Application Area 9: Appliance Design

A mechanical accelerometer is used to measure the acceleration of a rocket test sled. It contains a linear spring-damper system. See the image below. By tuning the stiffness and damping, the accelerometer can be designed to estimate the acceleration it feels within an acceptable time.

Figure 1.


Objective
We can compute the deviation of any system response x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=Hhaaaa@331E@ from its target value x * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=HhapaWaaW baaSqabeaapeGaa8Nkaaaaaaa@3415@ as follows:
S T E P ( t t a r g e t ) = { 0   i f   t t t a r g e t 1   i f   t > t t a r g e t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=nfacaWFub Gaa8xraiaa=bfadaqadaWdaeaapeGaa8hDa8aadaWgaaWcbaWdbiaa =rhacaWFHbGaa8NCaiaa=DgacaWFLbGaa8hDaaWdaeqaaaGcpeGaay jkaiaawMcaaiabg2da9iaa=bkadaGabaWdaeaafaqabeGabaaabaWd biaaicdacaWFGcGaa8xAaiaa=zgacaWFGcGaa8hDaiaa=bkacqGHKj YOcaWFGcGaa8hDa8aadaWgaaWcbaWdbiaa=rhacaWFHbGaa8NCaiaa =DgacaWFLbGaa8hDaaWdaeqaaaGcbaWdbiaaigdacaWFGcGaa8xAai aa=zgacaWFGcGaa8hDaiabg6da+iaa=bkacaWF0bWdamaaBaaaleaa peGaa8hDaiaa=fgacaWFYbGaa83zaiaa=vgacaWF0baapaqabaaaaa GcpeGaay5Eaaaaaa@6101@
f ( x ) = t s t a r t t e n d S T E P ( t t a r g e t ) *   | x x * | d t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=zgadaqada WdaeaapeGaa8hEaaGaayjkaiaawMcaaiabg2da9maawahabeWcpaqa aaqaaaqdbaWdbiabgUIiYpaaDaaaoeaacaWF0bWdamaaBaaabaWdbi aa=nhacaWF0bGaa8xyaiaa=jhacaWF0baapaqabaaapeqaaiaa=rha paWaaSbaaeaapeGaa8xzaiaa=5gacaWFKbaapaqabaaaaaaak8qaca WFtbGaa8hvaiaa=veacaWFqbWaaeWaa8aabaWdbiaa=rhapaWaaSba aSqaa8qacaWF0bGaa8xyaiaa=jhacaWFNbGaa8xzaiaa=rhaa8aabe aaaOWdbiaawIcacaGLPaaacaWFQaGaa8hOamaaemaapaqaa8qacaWF 4bGaeyOeI0Iaa8hEa8aadaahaaWcbeqaa8qacaWFQaaaaaGccaGLhW UaayjcSdGaa8hzaiaa=rhaaaa@5A37@
The problem is formulated as:
min    ( x x * ) 2 | steady state MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaqaaaaaaaaaWdbiaah2gacaWHPbGaaC OBaiaacckacaGGGcGaaiiOaiaacckacaGGOaWaaqGaa8aabaacbiWd biaa=HhacqGHsislcaWF4bWdamaaCaaaleqabaacbaWdbiaa+Pcaaa GccaGGPaWdamaaCaaaleqabaWdbiaaikdaaaaakiaawIa7a8aadaWg aaWcbaWdbiaa=nhacaWF0bGaa8xzaiaa=fgacaWFKbGaa8xEaiaa+b kacaWFZbGaa8hDaiaa=fgacaWF0bGaa8xzaaWdaeqaaaaa@4D4C@
such that    f( x )<1e3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaqaaaaaaaaaWdbiaadohacaWG1bGaam 4yaiaadIgacaGGGcGaamiDaiaadIgacaWGHbGaamiDaGqaaiaa=bka caWFGcGaa8hOaiaa=bkaieGacaGFMbWaaeWaa8aabaWdbiaa+Hhaai aawIcacaGLPaaacqGH8aapcaaIXaGaa4xzaiabgkHiTiaaiodaaaa@4736@
Design Variables
The stiffness of the spring, k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=Tgaaaa@3311@ , and its damping, c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qiqrFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeGabeqadiWa ceGabeqabeqadeqadmaakeaaieGaqaaaaaaaaaWdbiaa=ngaaaa@3309@ , are the design variables.
Results
By properly selecting the spring stiffness and damping, the accelerometer can be designed to respond in the desired time frame.
Figure 2.