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**Unformatted text preview: **Fall Tenn 200:5 MECH 330 — Machine Dynamics University of Victoria
Department of Mechanical Engineering Final Examination, Dec12th, 2006
MECH 330: Machine Dynamics Instructor: Dr. N. Dechev STUDENT NAME: 39 N: it: {DUNS hr! _[ ’ STUDENT NUMBER: 0 3 '“ a 3:? 83 Open TextBook Exam: Only the course textbook and a non-communicating calculator are permitted. Start time: 7:00 pm
End time: 10:00 pm There are a total of five (5) questions. Answer all questions. --------- ALL ANSWERS TO BE WRITTEN IN EXAMINATION BOOKLETS ONLY _--____._ Fall Term 2006 MECH 330 — Machine Dynamics Question #1: [20%]
Consider the mechanical system illustrated in Fig. Q1. All of the masses in the diagram are restricted to
horizontal motion along the x-axis. Please answer the following questions:
M ) How many degrees of freedom does this system have?
[/(Z) 011 the diagram, draw in a set of “generalized coordinates” with directions of action, to describe
the motion of the system. Remember to use the minimum number of generalized coordinates.
[/68 Derive the differential equations for this system using the “Free Body Diagram Method”, and
write them out in matrix format. Figure Q1. Question #2: [10%]
Consider an n-DOF mechanical system consisting of x masses, y springs and z dampers. The
mechanical system is subjected to an external excitation force in the form of Fosin(mt) acting on one of
the masses.
L/(a) How many natural frequencies does this system have?
L/(b) What is the signiﬁcance if the excitation frequency is equal to one of the natural frequencies?
do) Is it a beneﬁt or a problem if the scenario described in (b) occurs? Justify your answer. Fall Term 2006 MECH 330 — Machine Dynamics Question #3: [30%] Consider the motorcycle illustrated in Fig. Q3. It consists of the body of mass m, with the mass center located at point A, along with two sets of springs and dampers, k}, kg, (:1, (:2. Notice that the springs and dampers are at an angle with respect to the vertical.
a) Draw a ‘simpliﬁed’ block diagram to model the system in 2-DOF. Use one DOF to model the 1/ vertical bounce motion of the body, and the other DOF for the rotational pitch motion. You
can ignore the mass and elasticity of the tires. You can approximate the springs and dampers , by considering only their ‘vertical action” on the body. k/(b) Derive the differential equations for the system using the “Energy Method” using Lagrange’s
Equations These equations should be written in terms of L}, L2, in, 162, c 1, (:2, and m and your
chosen generalized coordinates. For the next part, assume the following parameters for the motorcycle. L; =0.75m, L2=l.25m, kg=5000 Nim, k2=3000 N/m, ci=1000 N's/m, c3=500 N'sfm, and m=500 kg. (0) Determine the natural frequencies and normal mode. shapes for this system. (d) If a vertical excitation force F (t)=100'sin(500t) N at L3 = 0.25 m is applied at point B, by how
much do the natural frequencies of the system change? Figure Q3. Fall Term 2006 MECH 330 - Machine Dynamics Question #4: [15%]
Fig. Q4 illustrates a diagram of a three-ﬂoor radio tower. Each ﬂoor is supported by four ‘ Posts”, each
of which have a length L and stiffness 160,050. Additionally, the middle ﬂoor of the tower is restrained
by two support wires of stiffness km”), angled at 70° to the ground, as shown. (a) Draw a ‘ simplified’ block diagram to model the system in 3-DOF. You can assume that motion of the ﬂoors will only occur in the y-direction.
(b) Use the ‘stiffness inﬂuence coefﬁcient method’ to determine the stiffness matrix for this
system. Post Cross—Section Figure Q4. Question #5: [25%]
Consider the radio tower of Figure Q4. Wind acting on the top ﬂoor can be approximated as a
sinusoidal force Fm =800'sin(30t) N. For the system, assume the following parameters:
m; =6000 kg , m2 =3000 kg , m3 =1000 kg
1415“,,”J =10 m , Erma”) =2x109N/m2
kmm = 1x106 N/m
The stiffness of a post in the ﬁxed-ﬁxed conﬁguration is given as: IZEH L3.
The moment of inertia of a post is given as: bid/12
Answer the following questions
6a) Write out the differential equations of this system in matrix format-
(13) Determine the natural frequencies and normal mode shapes of this system.
~/(’c) Determine the orthonormal mode shapes and the orthonormal modal matrix for the system.
(d) Use ‘Modal Analysis’ to ﬁnd the “time dependant response”, in the original generalized
coordinates (i.e. ynm for n=1 to 3). ...

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