonlinear springs in series with the aim of obtaining the equivalent spring and its characteristics. The case of two linear springs in series is presented first as a basis for the extension to the cases of two purely nonlinear springs in series and two or more equal Duffing springs in series, which both allow the exact determination of the equivalent spring. Then, the most general case of two nonlinear springs with odd-power terms and different coefficients is examined. The condition is derived in terms of their characteristics for which the exact solution for the equivalent spring can be obtained.

On the equivalent systems for concurrent springs and dampers – Part 2: Small out-of-plane oscillations

Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffness axes for which the potential energy does not contain mixed terms, are stated and proven. So far unknown invariants related to the sum of original and new stiffness coefficients are provided. In addition, the equivalent system of three mutually orthogonal dampers is obtained for any system of out-of-plane concurrent linear viscous. The theorem defining their damping coefficients and their directions, collinear with the principal damping axes for which the dissipative function does not contain mixed terms, is provided. The corresponding invariant for damping coefficients is presented, too. An ellipsoid of displacement and an ellipsoid of stiffness are discussed. Three illustrated examples are given.

Characterisation of tree vibrations based on the model of orthogonal oscillations

This study presents quantitative and qualitative insights into the analysis of data obtained by tracking the motion of reflective markers arranged along the trunk of a pole-like potted tree, which was recorded by a state-of-the-art infrared motion-tracking system. The experimental results showed in-plane damped trajectories of the markers with lateral displacements, i.e. out-of-plane vibrations of the tree under consideration. To explain such response and to determine the corresponding oscillatory characteristics, a completely new and original utilisation of the recorded in-plane damped trajectories is presented. The quantitative insight gained is based on the mechanical model that consists of two orthogonal springs and dampers placed in the plane where the motion takes place, and it is then directed towards the determination of the characteristics of the related orthogonal oscillations: two natural frequencies, the position of the principal axes to which they correspond, and two damping ratios. The qualitative insight gained involves analysing the shape and narrowness of the trajectory to assess how close-valued two natural frequencies are, and how small the overall damping is. The quantitative and qualitative methodologies presented herein are seen as beneficial for arboriculture, forestry and botany, but given the fact that orthogonal oscillations appears in many natural and engineering systems, they are also expected to be useful for specialists in other fields of science and engineering as well.

Corn seeding process fault cause analysis based on a theoretical and experimental approach

Abstract Contemporary trends have set the need for high precision systems in agricultural operations, but it is the seeding technology in the first place that must satisfy the toughest demands to achieve the highest possible profit. The improvement of seeding precision is impossible without knowledge of the working principles of all parts of a singulation mechanism and all factors which influence the seeding errors. The evolution of a seeder depended crucially on the improvement of the testing methodology and techniques. This paper presents a photo-electronic device which monitors the seed flow at free fall after leaving the seeding mechanism. An air blowing seeding mechanism was used to simulate the seeding. Its performances were tested with seven varieties of corn seed, two seeding plates, two different air flow rates, and four different speeds of revolution of the seeding plate. A full factorial design was used to determine the significance of influence of four input factors and interactions of two of them. Data validation was done by comparing the values calculated according to visual analysis of high-speed camera recordings and photo-electronic system data where a strong relationship was achieved (R2 > 0.99). It was determined that the speed of revolution of the seeding plate had the greatest influence on the varying distance between the consecutive seeds, while the seed variety exerted the least influence. The pressure of air flow and the variety of the seeds had the most significant influence on the variation of quality of feed index, although, the type and speed of revolution of the seeding plate were also statistically significant parameters. The occurrence of miss and multiple seed were caused by air flow pressure, variety of the seeds, type of the seeding plate, i.e., speed of revolution of the seeding plate. The applied optimization method can be a useful tool for finding the best possible combination of input parameters observed in the test in order to fulfill the criteria which vary depending on the user’s needs.

Sympodial tree-like structures: from small to large-amplitude vibrations

This study deals with tree-like structures that mimic a trunk with either first-order branches or both first and second-order branches of a sympodial tree. The corresponding mechanical model comprises physical pendula coupled with torsional springs and viscous dampers. Natural frequencies and modal shapes are obtained analytically and the effects of a branching angle and a stiffness ratio on their change is analysed. Then, the trunk is harmonically excited and the corresponding structural response is investigated, both for small undamped and damped vibrations, focusing on the concept of dynamic absorbers and the attenuation of the amplitude of the trunk and first-order branches as this attenuation is beneficial for practical engineering applications. The corresponding frequencies at which these amplitudes are equal to zero or in resonance are determined. Finally, free large-amplitude vibrations are examined. Conditions for internal resonances in the structure with first-order branching are determined analytically by using the method of multiple scales. Frequency spectra of large-amplitude vibrations are obtained numerically focusing on the influence of the branching angle on their content. Biomimetic potentials and advantages of designing these coupled branched structures are discussed through the whole study via the influence of the stiffness ratio of the springs, the branching angle and the hierarchy of branches on their response.

Stiffness properties of certain oscillatory systems: quantification and possibilities for corrections

This study is concerned with the use of energy methods to find the equivalent stiffness properties of certain vibration models. The first class of models are one-degree-of-freedom oscillatory systems that include several combinations of a main spring and an additional spring. The stiffness properties of the equivalent system as well as the condition for different types of the resulting oscillatory motion are determined. Then, two-degree-of-freedom systems comprising two pairs of mutually orthogonal or symmetrically attached springs are examined. The corresponding equivalent stiffness properties are obtained and the possibilities for corrections are also discussed.

Dynamic circle of plane motion

A rigid body in plane motion is considered. The location of all instantaneous points for which a general equation for moments is of the form equivalent to the one written with respect to the mass centre is determined. It is found that the curve containing all these points is a circle. Two examples are given to illustrate some applications of the findings.