Transverse Vibration of Strings

• Examines mechanical vibrations, specifically the transverse vibration of strings in continuous systems.

Wave Equation

• A one-dimensional taut elastic string of length l is subject to a transverse force, denoted as f(x, t), per unit length.
• The motion at any point is a function of its position (x) along the string and time (t).
• The deflection of the string is represented by w(x, t).
• w(x, t) represents the transverse displacement of the string.
• f(x, t) refers to the transverse force acting on the string.
• P(x) is the axial tension within the string.
• θ is the angle that the deflected string makes with the x-axis.
• A dynamic equilibrium expression relates the sum of forces in the z-direction (ΣFz) to mass times acceleration (ma).
• Small disturbance is assumed such that ds ≈ dx, simplifying the mass element in the dynamic equation.
• Small deformation implies sinθ ≈ tanθ ≈ ∂w/∂x and sin(θ + dθ) ≈ tan(θ + dθ) ≈ ∂w/∂x + ∂²w/∂x² dx.
• Additional substitution dP = (∂P/∂x) dx is used to simplify derived terms.
• By summing terms, making substitutions into the equilibrium expression, and dividing by dx, a dynamic equilibrium expression is derived.
• Assuming the tension is independent of x simplifies the equation further.
• Assuming free vibration, f(x,t) = 0, allows defining a new variable, c, representing wave speed.
• c² = P/ρ can be defined.
• The dynamic equilibrium expression can be expressed using c as either c²(∂²w/∂x²) = ∂²w/∂t².
• The equation (∂²w/∂x²) = ∂²w/∂t² is identified as the wave equation, where c is the wave speed through the medium.

Solution of the Wave Equation

• Employs the method of separation of variables to transform the partial differential equation into two ordinary differential equations to find the value of w(x, t).
• Assumes displacement can be represented by two functions: w(x,t) = W(x)T(t).
• Since each term on the left is only a function of x and the terms on the right are only a function of t, they both must be equal to a constant.
• Constant equals to -ω².
• Two ordinary differential equations (ODEs) are established.
• Both ODEs have harmonic solutions.
• Solutions are of the form W(x) = A cos(ω/c x) + B sin(ω/c x) and T(t) = C cos(ωt) + D sin(ωt).

Boundary Conditions

• There are four unknowns, A, B, C, and D, which call for knowledge about the conditions the string is experiencing.
• Conditions on the boundary are needed to find A and B.
• Initial conditions allow to find the constants C and D.
• To find boundary conditions, identification of the displacement degree-of-freedom is needed.
• The vertical force, is related to the tension in the string by, Fᵥ = Psinθ.
• For small displacements we can assume Fᵥ = Psinθ ≈ P(∂w/∂x).
• The primary boundary condition is therefore, w(x, t) or W(x) and the secondary boundary condition is Fᵥ ≈ P(∂w/∂x).
• Four boundary conditions on a string; free, pinned, elastic, damped, inertial, will be considered.
• Free end condition is characterized by a frictionless, massless slider, where F(0) = 0 and ∂w/∂x|x=0 = 0.
• Fixed(pinned) end has w(0, t) = 0.
• End attached to a spring; F(0) = kW(0, t) meaning P(∂w/∂x)|x=0 = kw(0, t).
• End attached to a damper; F(0) = c(∂w/∂t(0, t) meaning P(∂w/∂x)|x=0 = c(∂w/∂t).
• End attached to a mass; F(0) = M(∂²w/∂t²(0, t) meaning P(∂w/∂x)|x=0 = M(∂²w∂t²)*.
• Constants A and B can be determined from boundary conditions.
• Natural frequencies of the string is dependent on the boundary conditions, string properties: tension, density, length, and diameter.
• Mode shapes of the string corresponding to each of the natural frequencies are only a function of the boundary conditions, string properties: tension, density, length, and diameter.
• The coefficients, Bₙ are arbitrary.

Initial Conditions

• Continue with the pinned-pinned string to show how the initial conditions are evaluated.
• After evaluating the boundary conditions the response when the pinned-pinned string is vibrating at a single natural frequency, wₙ, may be written as, wₙ(x, t) = (Bₙsin(nπ/L x))(Ccosωt + Dsinωt).