# Analysis of Spring-Mass Systems and Damping Effects

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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

2.35)a foot pedal 4 a musical instrument is modeled by d sketch in figure p2.35:

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

2.82)wen excited harmonically @ 5 hz,d steady-st8 displacement of d mass is 5 cm.Calcul8 d amplitude

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 trans4ms 2 solve a homogeneous differential equ8on,consider d undamped single-degree-of-freedom system described by

3.1)calcul8 d solution 2 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,& k = 2000 n>mwith an impulse 4ce applied 2 it of 10,000 n 4 0.01 s.Compute d resulting response.

3.3)calcul8 d solution 2$x+ 2x# + 2x = δ(t - π)x(0) = 1 x# (0) = 0and plot d response.

3.4)calcul8 d solution 2$x+ 2x# + 3x = sin t + δ(t - π)x(0) = 0 x# (0) = 1 & plot d response.

3.5)calcul8 d response of a critically damped system 2 a unit impulse.

3.6)calcul8 d response of an overdamped system 2 a unit impulse

3.7)derive equ8on (3.6) frm equ8ons (1.36) & (1.38)

3.8)calcul8 d response & plot yur results 4 d case of an aluminum wing 2-m long with m = 1000 kg,ζ = 0.01,& i = 0.5 m4.Model f as 1000 n lasting 4 10–2s.

3.9)n>m.D cam strikes d valve 1ce evry 1 s.Calcul8 d vibr8on response,x(t),of d valve 1ce it hs been impacted by d cam.D valve

3.10)calcul8 d vibr8on of d mass m after d system falls & hits d ground.Assume dat d system is underdamped.

3.11)calcul8 d response of 3x$(t) + 12x# (t) + 12x(t) = 3δ(t) 4 zero initial conditions.D units R in new2ns.Plot d response

3.12)compute d response of d system 3x$(t) + 12x# (t) + 12x(t) = 3δ(t) subject 2 d initial conditions x(0) = 0.01 m & v(0) = 0.

3.13)calcul8 d response of d system 3x$(t) + 6x# (t) + 12x(t) = 3δ(t) - δ(t - 1) subject 2 d initial conditions x(0) = 0.01 m & v(0) = 1 m>s

3.14)compute & plot d response of d wheel system 2 an impulse of 5000 n over 0.01 s.Compare d undamped maximum amplitude 2 dat of d maximum amplitude of d damped system (use r = 0.457 m).

3.15)vibr8ng past d 0.01 m limit.If d damping in d aluminum is modeled as ζ = 0.05,approxim8ly how much time will pass be4 d camera vibr8on reduces 2 d required limit?

3.16).Findan expression 4 d value of d transverse mount stiffness,k,as a function of d rel8ve speed of d bird,v,d bird mass,

3.17)design a damper (i.E.,choose a value of d damping constant,c,such dat d part does nt deflect more dat 0.01 m)

3.18)calcul8 d analytical response of an overdamped single-degree-of-freedom system 2 an arbitrary nonperiodic excit8on

3.19)calcul8 d response of an underdamped system 2 d excit8on given in figure p3.19 where d pulse ends @ π s

3.20)k = 4 * 105 n>m,m = 1007 kg),find an expression 4 d maximum rel8ve deflection of d car’s mass versus d velocity of d car.

3.26) calcul8 d value of d overshoot (o.S.),4 d system of example 3.2.1.Note frm tp = π/ωd

3.27)ettling time of 3 s & a time 2 peak of 1 s.Calcul8 d appropri8 natural frequency & damping r8o 2 use in d design.

3.28)f0 = 30 n d natural period of d system (i.E.,t1 = π ωn).Recall dat k = 1000 n>m,ζ = 0.1,& ωn = 3.16 rad>s

3.29)mx$(t) + kx(t) = f0 sin ωt,x0 = 0.01 m & v0 = 0 compute d response of dis system 4 d values of m = 100 kg,k = 2500 n>m,ω = 10 rad>s,& f0 = 10 n

3.30)derive equ8ons (3.24),(3.25),& (3.26) & hence

3.31)calcul8 bn show dat bn = 0,n = 1,2,…,∞.Also verify d expression an by completing d integr8on indic8d.

3.32)determine d 4ier series 4 d rectangular wave illustr8d in figure p3.32

3.33)determine d 4ier series represent8on of d saw2oth curve illustr8d in figure p3.33

3.34)y(t) = 3e-t>2φ(t) m>s where φ(t) is d unit step function & m = 10 kg,ζ = 0.01,& k = 1000 n>m

3.35)calcul8 & plot d 2tal response of d spring–mass–damper system withm = 100 kg,ζ = 0.1,& k = 1000 n>m let t = 2π s

3.36)ωb = 3.162 rad>s with amplitude y = 0.05 m subject 2 initial conditions x0 = 0.01 m & v0 = 3.0 m>s.D system is m = 1 kg,c = 10 kg>s,& k = 1000 n>m

3.37)print d function & its 4ier series approxim8on 4 5,20,then 100 terms.

3.38)vibr8on 2olbox.Print d function & its 4ier series

approxim8on 4 5,20,& 100 terms

3.39)calcul8 d response of mx$+ cx# + kx = f0φ(t)where ϕ(t) is d unit step function 4 d case with x0 = v0 = 0

3.40)mx$(t) + cx# (t) + kx(t) = δ(t),x0 = 0,v0 = 04 d overdamped case (ζ 7 1).Plot d response 4 m = 1 kg,k = 100 n>m,& ζ = 1.5.

3.28)f0 = 30 n d natural period of d system (i.E.,t1 = π ωn).Recall dat k = 1000 n>m,ζ = 0.1,& ωn = 3.16 rad>s

3.29)mx$(t) + kx(t) = f0 sin ωt,x0 = 0.01 m & v0 = 0 compute d response of dis system 4 d values of m = 100 kg,k = 2500 n>m,ω = 10 rad>s,& f0 = 10 n

3.30)derive equ8ons (3.24),(3.25),& (3.26) & hence

3.31)calcul8 bn show dat bn = 0,n = 1,2,…,∞.Also verify d expression an by completing d integr8on indic8d.

3.32)determine d 4ier series 4 d rectangular wave illustr8d in figure p3.32

3.33)determine d 4ier series represent8on of d saw2oth curve illustr8d in figure p3.33

3.34)y(t) = 3e-t>2φ(t) m>s where φ(t) is d unit step function & m = 10 kg,ζ = 0.01,& k = 1000 n>m

3.35)calcul8 & plot d 2tal response of d spring–mass–damper system withm = 100 kg,ζ = 0.1,& k = 1000 n>m let t = 2π s

3.36)ωb = 3.162 rad>s with amplitude y = 0.05 m subject 2 initial conditions x0 = 0.01 m & v0 = 3.0 m>s.D system is m = 1 kg,c = 10 kg>s,& k = 1000 n>m

3.37)print d function & its 4ier series approxim8on 4 5,20,then 100 terms.

3.38)vibr8on 2olbox.Print d function & its 4ier series

approxim8on 4 5,20,& 100 terms

3.39)calcul8 d response of mx$+ cx# + kx = f0φ(t)where ϕ(t) is d unit step function 4 d case with x0 = v0 = 0

3.40)mx$(t) + cx# (t) + kx(t) = δ(t),x0 = 0,v0 = 04 d overdamped case (ζ 7 1).Plot d response 4 m = 1 kg,k = 100 n>m,& ζ = 1.5.

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