Sine-on-Random Fatigue Analysis

Sinusoidal vibration superimposed on random vibration and is a standard solution used in military applications.

A typical application for sine-on-random vibration is in aircrafts. During service, an aircraft experiences random vibration. On top of the random vibration, sinusoidal vibration at various frequencies come from turbine engines or rotor blades. Components in the aircraft are expected to survive for certain life cycles under sine-on-random vibration. This application is not limited to aircraft; any moving machines which contain high-speed rotating parts may experience sine-on-random vibration, regardless of severity.

Input loading for sine-on-random vibration is represented using a mixed mode: PSD of random load and a certain amplitude of sine load.

Damage Calculation

Damage calculation due to sine-on-random vibration is a similar procedure to regular random vibration fatigue (Refer to Random Response Fatigue Analysis). The difference due to the superimposition of sine loads changes how the spectral moments are calculated.

The moments are calculated as:

m n = ∑ k = 1 N f k n G k δ f + 1 2 ∑ i = 1 L f i n A i 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGTbWaaSbaaSqaaiaad6gaaeqaaO Gaeyypa0ZaaabCaeaacaWGMbWaa0baaSqaaiaadUgaaeaacaWGUbaa aOGaam4ramaaBaaaleaacaWGRbaabeaakiabes7aKjaadAgaaSqaai aadUgacqGH9aqpcaaIXaaabaGaamOtaaqdcqGHris5aOGaey4kaSYa aSaaaeaacaaIXaaabaGaaGOmaaaadaaeWbqaaiaadAgadaqhaaWcba GaamyAaaqaaiaad6gaaaGccaWGbbWaa0baaSqaaiaadMgaaeaacaaI YaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamitaaqdcqGHris5aa aa@4FF0@

Where,
n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@
Moment order.
f k MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbWaaSbaaSqaaiaadUgaaeqaaa aa@33BF@
Frequency values for Random Response.
G k MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGhbWaaSbaaSqaaiaadUgaaeqaaa aa@33A0@
Stress PSD response value at frequency f k MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbWaaSbaaSqaaiaadUgaaeqaaa aa@33BF@ .
N MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36C7@
Number of frequencies in stress PSD.
L MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C5@
Number of frequencies of sine tones.
f i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbWaaSbaaSqaaiaadMgaaeqaaa aa@33BD@
Sine-tone frequency values defined on the HARMO continuation line on FATLOAD.
A i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgaaeqaaa aa@3398@
Stress amplitude due to sine tone at the i-th frequency defined on the HARMO continuation line on FATLOAD.

The second term is a contribution from sine tones. The recommended Probability Density Function (PDF) to calculate number of cycles for sine-on-random is Dirlik. Refer to Random Response Fatigue Analysis.

Input

A random response analysis and a frequency response analysis are underlying subcases for sine-on-random fatigue. In a particular FATEVNT entry, a FATLOAD referencing the random response analysis and another FATLOAD referencing a frequency response analysis should be specified to activate sine-on-random fatigue.

The FATLOAD data referencing the frequency response analysis should also list frequencies (in Hz) and their amplitude factors in the HARMO continuation line.

As an example, consider SUBCASE 10 is a random analysis subcase, and SUBCASE 20 is a frequency response analysis subcase. The following setup showcases how sine-on-random fatigue is activated:
FATLOAD,100,,10
FATLOAD,200,,20
+,HARMO,1.0,0.1,15.0,1.0,20.0,1.1 
FATEVNT,1000,100,200

Where the three sine tone frequency values are 1.0, 15.0, and 20.0; their corresponding amplitude factors are 0.1, 1.0, and 1.1, respectively.

Output

General fatigue output for Damage and Life are supported. The damage output is multiplied by exposed time T defined on the FATSEQ Bulk Data Entry and reported.