# An aeroplane has a stall speed of 100 kt at a mass of 1000 kg. If the mass is increased to 2000 kg, the new value of the stall speed will be:

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3.49)a power line pole with a trans4mer is modeled bymx$+ kx = -y$ where x & y r as indic8d in figure p3.49.Assuming d initial conditions r zero,calcul8 d **response** of d rel8ve displacement (x – y) if d pole is subject 2 an earthquake-based excit8on of $y(t) = c aa1 0- tt0 b 0 …t 7t …2t02t0

3.50)calcul8 d response spectrum of an undamped **system** f(t) = cπt?????2 d 4cing function assuming d initial conditions r zero

3.51)using complex algebra,derive equ8on (3.89) frm (3.86) with s = jω

3.52)using d **plot** in figure p3.52,estim8 d system’s parameters m,c,& k,as well as d natural frequency

3.53)frm a compliance transfer function of a **spring**–**mass**–damper system d **stiffness** is determined 2 have a value of 0.5 n>m,a natural frequency of 0.25 rad>s,& a **damping** **coefficient** of 0.087 kg>s.Plot d inertance transfer function’s magnitude & phase 4 dis system.

3.54)frm a compliance transfer function of a spring–mass–damper system d stiffness is determined 2 have a value of 0.5 n>m,a natural frequency of 0.25 rad>s,& a damping coefficient of 0.087 kg>s.Plot d mobility transfer function’s magnitude & phase 4 d system.

3.55)calcul8 d compliance transfer function 4 a system described byd3x(t)dt3 + cd2x(t)dt2 +where f(t) is d input 4ce & x(t) is a displacement

3.56)calcul8 d frequency response function 4 d compliance 4 d system defined byd3x(t)dt3 + cd2x(t)dt2 +

3.57) plot d magnitude of d frequency response function 4 d system of problem 3.564 a = 1,b = 4,c = 11,d = 16,& e = 8

3.58) an experimental (compliance) magnitude plot is illustr8d in fig.P3.58.Determine ω,ζ,c,m,& k.Assume **dat** d units correspond 2 m>n along d vertical axis

3.59)show dat a critically damped system is bounded-input,bounded-output stable

3.60)show dat an overdamped system is bounded-input,bounded-output stable

3.61)is d solution of 2x$ + 18x = 4 cos 2t + cos t lagrange stable?

3.62)calcul8 d response of d system described by$x(t) + x# (t) + 4x(t) = ax(t) + bx# (t)4 x0 = 0,v0 = 1 4 d case dat a = 4 & b = 0.Is d response bounded?

3.63)a crude model of an aircraft wing can b modeled as 100x$(t) + 25x# (t) + 2000x(t) = ax# (t) here d fac2r a is determined by d aerodynamics of d wing & is proportional 2 d air speed.@ ?Value of d parameter a will d system start 2 flutter?

3.64)consider d inverted pendulum of figure p3.64 & compute d value of d stiffness k dat will keep d **linear** system stable.Assume dat d pendulum rod is massless.

3.65)numerically integr8 & plot d response of an underdamped system determined bym = 100 kg,k = 1000 n>m,& c = 20 kg>s,subject 2 d initial conditions of x0 = 0and v0 = 0,& d applied 4ce f(t) = 30φ(t -1).Then plot d exact response ascomputed by equ8on (3.17).Compare d plot of d exact solution 2 d numerical simul8on.

3.66)numerically integr8 & plot d response of an underdamped system determined bym = 150 kg & k = 4000 n>m subject 2 d initial conditions of x0 = 0.01 m andv0 = 0.1 m>s,& d applied 4ce f(t) = φ(t) = 15 (t -1),4 various values of d damping coefficient.Use dis“program” 2 determine a value of damping dat causes d transient term 2 die out within 3 seconds.Try 2 find d smallest such value of damping remembering dat added damping is usually expensive.

3.67)calcul8 d 2tal response of d base isol8on problem given in example 3.3.2,with ωb = 3 rad>s,m = 1 kg,c = 10 kg>s,k = 1000 n>m,& y = 0.05 m,subject 2 initial conditions x0 = 0.01 m & v0 = 3.0 m>s,by numerically integr8ngrather than using analytical expressions.Plot d response,reproduce figure 3.14,& compare d results 2 c dat dey r d same.

3.68)numerically simul8 d response of d system of a single-degree-of-freedom spring–mass system subject 2 d motion y(t) given in figure p3.68 & plot d response.D mass is 5000 kg & d stiffness is 1.5 * 103 n>m.

3.69)numerically simul8 d response of an undamped system 2 a step function with afinite rise time of t1 4 d case m = 1 kg,k = 1 n>m,t1 = 4 s,& f0 = 20 n.Thisfunction is described byf(t) = c fft001t 0 …t 7t …t1 t1plot d response

3.70)numerically simul8 d response of d system of problem 3.22 4 a 2-meter concrete wall with cross section 0.03 m2 & mass modeled as lumped @ d end of 1000 kg.Use f0 = 100 n,& plot d response 4 d case t0 = 0.25 s.

3.71)numerically simul8 d response of an undamped system 2 a ramp input of d 4mf(t) = f0 t,where f0 is a constant.Plot d response 4 3 periods 4 d casem = 1 kg,k = 100 n>m,& f0 = 50 n

3.72)compute & plot d response of d following system using numerical integr8on: 10x$(t) + 20x# (t) + 1500x(t) = 20 sin 25t + 10 sin 15t + 20 sin 2t with initial conditions of x0 = 0.01 m & v0 = 1.0 m>s

3.73)compute d response of d system in figure 3.26 4 d case dat d damping is

linear viscous,d spring is a nonlinear soft spring of d 4mk(x) = kx - k1x3d system is subject 2 an excit8on of d 4m (t1 = 1.5 & t2 = 1.6) f(t) = 1500[φ(t - t1) - φ(t - t2)] nand initial conditions of x0 = 0.01 m & v0 = 1.0 m>s.D system hs a mass of100 kg,a damping coefficient of 30 kg>s,& a linear stiffness coefficient of 2000 n>m.D value of k1 is taken 2 b 300 n>m3.Compute d solution & compare it 2 d linear solution (k1 = 0).Which system hs d largest magnitude?Compare yur solution 2 dat of example 3.10.1.

3.74)compute d response of a spring–mass system 4 d case dat d damping is linearviscous,d spring is a nonlinear soft spring of d 4m k(x) = kx - k1x3d system is subject 2 an excit8on of d 4m (t1 = 1.5 & t2 = 1.6)f(t) = 1500[φ(t - t1) - φ(t - t2)] nand initial conditions of x0 = 0.01 m & v0 = 1.0 m>s.D system hs a mass of100 kg,a damping coefficient of 30 kg>s,& a linear stiffness coefficient of 2000 n>m.D value of k1 is taken 2 b 300 n>m3.Compute d solution & compare it 2 thelinear solution (k1 = 0).How different r d linear & nonlinear responses?Repeatthis 4 t2 = 2.?Can u say regarding d effect of d time length of d pulse?

3.75)compute d response of a spring–mass–damper system 4 d case dat d damping is linear viscous,d spring stiffness is of d 4m

k(x) = kx - k1x2 d system is subject 2 an excit8on of d 4m (t1 = 1.5 & t2 = 2.5) f(t) = 1500[φ(t - t1) - φ(t - t2)] n & initial conditions of x0 = 0.01 m & v0 = 1 m>s.D system hs a mass of 100 kg,a damping coefficient of 30 kg>s,& a linear stiffness coefficient of 2000 n>m.D value of k1 is taken 2 b 450 n>m3.Which system hs d largest magnitude?

3.76)compute d response of a spring–mass–damper system 4 d case dat d damping is linear viscous,d spring stiffness is of d 4m k(x) = kx + k1x2 d system is subject 2 an excit8on of d 4m (t1 = 1.5 & t2 = 2.5) f(t) = 1500[φ(t - t1) - φ(t - t2)] n & initial conditions of x0 = 0.01 m & v0 = 1 m>s.D system hs a mass of 100 kg,a damping coefficient of 30 kg>s,& a linear stiffness coefficient of 2000 n>m.D value of k1 is taken 2 b 450 n>m3.Which system hs d largest magnitude?

3.77)compute d response of a spring–mass–damper system 4 d case dat d damping is linear viscous,d spring stiffness is of d 4m

k(x) = kx - k1x2 d system is subject 2 an excit8on of d 4m (t1 = 1.5 & t2 = 2.5) f(t) = 150[φ(t - t1) - φ(t - t2)] n & initial conditions of x0 = 0.01 m & v0 = 1 m>s.D system hs a mass of 100 kg,a damping coefficient of 30 kg>s,& a linear stiffness coefficient of 2000 n>m.D value of k1 is taken 2 b 5500 n>m3.Which system hs d largest transient magnitude?Which hs d largest magnitude in steady st8?

3.78)Compare the forced response of a system with velocity-squared damping, as defined inequation (2.129) using numerical simulation of the nonlinear equation, to that of the response of the linear system obtained using equivalent viscous damping, as defined by equation (2.131). Use the initial conditions x0 = 0.01 m and v0 = 0.1 m>s with a mass of 10 kg, stiffness of 25 N>m, applied force of the form (t1 = 1.5 and t2 = 2.5) F(t) = 15[Φ(t - t1) - Φ(t - t2)] N and drag coefficient of α = 25.

3.79)Compare the forced response of a system with structural damping (see Table 2.2) usingnumerical simulation of the nonlinear equation to that of the response of the linear system obtained using equivalent viscous damping as defined in Table 2.2. Use the initial conditions x0 = 0.01 m and v0 = 0.1 m>s with a mass of 10 kg, stiffness of 25 N>m, applied force of the form (t1 = 1.5 and t2 = 2.5) F(t) = 15[Φ(t - t1) - Φ(t - t2)] N

and solid damping coefficient of b = 8. Does the equivalent viscous-damping linearization overestimate the response or underestimate it?