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Tekniska Högskolan i Linköping, IEI/IKP/Tore Dahlberg TENTAMEN i Mekaniska svängningar och utmattning, 071012 kl 14-18 EXAMINATION in Mechanical Vibrations and Fatigue DEL 2 - (Problemdel med hjälpmedel) 6. Härled rörelseekvationen för en dämpad pendel l l enligt figur (massa M, längd L, gravitation g, dämpkonstant c). Pendeln gör små svängningar c runt sitt vertikala jämviktsläge. M Mg English: Deduce the equation of motion of a damped pendulum according to figure (mass M, length L, gravitation g, damping constant c). The pendulum performs small oscillations around its vertical equilibrium position. Lösning/Solution: The pendulum is subjected to two forces: the gravity force Mg and the damping force S d = c ⋅ l ˙φ . The equation of motion, J ¨φ=∑moment, becomes J ¨φ=−Mg sinφ−c ⋅ l ˙φ Using J = Ml 2 and sin φ = φ give Ml 2 ¨φ+c ⋅ l ˙φ+Mg φ=0 which is the equation asked for. 11
Tekniska Högskolan i Linköping, IEI/IKP Tore Dahlberg TENTAMEN i Mekaniska svängningar och utmattning, 071016 kl 14-18 EXAMINATION in Mechanical Vibrations and Fatigue DEL 2 - (Problemdel med hjälpmedel) 7. En stor plåt, belastad med en en-axlig spänning, spänning/stress (MPa) utsätts för en belastningssekvens enligt figur. 300 Denna sekvens upprepas. Materialet har en Wöhlerkurva som ges av sambandet 200 σ a =−50 logN + 400 (MPa) 100 där σ a är spänningsamplituden. Bestäm förväntat antal sekvenser till utmattningsbrott. 0 Använd Palmgren-Miners delskadehypotes. tid/time Inverkan av spänningens medelvärde får försummas. English: 7. A large plate, loaded in uni-axial tension, is subjected to a load sequence according to the figure. This sequence is repeated. The material has a Wöhler curve given by the equation σ a =−50 logN + 400 (MPa) where σ a is the stress amplitude. Determine the expected number of sequences to fatigue failure. Use the Palmgren-Miner damage accumulation rule. The influence of the stress mean value can be neglected. Lösning/Solution: Rain-flow count gives 1 cycle from 0 to 300 MPa, giving σ a = 150 MPa, 1 cycle from 50 to 300 MPa, giving σ a = 125 MPa, 2 cycles between 50 and 250 MPa, giving σ a = 100 MPa, and 1 cycle between 50 and 200 MPa, giving σ a = 75 MPa. These stress amplitudes give (from the Wöhler curve) N = 100 000, 316 228, 1000000, and 3 162 278 cycles, respectively The Palmgren-Miner damage accumulation rule gives 1 D = 100 000 + 1 316 228 + 2 1 000 000 + 1 3 162 278 = 1 64 600 Thus, failure is expected after approximately 64 000 sequences (giving 320 000 cycles). 12
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LinkÃ¶pings Universitet, HÃ¥llfasthetslÃ¤ra, IEI/IKP Tore Dahlberg ...

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