DrumSim

Simulates the surface of a perfectly elastic circular drum upon impingement by an arbitrary disturbance.

from drumsim import Mode, CircularDrum

Drum Vibrations: Analysis

Modal Dynamics and Relations

# Generate all modes up to (3,4)
for m in range(0,4):
    for n in range(1,5):
        Mode(m,n,radius=1,c=1).sim(fpath = './Modes')

The n parameter

Below we observe that as n increases, so too does the number of circular nodes, that is, circles where the drum displacement is stationary at all times.
In the (0,1) case, there exists only the one circular node at the fixed rim of the drum which explains why n must begin at 1: there is always the one circular node conferred by the boundary condition. This is analogous to the modes of a vibrating guitar string always having two nodes at both ends.
For (0,2), another circular node has been added close to half the radial lenght of the drum, resulting in outer and inner anti-oscillating regions.
Finally, (0,4) shows four concentric nodes - as expected.

(0,1)	(0,2)

(0,3)	(0,4)

The m parameter

Below we observe that m corresponds to the number of diametral nodes: lines along the diameter where the drum displacement is stationary at all times.
In the (0,1) case (above), there exists no linear node which explains why m may begin at 0.
For (1,1), a single diameter has been added resulting in anti-oscillating regions at either half of the drum.
(4,1) shows four (radially equidistant) diametric lines

(1,1)	(2,1)

(3,1)	(4,1)

General Solution: Axisymmetric Pertubration

Please note I've included a damping factor to make simulations more reflective of dissipation

To start off simple, let us observe solutions under initial conditions that possess radial symmetry
Consider a reflected quadratic u0(r) = -r(r-1) whose vertex is halfway radially outwards from the drum's centre

def u0(r, theta):
    return -r*(r-1)

For such an initial shape, the drum evolves accordingly:

drumquaddown = CircularDrum(radius=1, c=1, u0=u0, maxm=5,maxn=4)
drumquaddown.sim(fname="-r(r-1)", fpath="Final\General\Axisymmetric")

Initial	Evolution

Let us try with more initial conditions:

drumquadup = CircularDrum(radius=1, c=1, u0=lambda r,theta: r**2, maxm=5,maxn=4)
drumquadup.sim(fname="r^2", fpath="Final\General\Axisymmetric")
drumquadup.simInitial(fname="r^2", fpath="Final\General\Axisymmetric")

drumsin = CircularDrum(radius=1, c=1, u0=lambda r,theta: np.sin(2*np.pi*r), maxm=5,maxn=4)
drumsin.sim(fname="sin(2pir)", fpath="Final\General\Axisymmetric")
drumsin.simInitial(fname="sin(2pir)", fpath="Final\General\Axisymmetric")

drumlinear = CircularDrum(radius=1, c=1, u0=lambda r,theta: r, maxm=5,maxn=4)
drumlinear.sim(fname="r", fpath="Final\General\Axisymmetric")
drumlinear.simInitial(fname="r", fpath="Final\General\Axisymmetric")

drumcubic = CircularDrum(radius=1, c=1, u0=lambda r,theta: -(r-1/2)**3, maxm=5,maxn=4)
drumcubic.sim(fname="-(r-0.5)^3", fpath="Final\General\Axisymmetric")
drumcubic.simInitial(fname="-(r-0.5)^3", fpath="Final\General\Axisymmetric")

Initial	Evolution

While the patterns above are unique to their respective initial condition, we observe that they all share one main feature: they do not display any diametral nodes.
Recalling the nature of parameter m, we may infer that m=0 for all the modes comprising the superpositions of the drums above.
Indeed, we see that only modes of the form (0,n) contribute to the superposition.
We may conclude that axisymmetric initial conditions only have (0,n) modes contributing to their general solution.
To investigate why this may be the case, let us consider the non-axisymmetric case.

General Solution: Arbitrary Pertubration

For the arbitrary perturbation case, we'll define a function u_0 that simulates the striking of a drum at a localised point located at radial length rpos from the centre, and without loss of generality theta=0
Let us model the drum vibrations for such a localised strike at various lengths ranging [0, 1]

rpos = [0,0.25,0.5,0.75,1]

for i in rpos:
    def u0(r,theta):
        sigma = 0.05  # Localisation parameter
        impact = 10
        rpos = i
        angpoint = 0

        hdist = (r*np.cos(theta)-rpos*np.cos(angpoint))**2
        vdist = (r*np.sin(theta)-rpos*np.sin(angpoint))**2
        dist = hdist + vdist

        return  impact/(2*np.pi*sigma)*np.exp((-1/(2*sigma**2))*dist)
    
    drumstrike = CircularDrum(1, 1, u0, 5, 4)
    drumstrike.sim(fname="r={}m".format(i), fpath="Final\General\Strike")
    drumstrike.simInitial(fname="r={}m".format(i), fpath="Final\General\Strike")

Initial	Evolution

The degree to which a mode is excited upon a drum being struck at a point depends on how vigorously the mode naturally oscillates at that point.
This suggests that axisymmetric petrubrations excite only modes of form (0,n) because such pertubrations (by definition) cannot involve a stationary diameter, which is a condition required for any general (m,n) to be excited.