# Untitled 1

1.1)compute d magnitude of theres2ring 4ce if d mass of d pendulum is 2 kg & d length of d pendulum is 0.5 m.

1.2)compute d period of oscill8on of a pendulum of length 1 m @ d north pole where d acceler8on due 2 gravity is measured 2 b 9.832 m>s2

1.3)loaded with mass of 15 kg & d corresponding (st8c) displacement is 0.01 m.calcul8 d spring’s stiffness.

1.4)lot d data & calcul8 d spring’s stiffness.note dat d data contain some errors.also calcul8 d standard devi8on.

1.5)compute d amplitude of d res2ring 4ce if d mass of d pendulum is 2 kg & d length of d pendulum is 0.5 m

1.6)compute d angular natural frequency mass of d pendulumis 2 kg & d length is 0.5 m.?is d period of oscill8on in seconds

1.7)derive d solution of mx$ + kx = 0 & plot ω n = 2 rad>s,x0 = 1 mm,& v0 = 15 mm>s

1.8)solve mx$ + kx = 0 4 k = 4 n>m,m = 1 kg,x0 = 1 mm,& v0 = 0.plot d solution.

1.9)1 mm.d phaseshift frm t = 0 is measured 2 b 2 rad & d frequency is found 2 b 5 rad>s.x(t) = a sin (ωnt + ϕ)

1.10)determine d stiffness single-degree-of-freedom spring–mass system with a massof 100 kg such dat d natural frequency is 10 hz

1.11)determine ?effect gravity hs on d equ8on of motion & d system’s natural frequency

1.12)10 hz & amplitude 1 mm.calcul8 d maximum amplitude of d system’s velocity & acceler8on

1.13)show by calcul8on dat a sin(ωnt + ϕ) can b represented as a1 sin ωnt + a2 cos ωnt

1.14)4m x(t) = a1 sin ωnt + a2 cos ωnt,calcul8 d values of a1 & a2 in terms of d initial conditions x0 & v0

1.15)verify dat equ8on (1.10) s8sfies d initial velocity

1.16)a 0.5 kg mass is attached 2 a linear spring of stiffness 0.1 n>m mass of 50 kg & a stiffness of 10 n>m

1.17)by writing new2n’s law,ma = -kx,in differential 4m using a dx = v dv & integr8ng twice.

1.18)determine d natural frequency of d 2 systems illustr8d in figure p1.18

1.19)case k = 1000 n>m & m = 10 kg 4 2 complete periods x0 = 0 m,v0 = 1 m>s,b) x0 = 0.01 m,v0 = 0 m>s,& c) x0 = 0.01 m,v0 = 1 m>s

1.20)make a 3 dimensional surface plot of d amplitude a of an undamped oscilla2r given by equ8on (1.9) versus x0 & v0

1.22)a pendulum hs length of 250 mm.?is d system’s natural frequency in hertz?

1.23)is required 2 oscill8 1ce evry 2nd.?length should it b?

1.24)θ = θ,is reasonable 4 θ less than 10°.if a pendulum of length 0.5 m,hs an initial position of θ(0) = 0,?is d maximum value

1.25)its acceler8on is measured 2 have an amplitude of 10,000 mm>s2 with a frequency of 8 hz.compute d maximum displacement

1.26)derive d rel8onships given in window 1.4 4 d constants a1 & a2,used in d exp1ntial 4m of d solution,

1.27)d acceler8on of a machine part modeled as a spring–mass system compute d amplitude of d displacement of d mass

1.28)acceler8on amplitude of 8 mm>s2 & measured displacement amplitude of 2 mm.calcul8 d system’s natural frequenc

1.29)a spring–mass system hs a measured period of 5 s & a known mass of 20 kg.calcul8 d spring stiffness

1.30)plot d solution of a linear spring–mass system with frequency ωn = 2 rad>s,x0 = 1 mm,& v0 = 2.34 mm>s 4 @ least 2 periods

1.31)mass of 1 kg,stiffness of 4 n>m,& initial conditions of x0 = 1 mm & v0 = 0 mm>s 4 @ least 2 periods.

1.32)4 kg & d stiffness is 100 n>m.how much must d spring stiffness b changed in order 2 increase d natural frequency by 10%

1.33)length of 20 m,& d acceler8on due 2 gravity @ dat loc8on is known 2 b 9.803 m>s2

1.34)calcul8 d rms values of displacement,velocity,& acceler8on

1.35)compute d spring stiffness needed 2 keep d pendulum @ 1° m = 0.5 kg,g = 9.8 m>s2,l1 = 0.2 m,& l2 = 0.3 m

1.36)au2mobile is modeled as a 1000-kg mass supported by a spring of stiffness k =400,000 n>m.wen it oscill8s it does so with a maximum deflection of 10 cm

1.37)compute d frequency of vibr8on given dat d 2rsional stiffness is 2000 n m>rad & d wheel assembly hs a mass of 38 kg.take d distance x = 0.26 m

1.38)its acceler8on is measured 2 have an amplitude of 10,000 mm>s2 @ 8 hz.?is d machine’s maximum displacement?

1.39)initial velocity of 100 mm>s.it oscill8s with a maximum amplitude of 10 mm.?is its natural frequency

1.40)maximum amplitude 5 cm & a measured maximum acceler8on of 2000 cm>s2 ,calcul8 d natural frequency of d au2mobile.

1.42)x + 2x# + 2x = 0.compute d damping r8o & determine if d system is overdamped,underdamped,or critically damped.

1.43)onsider d system x$ + 4x# + x = 0 4 x0 = 1 mm,v0 = 0 mm>s.is dis system overdamped,underdamped,or critically damped?

1.44)compute d solution 2 x$ + 2x# + 2x = 0 4 x0 = 0 mm,v0 = 1 mm>s & write down d closed-4m expression 4 d response

1.45)derive d 4m of λ1 & λ2 given by equ8on (1.31) frm equ8on (1.28) & d definition of d damping r8o

1.46)use d euler 4mulas 2 derive equ8on (1.36) frm equ8on (1.35)

1.47)using equ8on (1.35) as d 4m of d solution of d underdamped system,calcul8 d values 4 d constants a1 & a2

1.48)calcul8 d constants a & ϕ in terms of d initial conditions & thus verify equ8on (1.38)

1.49)calcul8 d constants a1 & a2 in terms of d initial conditions & thus verify equ8ons (1.42) & (1.43)

1.50)calcul8 d constants a1 & a2 in terms of d initial conditions & thus verify equ8on (1.46)

1.51)using d definition of d damping r8o & d undamped natural frequency,derive equ8on (1.48) frm (1.47)

1.52)m = 1 kg,c = 2 kg>s,k = 10 n>m.calcul8 d value of ζ & ωn is d system overdamped,underdamped,or critically damped?

1.53)ωn = 2 rad>s & initial conditionsx0 = 1 mm,v0 = 1 mm,4 d following values of d damping r8o: ζ = 0.01,ζ = 0.2,ζ = 0.1,ζ = 0.4,& ζ = 0.8.

1.54)plot d response x(t) ωn = 2 rad>s,ζ = 0.1,& v0 = 0 4 d following initial displacements: x0 = 10 mm & x0 = 100 mm

1.55)calcul8 d solution 2 x$ + x# + x = 0 with x0 = 1 & v0 = 0 4 x(t)

1.56)100 kg,stiffness of 3000 n>m,& dampingcoefficient of 300 kg>s.calcul8 d undamped natural frequency,

1.57)ζ = 0.01.determine d equ8ons of motion & calcul8 an expression 4 d natural frequency & d damped natural frequency.

1.58)f 150 kg,stiffness of 1500 n>m,& damping coefficient of 200 kg>s.calcul8 d undamped natural frequency,does d solution oscill8?

1.59)100 kg mass,stiffness of 3000 n>m,& damping coefficient of 300 ns>m initial displacement of 0.1 m

1.60)system of 150 kg mass,stiffness of 1500 n>m,& damping coefficient of 200 ns>m is given an initial velocity of 10 mm> calcul8 d 4m of d response

1.61)choose d damping coefficient of a spring–mass–damper system with mass of 150 kgand stiffness of 2000 n>m such dat its response will die out after about 2 s

1.63)derive d equ8on of motion of d system in figure p1.63 & discuss d effect of gravity on d natural frequency & d damping r8o.

1.64)x(t) = e-ζωnta sin (ωdt + ϕ) where a & ϕ r given in terms of d initial conditions x0 = 0,& v0 ≠ 0

1.65)calcul8 d frequency of d compound pendulum of figure p1.65 if a mass mt is added 2 d tip,by using d energy method.

1.66)calcul8 d 2tal energy in a damped system with frequency 2 rad>s & damping r8o ζ = 0.01 with mass 10 kg 4 d case x0 = 0.1 m & v0 = 0

1.67)use d energy method 2 calcul8 d equ8on of motion & natural frequency of an airplane’s steering mechanism 4 d nose wheel of its landing gear.

1.68)derive d equ8on of motion using d energy method.then linearize d system

1.69)se d energy method 2 determine d equ8on of motion in θ & calcul8 d natural frequency of d system.

1.70)calcul8 d natural frequency of vibr8on of d smaller pipe (of radius r1) rolling back & 4th inside d larger pipe

1.71)ζ = 0.001.determine a damping coefficient & add a viscous-damping term 2 d pendulum equ8on

1.72)determine c 4 d rod if it is observed 2 have a damping r8o of ζ = 0.01

1.73)calcul8 d damped natural frequency if j = 1000 m2 # kg,c = 20 n # m # s>rad,& k = 400 n # m>rad.

1.74)model of an aircraft landing system.assume,x = rθ.?is d damped natural frequency?

1.75)k = 400,000 n>m,m = 1500 kg,j = 100 m2 # kg>rad,r =25 cm,& c = 8000 kg>s.calcul8 d damping r8o & d damped natural frequency

1.76)use lagrange’s 4mul8on model each of d brackets as a spring of stiffness k,& assume d inertia of d pulleys is negligible

1.77)use lagrange’s 4mul8on 2 calcul8 d equ8on of motion (t) by d “radius” r0 (i.e.,x = r0θ).jet

1.78)consider d inverted simple pendulum connected 2 a spring of figure p1.68.use lagrange’s 4mul8on 2 derive d equ8on of motion

1.79)use dis extended lagrange 4mul8on 2 derive d equ8on of motion of d damped au2mobile suspension driven by a dynamometer

1.80)use d energy method 2 calcul8 d system’s natural frequency of oscill8on 4 small angles θ(t).

1.81)derive d equ8on of motion using d lagrange 4mul8on,linearize d equ8on,& compute d system’s natural frequency

1.83)10 m/s,?is d maximum displacement of d mass @ d tip if d mass is 1000 kg & d bar is made of steel of length 0.25 m with a cross sectional area of 0.01 m2?

1.84)d mass is 1000 kg & d bar 0.25-m long,with a square cross section of 0.1 m on a side.d mass polar mo of inertia of d tip mass is 10 kg>m2 .compute both d 2rsion & longitudinal frequencies.

1.85)here l = 0.4 m,e = 20 * 1010 n>m2,& m = 100 kg.calcul8 d cross-sectional area dat should b used if d natural frequency is 2 b fn = 500 hz

1.86)let m b d mass of d diver (m = 100 kg) & l = 1.5 m.if d diver wishes 2 oscill8 @ 3 hz,?value of ei should d diving board m8rial have?

1.87)let k1 = k5 = k2 = 100 n>m,k3 = 50 n>m,& k4 = 1 n>m.?is d equivalent stiffness?

1.88)stiffness values of 10,100,& 1000 n>m.design a spring system using these values only,so dat a 100-kg mass is connected 2 ground with frequency of about 1.5 rad>s.

1.89)f k1 = k2 = 0.choose m & nonzero values of k3,k4,& k5 so dat d natural frequency is 100 hz

1.90)determine frm d plot (or by algebra) d percentage where d natural frequency changes by 1% & there4 d spring’s mass should nt b neglected.

1.91)calcul8 d natural frequency & damping r8o m = 10 kg,c = 100 kg>s,k1 = 4000 n>m,k2 = 200 n>m,& k3 = 1000 n>m.

1.92)calcul8 d natural frequency & damping r8o assume dat no friction acts on d rollers.

1.93)e = 2 * 1011 n>m2) & sizes d spring so dat d device hs a specific frequency.l8r,2 save w8,d spring is made of aluminum (e = 7.1 * 1010 n>m2

1.94)x – y plotter & reproduced in figure p1.94.d y coordin8 is d displacement in cm & d x coordin8 is time in seconds.

1.95)δ =1 nlnx0xnwhere xn is d amplitude of vibr8on after n cycles have elapsed.

1.96)derive d equ8on (1.78) 4 d trifalar suspension system

1.97)cantilevered beam of length 1 m & i = 10-9 m4 with a 6-kg mass attached @ its end.d system is given an initial displacement & found 2 oscill8 with a period of 0.5 s.calcul8 d modulus e

1.98)1000-kg car with stiffness of k = 400,000 n>m determine d damping coefficient if d displacement @ t1 is measured 2 b 2 cm & 0.22 cm @ t2.

1.99)a pendulum decays frm 10 cm 2 1 cm over 1 period.determine its damping r8o

1.100)d rel8onship btwn d log decrement δ & d damping r8o ζ is often approxim8d as δ = 2πζ.4 ?values of ζ would u consider dis a gud approxim8on

1.101)5000 n>m.it is observed dat during free vibr8on d amplitude decays 2 0.25 of its initial value after 5 cycles.calcul8 d viscous-damping coefficient,c.

1.102)tiffness 103 n>m attached 2 a 10-kg mass.place a dashpot parallel 2 d spring & choose its viscous

1.103)4 an underdamped system,x0 = 0 mm & v0 = 10 mm>s.determine m,c,& k such dat d amplitude is less than 1 mm

1.104)repeat problem 1.103 if d mass is restricted 2 b btwn 10 kg 6 m 6 15 k

1.105)shaft frm table 1.1 2 design a 1-m shaft with 2rsional stiffness of 105 n # m>rad.

1.106)consider designing a helical spring made of aluminum,such dat wen it is attached 2 a 10-kg mass d resulting spring–mass system hs a natural frequency of 10 rad/s.

1.107)k = 103 n>m).dis amounts 2 computing d length of d bar with its cross sectional area taking up about d same space @ d helical spring (r = 10 cm)

1.108)repeat problem 1.107 using plastic (e = 1.40 * 109 n>m2) & rubber (e = 7 *106 n>m2)

1.109)compute a design 4mula 4 d dimensions of d board (b,h,& l) in terms of d st8c deflection,d average diver’s mass,m

1.110)esirable damping r8o is ζ = 0.25.if d model hs a mass of 750 kg & a frequency of 15 hz,

1.111)write d equ8ons of motion in terms of d angle,w,d bar makes with d vertical

1.112)ssume dat a dashpot (of damping r8 c) also acts on d pendulum parallel 2 d 2 springs.how does dis affect d stability properties of d pendulum?

1.113)calcul8 d equ8ons of vibr8on & discuss values of d parameter rel8ons 4 which d system is stable.

1.114)jθ$+ (c – fd)θ#+ kθ = 0 nd fdθ#is d neg8ve damping provided by theaerodynamic 4ces

1.115)fixed amplitude,frequency,& phase as ζ changes thru d following set of values ζ = 0.01,0.05,0.1,0.2,0.3,& 0.4.x(t) = e-10ζt sin (1021 – ζ2t)

2.1) compute d magnitude of d 4ced response 4 d 2 cases ω = 2.1 rad>s &

ω = 2.5 rad>sec

2.2) compute d 2tal response of d system if d driving frequency is 2.5 rad>s & d

initial position & velocity r both zero

2.3)compute d response of a spring–mass system modeled by equ8on (2.2) 2 a 4ce of

magnitude 23 n,driving frequency of twice d natural frequency

2.5)calcul8 d natural frequency & d driving frequency of d system.

2.6)an airplane wing modeled as a spring–mass system with natural frequency 40 hz is driven harmonically by d rot8on of its engines @ 39.9 hz

2.7)compute d 2tal response of a spring–mass system with d following values: k 1000 n>m,m = 10 kg

2.8)compute d 2tal response of a spring–mass system with d following values: k 1000 n>m,m = 10 kg

2.9),write d equ8on of motion,& calcul8 d

response assuming dat d system is initially @ rest 4 d values k1 = 100 n>m,

2.10) write d equ8on of motion,& calcul8 d response assuming dat d system is initially @ rest 4 d values θ = 30°,k = 1000 n>m

2.11)compute d initial conditions such dat d response of mx+ kx = f0 cos ωt

2.12) ?is d effective stiffness of dis person in d longitudinal direction?(b) if d person,1.8 m

2.13)if d person in problem 2.12 is standing on a floor vibr8ng @ 4.49 hz with an amplitude

of 1 n (very small),

2.14)compute d maximum deflection of d hand end of d arm if d jackhammer applies a 4ce of 10 n @ 2 hz

2.15)find a design rel8onship 4 d spring stiffness,k,in terms of d rot8onal inertia,j;d magnitude of d applied mo

2.16)determine a single-degree-of-freedom model 4 d spar & compute its natural frequency

2.17)compute d response of a shaft-&-disk system 2 an applied mo of

2.18)consider a spring–mass system with zero initial conditions described by (t) + 4x(t) = 12 cos 2t,x(0) = 0,x# (0) = 0

2.19)consider a spring–mass system with zero initial conditions described by

2.20)calcul8 d constants a & ϕ 4 arbitrary initial conditions,x0 & v0,in d case of d 4ced response given by

2.21)consider d spring–mass–damper system defined by (use basic si units) 4x$(t) + 24x# (t) + 100x(t) = 16 cos 5t 1st,determine if d system is underdamped

2.22)show dat d following 2 expressions r equivalent:xp(t) = x cos (ωt – θ) & xp (t) = as cos ωt + bs sin ωt

2.23)calcul8 d 2tal solution of$x+ 2ζωnx# + ωn 2x = f0 cos ωt

2.24)a 100-kg mass is suspended by a spring of stiffness 30 * 103 n>m with a viscousdamping constant of 1000 ns>m

2.25)plot d 2tal solution of d system of problem 2.24 including d transient

2.26)a damped spring–mass system modeled by (units r new2ns)100x$(t) + 10x# (t) + 1700x(t) = 1000 cos 4t

2.27)calcul8 both d damped & undamped natural frequency of d system 4 small angles.

2.28)consider d pivoted mechanism of figure p2.27 with k = 4 * 103 n>m,l1 = 0.05 m,l2 = 0.07 m,l = 0.10 m

2.29)compute d response of a shaft-&-disk system 2 an applied mo of m = 10 sin 312t

2.30)compute d 4ced response of a spring–mass–damper system with d following values: c = 200 kg>s,k = 2000 n>m,m = 100 kg,subject 2 a harmonic

2.31)compute a value of d damping coefficient,c,such dat d steady-st8 response amplitude of d system

2.32) consider a spring–mass–damper system like d 1 in figure p2.31 with d following values: m = 100 kg,c = 100 kg>s,k = 3000 n>m,f0 = 25 n,& d driving

frequency ω = 5.47 rad>s.

2.33) compute d response of d system in figure p2.33 if d system is initially @ rest 4the values k1 = 100 n>m,k2 = 500 n>m

2.34)write d equ8on of motion 4 d system given in figure p2.34 4 d case dat

f(t) = f cos ω t & d surface is friction free

k = 2000 n>m,c = 25 kg>s,m = 25 kg,& f(t) = 50 cos 2πt n

2.36)distance r = 0.5 m.

compute d magnitude of d steady-st8 response if d measured damping r8o

of d spring system is ζ = 0.01

2.37)design a damper (dat is,choose a value of c) such dat d

maximum deflection @ steady st8 is 0.05 m

2.38)derive d 2tal response of d system 2 initial conditions x0 & v0 using d homogenous solution in d 4m

2.45)c = 50 kg>s,k = 1000 n>m,y = 0.03 m,& ωb = 3 rad>s,compute d magnitude of d particular solution.last,compute d transmissibility r8

2.46)4 a base motion system described by mx+ cx# + kx = cyωb cos ωbt + ky sin ωbtwith m = 100 kg,c = 50 n>m,y = 0.03 m,& ωb = 3 rad>s,find largest value

2.47)a machine weighing 2000 n rests on a support as illustr8d in figure p2.47.d support deflects about 5 cm as a result of d w8 of d machine

2.48)derive equ8on (2.70)x = y c 11 -1r+22(22+ζr)(22ζr)2 d 1>2

2.49)frm d equ8on describing figure 2.14,show dat d point (12,1) corresponds 2 d value tr 7 1 (i.e.,4 ol r 6 12,tr 7 1)

2.50)pure damping element.derive an expression 4 d 4ce transmitted 2 d support in steady st8.

2.51)damping coefficient of c = 231 kg>s,& a mass of 1007 kg.determine d amplitude of d absolute displacement of d au2mobile mass.

2.52)maximum amplitude of only2.5 mm (@ resonance).calcul8 d damping constant & d amplitude of d 4ceon d base.

2.53)@ whatspeed does car 2 experience resonance?calcul8 d maximum deflection of both cars @ resonance.

2.54)comparing d magnitude of ζ = 0.01,ζ = 0.1,& ζ = 0.2 4 d case r = 2.?happens if d road “frequency” changes?

2.55)calcul8 d damping coefficient,given dat d system hs a deflection (x) of 0.7 cm wen driven @ its natural frequency while d base amplitude (y)

is measured 2 b 0.3 cm

2.56)calcul8 d effect of d mass of d passengers onthe deflection @ 20,80,100,& 150 km>h.?is d effect of d added passengermass on car 2

2.57)choose values of c & k 4 d suspension system 4 car 2 (thesedan) such dat d amplitude transmitted 2 d passenger compartment is as small aspossible

2.58)compute d damping r8o needed2 keep d displacement magnitude transmissibility less than 0.55 4 a frequency r8o of r = 1.8

2.59)compare d maximum deflection 4 a wheel motion of magnitude 0.50 m & frequency of 35 rad>s 4 these 2 different masses

2.60)approxim8 d building mass by 105 kg & thestiffness of each wall by 3.519 * 106 n>m.compute d magnitude of d deflectionof d 2p of d building

2.73)calcul8 d approxim8 amplitude of steady-st8 motion assuming dat both d mass & d surface dat it slides on r made of lubric8d steel

2.74)nd coefficient of friction of 0.1 is driven harmonically @ 10 hz.d amplitude @ steady st8 is 5 cm.calcul8 d magnitude of d driving 4ce

2.75)if d mass is driven harmonically by a 90-n 4ce @ 25 hz,determine d equivalent viscous-damping coefficient if d coefficient of friction is 0.1

2.76)plot d free response of d system of problem 2.75 2 initial conditions of x(0) = 0

2.77)calcul8 how large d magnitude of d driving 4ce must b 2 sustain motion if d steel is lubric8d.how large must dis

2.78)calcul8 d phase shift btwn d driving 4ce & d response 4 d system of

2.79)calcul8 d energy loss & determine d magnitude & phase rel8onships 4 d 4ced response

2.80)4 5 different magnitudes.d measured quantities r is d damping viscous or coulomb?

2.81)calcul8 d equivalent loss fac2r 4 a system with coulomb damping

2.83)calcul8 d displacement 4 a system with air damping using d equivalent viscousdamping method

2.84)calcul8 d semimajor & semiminor axis of d ellipse of equ8on (2.119).then calcul8 d area of d ellipse

2.85)calcul8 d hysteretic damping coefficient.?is d equivalent viscous damping if d system is driven @ 10 hz

2.86)calcul8 d equivalent viscous-damping coefficient 4 a 20-hz driving 4ce.plot ceq,versus ω 4

2.87)calcul8 d nonconserv8ve energy of a system subject 2 both viscous & hysteretic damping.

2.88)derive a 4mula 4 equivalent viscous damping 4 d damping 4ce of d 4m,fd = c(x# )n,where n is an integer

2.89)determine an expression 4 thesteady-st8 amplitude under harmonic excit8on 4 a system with both coulomb & viscous damping present.

2.3.3)As an example of using Laplace transforms to solve a homogeneous differential equation, consider the undamped single-degree-of-freedom system described by

3.1)Calculate the solution to 1000x$(t) + 200x# (t) + 2000x(t) = 100δ(t), x0 = 0, v0 = 0

3.2)Consider a spring–mass–damper system with m = 1 kg, c = 2 kg>s, and k = 2000 N>mwith an impulse force applied to it of 10,000 N for 0.01 s. Compute the resulting response.

3.3)Calculate the solution to$x+ 2x# + 2x = δ(t – π)x(0) = 1 x# (0) = 0and plot the response.

3.4)Calculate the solution to$x+ 2x# + 3x = sin t + δ(t – π)x(0) = 0 x# (0) = 1 and plot the response.

3.5)Calculate the response of a critically damped system to a unit impulse.

3.6)Calculate the response of an overdamped system to a unit impulse

3.7)Derive equation (3.6) from equations (1.36) and (1.38)

3.8)Calculate the response and plot your results for the case of an aluminum wing 2-m long with m = 1000 kg, ζ = 0.01, and I = 0.5 m4. Model F as 1000 N lasting for 10–2s.

3.9)N>m. The cam strikes the valve once every 1 s. Calculate the vibration response, x(t), of the valve once it has been impacted by the cam. The valve

3.10)Calculate the vibration of the mass m after the system falls and hits the ground. Assume that the system is underdamped.

3.11)Calculate the response of 3x$(t) + 12x# (t) + 12x(t) = 3δ(t) for zero initial conditions. The units are in Newtons. Plot the response

3.12)Compute the response of the system 3x$(t) + 12x# (t) + 12x(t) = 3δ(t) subject to the initial conditions x(0) = 0.01 m and v(0) = 0.

3.13)Calculate the response of the system 3x$(t) + 6x# (t) + 12x(t) = 3δ(t) – δ(t – 1) subject to the initial conditions x(0) = 0.01 m and v(0) = 1 m>s

3.14)Compute and plot the response of the wheel system to an impulse of 5000 N over 0.01 s. Compare the undamped maximum amplitude to that of the maximum amplitude of the damped system (use r = 0.457 m).

3.15)vibrating past the 0.01 m limit. If the damping in the aluminum is modeled as ζ = 0.05, approximately how much time will pass before the camera vibration reduces to the required limit?

3.16). Findan expression for the value of the transverse mount stiffness, k, as a function of the relative speed of the bird, v, the bird mass,

3.17)Design a damper (i.e., choose a value of the damping constant, c, such that the part does not deflect more that 0.01 m)

3.18)Calculate the analytical response of an overdamped single-degree-of-freedom system to an arbitrary nonperiodic excitation

3.19)Calculate the response of an underdamped system to the excitation given in Figure P3.19 where the pulse ends at π s

3.20)k = 4 * 105 N>m, m = 1007 kg), find an expression for the maximum relative deflection of the car’s mass versus the velocity of the car.

3.21)Calculate and plot the response of an undamped system to a step function with a finite

rise time of t1 for the case m = 1 kg, k = 1 N>m, t1 = 4 s, and F0 = 20 N

3.22)of the wake from a passing boat impacts a seawall Calculate the response of the seawall–dike system to such a load

3.23)Plot the response for three periods for the case m = 1 kg,k = 100 N>m, and F0 = 50 N.

3.24)The machine has a mass of 5000 kg, and thesupport has stiffness 1.5 * 103 N>m. Calculate the resulting vibration of the machine

3.25)Calculate the analytical value of tp by noting that it occurs at the first peak, or critical point, of the curve.

3.26) Calculate the value of the overshoot (O.S.), for the system of Example 3.2.1. Note from tp = π/ωd3.27)ettling time of 3 s and a time to peak of 1 s. Calculate the appropriate natural frequency and damping ratio to use in the design.

3.28)F0 = 30 N the natural period of the system (i.e., t1 = π ωn). Recall that k = 1000 N>m, ζ = 0.1,and ωn = 3.16 rad>s

3.29)mx$(t) + kx(t) = F0 sin ωt, x0 = 0.01 m and v0 = 0 Compute the response of this system for the values of m = 100 kg, k = 2500 N>m, ω = 10 rad>s, and F0 = 10 N

3.30)Derive equations (3.24), (3.25), and (3.26) and hence

3.31)Calculate bn show that bn = 0, n = 1,2,…,∞. Also verify the expression an by completing the integration indicated.

3.32)Determine the Fourier series for the rectangular wave illustrated in Figure P3.32

3.33)Determine the Fourier series representation of the sawtooth curve illustrated in Figure P3.33

3.34)y(t) = 3e-t>2Φ(t) m>s where Φ(t) is the unit step function and m = 10 kg, ζ = 0.01, and k = 1000 N>m

3.35)Calculate and plot the total response of the spring–mass–damper system withm = 100 kg, ζ = 0.1, and k = 1000 N>m let T = 2π s

3.36)ωb = 3.162 rad>s with amplitude Y = 0.05 m subject to initial conditions x0 = 0.01 m and v0 = 3.0 m>s. The system is m = 1 kg, c = 10 kg>s,and k = 1000 N>m

3.37)Print the function and its Fourier series approximation for 5, 20, then 100 terms.

3.38)Vibration Toolbox. Print the function and its Fourier series

approximation for 5, 20, and 100 terms

3.39)Calculate the response of mx$+ cx# + kx = F0Φ(t)where ϕ(t) is the unit step function for the case with x0 = v0 = 0

3.40)mx$(t) + cx# (t) + kx(t) = δ(t), x0 = 0, v0 = 0for the overdamped case (ζ 7 1). Plot the response for m = 1 kg, k = 100 N>m, and ζ = 1.5.

3.41)Calculate the response of the underdamped system given by mx$+ cx# + kx = F0e-at using the Laplace transform method. Assume a 7 0

3.42)Solve the following system for the response x(t) using Laplace transforms: 100x$(t) + 2000x(t) = 50δ(t)

3.43) Use the Laplace transform approach to solve for the response x(t) + x(t) = sin 2t, x0 = 0, v0 = 1

3.44) Calculate the mean-square response of a system to an input force of constant PSD, S0, and frequency response function H(ω) = 10>(3 + 2jω).

3.45)m = 1007 kg, c = 2000 kg>s, k = 40,000 N>m Calculate the PSD of the response and the mean-square value of the response

3.46)compute autocorrelation Rxx(τ) for the deterministic signal A sin ωnt.

3.47)Rxx(τ) = 10 + 43 + 2τ + 4τ2 Compute the mean-square value of the signal.

3.48)Verify that the average x – x ∙ is zero by using the definition given in equation (3.47) to compute the average.