CHAPTER 3 PROBLEMS. δ = where A is the cross-sectional area, and E is the modulus of elasticity.

Transcription

1 CHPTER 3 PROLEMS d SPRING-MSS-DMPER PPLICTIONS Proble 3.1 The buoy shown in figure P3.1 has a irular ross-setion with diaeter d and has length L. Most of the weight of the buoy, w, is onentrated in the base. The buoy is released fro rest with one end in ontat with the surfae of the fluid with speifi weight of γ f. Hint: pply rhiedes s theore for buoyany. Tas: (a). Draw a free body diagra and obtain the equation of otion for the syste using Y as your oordinate. (b). Deterine the natural frequeny for the syste. (). Deterine the equilibriu position for the buoy. (d). Deterine the equation of otion for otion about the equilibriu position. Y L buoy Figure P3.1 EI Figure P3. Y Proble 3. body of ass is supported at the end of a antilever bea of length L and setion odulus EI, figure P3.. Reall that the defletion δ at the end of the bea is 3 PL defined by δ = where P is the load. 3EI (a). Draw a free body diagra and obtain the equation of otion for ass. (b). Neglet the ass of the bea and deterine the natural frequeny. E, Proble 3.3 body of ass is supported at the end of unifor bar of length L, figure P3.3. Reall that the defletion at the end of a unifor bar due to an axial load P is PL δ = where is the ross-setional area, and E is the odulus of elastiity. E (a). Neglet the ass of the bar and deterine the equation of otion using Y as your oordinate. (b). Deterine the natural frequeny of the syste. Proble 3.4 In figures P3.4, P3.4, and P3.4C the springs are initially unstrethed. Y Figure P3.3 (a). Selet a oordinate, draw the free body diagra, and obtain the equation of otion of the syste. (b). pply wor energy priniples and obtain the equation of otion. (). Calulate the natural frequeny of the syste using the following values: 1 =10 g, 1 =1000 N, =1000 N, 3 =1000 N. (d). Suppose the springs are initially strethed (or opressed), and onsider otion about equilibriu, does the differential equation hange? 3 1 Figure P3.4 Proble 3.5 The ass shown in figure P3.5 is wedged between the two springs. The springs are in opression at equilibriu, and are firly attahed to the walls and the ass. (a). Selet a oordinate, draw the free body diagra and obtain the equation of otion for the syste if the ass is displaed fro the equilibriu position and released fro rest. (b). pply wor energy priniples and obtain the equation of otion. (). Obtain the natural frequeny of the syste using the following values: 1 = 10 N/, = 15 N/, and = 50 g. 3 Figure P3.4 1 ass-less bar Figure P3.5 Figure P3.4C

2 Proble 3.6 The ass shown in figure P3.6 is held in plae by the two springs. The two springs are both in tension when the ass is at equilibriu and are attahed firly to the walls and the ass. Frition at the ontat between the ass and the inlined plane ay be assued to be negligible. (a). Selet a oordinate, draw the free-body diagra and obtain the equation of otion for the syste if the ass is displaed fro the equilibriu position and released. (b). pply wor energy priniples and obtain the equation of otion. (). Obtain the natural frequeny of the syste using the following values: 1 = 100 lb/in, = 50 lb/in, w = 50 lb, and α = 35. (d). Repeat part (a), but assue that the springs are initially unstrethed when the body is released fro rest. 1 w Figure P3.6 α Proble 3.7 For the syste illustrated in figure P3.7, the springs are undefleted. (a). Selet a oordinate, draw the free-body diagra, and derive the general differential equation of otion for Y. (b). Obtain the equation of otion for otion about the systes equilibriu position. (). For = 70 N/, = 100 g, = 80 Ns/, alulate the following: i) Undaped natural frequeny ω n, ii) Daping fator ζ, and iii) Daped natural frequeny ω d. (d). Repeat part (a) assuing that the body is displaed fro equilibriu. Proble 3.8 The spring shown in figure P3.8 is undefleted. Figure P3.7 (a). Draw the free-body diagra and state the differential equation of otion. (b). Using the given inforation: = 0 lb/in, W = 50 lb, α= 5º deterine the undaped natural frequeny. (). For = 3.74 lb se/ft deterine the daping fator, ζ, and the daped natural frequeny. (d). Drop the daping eleent and obtain the equation of otion by applying worenergy priniples. SPRING-MSS-DMPER SYSTEMS WITH TRNSIENT EXCITTION Undaped Systes Proble 3-9 onstant fore of.5 N ats upon the art shown in figure P3-9 for 0.15T n seonds, where T n is the period of the natural frequeny of the syste. The art has a ass of 1000 g, and the spring, a has a stiffness of 500 N/. W α Figure P3.8 (a). Selet oordinates, derive the equation of otion for the syste, and put into a for so that table.1 an be applied to find the partiular solution. (b). pplying the initial onditions X(0)=0; X ( 0) = 0, deterine the oplete solution for otion of the syste for the tie interval, 0< t < t f = T n. (). Deterine the pea response aplitude of the syste. f(t) Proble 3-10 The art shown in figure P 3-9 weighs 00 lb and is attahed to ground by a spring with a stiffness of ip/in. fore ats on the art that is desribed by the funtion, f(t)=-5t lb and ats on the art for 0.5 seonds. For t> 0.5 se, f (t) =0. Figure P3-9 (a). Selet a oordinate, derive the equation of otion for the syste, and put into a for so that table.1 an be applied to obtain the partiular solution. (b). Develop the oplete solution, satisfying the initial onditions X(0)=0; X ( 0) = 0, and deterine the pea response for a total tie period of 1 seond.

3 Proble 3-11 The tower shown in figure P3-11 has the sae differential equation odel as the art syste shown in figure Proble 3-9 where =. The fore ating on the tower an be odeled by f(t) = 75t-30.5t N with the fore ating for 1 seond. (a). Selet a oordinate and derive the equation of otion for the syste in a for that table.1 an be readily applied to obtain the partiular solution. (b). Develop the oplete solution that satisfies the initial onditions that the tower is undefleted and at rest. Solve for the horizontal displaeent of the tower for a 5 seond tie interval. Note: you will need to restate the equation of otion and restart the proble for t > 1 seond. Proble 3-1 The art shown in figure P3-1 is onneted to ground by a springdaper syste. fore with a onstant aplitude of 350 N ats on the art for t =.5ω d se. The art has a ass of 50 g. The spring has a stiffness of N/, and the syste has a daping ratio of The spring is initially undefleted. f(t) 3EI = 3 L M=10 Mg Figure P3-11 L=30, I= E=10GPa Tas (a). Selet a oordinate, draw a free-body diagra, and derive the equation of otion. (b). Solve the otion, applying the initial onditions that the syste is at rest and that the spring is initially undefleted. (). Deterine the tie differene between when the pea response ours and when the fore is no longer applied. Note: You will need to brea this solution into two periods overing: (i) the tie while the fore is applied, and (ii) The tie after the fore is turned off. f(t) Proble 3-13 The art shown in figure P3-1 weighs 150 lbs, = 000 lb/in, and = 6.0 in-se/lb. The fore f(t)= t lb is applied for 0.5 seonds. For t >0.5 se, f(t)=0. (a). Selet a oordinate, draw a free-body diagra, and derive the equation of otion for the fored and free response. (b). Solve for the physial response of the syste. pply the initial onditions that the spring is initially undefleted and the ass is at rest. (). Deterine when the pea aplitude of the syste ours. Note: You will need to brea this solution into two periods overing: (i) the tie while the fore is applied, and (ii) The tie after the fore is turned off. Proble 3-14 The art shown in figure P3-1 has a daped natural frequeny of 5.0 Hz, a ass of 150 g, and is onneted to ground by a spring with a stiffness of 150 N/. The art is ated upon by a fore defined by f(t)=-5t -15t N for a period of 1.5 se. fter this tie, f (t) =0. (a). Selet a oordinate, draw a free-body diagra, and derive the equation of otion. (b). pply the initial onditions that the spring is initially undefleted and that the art is at rest. Solve for X (t) using the results of table.. (). Deterine the pea aplitude for X (t). Note: You will need to brea this solution into two periods overing: (i) the tie while the fore is applied, and (ii) The tie after the fore is turned off. Daped Systes: Proble 3-15 The equation of otion for a base exitation proble as shown in figure P3-15 an be stated X + X + X = X + X, or as ( ) ( ) X = X b X + X b X. Subtrating X b fro both sides of the previous expression results in δ + δ + δ = X where b δ = X X is the relative displaeent of the ass with b respet to the base. Tas: ssue the syste has the following initial onditions: δ ( ) δ ( ) deterine the relative response of the syste for X = D. b 0 = 0 = 0, and Figure P3-1 ass-less bar} X Figure P3-15 X b (t) Proble 3-16 The art shown in figure P3-16 is onneted to ground by spring and daper, weights 50 lb, and has a daped natural frequeny of 9.96 Hz. assless plate at the end of the spring-daper obination has otion defined by x(t) =

4 1.5+.5t in. for 0.5 seonds. The daping ratio of the syste is found to be Spring has a stiffness of 15 ip/in and the daper has a value of 5 lb-se/in. (a). Selet a oordinate, draw a free-body diagra, and state the governing equation of otion. (b). pplying the initial onditions that the spring is initially unstrethed and that the ass is at rest solve for the otion of the syste. (). Deterine when the axiu veloity of the art ours after x(t) is reoved fro the ass-less bar. ass-less plate} x(t) Proble 3-17 The art shown in figure P3-16 is attahed to ground with a springdaper syste. Motion for the ass-less plate on the right is defined by x(t)=0t for t x = (π/ω n ). For t greater than t a, the plate is otionless. The properties for the oponents of the syste are defined by: = 100 g, = 1000 N/, = 500 N/, =.5 N-se/, and = 1.5 N-se/. Figure P3-16 (a). Selet a oordinate, draw a free-body diagra, and deterine the equation of otion. (b). pplying the initial onditions that the spring is initially unstrethed and that the ass is at rest deterine the otion. (). Deterine the art s pea response aplitude. Proble 3-18 The art shown in figure P3-16 weighs 500 lbs, = 000 lb/in, = 50 lb-se/in, = 00 lb/in, and = 15 lb-se/in. The plate s otion is defined by x(t) = 0.5t ft, where t is in seonds. The plate s otion last for t a =0.5 ses. (a). Selet a oordinate, draw a free-body diagra, and deterine the equation of otion (b). pplying the initial onditions that the spring is initially unstrethed and that the ass is at rest, solve for the otion of the syste. (). Deterine the pea aplitude of the syste after t a. SPRING-MSS-DMPER SYSTEMS WITH HRMONIC EXCITTION Proble 3.19 The syste shown in figure P3.19 is ated upon by the foring funtion shown. The paraeters for the syste are: =15 g, =75 N/, f 0 =750 N, ω=15.13 Hz. For otion about equilibriu, deterine the steady state aplitude and phase for: (a). = 50 Ns/. (b). = 0. (). Deterine the range of exitation frequenies for an aplitude of 0 or less using the daping fro part (a). f(t) = f 0 os ωt Proble 3.0 Referring to figure P3.19 and using the following paraeters: W = 175 lb, = 1.05(10 5 ) lb/ft, = 5 lbs/in, f 0 = 500 lb, ω = 4 Hz. Figure P3.19 (a). Deterine the steady state aplitude and phase. (b). Deterine the operating frequeny range to avoid so that the aplitude does not exeed 0.15 in. (). Deterine the required daping value at resonane suh that the pea aplitude does not exeed 0.3 in. Proble 3.1 Given: The differential equation of otion X + X + X = f0 sinω t with the paraeters: w = g = 10 lb, = 10 lb/in, and f 0 = 10 lb. Tas: For an exitation frequeny above 11 Hz (yles/se), selet the lowest possible value for suh that the aplitude of fored otion X(t) will be no ore than 0.3 inhes. lso, deterine the pea response aplitude at resonane for a daping value of 0.1 lb-se/in.

5 Proble 3. The syste shown in figure P3. is a onfiguration for seisi otion easureents. The syste is ounted on a struture that has a vertial vibration at 15 Hz and a double aplitude of. The sensing eleent has a ass =.0 g, and the spring has a stiffness = 1.75 N/. The otion of the ass relative to the instruent base is reorded on a revolving dru and shows a double aplitude of. during the steady-state ondition. Tas: Calulate the visous daping onstant. Hint: Note that the pen s otion easures relative otion between the ass and the base. Proble 3.3 vehile is oving along a path defined by π X Yb = sin L, with = 0.03, L = 0.6. The vehile has a onstant veloity v = X = v 0. For hr onveniene, assue that it starts at X = 0, then the position of the vehile an be defined by: hr ( hr )( 3600 se )( 1 ) X = X + Xt = 0 + v t X ( t ) = 0.78 v t. Tests show that the vehile has an undaped natural frequeny of 5 Hz (yles/se). The ass of the vehile is 750 g. Refer to figure P3.3. (a). Deterine the speed of the vehile that will ause the highest steady-state aplitude for the vehile? (b). Selet the daping value for the pea running speed in question (a) suh that the vehile s steady-state aplitude will be less than 0.1. Proble 3.4 Refer to proble 3.3 and apply the following inforation: Undaped natural frequeny of vehile, f n, is 5 Hz, weight of the vehile is 1750 lb, L = 1.5 ft, = 0.5 ft. (a). Deterine the speed at whih the pea aplitude will our? (b). Deterine the required daping value if the pea aplitude is to be 3.75 inhes. Proble 3.5 For the syste shown in figure P3.5 perfor the following tass: (a). Selet a oordinate, and obtain the equation of otion. (b). Deterine the steady state response for the given paraeters: 1 = 50 N/, = 400 N/, = 50 g, = 0.75 N-se/, = 15, ω = 0 Hz. (). Deterine an operating range so that a steady-state dynai aplitude of 30 is not exeeded Proble 3.6 The rotor of an eletri generator weighs 750 lbs and is attahed to a platfor weighing 7750 lbs. The otor has an eentriity of.5 ils. The otor and platfor an be odeled by the figure shown in figure P3.6. The equivalent stiffness of the platfor onneting the platfor to ground is 7.5(10 5 ) lb/in, and the equivalent daping of the platfor is 80 lb-se/in. The operating speed of the generator is 1800 rp. Neglet the frition between the platfor and ground. Y M Y OP struture Figure P3. v 0 Y OP / / Y b X ( ) π Y = sin X L Figure P3.3 X(t)=os(ωt) 1 Figure P3.5 b L (a). Deterine the pea displaeent of the platfor. (b). Deterine the pea response at resonane (r =1). (). Deterine the allowable aount of ibalane if the stiffness of the platfor is inreased by 5%. and the allowable pea response is to be under 4.9 ils. ssue that the ass of the platfor and the aount of available daping does not hange by an appreiable aount when the stiffness is inreased. a rotor platfor Proble 3.7 The odel of figure P3.6 has the following paraeters: ass of platfor, P, is 000 g, ass of rotor, R, is 700 g, daped natural frequeny of syste, ω n = Hz, and ζ (a). Deterine the allowable aount of ibalane so that the steady state aplitude does not exeed.5 at an operating speed of 600 rp. (b). Deterine the allowable ibalane if the stiffness is inreased by 35%. ssue that there is a negligible inrease in ass and daping when the stiffness is hanged. pply the allowable steady state aplitude fro part (a). Figure P3.6

6 (). Let the ibalane of the rotor be Deterine the range of operating speed to avoid so that the response aplitude does not exeed 3. ssue the syste now has a daping fator of 0.0. Use ω n = 9 Hz. Proble 3.8 s shown in figure P3.8, an industrial fan has an eletri-otor rotor that weighs 300 lbs and a base support with an estiated weight of 00 lbs. It has a steady operating speed of 3600 rp. Rap tests show that the fan assebly has a daped natural frequeny of Hz and a daping fator of 0.05 for otion in the horizontal plane. The assebly is uh stiffer for vertial otion. (a). Deterine the axiu ibalane displaeent a so that the steady state response of the assebly is below 50 ils when the running speed oinides with the daped natural frequeny. (b). Deterine the response for the ibalane found in (a) at the steady state running speed? (). The response resulting due to the ibalane in (b). is to be redued by 5%. Deterine the required ibalane to ahieve this. lso, deterine the required hange in stiffness to ahieve the sae effet if r > 1. Figure P3.8 otor rotor LINER MOTION WITH CONSTRINTS ppliations with Pulleys Proble 3.9 For the syste illustrated in figure P3.9 the tru is oving to the right with a onstant veloity of v 0 at X = h. For the general position shown, what is the tension in the able? Proble 3.30 The bodies shown in figure P3.30 are released fro rest fro the position shown. Neglet the inertia of the pulleys. (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). Derive the governing differential equation of otion by applying wor-energy priniples. (). fter 1 seond, what are the veloities of the bodies? (d). fter body C has oved 1 eter, what are the veloities of the two bodies? h r Figure P3.9 v O Proble 3.31 The syste illustrated in figure P3.31 is released fro rest in the position shown. Neglet the ass of pulleys. (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). What are the initial aelerations of the bodies? (). What are the veloities of the bodies after one seond? (d). What are the veloities of the bodies after body has oved 1 eter? Proble 3.3 The two blos shown in figure P3.3 are originally at rest. The ineti Coulob frition oeffiient µd = 0.. Neglet the inertias of the pulleys. Figure P3.30 C = (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). How fast are the blos going after 1 seond? (). How fast are the blos going after blo has oved 1 foot? w = 5 lb 0 º µ d =0.1 = 10 g b = 8 g Figure P º µ d 90º w = 350 lb Figure P3.3

7 Proble 3.33 The syste illustrated in figure P3.33 is released fro rest. The dynai oeffiient of frition is µ d = Neglet the ass of pulleys (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). Deterine the veloities of the bodies after blo C has oved down 8 in. (). Deterine the veloities of the bodies after 3 seonds. Proble 3.34 Two bodies and shown in figure P3.34 have asses of 45 g and 30 g, respetively. The ord onneting the bodies is inextensible. The syste is released fro rest. The oeffiient of ineti frition is µ d = 0.3. Neglet ass of pulleys. (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). Deterine the veloities of both bodies after body has oved 1 eter. (). Deterine the veloities of both bodies after 1 seond. 5º w = 1 lb w = 35 lb w C = 10 lb C Figure P3.33 Pulley ppliations with Springs and Dapers Proble 3.35 For the syste illustrated in figure P3.35, neglet the inertia of the pulleys. µ d =0.3 (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion in ters of the otion of ass. (b). Reove the daper and use onservation of energy to derive the equation of otion. (). Fro the results of part (a), deterine the natural frequeny and daping fator for the syste. Proble 3.34 Proble 3.36 For the proble illustrated in figure P3.36, =, and the spring is undefleted when X = 0. The syste is released fro rest at X = 0. Neglet the inertia of the pulleys. (a). For otion relative to the undefleted position, selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). Deterine the equilibriu ondition and the equation of otion for otion relative to the equilibriu position. (). pply onservation of energy and obtain the equation of otion. (d). Deterine the natural frequeny for the syste. C C Figure P3.35 Proble 3.37 For the syste illustrated in figure P3.37, the syste is in equilibriu. The inertia of the pulleys is negligible, and the spring reains intension during otion of the ass. (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). Derive the equation of otion using onservation of energy. (). What is the natural frequeny of the syste? Figure P3.36 Figure P3.37

8 Proble 3.38 The syste shown in figure P3.38 onsists of two asses onneted by a single inextensible ord. Mass 1 on the left is onneted to ground by a linear daper with daping oeffiient. nother ord extends fro the ground over a pulley at the botto of ass and ba down to ground through a linear spring with stiffness oeffiient. The syste is in equilibriu in the position shown, and the spring at reains in tension for otion of the two asses. The ass and inertia of the pulleys are negligible. 1 α (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion in ters of ass 1 s position. (b). Reove the daper and derive the equation of otion using onservation of energy. Figure P3.39 Proble 3.39 The syste shown in figure P3.39 onsists of two asses onneted by a single inextensible ord. Mass 1 is onneted to ground by a linear spring with stiffness oeffiient. Mass is onneted to ground by a daper with linear daping oeffiient. The spring is undefleted in the position shown. The ass and inertia of the pulleys are negligible. 1 (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion in ters of ass 1 s displaeent. (b). Deterine the equilibriu position and the equation of otion about the equilibriu position. (). Reove the daper and derive the equation of otion using onservation of energy. α Figure P3.40 Proble 3.40 The syste illustrated in figure P3.40 is in its equilibriu position with no frition between ass 1 and the ontat surfae. The inertia of the pulleys is negligible. (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion. (b). Eliinate the daping and derive the equation of otion using onservation of energy. (). What are the undaped and daped natural frequenies and the daping fator for the syste? Proble 3.41 The syste shown in figure P3.41 is in equilibriu and onsists of two asses onneted by a single inextensible ord. Mass 1 is onneted to ground by a linear daper with daping oeffiient. nother ord runs through fro the eiling through a pulley at the top of ass and ba to the eiling through a linear spring with stiffness oeffiient. The spring reains in tension during otion of the two asses. The inertia of the pulleys is negligible. (a). Selet oordinates, draw the free body diagras, state the ineati onstraint equation(s), and derive the equation of otion in ters of ass 1. (b). What are the undaped and natural frequenies and what is the daping fator? (). Reove the daper and derive the equation of otion using onservation of energy. 1 Figure P3.41 PRTICLE KINETICS Polar Coordinate ppliations Proble 3.4 partile of ass has the veloity vo = gr as it approahes the ylindrial surfae shown in figure P3.4. The surfae is fritionless. (a). Draw a free-body diagra, and derive the equation of otion for the oordinate θ. (b). Find the noral reation fore as a funtion of θ only. For what value of θ will the partile leave the surfae? v O Figure P3.4 r θ

9 Path Coordinate ppliations Proble 3.43 ass of 1g is fored to follow a path defined by Y = (1-X ) where X and Y are in eters, figure P3.43. The ass is onneted to a spring at O that has an undefleted length of 1 eter and a stiffness of 3 N/. The ass is released fro rest at ( X = 0, Y=.) (a). Draw a free-body diagra and find the reation fore ating on the body at. (b). Find its veloity at ( X = 1, Y = 0.). Proble 3.44 The box illustrated in figure P3.44 is released fro rest at Y = 10. It slides down the plane without frition. Y O Y = (1-X ) X When X = 5, perfor the following tass: (a). Deterine the veloity of the box in ters of path oordinates. (b). Draw a free-body diagra of the box and deterine the aeleration path oponents. (). Deterine the oponents of v and a in the X, Y and r, θ oordinate systes. MDOF EQUTIONS OF MOTION Proble 3.45 Derive the differential equation of otion for the three-ass syste illustrated in figure P3.45 and put your equations in atrix forat. Proble 3.46 Cart 1 of ass 1 in figure P3.46 is traveling to the right at a onstant speed V 1 before olliding with art. The oeffiient of restitution for the ollision between arts 1 and is e = Syste properties are: 1 = 10 g, V 10 = 5 /s, = 15 g, 3 = 7.5 g, 1 = 1500 N/. y () 10 5 Figure P3.43 XY=10 (a). Deterine the veloity of ass iediately following the ollision. (b). Solve for the response of the arts after the ollision. r θ Proble 3.47 The unattahed end of spring 5 in figure P3.47 is osillating with otion defined by X 5 (t)= D os(ωt), and ass has a fore f(t) = f O sin(ωt) ating on it. (a). Selet oordinates and draw the required free body diagras. (b). Derive the differential equation of otion for the four-ass syste illustrated in figure P3.47 and put your equations in atrix forat. 0 x ( ) Figure P3.44 X 1 X X Figure P3.45 f(t) V 1 X 1 X X 5 (t) Figure P3.46 Figure P3.47

10 Proble 3.48 three-ass syste is illustrated in figure P3.48. Note that the lefthand wall is osillating with otion defined by X 0 = os(ω t). (a). Selet oordinates and state the nature of the assued otion. (b). Draw the required free-body diagras. (). Derive the differential equations of otion and present equations in atrix forat X 0 (t) f f 1 3 f Figure P3.49 Proble 3.49 Derive the differential equation of otion for the three-ass syste illustrated in figure P3.49 and put your equations in atrix forat. Figure P3.48 (a). Selet oordinates and state the nature of the assued otion. (b). Draw the required free-body diagras. (). Derive the differential equations of otion and present equations in atrix forat. Proble 3.50 Note that the left-hand wall of negligible ass in figure P3.50 is osillating with otion defined by X 0 = os(ω t). (a). Selet oordinates and state the nature of the assued otion. (b). Draw the required free-body diagras. (). Derive the required differential equations and present equations in atrix forat. X 0 (t) 1 1 f 1 3 f 3 f Proble 3.51 The unattahed end of spring 5 in figure P3.51 is osillating with otion defined by X 5 (t)= D os(ω t), and ass has a fore f(t) = f O sin(ω t) ating on it. Figure P3.50 (a). Selet oordinates and state the nature of the assued otion. (b). Draw the required free body diagras. (). Derive the differential equation of otion for the four-ass syste and put your equations in atrix forat. f(t) X 5 (t) x ) 5 (t 5 Figure P3.51

11 MDOF RESPONSE PPLLICTIONS Proble 3.5 vibrating syste is defined by the following atrix differential equation 1 0 X X1 {} = X + 3 X with: 1 =1.5 g, =.5 g, 1 =5 N/, =10 N/, and 3 =5 N/. The initial onditions are: X10 = 0.05, X0 = 0.075, X 10 = X 0 = 0. (a). Calulate the eigenvalues and natural frequenies. (b). Calulate the eigenvetors. Draw a plot of your eigenvetors showing the relative otion of the asses. * T M * = I and onfir that your (). Noralize the eigenvetors suh that [ ] [ ][ ] [ ] noralized atrix of eigenvetors satisfies [ *] T [ ][ *] [ ] K = Λ where [Λ] is a diagonal atrix the eigenvalues opleting the diagonal. (d). Obtain the unoupled equations of otion. (e). Obtain the unoupled odal oordinate tie response for the syste. (f). Obtain the physial oordinate tie response for the syste. Proble 3.53 The double pendulu syste is defined by the atrix differential equation, ( 1 + ) l1 ll 1 θ1 ( w1+ w) l1 0 θ1 + = {} 0 ll 1 l θ 0 wl θ with : 1 = 0.5 g, = 0.5 g, l 1 =100, l = 50, θ0 = 5, θ10 = θ10 = θ0 = 0. (a). Calulate the eigenvalues and natural frequenies. (b). Calulate the eigenvetors. Draw a plot of your eigenvetors showing the relative otion of the asses. (). Noralize the eigenvetors suh that [ * ] T [ M][ * ] = [ I] and onfir that your * T * Λ is noralized atrix of eigenvetors satisfies [ ] [ K][ ] = [ Λ ], where [ ] diagonal atrix with the eigenvalues as its diagonal atrix. (d). Obtain the unoupled odal oordinate equations of otion. (e). Obtain and solve for the odal oordinate tie response. (f). Obtain the physial oordinate tie response for the syste. Proble 3.54 For a vibrating syste defined by the atrix differential equation: I o + R R θ R R θ + = 0 R X R X with: R = 0.5, = 1.5 g, 1 = 3 g, 1 = 5000 N/, = 4750 N/ and 1 Io = R. (a). Calulate the eigenvalues and natural frequenies. (b). Calulate the eigenvetors, and plot your eigenvetors showing the relative displaeent of the asses. (). Noralize the eigenvetors suh that [ * ] T [ M][ * ] = [ I] and onfir that your noralized atrix of eigenvetors satisfies [ * ] T [ K][ *] = [ Λ ], where [ Λ ] is diagonal atrix with the eigenvalues as its diagonal atrix. Proble 3.55 The equation of otion for a vibrating syste is given by the following atrix differential equation: X X X X = f0sinωt X 3 0 X 3 0

12 The given syste properties: 1 = g, = 3 g, 3 = g, 1 = 100 N/, = 150 N/, f 0 = 0 N, and ω = 15 Hz and the initial onditions X = 0.05, X = 0.075, X = 0.0 X = X = X = 0. are: (a). Deterine the eigenvalues and noralized eigenvetors for the syste. (b). Obtain the unoupled odal oordinate tie response for the syste. (). Obtain the physial oordinate tie response of the syste. (d). Obtain the steady state response oordinate of the bodies. Proble 3.56 The atrix equation of otion for a two ass syste is where 1 = g, =4 g, 1 =0, = 3 =000 N/. The eigenvalues for the syste are found to be ω n1=633.97(rad/se) and ω n=3667(rad/se). The noralized atrix of eigenvetors is [ * ] = The syste starts fro rest with X 1 (0)=X (0)=0. The first and seond odes have 5% of ritial daping. 1 0 x1 1+ x1 000t 0 + x = + 3 x 0 Figure P3.57 (a). Sate the odal oordinate differential equations and their tie solutions. (b). State the tie solution to the physial oordinates. Proble 3.57 For the odel stated in figure P3.57 use the following data to deterine the free otion of the two bodies: 1 =0.5 g, =1 g, 1 = =1 N/, 3 =0; x (0)=1, x1( 0) = x 1( 0) = x ( 0) = 0, f 1 (t)=f (t)=0. Proble 3.58 For the odel stated in figure P3.57 use the following data to deterine the free otion of the two bodies: 1 =0.5 g, =1 g, 1 =0, =1 N/, 3 =0;, x1( 0) = x( 0) x 1( 0) = 0; x ( 0) = 1, se f 1 (t)=f (t)=0. Proble 3.59 Develop the transient solution for the odel stated in figure P3.57 using the data fro Proble 3.57 with the initial onditions: x1( 0) = x( 0) = x 1( 0) = x ( 0) = 0, and the onstant applied fore vetor f 1 (t)=1 N; f (t)=0. Proble 3.60 Develop the transient solution for the odel stated in figure P3.57 along with the syste paraeters defined in Proble 3.57 and the initial onditions: x1( 0) = x( 0) = x 1( 0) = x ( 0) = 0, the onstant applied fore vetor f 1 (t)=1 N; f (t)=0, and the following two sets of odal daping fators: (a) ζ 1 =ζ =0; (b) ζ 1 =0.0, ζ =0.1. Plot the solutions for both the odal and physial oordinates versus tie so that 4 yles of osillations our. Disuss your results. Proble 3.61 Develop the transient solution for the odel stated in figure P3.57 with the initial onditions: x1( 0) = x( 0) = x 1( 0) = x ( 0) = 0, the onstant applied fore vetor f 1 (t)=1 N; f (t)=0, the following odal daping fators: ζ 1 =ζ =0.15, and the paraeters defined in Proble Plot the solutions for both the odal and physial oordinates versus tie so that 0 yles of osillations our. Disuss your results. Proble 3.6 vibrating syste is defined by the following atrix differential equation 1 0 X X1 f1 () t 0 + X = X f () t with: 1 = g, = g, 1 = = N/, f 1 (t)=3os(t), f (t)=5os(t). The initial onditions are: X10 = 0.05, X0 = 0.075, X 10 = X 0 = 0. (a). Calulate the eigenvalues and natural frequenies. (b). Calulate the eigenvetors. Draw a plot of your eigenvetor showing the relative otion of the asses.

13 (). Noralize the eigenvetors suh that [ * ] T [ M][ * ] [ I] noralized atrix of eigenvetors satisfies [ *] T [ ][ *] [ ] diagonal atrix the eigenvalues opleting the diagonal. (d). Obtain the unoupled equations of otion. (e). Obtain the unoupled odal tie response for the syste. (f). Obtain the physial tie response for the syste. (g). Obtain the steady state response for the syste. = and onfir that your K = Λ where [Λ] is a 1 F(t)=f 0 osωt 1 Figure P Proble 3.63 The syste shown in figure P3.63 has the following properties: w 1 = 10 lb, w = 4 lb, 1 = 5 lb/in, = 15 lb/in, 3 = 30 lb/in, 4 = 35 lb/in, f 0 = 10 lb, and ω = 30 Hz. The initial onditions for the proble are: X = 1in, X = 0.5in, X = X = (a). Selet oordinates, define the nature of the assued otion, and obtain the differential equation of otions, and present the in atrix for. (b). Deterine the eigenvalues and natural frequeny of the syste. (). Obtain the eigenvetors for the syste. (d). Obtain the odal differential equation of otion. (e). Obtain the tie response for the physial oordinates. (f). Obtain the steady-state response for the syste. Proble 3.64 The syste shown in figure P3.64 has the following properties: 1 = 100 g, = 5 g, 1 = 750 N/, = 150 N/, 3 = 150 N/, 4 = 50 N/, 5 = 50 N/, X0(t) = X 0 sin ωt, X 0 = 10, and ω = 60 Hz. The initial onditions for the proble X = 10, X = 0, X = X = 0. are: (a). Selet oordinates, define the nature of the assued otion, and obtain the differential equation of otions, and present the in atrix for. (b). Deterine the eigenvalues and natural frequenies of the syste. (). Obtain the eigenvetors for the syste. (d). Obtain the odal differential equation of otion. (e). Obtain the response of the syste in physial oordinates. (f). Obtain the steady state response of the syste. X 0 (t) Figure P3.64 F 1.75 g µ d = º 0.75 = 1.75 N/ WORK-ENERGY PPLICTIONS Proble 3.65 The ass illustrated in figure P3.65 has zero veloity at and is ated on by its weight and a horizontal fore of 40 N. (a). What is the veloity of the ass at the tie it enounters the spring? (b). What is the axiu defletion of the spring? Figure P in F Proble 3.66 The 7.5 lb slider oves along the sooth bar under the influene of the spring and the onstant external fore F of 500 lbs, figure P3.66. The spring is undefleted in position and has a stiffness oeffiient = 110 lb/in. The slider has zero veloity at point. 1 in Tas: What is its veloity at point? Proble 3.67 For the syste illustrated in figure P3.67, the ass starts at point with zero veloity. It is attahed to point with a spring of stiffness 0 N/. The spring is undefleted at point. 8 in Figure P N 0.5 Tas: Deterine the veloity of the ass when it reahes point g Figure P3.67

14 Proble 3.68 t tie t =0, the blo illustrated in Figure Proble 3.68 is supported at rest in the botto position. The blo has a ass of 10 g. n undefleted spring is onneted to the blo with a stiffness oeffiient of 1 N/. fore of 100 N is applied to the blo ausing it to rise to point. Tas: What is the blo s veloity at point? 4 α LINER CONSERVTION OF MOMENTUM PPLICTIONS Proble 3.69 Cart 1 of ass 1 in figure P3.69 starts fro rest at a distane X 10 fro art of ass and rolls down the inline and ipats art, whih is in equilibriu prior to ipat The oeffiient of restitution between the arts is e. Mass is in equilibriu at the tie of ipat. (a). Deterine the veloity of art iediately following the ollision. (b). Obtain the equation of otion for art following the initial ollision and prior to another subsequent ollisions. (). Deterine the axiu defletion of the spring. Support 3 Figure P3.68 f = 100 N Proble 3.70 The bullet, w b, is traveling at a onstant speed of v 0 when it ipats and ebeds within the art, w, figure P3.70. The ass of the buper that ipats the wall is negligible. 1 X 10 (a). Deterine the veloity of the art iediately after ipat. (b). Derive the response for the otion for the blo after the buper aes ontat with the wall. (). Deterine the required spring onstant so that the art oes to rest in a distane of one inh. Use the following syste properties: w = 5 lb; v 0 = 80 ft/se., w b =0.9 oz. Note: neglet daping. (d). Deterine the required aount of daping if the stopping tie fro part (b) is to be redued by 15%. CONSERVTION OF MOMENT OF MOMENTUM PROLEMS Proble 3.71 The ass illustrated in figure P3.71 has an initial angular veloity of ω O at radius R O. The fore is slowly inreased until a new onstant radius of R O / is ahieved. α Figure P3.69 v 0 w w b Tas: Deterine the new angular veloity and how uh wor has been done. Figure P3.70 Proble 3.7 partile oves on the inside surfae of a sooth onial shell and is given an initial veloity v 0 tangent to the horizontal ri of the surfae at, figure P 3.7. s the partile slides past point, a distane z below perfor the following tass: (a). Deterine v 0 and the iruferential oponent of veloity. (b). Deterine v= v, the agnitude of the veloity of the partile. v 0 r 0 Z r0 ω r h h Figure P 3.7 F Figure P3.71

15 Proble 3.73 sall ass partile is given an initial veloity v 0 tangent to the horizontal ri of a sooth heispherial bowl at a radius r 0 fro the vertial enterline, as shown at point in figure P3.73. s the partile slides past point, a distane h below and a distane r fro the vertial enterline, its veloity v aes an angle θ with the horizontal tangent to the bowl through. Tas: Deterine θ. Proble 3.74 s shown in figure P3.74, a partile initially rotates at 30 rad/se along a fritionless surfae at a distane of ft fro the enter. flexible ord restrains the partile. The ord is given a onstant downward veloity of 5 ft/se. v 0 r 0 O h r v θ O r Perfor the following tass when the partile is 1 ft fro the enter: (a). Calulate the veloity vetor. (b). Calulate the wor done on the partile. Proble 3.75 s illustrated in figure P3.75, a sall ball weighing lbs ball is onneted to bearings on the shaft by light inextensible ords having a length l of ft and is rotating about a vertial axis at a speed ω 1 of 15 rad/se. The angle θ 1 is 30. earing is oved up 6 in. (a). Deterine the angular speed ω. (b). Deterine the wor done on the syste. θ v Side view Figure P3.73 Z 1 ft 30 d/ ft θ 1 l 5 ft/se Figure P3.74 ω 1 θ 1 l X Figure P3.75 Y