Loudspeaker Cabinet Bracing: Finite Element Analysis - Part I
Finite element analysis of the problem was conducted using the program SAP2000 Advanced version 14.2.4, developed by Computers and Structures, Incorporated. SAP2000 is general-purpose finite element software package that is commonly used in structural engineering practice to model the mechanical behavior of physical systems. The software is capable of static and dynamic analysis in three-dimensional space including geometric and material nonlinearity. The dynamic capabilities include modal, steady state harmonic, response spectrum, power spectral density, and explicit time history analysis for a known excitation function. For our purposes, we will use the static, modal, and steady state harmonic analysis capabilities.
A series of five models of the panel were prepared with a varying number of two inch deep by ¾” thick braces at uniform spacing intervals matching the sub-panel configurations listed in the discussion above. The MDF panel and braces were discretized into one in square finite elements based on the Mindlin-Reissner thick-plate shell formulation that accounts for transverse shear deformation in the plane of the elements. The models were restrained against linear movement, but not rotation, at the panel edges.
Finite element model of the panel with lateral edge restraints.
These panel models are designed to address the partial restraint of the panel at the braces and the partial rotational restraint from the continuity of the sub-panels across the braces for case of the isolated cabinet side panel. The primary differences of the isolated panel model from an actual cabinet is from:
Partial lateral and rotational edge restraint from the perpendicular panels.
Partial restraint of the brace ends from adjacent braces around the cabinet.
Variations in the edge restraint due to different panel dimensions and the driver openings.
The additional mass of the drivers on the cabinet front affecting one side panel edge.
Partial support from the floor mounting to the bottom panel affecting one side panel edge.
In addition to shifting the numerical results, these conditions would lead to behavior that included additional lateral modes with displacement of the panel edges, increased asymmetry in the mode shapes, and an increased number of clustered modes at similar frequencies.
For the steady state harmonic analysis, we will use frequency independent hysteretic damping at a ratio of 0.5% of critical damping. The panel was then loaded with uniformly applied pressure over the panel surface at a deviation of 20 Pascals (0.4177 psf) from ambient atmospheric pressure, equivalent to an SPL of 120 dB. This load was applied as a harmonic function of time over a frequency sweep from 20 to 700 Hz at 5 Hz intervals.
The reason we are interested in the modal behavior is that higher modes take more energy to excite than lower modes. Stated in a practical sense, for a given amount of force driving the cabinet, a higher mode will vibrate at smaller amplitude, which will reduce unwanted acoustic output from the system.
For the unbraced panel, we will first look at the undamped mode shapes of the first eight modes from an eigenvalue analysis. While mode shapes are non-dimensional in nature, a uniform scale factor of 0.01 was used for reference in the modal plots.
Note that in the plots, the z axis defines the panel length, the y axis defines the panel width, and the x axis defines displacements normal to the plane of the panel.
Mode shape with no braces, mode 1 at f = 114.05 Hz.
The first mode is the primary mode, which is a combined longitudinal and transverse width mode. This coupled modal behavior matches the static load deformation, having no inflection points along either the length or width of the panel.
Mode shape with no braces, mode 2 at f = 170.92 Hz.
The first harmonic is the second longitudinal mode with an inflection point at the center along the panel length.
Mode shape with no braces, mode 3 at f = 265.68 Hz.
Mode 3 is the third longitudinal mode with inflection points at third points along the length of the panel.
Mode shape with no braces, mode 4 at f = 396.30 Hz.
This mode is the second transverse mode along the width of the panel. A view along the width, y axis, rather than the length, z axis, would look similar to mode 2.
Mode shape with no braces, mode 5 at f = 397.81 Hz.
This mode behaves as the fourth longitudinal mode of the panel.
Mode shape with no braces, mode 6 at f = 451.21 Hz.
Mode 6 is another coupled mode that consists of a combined second mode in both the length and width, both having a single internal inflection point.
Mode shape with no braces, mode 7 at f = 542.60 Hz.
Mode 7 is also a coupled mode and is combined from the third mode along the length with the second mode along the width of the panel.
Mode shape with no braces, mode 8 at f = 566.53 Hz.
The eighth mode displays uncoupled modal behavior that is the fifth longitudinal mode.
Looking at the diagrams above, it should be obvious that modal behavior is directly analogous to transverse wave propagation through the cabinet panel. We are simply looking at a half wave and the corresponding harmonics.
Now we will look at the mode shapes for the panels with various numbers of braces of a constant stiffness based on the 2 inch by ¾ inch cross-section. The modal behavior of the panel will be complicated by the additional mass of the stiffeners that not only add localized mass to the panel, but the mass is eccentric to the plane of the panel, which will introduce some rotational effects within the interior of the panel area. This effect will add some complexity to the modal behavior of the system.
Mode shape with 1 - 2” x ¾” brace, mode 1 at f = 157.69 Hz.
A single centerline brace attempts to enforce the second longitudinal mode of the unbraced panel, but with insufficient stiffness to fully force an inflection point and the corresponding second longitudinal mode of the unbraced panel. Never the less, the natural frequency of the panel is significantly increased from the unbraced response at 114.1 Hz.
Mode shape with 3 2” x ¾” braces, mode 1 at f = 207.62 Hz.
The three brace configuration moves further from the matching the corresponding longitudinal mode of the unbraced panel, falling short of even the third longitudinal mode at 265.7 Hz. Once again, the natural frequency is substantially increased, above both the previous braced mode and the second longitudinal unbraced mode at 170.9 Hz.
Mode shape with 4 2” x ¾” braces, mode 1 at f = 223.28 Hz.
We are now at the point where the need for additional brace stiffness is evident as the four brace configuration is barely able to generate noticeable local restraint at the brace points. This mode has a higher frequency than the three brace configuration, but falls short of the third unbraced modal frequency, showing behavior that is trending towards global rather than local restraint from the additional braces.
Mode shape with 9 2” x ¾” braces, mode 1 at f = 273.60 Hz.
The nine brace configuration effectively behaves as a global, higher frequency fundamental mode. The frequency is higher than that of previous four brace configuration, but just barely clears the frequency of the third longitudinal mode of the unbraced panel at 265.7 Hz.
Next, we will look at the deformation of the panel with various braces near peak resonance from the harmonic loading. The plots are all scaled at a consistent factor of 2500.
Deformed shape with no braces at f = 115 Hz.
The unbraced panel clearly shows deformation consistent with the corresponding fundamental mode shape.
Deformed shape with 1 - 2” x ¾” brace at f = 160 Hz.
The deformation under harmonic load of the single brace configuration again matches the mode shape with the brace providing some localized restraint. Also, note the significant decrease in deformation compared with the unbraced panel.
Deformed shape with 3 2” x ¾” braces at f = 210 Hz.
The trend is continued for comparison both with the corresponding mode shape and with the decrease in deformation from the previous brace configuration.
Deformed shape with 4 2” x ¾” braces at f = 225 Hz.
With an additional brace, the trend from local restraint behavior towards global restraint behavior continues to progresses from the previous case.
Deformed shape with 9 - 2” x ¾” braces at f = 275 Hz.
The effectiveness of the braces to locally restrain the panel continues to decrease with nine braces even as the global deformation is restrained further.
At this point, the behavior of the braced system is effectively that of the fundamental mode of the unbraced system, but with a higher frequency and reduced displacement, as shown in the following plots.
Displacement, panel center, for various numbers of 2” braces.
Displacement, sub-panel center, for various numbers of 2” braces.
The plots are based on displacement at the center of the panel or sub-panels formed between braces nearest the center of the panel.
Plotting the maximum displacement clearly shows both the decrease in displacement and the relative decrease in effectiveness of each additional brace with a constant stiffness.
There is also some shifting around of the relative effectiveness between different numbers of braces. This is another indication of the decreasing effectiveness of more bracing without increasing the stiffness of the braces as the system transitions from local to global behavior with mode shapes similar to the unbraced fundamental, but with increased frequencies and reduced displacement. This shifting of displacement amplitudes could easily be misinterpreted as the additional bracing is not effective, but it is only an indication of the transition that is occurring.
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Recent Forum Posts:
Twexcom, post: 926039Run the vertical brace from the right side of the cabinet to the left side.
I think one vertical brace would resonate as well, because it is still wood, it would just be like another wall.
Now there is no front-to-back bracing except at the sides, and the right and left walls are divided into two smaller right and left walls which are still resonant. (and yes, the brace itself could).
Run it front to back and the same problems exist as above, only rotated 90 degrees.
Rickster71, post: 854260
I wonder what the reasoning is (or difference) behind so many horizontal braces, instead of one full length vertical brace?
I think one vertical brace would resonate as well, because it is still wood, it would just be like another wall.
I'm not certain, though.
CaliMon, post: 881082
Okay, so who EbenLee? Never heard of them.
They've been around for less then 2 years (I've been working with them for about a year now). They're a small shop that specializes in hand-made and custom speakers. Everything they do is geared towards sonic purity, structure integrity and exquisite finishes. EbenLee's website is here.
They're the only company I know of who are doing all the things Selb4itkicksit mentioned, so I assume that's who he's alluding to.
Selb4itkicksit, post: 880900
Okay, I'm checking out a new brand that looks interesting. I need some small speakers for a den and the ones I'm looking at are made out of Baltic birch ply also using the layer cabinet contruction that was mentioned earlier in this thread.
Whats the advantage of baltic birch ply and a stacked cabinet? Is it worth the extra costs?
From EbenLee Audio perhaps?