Membranas elásticas

The vibrating-membrane problem - based on basic principles and simulations
Hermann Härtel 1 and Ernesto Martin2
1
2

IPN - Institute for Science Education, D-24098 Kiel Universidad de Murcia, E-30071 Espinardo, Murcia

Abstract
Rectangular and circular membranes have been modelled as discrete arrays of mass points connected by massless springs. Based on Newton’s principles and Hooke’slaw, the movement of such membranes has been simulated. All vibrational modes, as known from closed form solutions of the corresponding wave equations, can be excited, with deviations from theoretical values of no more than a few percent. This approach can be used to develop an intuitive understanding of vibrating membranes. The phenomenon of regular vibrational modes provides a suitable startingpoint for a thorough mathematical treatment. In a more general sense this topic demonstrates the possibility that elasticity is no longer a matter of high mathematical demand. The true nature of the “rigid body” as an unrealistic but perfect model can convincingly be demonstrated.

1.

Introduction
∇ is the Laplace operator,
2

As has been demonstrated recently, the vibrating-membrane problemcan be used as a rather appropriate example to demonstrate the power of computer algebra systems (CAS) like Axiom Maple, Mathematica, Derive etc. [1]. This approach, however, depends on a well-developed mathematical ability on the part of the learner and on his or her willingness to accept such an abstract and demanding path of explanation, where the solution of differential equations serves as adescription of real world phenomena, in this case the vibrating modes of an elastic membrane. In the following we would like to show that the same results can be achieved with much less mathematical effort and in a more direct fashion, based only on Newton’s principles and linear elastic forces.

∇ =

2

∂ ∂x

2 2

+

∂ ∂y

2 2

in rec-

2.

Theoretical Background

Oursystem consists of a plane membrane, in principle of any shape, homogeneously stretched by a tension T, given as force per unit length. The membrane has a mass µ per unit area and the boundary is clamped. For small vibrations and in the absence of external forces the wave equation, describing the motion of the different points (coordinates x, y in the plane of the membrane), is [2]: µ ∂ s 1 ∂ s ∇2s =-- ⋅ = ----- ⋅ 2 2 2 T ∂t v ∂t
2 2

tangular co-ordinates x, y, and v ≡ T ⁄ µ is the velocity of the waves in the elastic membrane. We have denoted by s(x,y,t) the transverse displacement of any point relative to the position when the membrane is at rest. For membranes held along the edge (s=0, as boundary condition), we have to find standing-wave solutions of the wave equation which have nodesalong the boundary of the membrane. For simple shapes (rectangular or circular membranes), the standing wave solutions or normal modes of vibration are usually worked out using a set of curvilinear coordinates in which the edge of the membrane forms one of the coordinate axes. In many cases we can use separation of variables which simplifies the problem. In the following the main characteristicsof the modes for the rectangular and circular membranes are described. With our simulation tool xyZET [3] we can in principle experiment with membranes of any shape. The results in this article, however, are restricted to rectangular and circular geometries which allows us to compare our simulated results with theoretical solutions of the related wave equation.

Rectangular Membrane (bordersfixed: s=0 in x=0,a and y=0,b) By separating the variables (s=X(x)Y(y)exp(iωt), the standing wave modes for this case can be expressed as follows: sin ( k x x ) ⋅ sin ( k y y ) , multiplied by a harmonic time dependence sin(ω0t), where the resonance frequency, ω0, will depend on the mode of vibration (values of kx, ky) ω0 2  ------ = k 2 + k 2 . x y  v The boundary conditions require that kx, ky...