Review of Elementary concepts

among the endless possible motions of matter, periodic motion plays a particularly important role, not only because it occurs so widely in nature and everyday life, in man as well as machine, from the beat of the pulse to the motion of the planets, but also because of its basic role in physics; the concept and measurement of time are intimately linked to it.
a periodic motion of a particle is one which repeats itself over and over again. the motion is bounded, which means that it is confined to a finite region of space. this confinement is the result of the interaction of the particle with other particles, including matter in bulk. thus, in planetary motion the interaction force on it is the collective effect of all the water 'particles'.
the periodic motions of the moon and the planets have played important roles in the development of physics. kepler's discovery that the planetary orbits are elliptical with the sun at the focus, that the sector velocity of the planet is constant, and that the period is proportional to the 3/2 power of the major axis of the orbit, supported the hypothesis that the gravitational force is central and varies as the inverse square of the sun-planet distance.
another important example is the motion of a charged particle in a unuform magnetic field. in a plane normal to the magnetic field, the orbit will be circular with constant speed, and if the particle has a magnetic moment, it will also precess with a frequency proportional to the magnetic field. when applied to the nucleus, the measurement of this precession or gyro frequency is often used as a means of measuring the magnetic field.

Theoretical System

Scientific theories are perpetually changing. This is not due to mere chance but might well be expected, according to our characterization of empicrical science.
perhaps this is why, as a rule, only branches of science-and these only temporarily-ever acquire the form of an elaborate and logically well-constructed system of theories. in spite of this, a tentative system can usually be quite well surveyed as a whole, with all its important consequences. this is very necessary; for a severe test of a system presupposes that it is at the time sufficiently definite and final in form to make it impossible for new assumptions to be smuggled in. in other words, the system must be formulated sufficiently clearly and definitely to make every new assumption easily recognizable for what it is: a modification and therefore a revision of the system.
this, I believe, is the reason why the form of a rigorous system is aimed at. it is the form of a so-called 'axiomatized system'-the form which Hilbert, for example, was able to give to certain branches of theoretical physics. the attempt is made to collect all the assumption which are needed, but no more, to form the apex system. they are usually called the 'axioms' (or 'postulates', or, 'primitive propositions'; no claim to truth is implied in the term 'axiom' as here used). the axioms are choosen in such a way that all the other statements belonging to the theoretical system can be derived from the axioms by purely logical or mathematical transformations.
a theoretical system amy be said to be axiomatized if a set of statements, the axioms, has been formulated which satisfies the following four fundamental requirements. (a) the system of axioms must be free from contradiction (whether self-contradiction or mutual contradiction). this is equivalent to the demand that not every arbitrarily chosen statement is deducible from it. (b) the system must be independent, i.e. it must not contain any axiom deducible from the remaining axioms. (in other words, a statement is to be called an axiom only if it is not deducible within the rest of the system.) these two conditions concern the axiom system to the bulk of the theory, the axioms should be (c) sufficient for deduction of all statements belonging to the theory which is to be axiomatized, and (d) necessary, for the same purpose; which means that they should contain no superfluous assumptions.

APPLICATION OF INSTRUMENTED FALLING DART IMPACT TO THE MECHANICAL CHARACTERIZATION OF THERMOPLASTIC FOAMS

J. I. VELASCO*, A. B. MARTINEZ

Departament de Ciencies dels Materials i Enginyeria Metal.lurgica, Escola Tecnica Superior d’Enginyers Industrials de Barcelona (ETSEIB), Universitat Politecnica de Catalunya (UPC), Avda. Diagonal 647, 08028 Barcelona, Spain. E-mail : JVELASCO@CMEM.UPC.ES

D. ARENCON

Centre Catala del Plastic (CCP), Vapor Universitari de Terrassa. C/Colom 114, 08222 Terrassa, Spain

M. A. RODRIGUEZ-PEREZ, J. A. DE SAJA

Departamento de Fisica de la Materia Condensada, Cristalografia y Mineralogia, Universidad de Valladolid, Facultad de Ciencias. Prado de la Magdalena s/n, 47011 Valladolid, Spain.

The applicability of instrumented falling weight impact techniques in characterizing mechanichally thermoplastic foams at relatively high strain rates is presented. In order to try simulating impact loading of foams against sharp elements, an instrumented dart having a hemispherical headstock was employed in the test. Failure strength and toughness values were obtained from high-energy impact experiments, and the elastic modulus could be measured from both flexed plate and indentation low-energy impact tests. The result indicate a dependence of the failure strength, toughness, and the elastic modulus on the foam density, the foaming process, and the chemical composition. This influence was found to be similar to that of pure nonfoamed materials and also to that observed from low-rate compression tests. The results also indicate that the indentation low-energy impact tests were more accurate in obtaining right values of the elastic modulus than the flexed plate low-energy impact tests usually used to characterize rigid plastics. The foam indentation observed with this test configuration contributes to obtaining erroneous values of the elastic modulus if only a simple flexural analysis of plates is applied.

© 1999 Kluwer Academic Publishers

Traditionally, impact tests have been employed to measure the ability of a sample or a finished part to absorb a shock or impact. Falling weight impact tests stand out among the different types of impact tests because the simply supported or fixed (clamped) sample receives the collision of a mass falling from determined height. These tests have advantage of multiaxiality and the possibility of working with finished articles, if they are properly fixed.

The noninstrumented (analogic) impact tests give neither qualitative information about the energy required for the fracture initiation nor the information about the mechanical behavior of the material. These tests give only statistical plots as a relationship between the probability of failure or survival of the sample according the strictness of the test.

. LINEAR DIFFERENTIAL EQUATIONS

For equations with one main condition

(Those linear), you have permission

To take your solutions,

With firm resolutions,

And add them in superposition. *

Let’s say a little more about the solution in eq. (3.2). If a is negative, then let’s define a = -cu², where cu is a real number. The solution now becomes x(t) = Ae^iojt + Be-^iojt. Using e^ie = cos 6 + i sin 0, this can be written in terms of trig functions, if desired. Various ways of writing the solution are:

x(t) = Ae^' + Be-™'

x(t) = C cos cut+ D sin cut,

x(t) = Ecos(uut + 4>₁),

x(t) = Fsin(cut + 4>₂). (3.3)

The various constants here are related to each other. For example, C = E cos f₁ and D = -E sin f₁, which follow from the cosine sum formula. Note that there are two free parameters in each of the above expressions for x(t). These parameters are determined from the initial conditions (say, the position and speed at t = 0). Depending on the specifics of a given problem, one of the above forms will work better than the others.

If a is positive, then let’s define a = cu², where cu is a real number. The solution in eq. (3.2) now becomes x(t) = Ae^ut + Be-"*. Using e^e = cosh(9 + sinh<9,>

x(t) = Ae^ + Be-^

x(t) = C cosh cut+ Dsinh cut,

x(t) = Ecosh(cut + 4>₁),

x(t) = Fsinh(cut + 4>₂). (3.4)

Again, the various constants are related to each other. If you are unfamiliar with the hyperbolic trig functions, a few facts are listed in Appendix A.

Remarks: Although the solution in eq. (3.2) is completely correct for both signs of a, it is generally more illuminating to write the negative-a solutions in either the trig forms or the e^±iujt exponential form where the i’s are explicit.

As in the first example above, you may be concerned that although we have found two solutions to the equation, we might have missed others. But the general theory of differential equations says that our second-order linear equation has only two independent solutions. Therefore, having found two independent solutions, we know that we’ve found them all. *

The usefulness of this method of guessing exponential solutions cannot be overempha­sized. It may seem somewhat restrictive, but it works. The examples in the remainder of this chapter should convince you of this.

This is our method, essential,

For equations we solve, differential.

It gets the job done,

And it’s even quite fun.

We just try a routine exponential.

Oscilations part 1

Linear differential equations

A linear differential equation is one in which x and its time derivatives enter only through their first powers. An example is 3x¨+7x˙ +x = 0. An example of a nonlinear differential equation is 3x¨ + 7x˙² + x = 0.

If the right-hand side of the equation is zero, then we use the term homogeneous differential equation. If the right-hand side is some function of t, as in the case of 3x¨ - 4x˙ = 9t² - 5, then we use the term inhomogeneous differential equation. The goal of this chapter is to learn how to solve these two types of equations. Linear differential equations come up again and again in physics, so we had better find a systematic method of solving them.

The techniques that we will use are best learned through examples, so let’s solve a few differential equations, starting with some simple ones. Throughout this chapter, x will be understood to be a function of t. Hence, a dot will denote time differentiation.

Example 1 (x˙ = ax): This is a very simple differential equation. There are two ways (at least) to solve it.

First method: Separate variables to obtain dx/x = adt, and then integrate to obtain lnx = at + c. Exponentiate to obtain

x = Ae^at, (3.1)

where A = e^c is a constant factor. A is determined by the value of x at, say, t = 0.

Second method: Guess an exponential solution, that is, one of the form x = Ae^at. Substitution into x = ax immediately gives a = a. Therefore, the solution is x = Ae^at. Note that we can’t solve for A, due to the fact that our differential equation is homogeneous and linear in x (translation: A cancels out). A is determined from the initial condition.

This method may seem a bit silly, and somewhat cheap. But as we will see below, guessing these exponential functions (or sums of them) is actually the most general thing we can try, so the method is indeed quite general.

Remark: Using this method, you may be concerned that although we have found one solution, we might have missed another one. But the general theory of differential equations says that a first-order linear equation has only one independent solution (we’ll just accept this fact here). So if we find one solution, then we know that we’ve found the whole thing. *

Example 2 (x = ax): If a is negative, then this equation describes the oscillatory motion of, say, a spring. If a is positive, then it describes exponentially growing or decaying motion. There are two ways (at least) to solve this equation.

First method: We can use the separation-of-variables method of Section 2.3 here, because our system is one in which the force depends on only the position x. But this method is rather cumbersome, as you found if you did Exercise 2.10 or 2.11. It will certainly work, but in the case where our equation is a linear function of x, there is a much simpler method:

Second method: As in the first example above, we can guess a solution of the form x(t) = Ae^at and then find out what a must be. Again, we can’t solve for A, because it cancels out. Plugging Ae^at into i¨ = ax gives a = ±^. We have therefore found two solutions. The most general solution is an arbitrary linear combination of these,

x(t) = Ae^¹ + Be-^\ (3.2)

as you can quickly check. A and B are determined from the initial conditions.

Very Important Remark: The fact that the sum of two different solutions is again a so­lution to our equation is a monumentally important property of linear differential equations. This property does not hold for nonlinear differential equations, for example x² = x, because the act of squaring after adding the two solutions produces a cross term which destroys the equality, as you should check.

This property is called the principle of superposition. That is, superimposing two solutions yields another solution. This quality makes theories in physics that are governed by linear equations much easier to deal with than those that are governed by nonlinear ones. General Relativity, for example, is permeated with nonlinear equations, and solutions to most General Relativity systems are extremely difficult to come by.