Loudspeaker Cabinet Bracing: A Little Theoretical Background

On a fundamental level, stiffness is an object’s resistance to distortional movement caused by a force.  Stiffness is the mathematical inverse of compliance, which is more commonly used in audio, and as such, it is defined in units of force divided by length.

The physical parameters that go into determining stiffness are a complex function of many factors.  Material properties, the physical dimensions of the object’s cross section, and the distance it spans while supporting the force all come into play along with how the load is applied and how the object is supported.

The reason we are concerned with the stiffness of a loudspeaker cabinet is the relation between sound and movement.  Sound pressure level is a function of movement within the medium that is transmitting a sound wave, which in our case is the air.  Particle displacement can be related to sound pressure with the following one-dimensional equation:

Where omega is the circular natural frequency, rho is the density of the transmission medium, c is the speed of sound in that medium, and x is the displacement.  Circular natural frequency can be related to the natural frequency, in Hz:

This general relationship states that greater particle movement leads to greater sound pressure.  Sound pressure is also related to particle velocity and acceleration, both of which are related as derivatives of position with respect to time.

Relating the fluid displacement to the displacement of the cabinet becomes a problem of fluid-structure interaction.  For lightly damped structures, sound pressure peaks from mechanical vibration occur at frequencies very near the structural modes.  The complex pressure field in three dimensional space can be related to the vibration of a rectangular plate of length a and width b in a spherical coordinate system:

Fluid-structure interaction is a complex problem to solve that involves linking mathematics between two traditionally separate braches of physics, solid mechanics and fluid mechanics.  For our problem, the solution to the fluid-structure interaction problem is further complicated by two different volumes of air internal and external to the cabinet, three interaction surfaces, and both direct and indirect sources of vibration from the driver to the cabinet.  A full study of a multi-physics phenomenon such as this is beyond the scope of the current discussion.  What we are interested in is establishing the general trend for the magnitude of unwanted acoustic output as a direct relation to the amplitude of cabinet vibration.  To that end, we will not consider certain aspects of fluid-structure interaction including the effect that internal cabinet pressure will have on the vibration of the cabinet.

We will now proceed by addressing the dynamic behavior of the cabinet.  First, we need to define the mechanical behavior of the cabinet by establishing the equation of motion for the system:

Where m is mass, c is viscous damping, k is stiffness, F(t) is a forcing function that varies over time t, x is the displacement of the system, and the single and double dot over the x indicate the first and second derivatives of the displacement with respect to time, namely velocity and acceleration, respectively.  The stiffness term is the mathematical inverse of compliance, or flexibility, which is more commonly used in the audio.

Rearranging terms and setting the force to zero, we can solve for the natural frequency of the system:

These other terms are the undamped circular frequency and the damping ratio respectively:

When damping is included, we can redefine the damped circular frequency as:

Of note, the Q factor that is often used in audio can be defined as:

Thus, the fundamental natural frequency, in Hz, of an object can be determined from knowledge of the stiffness and mass of an object as follows.

There are, of course, additional natural frequencies of vibration, harmonics, which can be defined over the geometric field of stiffness and mass for physical objects.  Each frequency is associated with a unique deformed shape known as mode shape and within these mode shapes, are certain locations that effectively have no movement, which are known as inflection points.  The solution for the frequencies and mode shapes of a physical object are known as modal analysis or eigenvalue/ eigenvector analysis.

Modal analysis is non-dimensional, meaning that it will determine the relative deformation within the panel at each resonant frequency, but not the actual magnitudes of the deformations.  Effectively, modal analysis determines how an object wants to vibrate when subjected to an arbitrary impulse and then allowed to vibrate freely without the continuing application of force.  To visualize modal behavior, think of a struck tuning fork or a plucked string on a musical instrument.

The behavior of the cabinet panels can be described using plate theory.  For simplicity, we will look at an individual panel of a cabinet along with any braces used to reinforce that panel while under a uniform surface load from pressure.

For a rectangular plate with edge dimensions a and b, and thickness h, under uniform load q, we can determine the static load deflection w from the following fourth order partial differential equation in the xy plane:

The term D is the flexural rigidity of the plate with a thickness of h:

The variables E and v are the Modulus of Elasticity and Poison’s Ratio, respectively, for the material.

The differential equation for the plate can then be solved for different support boundary conditions.  For our situation with a speaker cabinet panel that is supported on four edges, we can determine an upper and lower bound on the amount of deflection due to a load, and therefore the stiffness, by assuming a case where the edges are completely free to rotate and where the edges are completely restrained from rotating.  In reality, panel edges are partially restrained by the stiffness of the other panels that make up the cabinet, so we know the real behavior is somewhere between these bounds.

Rotationally unrestrained edges: 

Rotationally restrained edges: 

The terms E_m and H_m are the bending moments developed along the edge due to the rotational restraint with:

A uniformly loaded plate will produce either triangular or trapezoidal shaped reaction forces on supporting edges depending on the aspect ration of the sides.  When the sides are not equal, the shorter edge will always see a 45^o triangular reaction and the longer side will see a trapezoidal reaction unless all the sides are equal.

The behavior of the braces can be described using beam theory.  The differential equation for a beam is also fourth order with length along the x axis:

The term EI is the flexural rigidity of the beam, comprised of the material modulus of elasticity, E, and the moment of inertia of the cross section, I, which for a rectangular beam of a cross section width b and depth d is:

We can then solve for the loading and boundary conditions as we did before with the plate.

For an unrestrained beam of length l under a triangular load of q(x): 

For a restrained beam of length l under a triangular load of q(x): 

Note that q_o is the peak magnitude of the triangular load q(x) at the center of the beam.

Considering the original equation of motion under static load conditions, where the first and second derivatives of displacement, velocity and acceleration, are insignificant, it reduces to F = kx.  Rearranging this equation to solve for the stiffness, we get k = F/x, the applied force over the resulting deflection.