shi20396 chương 5

For a torsion bar, kT = T/θ = Fl/θ, and so θ = Fl/kT . For a cantilever, kC = F/δ, δ = F/kC. For the assembly, k = F/y, y = F/k = lθ + δ | 5-1 Chapter 5 a F k1 k2 k3 y k F y F F F y F k2 k3 so 1 Ans. 1 k1 1 k2 1 k3 b - JNJhr k1 - JUIJr1 k2 - JUIJr k3 F y c -NJNJri k2 k1 k3 F k y k2 y k3 y k F y ki k2 k3 Ans. 1 _ 1 1 k k1 k2 k3 k 1 1 1 k1 k2 k3J 5-2 For a torsion bar kT T 6 Fl 6 and so 6 Fl kT. For a cantilever kC F 8 8 F kC. For the assembly k F y y F k 16 8 _ F _ Fl2 F y k kT kC 1 So Or k --------------- Ans. l 1 kT 1 kC 5-3 For a torsion bar k T B GJ 1 where J nd4 32. So k nd4G 32l Kd4 springs 1 and 2 are in parallel so k k1 k2 Kd K - l1 l2 Kd4 l 1 x And Then 6 T- _ - k Kd4 1 J- x l x T k 6 Kd- 6 Kddl l x x Chapter 5 107 Thus 7 Kd4 e t2 Kd46 x l x If x l 2 then T 72. If x l 2 then T 72 Using t 16T nd3 and 6 32Tl Gnd4 gives n d 3t T -------- 16 and so 32l n d3r 2l Tall 6all ----7 ---- ----- Gnd4 16 Gd Thus if x l 2 the allowable twist is 2x Tall A 6aii xtt Ans. Gd . Since k Kd4 1 J x l x _nGd4 1 1 32 x l x Then the maximum torque is found to be Tmax 1 _L_ Ans. 16 x l x 5-4 Both legs have the same twist angle. From Prob. 5-3 for equal shear d is linear in x. Thus d1 Ans. 4 d_ n G k ------ 32 . 2 Tall . 6all -------- Ans. Gd2 32l 2 Ans. Tmax k6all Ans. F A nr2 n r1 x tan a 2 Fdx Fdx AE En r1 x tan a 2 F fl dx n E 0 r1 x tan a 2 F 1 V --- I-------------------I n E tan a r1 x tan a 0 F 1 n E r1 r1 l tan a 108 Solutions Manual Instructor s Solution Manual to Accompany Mechanical Engineering Design Then l F n Er1 r1 l tan a k 1 EA1 l 2l 1 ----tan a di Ans. 5-6 w is cable s weight per foot Enlarged free body of length dx T F T dT w dx T 0 dT T w dx Solution is T wx c w dx T dT T x o P wl c T w x P wl T P w l x The infinitesmal stretch of the free body of original length dx is s Tdx d 8 ---- AE P w l x -------------dx AE Integrating 1 P w l x dx o AE 8 P Ans. AE 2AE 5-7 M wlx wl2 2 2 wx2 2 Eid-y dx wlx 2 2 wl2 x 3 w x3 y Ci 6 0 at x 0 a C1 0 dx Ely wlx 3 wl2 x 2 6 4 4 w x 4 _ IT C2 y 0 at x 0 a C2 0 2 w x2 y 4lx 6l2 24 El x2 .