18.12. Application in Beam Vibrations (Wave Equation)

Wave Equation: ∂²u/∂t² = c² ∂²u/∂x² Boundary Conditions: u(0,t)=u(L,t)=0 Initial Conditions: • Displacement: u(x,0)=f(x) • Velocity: u_t(x,0)=g(x)

In this section, we are introduced to the wave equation, which is a partial differential equation that describes the behavior of waves, such as vibrations in beams. The equation can be represented as: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ where $u$ represents the displacement of the beam at point $x$ and time $t$, while $c$ is the wave speed.

The boundary conditions specify that the displacement at the ends of the beam, points 0 and L, must be zero ($u(0,t) = u(L,t) = 0$), indicating that the beam is simply supported at these points.

Initial conditions provide specific starting parameters for the displacement and velocity at time $t=0$. For example, the displacement is given as $u(x,0) = f(x)$, which defines the shape of the beam at the beginning. The velocity condition is defined as $u_t(x,0) = g(x)$, specifying how fast each point on the beam is moving initially.

Imagine a guitar string when you pluck it. The waves created on the string can be described using the wave equation. The ends of the string are fixed in position (just like our boundary conditions), meaning they can't move. At the moment you pluck the string (time t=0), the initial shape of the string defines how it will vibrate thereafter, similar to how the initial conditions define the starting properties of our beam.

Solution using Separation of Variables + Fourier Series: X∞ (cid:20) (cid:18) nπct(cid:19) (cid:18) nπct(cid:19)(cid:21) (cid:16)nπx(cid:17) u(x,t)= A cos +B sin sin n=1 • A_n, B_n are computed from f(x), g(x) • Models transverse vibration of simply supported beam

The solution to the wave equation can be approached using the method of separation of variables combined with Fourier series. This allows us to express the displacement $u(x,t)$ as a sum of harmonic functions, leading to: $u(x,t) = \sum_{n=1}^{\infty} \left( A_n \cos\left(\frac{n\pi ct}{L}\right) + B_n \sin\left(\frac{n\pi ct}{L}\right) \right) \sin\left(\frac{n\pi x}{L}\right)$

Here, $A_n$ and $B_n$ are coefficients computed from the initial conditions $f(x)$ and $g(x)$. This series representation effectively models the transverse vibrations of the beam, where each term represents a mode of vibration determined by the beam's properties and boundary conditions.

Think of a trampoline. When you jump on it, the surface vibrates up and down in waves. Each wave corresponds to a different mode of vibration, just like each term in our series solution. The coefficients $A_n$ and $B_n$ help tell us how deeply each mode vibrates based on how hard you jumped (the initial displacement) and how fast you bounced (the initial velocity).

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