درس ارتعاشات مکانیکی
:تمرین سری دوم دکتر عبدالرضا اوحدی:استاد درس
ارتعاشات آزاد سيستم هاي يك درجه آزادي الیاس برهمند:تدریسیار
0413/8/5 :موعد تحویل دانشکده مهندسی مکانیک
اجباري: A سري
1- A uniform bar of length L and weight W is suspended symmetrically by two string، as
shown in figure. Set up the differential equation of motion for small angular oscillations
of the bar about the vertical axis O-O and determine its period.
2- Find the natural frequency of the following system. Consider the rod to be rigid, with
the mass and the mass and length of the rod as m3 and L ،respectively.( Use both the
Newton and energy methods to calculate the natural frequency)
3- A time recorder, as shown in the figure, operates using a pendulum with a length 𝐿 and
a period of 2 seconds. A platinum wire connected to the pendulum closes the electrical
timing circuit through a drop of mercury when it reaches the lowest point of its swing.
a) What should the length 𝐿 of the pendulum be?
b) If the platinum wire is in contact with the mercury for a distance of 0.3175 cm during
the swing, determine the amplitude of oscillation 𝜃 so that the contact time is limited to
0.01 seconds. (Assume the velocity during contact is constant and the amplitude of
oscillation is small.)
4- A piston of mass 4.53 kg is traveling in a tube with a velocity of 15.24 m/s and engages
a spring and damper، as shown in fig. Determine the maximum displacement of the
piston after engaging the spring-damper. How many seconds does it take?
5- The free-vibration responses of an electric motor of weight 500 N mounted on different
types of foundations are shown in Figs. 2.107(a) and (b). Identify the following in each
case: (i) the nature of damping provided by the foundation, (ii) the spring constant and
damping coefficient of the foundation, and (iii) the undamped and damped natural
frequencies of the electric motor.
6- A mass moves in a fluid against sliding friction as illustrated in the figure below. Model
the damping force as a slow fluid (i.e., linear-viscous damping) plus Coulomb friction
because of the sliding, with the following parameters: m = 250 kg, μ = 0.01, c = 25
kg/s, and k = 3000 N/m. Using Matlab software (a) Compute and plot the response to
the initial conditions: x0 = 0.1 m, v0 = 0.1 m/s. (b) Compute and plot the response to
the initial conditions: x0 = 0.1 m, v0 = 1 m/s. How long does it take for the vibration to
die out in each case?
7- For what value of c is the damping ratio of the system of figure equal to 1.25?
8- The characteristic equation of a single-degree-of-freedom system, given by the
following equation:
𝑠 2 + 𝑎𝑠 + 𝑏 = 0
where a = c/m and b = k /m can be considered as the parameters of the system. Identify
regions that represent a stable, unstable, and marginally stable system in the parameter
plane—i.e., the plane in which a and b are denoted along the vertical and horizontal axes
respectively.
9- Figure shows a uniform rigid bar of mass m and length l, pivoted at one end (point O)
and carrying a circular disk of mass M and mass moment of inertia J (about its rotational
axis) at the other end (point P). The circular disk is connected to a spring of stiffness k
and a viscous damper of damping constant c as indicated.
a. Derive the equation of motion of the system for small angular displacements of the
rigid bar about the pivot point O.
b. Derive conditions corresponding to the stable, unstable, and marginally stable
behavior of the system.
امتيازي: B سري
1- Figure 2.82 shows a metal block supported on two identical cylindrical rollers rotating
in opposite directions at the same angular speed. When the center of gravity of the block
is initially displaced by a distance x, the block will be set into simple harmonic motion.
If the frequency of motion of the block is found to be ω، determine the coefficient of
friction between the block and the rollers.
2- Determine the differential equation of motion for free vibration of the system shown in
Fig. using virtual work.
3- One end of a uniform rigid bar of mass m is connected to a wall by a hinge joint O, and
the other end carries a concentrated mass M, as shown in Fig. The bar rotates about the
hinge point O against a torsional spring and a torsional damper. It is proposed to use
this mechanism, in conjunction with a mechanical counter, to control entrance to an
amusement park. Find the masses m and M, the stiffness of the torsional spring and the
damping force necessary to satisfy the following specifications: (1) A viscous damper
or a Coulomb damper can be used. (2) The bar has to return to within 5 of closing in
less than 2 sec when released from an initial position of u = 75°.
4- A piece with mass 𝑚 is placed horizontally on a semi-cylinder in such a way that the
contact point is below the center of mass 𝐺. Assuming no slipping occurs, derive the
differential equation in terms of 𝜃.
5- A foot pedal mechanism for a machine is crudely modeled as a pendulum connected to
a spring as illustrated in the following figure. The purpose of the spring is provide a
return force for the pedal action. Compute the spring stiffness needed to keep the
pendulum at 1° from the horizontal and then compute the corresponding natural
frequency. Assume that the angular deflections are small, such that the spring deflection
can be approximated by the arc length; that the pedal may be treated as a point mass;
and that pendulum rod has negligible mass. The pedal is horizontal when the spring is
at its free length. The values in the figure are m = 0.5 kg, g = 9.8 m/s2, l1 = 0.2 m, and
l2 = 0.3 m.
