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Tables & Breadboards

ScienceDesk™ resonance peak BreadboardsBreadboard ℘ is the damping ratio andk is the stiffnessFrom this equation, it can be seen thatwhen the frequency of the forcing func- tion is close to the resonant frequency, thecompliance is determined solely by thedamping term and can be quite large.
Supports 2 and is represented by astraight line with a slope of -2. It repre-sents the ultimate design goal when manu- facturing optical tables - the nearer the ac- tual curve fits the straight line the better the dynamic stiffness.A compliance curve of a real table can nowbe examined. The figure below shows atypical compliance curve measured at the corner of a Thorlabs tabletop.
Optical Tables Table Supports stiffnessregion1k COMPLIANCE Non-Magnetic mass dominated region peak height determinedby damping 1 cf 1mf 2 f n FREQUENCY Compliance versus frequency for a onedegree of freedom system Low Frequency Compliance
10 -3 At zero and low frequencies, the stiffnessterm dominates the compliance equation.When a low frequency forcing vibration is applied to the unattached end of our bar, it bends in response. The amount of de- flection is determined by the stiffness ofthe bar which ultimately depends on itsshape, the tensile modulus of elasticity (Young's modulus) of the bar material, and the method of mounting and/or constrain- ing the bar.Any solid body has a fixed equilibrium,and when forces are applied it can be de- formed from this equilibrium shape. The potential energy of the body rises and this is manifested as resistive forces, which actto restore the equilibrium shape.When the bar is deflected, the restoringforces return it to its equilibrium (linear) po- sition. However, the momentum of the bar causes it to overshoot this position.Restoring forces now act in the opposite di-rection to return the bar to its equilibrium position and again, momentum causes an overshoot beyond linearity. This oscillationof the bar is an example of a simple harmon-ic oscillator, and in the absence of damping, this oscillation would persist forever. When a body is vibrating at its resonantfrequency, the energy in the vibration al- ternates between potential and kinetic. Atthe maximum deflection or displacement,the velocity is zero, the acceleration is maximum, the potential energy is maxi- mum and the kinetic energy is zero. At the equilibrium, or zero displacement, the op-posite is true.
10 -3 10 -4 10 -4 A typical compliance curve of a highperformance optical table
10 -5 10 -5 10 -6 Several aspects of this curve merit specialcomment. The initial portion of the plot, i.e. before the first table resonance, is de-termined primarily by the table supports,not the table itself. Also, notice how the peaks at complianceresonances decrease in size towards higher frequency. As the frequency increases, thedenominator in the compliance expressionbecomes large and therefore the compli- ance is reduced. This means that as the fre- quency increases, a given excitation force produces a smaller amplitude excitation inthe table. The low frequency peaks are the most im-portant for two reasons. First, these are the largest peaks, corresponding to the weakest points in the compliance spectrum. Second, typical vibrations from laboratoryequipment are usually below 150 Hz. Thepeaks should be at the highest frequency possible in order to keep the compliance in the 0 to 150 Hz region as low as possible.
-6 COMPLIANCE (in./lb) -7 10 COMPLIANCE (mm/N) ideal bodyreal body 10 -7 10 -8 10100100010 FREQUENCY (Hz) Note: in./lb = 5.7 mm/N High Frequency Compliance At higher frequencies, the compliance istotally dominated by the mass (inertia) of the table. From the previous equations, thecompliance at high frequencies is given by: Where m is the effective mass and ƒ is thefrequency of the forcing vibration.In real systems the mass term in the gener-al equation for compliance can becomequite complex. In certain very simple vibrational systems, such as our vibrating bar, it is fairly easy to evaluate the mass involved in the vibration. However, in the case of a structure such as an optical tablewith several types of bending and flexuralresonant modes of vibration, different points on the table are undergoing differ- ent amplitudes of vibration. At nodal points, there is no vibrational amplitude atall for that specific vibrational node. Theeffective mass involved in the vibration is therefore a complicated function best de- termined by computer programs. Compliance of a Real Table Summary of Key Points Compliance at Resonance The concept of an ideal rigid body is use-ful when considering optical table perfor- mance. This theoretical structure does notresonate and therefore has no compliance peaks. When plotted on a log:log scale, an ideal rigid body has a compliance propor- tional to 1/ƒ
â–  More sensitive setups require tabletopswith a lower compliance. When the forcing vibration is at the reso- nant frequency, each maximum in the ve-locity of the forcing vibration coincides with a maximum in the acceleration of the excited vibration. This adds to the acceleration of the bar, which thereby ac- cumulates vibrational energy and actually amplifies the forcing vibration. This is in accordance with the previous equation for compliance, which can be solved to show: Where:ƒ is the frequency of the forcing function, ƒnis the natural resonant frequency of thesystem,
â–  Below the first resonant frequency, Sales: 973-579-7227
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