2. Förklara vad en egenfrekvens är. English: Explain what en eigenfrequency is.

Transkript

1 Linköpings Universitet, Hållfasthetslära, IEI/IKP TENTAMEN i Mekaniska svängningar och utmattning, TMMI kl L Ö S N I N G A R ---- SOLUTIONS 1. Ange sambanden mellan vinkelfrekvens ω, cyklisk frekvens f och svängningstid (periodtid) T för en svängning. English: Give the relationships between the angular frequency ω, the cyclic frequency f and the period time T of a vibration. Lösning: ω=2πf and f = 1 T 2. Förklara vad en egenfrekvens är. English: Explain what en eigenfrequency is. En frekvens ett system svänger med om systemet lämnas att svänga fritt. A frequency a system will vibrate with if it is left to vibrate freely. 8

2 Tekniska Högskolan i Linköping, IEI/IKP TENTAMEN i Mekaniska svängningar och utmattning, kl DEL 1 - (Teoridel utan hjälpmedel) 3. spänning/stress (MPa) Hur många cykler erhålls ur vidstående svängningssekvens? Ange medelvärde och 300 amplitud för varje cykel. Använd rain-flow 200 count-metoden English: How many cycles is obtained from the 100 sequence given? Give mean value and amplitude of each cycle. Use the rain-flow 0 tid/time count method. There will be four cycles: two cycles with stress σ=150 ± 150 MPa, one cycle with stress σ=150 ± 50 MPa, and one cycle with σ=100 ± 100 MPa. 4. Ange Neubers hyperbel: skriv upp ekvationen och förklara de ingående storheterna. Förklara även hur den används. English: Give the Neuber hyperbola: write down the equation and explain the different factors in it. Also, explain how it is used. The Neuber hyperbola reads ε σ= K 2 2 f σ E where ε and σ is strain and stress (at point of stress concentration), K f is the fatigue notch factor (K t, the stress concentration factor, is sometimes used here), and σ is the stress far away from the stress concentration). The intersection of the Neuber hyperbola and the material relationship (for example the Ramberg-Osgoods relation) gives the stress and the strain at the point of stress concentration. 9

3 Tekniska Högskolan i Linköping, IEI/IKP TENTAMEN i Mekaniska svängningar och utmattning, kl En massa M hängs upp i tre fjädrar (styvhet k k 1 = k, k 2 =2k och k 3 =3k). 1 k 2 k1 k (a) Bestäm egenvinkelfrekvensen för systemet M 3 k2 om massan och fjädrarna monteras enligt figur k3 M (a). (a) (b) (b) Vad blir egenvinkelfrekvensen om massan och fjädrarna monteras enligt figur (b)? English: 5. A mass M is mounted with three springs (stiffness k 1 = k, k 2 =2k, and och k 3 =3k). (a) Determine the (angular) eigenfrequency of the system if the mass and the springs are mounted as shown in figure (a). (b) What will the eigenfrequency be if mass and springs are mounted as in figure (b). (a) The equation of motion of the mass is Mẍ = F 1 F 2 + F 3 (a) where F 1 = k 1 x F 2 = k 2 x and F 3 = k 3 x This gives Mẍ +(k 1 + k 2 + k 3 )x = 0 which gives (b) The equation of motion of the mass now becomes Mẍ = F For the two springs in series one obtains (same force F in the two springs) Enter this into (b). It gives and the eigenfrequency becomes ω e = (k 1 + k 2 + k 3 ) = 6 k M M = 2.45 k M x = x 1 + x 2 = F k 1 + F k 2 giving k 1k 2 k 1 + k 2 x = F Mẍ + k 1 k 2 k 1 + k 2 + k 3 x = 0 ω e = k 1 k 2 (k 1 + k 2 ) + k 3 Thus, the eigenfrequency in case (b) goes down, but not that much, because the stiffest spring k 3 will dominate the behaviour of the mass in both cases. (b) 1 M = 11 3 k M = 1.91 k M 10

4 Tekniska Högskolan i Linköping, IEI/IKP/ TENTAMEN i Mekaniska svängningar och utmattning, kl 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. 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

5 Tekniska Högskolan i Linköping, IEI/IKP TENTAMEN i Mekaniska svängningar och utmattning, kl 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 (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 (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. 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 = , , , and cycles, respectively The Palmgren-Miner damage accumulation rule gives 1 D = = Thus, failure is expected after approximately sequences (giving cycles). 12

6 Tekniska Högskolan i Linköping, IEI/IKP TENTAMEN i Mekaniska svängningar och utmattning, kl Ett material utsätts för en lastsekvens som töjning/ strain ger töjningar enligt figur. Använd Morrows 0,003 ekvation för att beräkna förväntat antal 0,001 lastsekvenser till brott. Inverkan av spänningens tid/time medelvärde får försummas. Materialdata: E = 200 GPa, ν = 0,3, σ U = σ B = - 0, MPa, Ψ = 0,65, σ f = 900 MPa, ε f = 0,26, b = 0,095, och c = 0,47. English: 8. A material is subjected to a load sequence giving the strain shown in the figure. Use the Morrow relationship to determine the expected number of load sequences to fatigue failure. The influence of the mean stress can be neglected. Material properties: E = 200 GPa, ν = 0.3, σ U = 700 MPa, Ψ = 0.65, σ f = 900 MPa, ε f = 0.26, b = 0.095, and c = The diagram gives four cycles: - One cycle with strain range Δε = 0,0055, giving strain amplitude ε a = 0, According to Morrow one obtains (neglecting the mean stress) ε a = σ f E (2N)b +ε f (2N) c giving 0, = (2N) 0, , 26 (2N) 0,47 Solving for N gives N = cycles (2N is load reversals to failure). - One cycle with strain range Δε = 0,0025, giving ε a = 0, Morrow gives 0, = (2N) 0, , 26 (2N) 0,47 Solving for N gives N = cycles - Two cycles with strain range Δε = 0,003, giving ε a = 0,0015. Morrow gives 0, 0015 = (2N) 0, , 26 (2N) 0,47 Solving for N gives N = cycles. Palmgren-Miner now gives 1 D = = Expected number of sequences to failure is