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