Page 1 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 1: Transverse Vibrations of Strings Contents: 1. Introduction 2. Mathematical Model 3. Examples Keywords: Transverse vibrations, Taut string, Hanging chain, equation of motion, non-homogeneous boundary condition Page 2 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 1: Transverse Vibrations of Strings Contents: 1. Introduction 2. Mathematical Model 3. Examples Keywords: Transverse vibrations, Taut string, Hanging chain, equation of motion, non-homogeneous boundary condition Transverse Vibrations of Strings 1 Introduction A string is a one dimensional elastic continuum that does not transmit or resist bending moment. Elements that may be modeled as taut strings: Strings in stringed musical instruments such as sitar, guitar or violin Cables in a cable-stayed bridge or cable-car High tension wires 2 Mathematical Model Assumptions in modeling: Motion is planar Slope of the string is small Longitudinal motion is negligible Tension does not change with displacement of the string 2 Page 3 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 1: Transverse Vibrations of Strings Contents: 1. Introduction 2. Mathematical Model 3. Examples Keywords: Transverse vibrations, Taut string, Hanging chain, equation of motion, non-homogeneous boundary condition Transverse Vibrations of Strings 1 Introduction A string is a one dimensional elastic continuum that does not transmit or resist bending moment. Elements that may be modeled as taut strings: Strings in stringed musical instruments such as sitar, guitar or violin Cables in a cable-stayed bridge or cable-car High tension wires 2 Mathematical Model Assumptions in modeling: Motion is planar Slope of the string is small Longitudinal motion is negligible Tension does not change with displacement of the string 2 (a) l x z w(x; t) ; A; T (b) x x + dx ds T(x; t) (x; t) T(x + dx; t) (x + dx; t) p(x; t)ds n(x; t)ds Figure 1: (a) A taut string (b) An innitesimal string element Consider a string of density , area of cross-section A(x), length l and under a tension T(x; t) subject to distributed forces (force per unit length) p(x; t) in the transverse direction, and n(x; t) in the axial direction . Let w(x; t) be the transverse displacement eld variable. Longitudinal dynamics: Balance of forces in the longitudinal direction of the sting element in Fig. 1(b) reads T(x + dx; t) cos[(x + dx; t)] T(x; t) cos[(x; t)] + n(x; t)dx = 0 ) [T(x; t)] ;x =n(x; t) ; (1) where [] ;x = @=@x. Here, we have assumed ds dx and cos 1. 3 Page 4 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 1: Transverse Vibrations of Strings Contents: 1. Introduction 2. Mathematical Model 3. Examples Keywords: Transverse vibrations, Taut string, Hanging chain, equation of motion, non-homogeneous boundary condition Transverse Vibrations of Strings 1 Introduction A string is a one dimensional elastic continuum that does not transmit or resist bending moment. Elements that may be modeled as taut strings: Strings in stringed musical instruments such as sitar, guitar or violin Cables in a cable-stayed bridge or cable-car High tension wires 2 Mathematical Model Assumptions in modeling: Motion is planar Slope of the string is small Longitudinal motion is negligible Tension does not change with displacement of the string 2 (a) l x z w(x; t) ; A; T (b) x x + dx ds T(x; t) (x; t) T(x + dx; t) (x + dx; t) p(x; t)ds n(x; t)ds Figure 1: (a) A taut string (b) An innitesimal string element Consider a string of density , area of cross-section A(x), length l and under a tension T(x; t) subject to distributed forces (force per unit length) p(x; t) in the transverse direction, and n(x; t) in the axial direction . Let w(x; t) be the transverse displacement eld variable. Longitudinal dynamics: Balance of forces in the longitudinal direction of the sting element in Fig. 1(b) reads T(x + dx; t) cos[(x + dx; t)] T(x; t) cos[(x; t)] + n(x; t)dx = 0 ) [T(x; t)] ;x =n(x; t) ; (1) where [] ;x = @=@x. Here, we have assumed ds dx and cos 1. 3 Transverse dynamics: Newton's second law for the string element is given by Adxw ;tt = T(x + dx; t) sin[(x + dx; t)] T(x; t) sin[(x; t)] + p(x; t)dx ) A(x)w ;tt [T(x; t)w ;x ] ;x = p(x; t) (2) where sin tan = w ;x . The linear partial dierential equation (2), along with (1), represents the dynamics of a taut string. The complete description of the free vibra- tion problem requires specication of two boundary conditions (w(0; t) and w(l; t)), and two initial conditions (w(x; 0) and w ;t (x; 0)). 3 Examples Uniform taut string: For a uniform taut unforced string (see Fig. 1(a)), A(x) = A and n(x; t) = 0 (implying T ;x = 0). The equation of motion simplies to w ;tt c 2 w ;xx = 0 (3) where c = p T=A is a constant having the dimension of speed. Since there is no transverse displacement at the boundaries, the appropriate boundary conditions are given by w(0; t) = 0 and w(l; t) = 0. A string with a free/sliding boundary condition is shown in Fig. 2. At the right boundary, the massless and frictionless roller cannot support any 4 Page 5 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 1: Transverse Vibrations of Strings Contents: 1. Introduction 2. Mathematical Model 3. Examples Keywords: Transverse vibrations, Taut string, Hanging chain, equation of motion, non-homogeneous boundary condition Transverse Vibrations of Strings 1 Introduction A string is a one dimensional elastic continuum that does not transmit or resist bending moment. Elements that may be modeled as taut strings: Strings in stringed musical instruments such as sitar, guitar or violin Cables in a cable-stayed bridge or cable-car High tension wires 2 Mathematical Model Assumptions in modeling: Motion is planar Slope of the string is small Longitudinal motion is negligible Tension does not change with displacement of the string 2 (a) l x z w(x; t) ; A; T (b) x x + dx ds T(x; t) (x; t) T(x + dx; t) (x + dx; t) p(x; t)ds n(x; t)ds Figure 1: (a) A taut string (b) An innitesimal string element Consider a string of density , area of cross-section A(x), length l and under a tension T(x; t) subject to distributed forces (force per unit length) p(x; t) in the transverse direction, and n(x; t) in the axial direction . Let w(x; t) be the transverse displacement eld variable. Longitudinal dynamics: Balance of forces in the longitudinal direction of the sting element in Fig. 1(b) reads T(x + dx; t) cos[(x + dx; t)] T(x; t) cos[(x; t)] + n(x; t)dx = 0 ) [T(x; t)] ;x =n(x; t) ; (1) where [] ;x = @=@x. Here, we have assumed ds dx and cos 1. 3 Transverse dynamics: Newton's second law for the string element is given by Adxw ;tt = T(x + dx; t) sin[(x + dx; t)] T(x; t) sin[(x; t)] + p(x; t)dx ) A(x)w ;tt [T(x; t)w ;x ] ;x = p(x; t) (2) where sin tan = w ;x . The linear partial dierential equation (2), along with (1), represents the dynamics of a taut string. The complete description of the free vibra- tion problem requires specication of two boundary conditions (w(0; t) and w(l; t)), and two initial conditions (w(x; 0) and w ;t (x; 0)). 3 Examples Uniform taut string: For a uniform taut unforced string (see Fig. 1(a)), A(x) = A and n(x; t) = 0 (implying T ;x = 0). The equation of motion simplies to w ;tt c 2 w ;xx = 0 (3) where c = p T=A is a constant having the dimension of speed. Since there is no transverse displacement at the boundaries, the appropriate boundary conditions are given by w(0; t) = 0 and w(l; t) = 0. A string with a free/sliding boundary condition is shown in Fig. 2. At the right boundary, the massless and frictionless roller cannot support any 4 l x z; w ; A; T w(0; t) = 0 Tw ;x (l; t) = 0 Figure 2: Taut string with geometric and natural boundary conditions transverse force (given by T sin (l; t)). Therefore, the boundary condition at the roller is Tw ;x (l; t) = 0 (taking sin tan = w ;x ). A condition on the geometry/kinematics at a boundary is know as a ge- ometric/essential boundary condition (the left end boundary condition in Fig. 2). A condition on the force at a boundary is know as a dynamic/natural boundary condition (the right end boundary condition in Fig. 2). The hyperbolic partial dierential equation (3) is known as the linear one- dimensional wave equation, and c is known as the wave speed. In the case of a taut string, c is the speed of transverse waves on the string. This implies that a disturbance created at any point on the string propagates with a speed c. It should be clear that the wave speed c is distinct from the transverse material velocity (i.e., the velocity of the particles of the string) which is given by w ;t (x; t). Uniform hanging string/chain: For a hanging string (see Fig. 3), n(x; t) = Ag. From (1), the expression of 5Read More

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