ASSIGNMENT 2

ENME 599
ASSIGNMENT 2 - 3 problems
NOTE: The TA will only mark one or two questions (random selection). Write your name on
each page. DO NOT show your student ID number. Make sure you include all procedures
(“by hand” or computer calculations). Put all the necessary units. Highlight the answers.
Due date: Feb. 24. Submit in the drop off box on the 3rd floor of MEB by noon.
Problem 1.
Consider the kinematically excited system shown below, with the values m = 8 kg, c = 30 (N
s)/m, k1 = 5,000 N/m, and k2 = 2,000 N/m. The forcing quantity is a displacement of the
base, yb(t, ∆t), defined as follows
⎧0
⎪
⎪
⎪
π (t - 0.5 Δt)
yb(t, ∆t) = ⎨ 0.5 y b0 [ sin(
) +1 ]
Δt
⎪
⎪
⎪ y
⎩ b0
for t ≤ 0
for 0 < t ≤ Δt
for t > Δt
where yb0 = 0.005 m and ∆t = 10-3 sec. Note that this function, i.e., yb(t, ∆t), approximates
the “unit step” function as ∆t → 0. You are encouraged to plot and analyze this function and
€
its 1st time derivative for different values of ∆t in the range 10-2 - 10-5 sec.
i. Draw free body diagrams for the mass m and the point A
ii. Derive a single differential equation of this system, in the symbolic form, for the case
when (a) the dependent variable is the displacement of the mass yi(t) in the “inertial”
coordinates, and (b) the dependent variable is the displacement yni(t) in the “noninertial” coordinates. In each case the independent variable is the base excitation
yb(t, ∆t).
Instructor:
S. Spiewak, [email protected]
Department of Mechanical and Manufacturing Engineering, University of Calgary
Feb. 2 , 2014
ENME 599
Note 1: It may be convenient to use an auxiliary variable yAi(t) = yAni(t) + yb(t) in the
process of deriving the needed single differential equations.
Note 2: If needed, use the “p method” to combine 2 differential equations into one,
as done in the Assignment #1.
iii. Find numerical values of the natural frequency, f0, and the damping ratio, ζ, for the
case (a) and (b). If these quantities are the same in each case (difference less than
0.01%), clearly state this observation in the submitted work. Recall that if the characteristic equation is of the order higher than 2, pairs of complex conjugate roots define
the damping ratios and natural frequencies at each specific “vibration mode”.
Note 1: It is recommended to inspect a structure of the characteristic equation for
the two cases under investigation.
iv. Classify this system from the viewpoint of its damping (i.e., underdamped, overdamped, or critically damped)
v. Write differential equations of this system in the numerical form for the case (a) and
(b). You can represent each of the cases (a) and (b) either by 2 equations derived directly from the free body diagrams, or by one equation derived in item ii
vi. Solve these equations numerically for the case (a) and (b). The initial displacement
of the mass is yi0 = 0 m and yni0 = 0.0 m, and the initial velocity is vi0 = 0 m/s, and
vni0 = 0 m/s. Furthermore the initial position of point A is 0 (for inertial and noninertial coordinates case). Plot the results on one plot for the time interval t ∈ {0,3}
seconds.
vii. Calculate and plot a displacement of the point A, yAni(t), in the non-inertial coordinates for the same time interval
Problem 2.
Consider a mechanical system comprising a mass m = 1.5 kg attached to a stiff, massless
bar whose length is L = 0.8 m. As shown in the figure below, the bar is supported at its ends
by two springs whose stiffness is kL = kR = k = 75,000 N/m. The mass is attached at a = 0.5
m from the left end of the bar. Assume that we are interested only in vertical displacements
of the mass and bar ends, and that these displacements are very small as compared with
the length of the bar.
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ENME 599
i. Draw a free body diagram of this massless bar with the attached mass m
ii. Derive a single differential equation of this system, in the symbolic form, for the case
when the dependent variable is the vertical displacement of the mass ym(t) and independent variable is the force Fex(t)
iii. Find numerical values of the natural frequency, f0, and the damping ratio, ζ.
iv. Classify this system from the viewpoint of its damping (i.e., undamped,
underdamped, overdamped, or critically damped)
v. Write the differential equation of this system in numerical form
vi. Solve this equation numerically for the given physical properties. The initial condi-3
tions are yL(0) = 3*10 m, yR(0) = 0, and both velocities are 0. The external force is
Fex(t) = 0. Plot the result for the time interval equal to approximately 5 periods of
oscillation.
Problem 3.
Consider a mechanical system comprising a mass m = 1 kg attached to a stiff, massless bar
whose length is L = 1.25 m. As shown in the figure below, the bar is supported at its ends by
two pairs of spring-damper units. The stiffnesses are kL = 12,000 N/m and kR = 3,000 N/m.
The dampers are cL = 100 (N s)/m and cR = 50 (N s)/m. The mass is attached at a = 0.25 m
from the left end of the bar. Assume that we are interested only in vertical displacements of
the mass and bar ends, and that these displacements are very small as compared with the
length of the bar.
i. Draw a free body diagram of this massless bar with the attached mass m
ii. Derive a single differential equation of this system, in the symbolic form, for the case
when the dependent variable is the vertical displacement of the mass ym(t) and independent variable is the force Fex(t)
iii. Find numerical values of the natural frequency, f0, and the damping ratio, ζ.
continued ...
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ENME 599
iv. Classify this system from the viewpoint of its damping (i.e., undamped, underdamped, overdamped, or critically damped)
v. Write the differential equation(s) of this system in numerical form
vi. Solve this equation numerically for the given physical properties. The initial condi-3
tions are yL(0) = 10 m, yR(0) = 0, both velocities are 0, and both accelerations are
0. The external force is Fex(t) defined below. Plot the result for the time interval
equal to approximately 5 periods of oscillation.
⎧0
⎪
⎪
⎪
π (t - 0.5 Δt)
Fex(t, ∆t) = ⎨0.5 F0 [ sin(
) +1 ]
Δt
⎪
⎪
⎪F
⎩ 0
for t ≤ 0
for 0 < t ≤ Δt
for t > Δt
where F0 = 1 N and ∆t = 10-3 sec. Note that this function, i.e., Fex(t, ∆t), approximates
€ the “unit step” function as ∆t → 0.
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