Tension in Strings

4 Key excerpts on "Tension in Strings"

• 

  eBook - PDF

  Formulas for Mechanical and Structural Shock and Impact

  • Gregory Szuladzinski(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  315 9 Cables and Strings THEORETICAL OUTLINE From an engineering viewpoint, a cable or a string is a very simple structural element, which resists loads applied only along its axis. In a common parlance,* there are some dif-ferences between the two objects, but here both names are used interchangeably, to designate elements, which have no bending stiffness. The above description of string properties should be augmented by the following: If a string is preloaded to some level of tension, it will also resist lateral forces, owing to the geometric stiffness, as described in Chapter 2. This is how musical string instruments work, although they were invented long before the terms used here were conceived. A structural beam offers two components of resistance to deformation: bending and stretching. The first of those is related to what we understand as a “beam proper” while the second is associated with a cable-like action. Under some circumstances, the second component becomes predominant, which enables us to simplify such a beam to a cable. This makes the following study of string-like action so much more important. S TATICS OF C ABLES While a string is a simple structural element, the lack of lateral stiffness puts it into a mod-erate deflection range, which makes the description of its response a nonlinear procedure. Consider a cable, initially in the form of a straight line, spanning the distance between two points. Two types of distributed load may be involved, as Figure 9.1 illustrates. The first one is the action of forces, like gravity, which retain their original direction as in Figure 9.1a while the second is a uniform pressure, changing direction according to the deflected position (Figure 9.1b). The difference between Figure 9.1a and b is not significant until deflections are quite visible.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Fourier Series and Orthogonal Functions

  • Harry F. Davis(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  y ) in the plane.

  We must explain what we mean by the term tension. We assume that the membrane is stretched even when it is in its equilibrium position. Consider a line segment in the xy plane. With the above assumptions, the force which the particles on one side of the line segment exert on the particles to the other side is in a direction perpendicular to the line segment. The magnitude of this force per unit length (length measured along the line segment) is called the tension and is denoted T. We assume that the deflections are sufficiently small that we can consider T to be a constant.

  We now derive the differential equation governing vibrations of the membrane by a procedure somewhat similar to that we used for the vibrating string. In particular, we assume the angles involved to be so small we can replace their sines by their tangents. Consider a simple closed curve C in the xy plane (there is no loss in generality here if we take C to be a circle). Let F denote the scalar component, in the positive z direction, of the resultant force acting on that portion of the membrane enclosed by C. One easily sees that this is given by

  (1)

  where ds is the element of arc length on C and ∂w/∂n is the directional derivative of w (x ,y ,t ) in a direction normal to C (outward from the interior enclosed by C ). Indeed, T ds is the element of force acting across ds, and ∂w/∂n is the tangent of the angle between this force and the xy plane. Of course, F is a function of t.

  We now make use of the two-dimensional analog of (9), Section 5.4, which is called Green’s theorem in the plane. This is readily derived (Exercise 9) by applying (9), Section 5.4, to a cylindrical surface, assuming the functions depend only on x and y,

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Boundary and Eigenvalue Problems in Mathematical Physics

  • Hans Sagan(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  II

  REPRESENTATION OF SOME PHYSICAL PHENOMENA BY PARTIAL DIFFERENTIAL EQUATIONS

  §1. THE VIBRATING STRING

  1. Vibrations of a Stretched String—Vectorial Approach

  We are going to derive in this subsection the equation for the vertical displacements of a perfectly flexible string stretched between two fixed points. As to choice of coordinate system, a discussion of the physical properties of the string, and the neglect of quantities which are small of higher order, we refer the reader to Chapter I, §2.1 (p. 15).

  Since the string is stretched between two fixed points, there is a certain tension τ exerted upon the end points. We assume the string to be perfectly flexible; i.e., only tensile forces can be transmitted in the direction of its tangent line. It follows that the tension acts with the same magnitude upon any part of the string in the direction of the tangent.

  In order to obtain an equation for the vertical displacements, we have to establish according to the principle of vectorial mechanics the resultant of all external forces exerted upon any element of the string in the u-direction and equate this to the inertial force of the respective element in the indirection. Considering Fig. 3 , which represents a section of the string, we find that the u-component of the tension τ at P1 , the beginning point of the considered element of length Δs, amounts to −τ sin α1 and at P2 , the end point of the considered element, amounts to τ sin α2 where τ = |τ|. Thus, the resultant of all external forces exerted upon the element of length Δs in the u-direction τ( u ) is given by

  Since

  where we write u(x) instead of u(x, t) and du/dx instead of ∂и/∂x, (because t does not change its value in this analysis), we have according to

  Fig. 3

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Computational Physics

  Problem Solving with Python

  • Rubin H. Landau, Manuel J Páez, Cristian C. Bordeianu, Manuel J. Páez(Authors)
  • 2015(Publication Date)
  • Wiley-VCH

    (Publisher)

  In Section 21.4.1, we will solve for the tension in a string as a result of gravity. Readers interested in an alternate easier problem that still shows the new physics may assume that the density and tension are proportional: ρ(x) = ρ 0 e αx , T (x) = T 0 e αx . (21.32) 501 21.4 Strings with Variable T ension and Density While we would expect the tension to be greater in regions of higher density (more mass to move and support), being proportional is clearly just an approximation. Substitution of these relations into (21.31) yields the new wave equation:  2 y(x , t )  x 2 + α  y(x , t )  x = 1 c 2  2 y(x , t )  t 2 , c 2 = T 0 ρ 0 . (21.33) Here c is a constant that would be the wave velocity if α = 0. This equation is similar to the wave equation with friction; only now the first derivative is with respect to x and not t . The corresponding difference equation follows from using central-difference approximations for the derivatives: y i , j +1 = 2 y i , j − y i , j −1 + αc 2 (Δt ) 2 2Δx [ y i+1, j − y i , j ] + c 2 c ′ 2 [ y i+1, j + y i−1, j − 2 y i , j ] , y i ,2 = y i ,1 + c 2 c ′ 2 [ y i+1,1 + y i−1,1 − 2 y i ,1 ] + αc 2 (Δt ) 2 2Δx [ y i+1,1 − y i ,1 ] . (21.34) 21.4.1 Waves on Catenary Up until this point we have been ignoring the effect of gravity upon our string’s shape and tension. This is a good approximation if there is very little sag in the string, as might happen if the tension is very high and the string is light. Even if there is some sag, our solution for y(x , t ) could still be used as the disturbance about the equilibrium shape. However, if the string is massive, say, like a chain or heavy cable, then the sag in the middle caused by gravity could be quite large (Figure 21.5), and the resulting variation in shape and tension needs to be incor- porated into the wave equation.

  Sign up to read

  Learn more about book