Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.5 (1.66 through 1.74) 1.66 A helicopter landing gear consists of a metal framework rather than the coil spring based suspension system used in a fixedwing aircraft. The vibration of the frame in the vertical directio
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3 61 Problems and Solutions for Section 3.7 (3.45 through 3.52) 3.45 Using complex algebra, derive equation (3.89) from (3.86) with s = j. Solution: From equation (3.86):
Hs=
Substituting s = j! yields
H j! =
()
1 ms + cs + k
2
()
1 m j!
(
)
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3 68 Problems and Solutions Section 3.8 (3.53 through 3.56) 3.53 Show that a critically damped system is BIBO stable. Solution: For a critically damped system
h t !" =
(
)
1 !# t !" t ! " e n( ) m
(
)
Let f(t) be bounded by the finite constan
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3Problems and Solutions from Section 3.9 (3.573.64)
72
3.57*. Numerically integrate and plot the response of an underdamped system determined by m = 100 kg, k = 1000 N/m, and c = 20 kg/s, subject to the initial conditions of x0 = 0 and v0 = 0, and
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3 89 Problems and Solutions Section 3.10 (3.65 through 3.71) 3.65*. Compute the response of the system in Figure 3.26 for the case that the damping is linear viscous and the spring is a nonlinear soft spring of the form k(x) = kx ! k1 x 3 and the sy
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
2 1 Problems and Solutions Section 2.1 (2.1 through 2.15) 2.1 To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation (2.2) with = 2 rad/s, given by equation (2.11) for a variety of values
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 2.2 (2.16 through 2.31) 2.16 Calculate the constants A and ! for arbitrary initial conditions, x 0 and v 0 , in the case of the forced response given by Equation (2.37). Compare this solution to the transient response o
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
2 24 Problems and Solutions Section 2.3 (2.32 through 2.36) 2.32 Referring to Figure 2.10, draw the solution for the magnitude X for the case m = 100 kg, c = 4000 N s/m, and k = 10,000 N/m. Assume that the system is driven at resonance by a 10N for
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
2 27 Problems and Solutions Section 2.4 (2.37 through 2.50) 2.37 A machine weighing 2000 N rests on a support as illustrated in Figure P2.37. The support deflects about 5 cm as a result of the weight of the machine. The floor under the support is so
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
2 39 Problems and Solutions Section 2.5 (2.51 through 2.58) 2.51 A lathe can be modeled as an electric motor mounted on a steel table. The table plus the motor have a mass of 50 kg. The rotating parts of the lathe have a mass of 5 kg at a distance 0
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
242 Problems and Solutions Section 2.6 (2.59 through 2.62) 2.59 Calculate damping and stiffness coefficients for the accelerometer of Figure 2.23 with moving mass of 0.04 kg such that the accelerometer is able to measure vibration between 2 0 and 50
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
244 Problems and Solutions Section 2.7 (2.63 through 2.79) 2.63 Consider a springmass sliding along a surface providing Coulomb friction, with stiffness 1.2 10 4 N/m and mass 10 kg, driven harmonically by a force of 50 N at 10 Hz. Calculate the ap
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
2 60
Problems and Solutions Section 2.8 (2.80 through 2.86) 2.80*. Numerically integrate and plot the response of an underdamped system determined by m = 100 kg, k = 20,000 N/m, and c = 200 kg/s, subject to the initial conditions of x0 = 0.01 m and
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
2 69
Problems and Solutions Section 2.9 (2.87 through 2.93) 2.87*. Compute the response of the system in Figure 2.34 for the case that the damping is linear viscous and the spring is a nonlinear soft spring of the form k(x) = kx ! k1 x 3 and the sy
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.1 (1.1 through 1.19) 1.1 The spring of Figure 1.2 is successively loaded with mass and the corresponding (static) displacement is recorded below. Plot the data and calculate the spring's stiffness. Note that the data
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions for Section 1.2 and Section 1.3 (1.20 to 1.51) Problems and Solutions Section 1.2 (Numbers 1.20 through 1.30) 1.20* Plot the solution of a linear, spring and mass system with frequency n =2 rad/s, x0 = 1 mm and v0 = 2.34 mm/s,
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.4 (problems 1.52 through 1.65) 1.52 Calculate the frequency of the compound pendulum of Figure 1.20(b) if a mass mT is added to the tip, by using the energy method. Solution Using the notation and coordinates of Figur
Korea Advanced Institute of Science and Technology
Mechanical Vibration
MECHANICAL MAE351

Spring 2009
1
Normalized Magnitude of Primary Mass
1 Xk 1 .
2
a
2
2
p
. 1
a
2
F0 = 1 N (unit force)
.
2
2 Xk 1
0
0.5
1 a
1.5
2
/ 1
/ 2
Absorber zone
2
Comparing these 3 roots to the plot yields that:
= 0.3929, 1.1180, 1.138 a
Xk F0
0.3929 < < 1.1180 a 7.409
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3 57 Problems and Solutions Section 3.6 (3.43 through 3.44) 3.43 A power line pole with a transformer is modeled by
m! + kx = ! ! x y
where x and y are as indicated in Figure 3.23. Calculate the response of the relative displacement (x y) if the p
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3 54 Problems and Solutions Section 3.5 (3.39 through 3.42) 3.39 Calculate the meansquare response of a system to an input force of constant PSD, S0, 10 and frequency response function H ! = 3 + 2 j!
()
(
)
Solution: Given: S ff = S0 and H ! =
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
3 50 Problems and Solutions for Section 3.4 (3.35 through 3.38) 3.35 Calculate the response of ! m! + cx + kx = F0 !(t) x where (t) is the unit step function for the case with x0 = v0 = 0. Use the Laplace transform method and assume that the system
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.6 (1.75 through 1.81) 1.75 Show that the logarithmic decrement is equal to
!=
1 x0 ln n xn
where xn is the amplitude of vibration after n cycles have elapsed. Solution:
'() t ! ! xt $ Ae n sin ) d t + * # ln # & =
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.7 (1.82 through 1.89) 1.82 Choose a dashpot's viscous damping value such that when placed in parallel with the spring of Example 1.7.2 reduces the frequency of oscillation to 9 rad/s. Solution: The frequency of oscill
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.8 (1.90 through 1.93) 1.90 Consider the system of Figure 1.90 and (a) write the equations of motion in terms of the angle, , the bar makes with the vertical. Assume linear deflections of the springs and linearize the
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.9 (1.94 through 1.101) 1.94* Reproduce Figure 1.38 for the various time steps indicated. Solution: The code is given here in Mathcad, which can be run repeatedly with different t to see the importance of step size. Ma
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions Section 1.10 (1.102 through 1.114) 1.102 A 2kg mass connected to a spring of stiffness 103 N/m has a dry sliding friction force (Fc) of 3 N. As the mass oscillates, its amplitude decreases 20 cm. How long does this take? Solut
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions for Section 4.1 (4.1 through 4.16) 4.1 Consider the system of Figure P4.1. For c1 = c2 = c3 = 0, derive the equation of motion and calculate the mass and stiffness matrices. Note that setting k3 = 0 in your solution should resu
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions for Section 4.2 (4.19 through 4.33) 4.19 Calculate the square root of the matrix
" 13 !10 % M=$ ' # !10 8 &
" " a !b % 1/ 2 1/ 2 $ Hint: Let M = $ ' ; calculate M !b c & # #
(
)
2
% and compare to M.' &
Solution: Given:
"
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions for Section 4.3 (4.34 through 4.43) 4.34 Solve Problem 4.11 by modal analysis for the case where the rods have equal stiffness
! (i.e., k1 = k2 ), J1 = 3J 2 , and the initial conditions are x(0) = !0 1# and x 0 = 0. " $
T
()
Korea Advanced Institute of Science and Technology
MECHANICAL VIBRATION
MECHANICAL MAE351

Spring 2009
Problems and Solutions for Section 4.4 (4.44 through 4.55) 4.44 A vibration model of the drive train of a vehicle is illustrated as the threedegreeoffreedom system of Figure P4.44. Calculate the undamped free response [i.e. ! M(t) = F(t) = 0, c1 =