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336

CHAPTER 3 HARMONICAlLY EXCITED VIBRATION match the items in the two columns below:

1. Magnification factor of an undamped system

a.

2.... 00" -

2. Period of beating

£I)

1 + (2'r)2 [ b. (I _ r2)2 + (~r)'

J1/2

3. Magnification factor of a damped system 4. Damped frequency

I d'-2

S. Quality factor

e.wn~

1- r

6. Displacement transmissibility

3.6 Match the following equations of motion: 1. m"i + ci + kz = -my 2. M x' + + kx = meCli sin wt 3. + kx ± pN = F(t) 4. + k(1 + i{J)x = Fo sin wt S. + eX + kx = Fosinwt

mx mx mx

ex

a. System with Coulomb damping b. System with viscous damping c. System subject to base excitation

d. System with hysteresis damping e. System with rotating unbalance

I PROBLEMS Section 3.3 Response of an Undamped System Under Harmonic Force 3.1

A weight of 50 N is suspended from a spring of stiffness 4000 N/m and is subjected to a har· monic force of amplitude 60 N and frequency 6 Hz. Find (a) the extension of the spring due to the suspended weight, (b) the static displacement of the spring due to the maximum applied force, and (c) the amplitude of forced motion of the weight.

3.2

A spring-mass system is subjected to a harmonic force whose frequency is close to the natural frequency of the system.1fthe forcing frequency is 39.8 Hz and the natural frequency is 40.0 Hz, determine the period of beating.

3.3

Consider a spring-mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic force F(t) = 400 cos lOt N. Find and plot the total response of the system under the following initial conditions:

B. Xo = 0.1 m,.:to = 0 Xo = 0,;'0 = IOmls e. Xo = 0.1 m,;,o = IOmls

b.

PROBLEMS 3.4

337

Consider a spring·mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic force F(t) = 400 cos 20t N. Find and plot the total response of the system under the following initial conditions:

a. Xo = 0.1 In, Xo = 0 b. Xo = 0, Xo = IOmis c. Xo = O.lm,xo = IOmis 3.S

Consider a spring-mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic force F(t) = 400 cos 20.1t N. Find and plot the total response of the system under the fol-

lowing initial conditions: a. Xo = 0.1 In, :to = 0 b. Xo = 0, Xo = IOmis c. Xo = O.lm,xo = IOmis 3.6

Consider a spring-mass system, with k = 4000 N/m and m = 10 kg, subject to a harmonic force F(t) = 400 cos 30t N. Find and plot the total response of the system under the following initial conditions:

a. Xo = 0.1 In, Xo = 0 b. Xo = 0, Xo = IOmis c. Xo = O.lm,:to = IOmis 3.7

A spring-mass system consists of a mass weighing 100 N and a spring with a stiffness of 2000 N/m. The mass is subjected to resonance by a harmonic force F(t) = 25 cos wt N.

t

Find the amplitude of the forced motion at the end of (a) cycle, (b) 2 ~ cycles, and (c) 5~ cycles.

3.8

A mass m is suspended from a spting of stiffness 4000 N/m and is subjected to a harmonic force having an amplitude of 100 N and a frequency of 5 Hz. The amplitude of the forced motion of the mass is observed to be 20 mm. Find the value of In.

3.9

A spring-mass system with m = 10 kg and k = 5000 N/m is subjected to a harmonic force of amplitude 250 N and frequency w. If the maximum amplitude of the mass is observed to be 100 mm, find the value of w.

3.10 In Fig. 3.I(a), a periodic force F(t) = Fo cos wt is applied at a point on the spring that is located at a distance of 25 percent of its length from the fixed support. Assuming that c = 0, find the steady-state response of the mass In. 3.11

A spring-mass system, resting on an inclined plane, is subjected to a harmonic force as shown in Fig. 3.38. Find the response of the system by asswning zero initial conditions.

3.12 The natural frequency of vibration of a person is found to be 5.2 Hz while standing on a horizontal floor. Asswning damping to be negligible, determine the following:

a. If the weight of the person is 70 kg,. determine the equivalent stiffness of his body in the vertical direction. b. If the floor is subjected to a vertical harmonic vibration of frequency of 5.3 Hz and amplitude of 0.1 m due to an unbalanccdrotating machine operating on the floor, determine the vertical displacement of the person.

338

CHAPTER 3 HARMONICAlLY EXCITED VIBRATION

k

~

4,000 N/m

FIGURE 3.38

3.13 Plot the forced-vibration response of a spring-mass system given by Eq. (3.13) for the following sets of data:

a. Set 1: 88t = 0.1, W = 5,0011 = 6, Xo = 0.1, Xo = 0.5 b. Set 2: 8" ~ 0.1, w ~ 6.1, w. ~ 6, Xo ~ 0.1, Xo ~ 0.5 Co Set 3: 8" ~ 0.1, w ~ 5.9, w. ~ 6, Xo ~ 0.1, Xo ~ 0.5 3.14 A spring-mass system is set to vibrate from zero initial conditions under a harmonic force. The response is found to exhibit the phenomenon of beats with the period of beating equal to 0.5 s and the period of oscillation equal to 0.05 s. Find the natural frequency of the system and the frequency of the harmonic force. 3.15 A spring-mass system, with m ~ 100 kg and k ~ 400 N/m, is subjected to a harmonic force f(t} ~ Focos wt withFo ~ ION. Find the response of the system when wis equal to (a) 2 rad/s, (b) 0.2 rad/s, and (c) 20 radls. Discuss the results. 3.16 An aircraft engine has a rotating unbalanced mass m at radius r. If the wing can be modeled as a cantilever beam of uniform cross section a X b, as shown in Fig. 3.39(b), determine the maximum deflection of the engine at a speed of N rpm. Assume damping and effect of the wing between the engine and the free end to be negligible. 3.17 A three-bladed wind turbine (Fig. 3.4O(a» has a small unbalanced mass m located at a radius r in the plane of the blades. The blades are located from the central vertical (y) axis at a distance R and rotate at an angular velocity of w. If the supporting truss can be modeled as a hollow steel shaft of outer diameter 0.1 m and inner diameter 0.08 m, determine the maximum stresses developed at the base of the support (point A). The mass moment of inertia of the turbine system about the vertical (y) axis is 10. Assume R ~ 0.5 m, m ~ 0.1 kg, r ~ 0.1 m, 10 ~ 100 kg_m2, h ~ 8 m, and w ~ 31.416 rad/s.

PROBLEMS

I' L----ItL J!-----------..

fE

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