TIME REPRESENTATION IN SCIENTIFIC DIAGRAMS: GALILEO GALILEI’S SOLUTION TO THE PROBLEM OF THE MOTION OF A FALLING BODY

Рассматриваются способы научного изображения темпоральных явлений на примере чертежей Галилео Галилея, при помощи которых он описывает и исследует равноускоренное движение. Для анализа применяется концептуальная рамка теории изобразительной и неизобразительной репрезентации Грегори Карри. Показано, что в случае научных диаграмм и графиков, представляющих время как одно из измерений пространства, основанием для геометрической изобразимости времени становится полагаемый изоморфизм между временем как континуумом мгновений и линией как континуумом точек. Парадигму такого структурного сопоставления мы находим в математическом мышлении Галилея, наиболее ярко проявляющемся в доказательстве формулы равноускоренного движения, представленном в «Беседах и математических доказательствах». The textbook narrative of the scientific revolution of the 17th century says that the early modern transformation of physics and mechanics was grounded in mathematization, that is, the application of mathematical principles and procedures to physical entities and events. However, such a transformation faces a major obstacle: compared to geometry, mechanics includes an additional dimension, namely, time. When temporality of motion is to be represented geometrically, a question arises on how a temporal succession can be expressed by a static image. The problem of representation of temporal events is not limited to science. In my paper, I apply a conceptual tool elaborated by Gregory Currie for the analysis of temporal representations in art, especially in cinema, to the analysis of scientific diagrams. In his book Image and Mind. Film, Philosophy, and Cognitive Science (1995), Currie distinguishes depictive and nondepictive representations, arguing that depictive representation requires similarity and homomorphism between an object ant its representation. Thus, it seems that any non-temporal image of temporal processes would lack the required similarity and cannot be a depictive representation. However, taking into account explanations given by Galileo Galilei for his famous diagrams of accelerated motion, I argue that the representation of time in scientific diagrams as a geometrical line is grounded in isomorphism between time as a continuous structure and continuous structure of a geometrical line. The main temporal process studied by mechanics is motion. Motion can be represented in two main ways: as a trajectory of a body over some period of time or as a functional relation of various parameters of motion (speed, path, acceleration) versus time. In the latter case, time is usually represented in a diagram as a geometrical line. We can find the origin of this type of representation in the late medieval doctrine of ‘intensio et remissio qualitatum’, intension and remission of qualities, in the context of which first diagrams representing intensity and extension of velocity of nonuniform motion as a changing quality over time were produced (Nicolas Oresme). We can find very similar graphical schemes in Galileo Galilei’s works, especially in Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638). In this work, Galileo announces with all clarity that he considers time to be the same aggregate of temporal moments as a line is an aggregate of points: every moment of time has a corresponding point on the geometrical line. This allows us to establish a homomorphic similarity between temporal duration and spatial (geometrical) extension. Thus, the essential requirement for depictive representation is met. Concluding, I have to point out that the homomorphic relation in this case is established between not real but abstract entities. The visible line itself is a representation of non-visible abstract geometrical line; in the same way, time consisting of non-divisible moments is just an abstract construction which refers to physical of psychological time-duration. However, the established relation between abstract time and abstract geometrical lines is a grounding event of the modern physical science.

Dynamic Behavior of a Bidirectional Functionally Graded Sandwich Beam under Nonuniform Motion of a Moving Load

A bidirectional functionally graded Sandwich (BFGSW) beam model made from three distinct materials is proposed and its dynamic behavior due to nonuniform motion of a moving point load is investigated for the first time. The beam consists of three layers, a homogeneous core, and two functionally graded face sheets with material properties varying in both the thickness and longitudinal directions by power gradation laws. Based on the first-order shear deformation beam theory, a finite beam element is derived and employed in computing dynamic response of the beam. The element which used the shear correction factor is simple with the stiffness and mass matrices evaluated analytically. The numerical result reveals that the material distribution plays an important role in the dynamic response of the beam, and the beam can be designed to meet the desired dynamic magnification factor by appropriately choosing the material grading indexes. A parametric study is carried out to highlight the effects of the material distribution, the beam layer thickness and aspect ratios, and the moving load speed on the dynamic characteristics. The influence of acceleration and deceleration of the moving load on the dynamic behavior of the beam is also examined and highlighted.

VIBRATION OF FGSW BEAMS UNDER NONUNIFORM MOTION OF MOVING LOAD USING AN EFFICIENT THIRD-ORDER SHEAR DEFORMATION FINITE ELEMENT FORMULATION

Vibration of functionally graded sandwich (FGSW) beams under nonuniform motion of a moving load is studied using a third-order shear deformation finite element formulation. The beams consists three layers, a homogeneous ceramic core and two functionally graded faces. Instead of the rotation, the finite element formulation is derived by using the transverse shear rotation as a unknown. Newmark method is used to compute the dynamic response of the beams. Numerical result reveals that the derived formulation is efficient, and it is capable to give accurate vibration characteristics by a small number of the elements. A parametric study is carried out to illustrate the effects of the material distribution, layer thickness ratio and moving load speed on the dynamic behavior of the beams. The influence of acceleration and deceleration of the moving load on the vibration of the beams is also examined and discussed.

VIBRATION OF FGSW BEAMS UNDER NONUNIFORM MOTION OF MOVING LOAD USING AN EFFICIENT THIRD-ORDER SHEAR DEFORMATION FINITE ELEMENT FORMULATION

Vibration of functionally graded sandwich (FGSW) beams under nonuniform motion of a moving load is studied using a third-order shear deformation finite element formulation. The beams consists three layers, a homogeneous ceramic core and two functionally graded faces. Instead of the rotation, the finite element formulation is derived by using the transverse shear rotation as a unknown. Newmark method is used to compute the dynamic response of the beams. Numerical result reveals that the derived formulation is efficient, and it is capable to give accurate vibration characteristics by a small number of the elements. A parametric study is carried out to illustrate the effects of the material distribution, layer thickness ratio and moving load speed on the dynamic behavior of the beams. The influence of acceleration and deceleration of the moving load on the vibration of the beams is also examined and discussed.

A Single Image Deblurring Algorithm for Nonuniform Motion Blur Using Uniform Defocus Map Estimation

One of the most common artifacts in digital photography is motion blur. When capturing an image under dim light by using a handheld camera, the tendency of the photographer’s hand to shake causes the image to blur. In response to this problem, image deblurring has become an active topic in computational photography and image processing in recent years. From the view of signal processing, image deblurring can be reduced to a deconvolution problem if the kernel function of the motion blur is assumed to be shift invariant. However, the kernel function is not always shift invariant in real cases; for example, in-plane rotation of a camera or a moving object can blur different parts of an image according to different kernel functions. An image that is degraded by multiple blur kernels is called a nonuniform blur image. In this paper, we propose a novel single image deblurring algorithm for nonuniform motion blur images that is blurred by moving object. First, a proposed uniform defocus map method is presented for measurement of the amounts and directions of motion blur. These blurred regions are then used to estimate point spread functions simultaneously. Finally, a fast deconvolution algorithm is used to restore the nonuniform blur image. We expect that the proposed method can achieve satisfactory deblurring of a single nonuniform blur image.

Generalization of Multispeed Gear Pitch Curves Design

Defined by various geometries and kinematics characteristics, noncircular gears represent special mechanical systems able to produce nonuniform motion of the driven element. Multispeed gears are a less investigated category of noncircular gears, used to combine constant and variable speed, recommended for industrial applications that require uniform speeds for different parts of a cycle, with a smooth transition between the parts. Due to the motion requirements, the design of a multispeed gear train has as input data either the driven gear law of motion or the gear transmission ratio variation. This paper presents a specific original algorithm for the generalization of the multispeed gear pitch curve modeling, based on the definition of the gear instantaneous transmission ratio. The next step of the multispeed gear design, the teeth generation, accessible through solid modeling or analytical procedure, will be approached in a future study.