MAE 653-1 Advanced VibrationSpring 2010
Lesson #5 Tuesday, Jan. 26, 2010Textbook Sections 2.7
FREE VIBRATON WITH NON-VISCOUS DAMPINGHermann E. Alcazar
5. Free vibration with non-viscous dampingMAE 653 - Advanced
Vibrations/Fall 2010
2.7 Free vibration with Coulomb damping. Mathematical
background.In general a second-order nonhomogeneous linear
differential equation with constant coefficients has the form,where
a, b, and c are constants and G is a continuous function. Its
related homogeneous equation is,The general solution of the
nonhomogeneous differential equation can be written as,
where xh is the general solution of the homogeneous equation and
xp is a particular solution.. The particular solution can be found
using the method of undetermined coefficients or the method of
variation of parameters. In the method of undetermined coefficients
the shape of the particular solution is assumed, and its
coefficients are obtained after replacing into the differential
equation.If G(x)=ekx P(x), where P is a polynomial of degree n, try
yp(x)=ekx Q(x), where Q is a n-degree polynomial.If G(x)=ekx P(x)
cos mx or G(x)=ekx P(x) sin mx, where P, Q, and R are n-degree
polynomials, try yp(x)=ekx Q(x) cos mx + ekx R(x) sin mx.If any
term of yp is a solution of the homogeneous equation, multiply yp
by x (or by x2 if necessary)
5. Free vibration with non-viscous dampingMAE 653 - Advanced
Vibrations/Fall 2010
2.7 Free vibration with Coulomb damping. Equation of
motionCoulombs law or dry friction states that, when two bodies are
in contact, the force required to produce sliding is proportional
to the normal force acting in the plane of contact.Case 1:Case
2:Finally:
5. Free vibration with non-viscous dampingMAE 653 - Advanced
Vibrations/Fall 2010
2.7 Free vibration with Coulomb damping. Solution (1)If a cosine
function is considered as the solution, for an initial zero
velocity, thenThe amplitudes after the first half cycles are
(x0-2xd).In each cycle, the amplitude reduces in 4N/kMotion will
stop at the end of the r-th half cycle, when the force exerted by
the spring (kx) is less than the friction force (N). The amplitude
falls into de dead zone.
5. Free vibration with non-viscous dampingMAE 653 - Advanced
Vibrations/Fall 2010
2.7 Free vibration with Coulomb damping. Solution (2)For an
initial zero velocity, then
5. Free vibration with non-viscous dampingMAE 653 - Advanced
Vibrations/Fall 2010
Homework # 2Due, Thursday February 04, 2010
Solve the following problems from the textbook Mechanical
Vibrations by Rao: 2.46-2.47 Figure 2.75 2.62 2.74 2.96 2.100
Figure 2.95. Plot the solution (in a single graph, tag all
characteristic values) considering m=4 kg, k=1250 N/m, x0=100 mm,
and v0=-10 m/s for each of the following cases: (a) c = 0 (b) c =
100 N-s/m (c) c = 200 N-s/m (d) c = 400 N-s/m 2.121Plot the
solution (tag all characteristic values). Determine the number of
cycles of motion and the final static position. Include the used
Matlab code.
5. Free vibration with non-viscous dampingMAE 653 - Advanced
Vibrations/Fall 2010
Project: Build, test, acquire and analyze data from an
experiment at the vibrations laboratory. The topic is open and will
be defined by the group in coordination with Dr. Mourad Riad or Mr.
Yan Luo. Groups of 2 (3 only depending on the complexity of the
proposed experiment).Abstract, due Thursday February 11. In the
format of research papers.Presentation, dead week at the vibrations
laboratory.Report, after the presentation. In the format of
research papers.Take-home questions: The questions for the midterm
and final exams will use the concepts of vibration criteria,
balancing of rotating machines, critical speed on shafts, balancing
of reciprocating engines, vibration isolation, and vibration
absorbers (Ch. 9). The knowledge of these topics are your
responsibility.
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