Hambric Vibroacoustic Demonstrators

Coupled oscillator theory can be used to analyze many interesting structural dynamic problems. The low-frequency vibrations of more complex interconnected structures can be simulated easily if the modal masses, resonance frequencies, and loss factors are reasonably understood. Sometimes referred to as lumped parameter modeling, this is a powerful and insightful methodology. You can use coupled oscillator theory to emulate the behavior of singly and doubly isolated mounts, accelerometers mounted to structures, and double-glazed sound isolation systems (windows, walls, and other barriers).

Oscillator 1 (colored blue) is connected to ground via spring 1 and oscillator 2 (colored red) is connected to oscillator 1 via spring 2. Both springs have internal loss factors - no viscous dashpots are used in this simulation as most structural damping is actually strain-based. The oscillator sizes and spring thicknesses change as you adjust the mass and spring sliders. You may reduce spring constant 1 to zero if you like, producing another important system - a mass-spring-mass (you cannot, however, set spring constant 2 to zero).

This demonstrator shows many quantities, starting with the forced vibration response of both oscillators, shown as accelerances (a/F) in the left plots. Drive point and transfer accelerances are shown for drives on oscillator 1 (bottom) and oscillator 2 (top). Note that the transfer accelerances for both drives are identical (a1/F2 = a2/F1) - this is the principle of reciprocity used in experimental modal analysis. The high frequency drive point accelerances converge to the inverse of the oscillator masses (recall that F=ma!) The peaks in the accelerances are not at the resonance frequencies of the individual oscillators (shown in the title above the plots). When the oscillators are coupled, however, the combined resonance frequencies depend on the overall system (these are also shown in the top title so you can compare them to the uncoupled frequencies).

To examine how the oscillators move relative to each other at and away from resonance animate them. There are animation buttons for drives on both oscillators and the behavior can be quite different. Change frequency by either using the frequency slider or clicking on any of the accelerances and animate again. I recommend starting at very low frequency, then increasing frequency to the lowest resonance, then to the highest, and finally to very high frequencies. Notice how at some frequencies oscillators are strongly coupled, vibrating either in-phase or out-of-phase with each other. At other frequencies, the oscillators are completely isolated from each other.

To examine a mass-spring-mass system reduce the oscillator 1 spring constant to 0. The peak in the vibration responses is at the so-called mass-spring-mass resonance frequency (which I show in the bottom of the title when k1=0). This is an important pass-band in double glazed window systems, where the lumped mass of the first panel oscillates against the spring constant associated with the air gap between panels, which in turn is coupled to the lumped mass of the second panel. Try animating the motion of this system at the coupled resonance, as well as at the accelerance dips for both drive locations.

The force transmitted to the base of oscillator 1 is an important parameter in isolation mount design. Force transmissibility is shown in the center plots for drives on both oscillators. Below the first resonance frequency the system response is purely static and the force transmissibility is unity. Both resonances amplify the transmitted forces depending on the damping. Above both system resonances the isolation improves with increasing frequency. You can reduce the top (oscillator 2) mass and increase its stiffness to emulate a single degree of freedom isolation system (notice how both force transmissibilities become identical). Next, set the oscillator 2 mass to that of a system you'd like to isolate and adjust the oscillator 1 mass (sometimes called a raft) and the two stiffnesses to optimize a doubly mounted system.

Power flows from the driven oscillator to the coupled oscillator depending on the proximity of the oscillator resonance frequencies, their internal damping, and the strength of the coupling between them. I normalize the power dissipated within each oscillator by the power input to the driven oscillator and show it as a percentage. Examine the powers within each oscillator as you adjust their internal damping and shift their resonance frequencies - the behaviors depend strongly on the individual uncoupled resonance frequencies and the ratio of the damping between oscillators. This sort of analysis is the basis of Statistical Energy Analysis (SEA) which relates the flow of energy/power to the damping within groups of modes and their coupling.