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DepartamentodeEngenhariaMecânicaÁreaCientíficadeMecânicadosMeiosSólidos
RMM–2015/16
MecânicaAplicada
Cap.191-Vibraçõesmecânicas
Problema19.38
Abarra𝐴𝐶de45 𝑁estápresaamolascomconstantes𝑘 = 750 𝑁/𝑚em𝐵e𝑘 = 930 𝑁/𝑚 em𝐶 , que podem atuar em tração ou compressão. Se aextremidade𝐶forligeiramentebaixadaelibertada,determine:
(a) afrequênciadevibração;(b) a amplitude domovimento do ponto𝐶, sabendo que a velocidade
máximadessepontoéde0,81 𝑚/𝑠.
Problema19.40
Duasbarrasuniformes,idênticas,de3kgestãoarticuladasnopontoBepresas a umamola de constante k=4 kN/m. As barras são guiadas porpequenas rodas de massa desprezávele o sistema está em equilíbrioquandoasbarrasestãonaposiçãohorizontal,comomostradonafigura.SabendoqueopontoBébaixadoem20mmelibertado,determine:
(a) afrequênciadavibraçãoresultante;(b) aintensidadedaaceleraçãoangularmáximadabarraAB.
Problema19.41
Uma barra uniforme𝐴𝐵de1,2 𝑘𝑔está ligada a uma articulação em𝐴e aduasmolas,cadaumadeconstante𝑘 = 450 𝑁/𝑚.
(a) Determineamassa𝑚dobloco𝐶paraaqualoperíododepequenasoscilaçõeséde0,6 𝑠.
(b) Se a extremidade𝐵é baixada em60 𝑚𝑚e libertada, determine avelocidademáximadobloco𝐶.
Problema19.57
Duas hastes uniformes, cada uma de peso𝑊 = 108 𝑁 e comprimento𝐿 = 100 𝑐𝑚, são soldadas juntas para formar a montagem mostrada nafigura. Sabendo que a constante de cada mola é𝑘 = 750 𝑁/𝑚e que aextremidade𝐴recebeumpequenodeslocamentoeé libertado,determineafrequênciadomovimentoresultante.
1Osproblemasapresentadosreferem-seaolivro“MecânicaVetorialparaEngenheiros–Dinâmica,FerdinandP.Beer,E.RussellJohnstonJr.,WilliamE.Clausen,7ªEdMcGraw-Hill”
PROBLEMS
1234
19.37 The 5-kg uniform rod AC is attached to springs of constant k 5 500 N/m at B and k 5 620 N/m at C, which can act in tension or compression. If the end C of the rod is depressed slightly and released, determine (a) the frequency of vibration, (b) the ampli-tude of the motion of point C, knowing that the maximum velocity of that point is 0.9 m/s.
19.38 The uniform rod shown weighs 15 lb and is attached to a spring of constant k 5 4 lb/in. If end B of the rod is depressed 0.4 in. and released, determine (a) the period of vibration, (b) the maxi-mum velocity of end B.
AB
C
1.4 m
0.7 m
Fig. P19.37C
A B
30 in.b = 18 in.
Fig. P19.38
19.39 A 30-lb uniform cylinder can roll without sliding on a 15°-incline. A belt is attached to the rim of the cylinder, and a spring holds the cylinder at rest in the position shown. If the center of the cylinder is moved 2 in. down the incline and released, determine (a) the period of vibration, (b) the maximum acceleration of the center of the cylinder.
19.40 A 15-lb slender rod AB is riveted to a 12-lb uniform disk as shown. A belt is attached to the rim of the disk and to a spring which holds the rod at rest in the position shown. If end A of the rod is moved 0.75 in. down and released, determine (a) the period of vibration, (b) the maximum velocity of end A.
O
B
A
k = 30 lb/in.
15°
5 in.
Fig. P19.39
D
AC
B
36 in.
10 in.
k = 30 lb/in.
Fig. P19.40
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1237Problems 19.52 A compound pendulum is defined as a rigid slab which oscillates about a fixed point O, called the center of suspension. Show that the period of oscillation of a compound pendulum is equal to the period of a simple pendulum of length OA, where the distance from A to the mass center G is GA 5 k2/r. Point A is defined as the center of oscillation and coincides with the center of percus-sion defined in Prob. 17.66.
O
G
A
r
Fig. P19.52 and P19.53
C
G
k
250 mm40 mm
B
A
Fig. P19.55
A
C
G
Bkk
Fig. P19.56
A
B DC
k k
L2
L2
L
Fig. P19.57
19.53 A rigid slab oscillates about a fixed point O. Show that the smallest period of oscillation occurs when the distance r from point O to the mass center G is equal to k.
19.54 Show that if the compound pendulum of Prob. 19.52 is suspended from A instead of O, the period of oscillation is the same as before and the new center of oscillation is located at O.
19.55 The 8-kg uniform bar AB is hinged at C and is attached at A to a spring of constant k 5 500 N/m. If end A is given a small displace-ment and released, determine (a) the frequency of small oscilla-tions, (b) the smallest value of the spring constant k for which oscillations will occur.
19.56 A 45-lb uniform square plate is suspended from a pin located at the midpoint A of one of its 1.2-ft edges and is attached to springs, each of constant k 5 8 lb/in. If corner B is given a small displacement and released, determine the frequency of the resulting vibration. Assume that each spring can act in either tension or compression.
19.57 Two uniform rods, each of mass m 5 12 kg and length L 5 800 mm, are welded together to form the assembly shown. Knowing that the constant of each spring is k 5 500 N/m and that end A is given a small displacement and released, determine the frequency of the resulting motion.
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87,5 𝑐𝑚175 𝑐𝑚