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A compliant support does not reduce every vibration reaching a ceramic circuit. It can amplify motion near a support resonance while attenuating higher-frequency base motion. The circuit may also move farther relative to its enclosure even when its absolute acceleration decreases. Support selection therefore needs two outputs: transmitted motion and the travel consumed between the circuit and surrounding package features.
Key design decisions
- Compare the excitation spectrum with the installed support modes before calling a softer attachment an isolator.
- Evaluate relative motion and available clearance separately from absolute acceleration.
- Keep damping, stiffness, cable constraints and sensor mounting associated with the actual assembled configuration.
Distinguish movement of the base, circuit and internal features
Define y as displacement of the package base and x as displacement of an approximately rigid supported circuit assembly along one evaluated axis. Their difference, x − y, is the travel taken up by the supports and neighboring clearances. A component or wire loop can have additional local motion that this rigid-assembly description does not include.
The input must be identified as base motion. A cable pulling directly on the circuit is a different excitation and can bypass the intended isolation path. Likewise, a flexible ceramic plate or a circuit with several coupled rocking modes cannot be represented adequately by one translating mass. Begin with the simplified model only where the measured or modeled mode shape supports that approximation.
Define stiffness and damping about the installed state
For a linear translating mass m supported by stiffness k and viscous damping c, the equation is m x'' + c(x' − y') + k(x − y) = 0. The undamped natural frequency is fn = sqrt(k/m)/(2 pi), and damping ratio is zeta = c/(2 sqrt(k m)). Stiffness is the incremental response about the installed preload, not automatically an adhesive's bulk modulus or a pad's advertised hardness.
Mounting material, contact area, compression, temperature and frequency can change the effective properties. The cable and thermal connections may contribute parallel stiffness or damping. Record those paths with the supports rather than fitting a convenient stiffness to a bare circuit and carrying it unchanged into the closed package.
Calculate absolute and relative response separately
For steady sinusoidal base motion at frequency f, let r = f/fn. Define D = sqrt[(1 − r²)² + (2 zeta r)²]. The absolute displacement amplitude ratio is sqrt[1 + (2 zeta r)²]/D. The relative-displacement amplitude divided by base amplitude is r²/D. At one sinusoidal frequency, the absolute acceleration ratio equals the absolute displacement ratio.
Do not calculate relative-travel amplitude by subtracting the two displacement amplitudes. Circuit and base motions generally have different phase angles. Their time-domain difference or complex transfer function is needed. The separate relative-response expression accounts for that phase relationship and can give a much larger travel than an amplitude-only subtraction would suggest.
Tabs = sqrt[1 + (2 zeta r)²]/D; Trel = r²/D; D = sqrt[(1 − r²)² + (2 zeta r)²]
- Tabs: absolute circuit-motion amplitude divided by base-motion amplitude.
- Trel: relative circuit-to-base travel amplitude divided by base-motion amplitude.
- r = f/fn and zeta is the viscous damping ratio.
Linear, steady sinusoidal base excitation of one translating degree of freedom. No contact loss, travel stops, material nonlinearity, ceramic bending or coupled rotation is included.
Compare the same support at three excitation frequencies
Assume fn = 50 hertz and zeta = 0.10. At ten hertz, r is 0.2: the absolute motion ratio is about 1.042 and the relative ratio is about 0.0416. The assembly largely follows the base, with slightly amplified absolute motion. At 50 hertz, the absolute ratio rises to about 5.099 and the relative ratio to five.
At 150 hertz, r is three: the absolute ratio falls to about 0.1454 while the relative ratio is about 1.122. This is useful attenuation of absolute base motion, but the circuit-to-enclosure travel remains important. The support does not become universally better or worse; its suitability depends on which frequencies and clearances matter in the actual installation.
| Excitation frequency | Absolute motion ratio | Relative travel ratio | Interpretation |
|---|---|---|---|
| 10 Hz | 1.042 | 0.0416 | Circuit nearly follows base |
| 50 Hz | 5.099 | 5.000 | Large response near support resonance |
| 150 Hz | 0.1454 | 1.122 | Reduced absolute motion but continuing relative travel |
Convert the motion into a clearance requirement
Suppose the assumed 50-hertz sinusoidal base acceleration has a peak amplitude of 0.20 g, using g = 9.81 meters per second squared. Base displacement amplitude is acceleration divided by (2 pi f)², approximately 0.0199 millimeter. Multiplying by the relative ratio of five gives approximately 0.0994 millimeter of relative peak travel. Peak-to-peak travel is twice that value.
Compare motion against available clearance around the installed equilibrium, including assembly tolerances and thermal movement. For a vertically loaded linear support with the same natural frequency, static gravitational deflection is g/(2 pi fn)², also approximately 0.0994 millimeter in this example. Do not count that static shift twice if the drawing clearance is already measured in the installed, weight-loaded state. Contact with a stop changes the model and can create local impact loads.
Avoid treating added damping as identical to softer support
Increasing damping ratio from 0.10 to 0.30 while preserving fn reduces the absolute response at r = 1 from about 5.099 to 1.944. However, at r = 3 the absolute ratio increases from about 0.1454 to 0.2511. Damping suppresses the resonance response but can transmit more high-frequency base motion through its velocity-dependent path.
A material change may alter damping and stiffness together, so comparing material names is insufficient. Measure the installed frequency response and preserve the temperature, preload and excitation level. If response changes with amplitude, investigate contact, friction or nonlinear material behavior instead of assuming one constant damping ratio represents the complete operating range.
Allow for shifts in the installed natural frequency
A resonance outside the dominant excitation band at nominal properties may move into that band at tolerance or temperature limits. For an assumed nominal 50-hertz mode, a stiffness range of minus or plus 20 percent and a mass range of minus or plus ten percent gives frequency endpoints of approximately 42.64 and 57.74 hertz. These follow from the square root of stiffness divided by mass, not an arithmetic average of percentage errors.
The endpoints are a sensitivity exercise, not a material tolerance specification. Determine actual property variation and identify whether it changes the mode shape as well as frequency. Added components, replacement cables, adhesive coverage or a new thermal strap can alter the dynamic boundary while the ceramic outline stays unchanged. Compare the frequency envelope with actual service excitation rather than checking only the nominal resonance.
Verify the response without letting the instrument define it
Measure base and circuit response with known positions, axes and synchronization. Keep fixture motion distinct from the intended package-base input. The sensor's own mounting affects its frequency response, and its attachment must not impose a new rigid path or unsupported local load on the ceramic. A calibration performed on a rigid metal block does not remove an installation-dependent measurement error.
Use an appropriately reviewed attachment or noncontact measurement approach for the circuit surface. Do not drill the ceramic or remove protective layers merely to follow a generic accelerometer mounting instruction. Retain sensor construction, attachment, cable restraint and useful measurement bandwidth with the response data. Also inspect the package for contact marks or intermittent electrical behavior that would contradict the assumed contact-free linear motion.
Specify the support by response and travel, not softness alone
The comparison should state the installed mode range, damping basis, absolute response at relevant frequencies and relative travel consumed at nearby features. Include the assembly orientation and whether specified vibration values are peak, peak-to-peak or root mean square. A steady sinusoidal model does not directly establish random-vibration, shock or fatigue performance.
Select support changes together with electrical connections, thermal paths and available motion. If isolation requires travel that the enclosure cannot provide, increasing compliance alone is not a complete solution. The final package evaluation must show both an acceptable response and an intact supported ceramic interface under the agreed loading conditions.
Provide the installed support and vibration boundary
Send the package geometry and excitation definition so absolute response and relative travel can be assessed together.
- Circuit assembly mass, support locations, motion axes, preload, cable restraint and thermal-connection geometry.
- Measured or justified incremental stiffness and damping across temperature and relevant loading amplitude.
- Base-excitation frequencies or spectrum with clear peak, peak-to-peak or RMS conventions and test orientation.
- Available installed clearances, component and lid tolerances, thermal movement and travel-stop geometry.
- Paired base and circuit response records, sensor positions and mounting, fixture behavior and pre/post electrical observations.
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