Problem 1
a)
This is the force diagram of this system, as shown in the following figure.
b)
According to Newton’s second law (positive downward):
M x¨ = M g+K x +K (x −x )+B (x˙ −x˙ )−f (t)
1 1 1 1 1 2 1 2 1 1 2 a
After rearrangement:
M x¨ +B x˙ −B x˙ +(K +K )x −K x = M g−f (t)
1 1 1 1 1 2 1 2 1 2 2 1 a
According to Newton’s second law (positive downward):
M x¨ = M g+K (x −x )+K x +B (x˙ −x˙ )+B x˙ −f (t)
2 2 2 2 2 1 3 2 1 2 1 2 2 b
After rearrangement:
M x¨ −B x˙ +(B +B )x˙ −K x +(K +K )x = M g−f (t)
2 2 1 1 1 2 2 2 1 2 3 2 2 b
From the above equations of motion, we can solve for x¨ and x¨ :
1 2
M g−f (t)−B x˙ +B x˙ −(K +K )x +K x
1 a 1 1 1 2 1 2 1 2 2
x¨ =
1
M
1
M g−f (t)+B x˙ −(B +B )x˙ +K x −(K +K )x
2 b 1 1 1 2 2 2 1 2 3 2
x¨ =
2
M
2
1
Now, we can write the state variable equations:
x˙ = x
1 3
x˙ = x
2 4
M g−f (t)−B x +B x −(K +K )x +K x
1 a 1 3 1 4 1 2 1 2 2
x˙ =
3
M
1
M g−f (t)+B x −(B +B )x +K x −(K +K )x
2 b 1 3 1 2 4 2 1 2 3 2
x˙ =
4
M
2
In matrix form, the state equations can be written as:
x˙ 0 0 1 0 x 0 0 0
1 1
x x˙ ˙2 3 = −K10 M+ 1K2 MK0 2
1
−0 MB1
1
MB1 1
1
x x2 3 + −0 M1
1
0 0 (cid:20) f fa b(( tt ))(cid:21) + g0
x˙
4
MK2
2
−K2 M+ 2K3 MB1
2
−B1 M+ 2B2 x
4
0 − M1
2
g
Problem 2
a)
This is the force diagram of this system, as shown in the following figure.
The equation of motion for M :
1
M x¨ = −K (x −x )−B (x˙ −x˙ )
1 1 1 1 2 1 1 3
2
The equation of motion for M :
2
M x¨ = K (x −x )−K (x −x )−B (x˙ −x˙ )
2 2 1 1 2 2 2 3 2 2 3
The equation of motion for M :
3
M x¨ = B (x˙ −x˙ )+B (x˙ −x˙ )+F(t)+K (x −x )
3 2 1 1 3 2 2 3 2 2 3
b. Simulink Dynamic Model
Now we will establish the Simulink model using the given parameters:
• M = M = 3 kg; M = 5 kg
1 2 3
• K = K = 125 N/m; B = B = 15 N.s/m
1 2 1 2
• F(t) = 100∗[H(t−1)−H(t−10)] N, where H is the Heaviside step
function
In Simulink, we can use the integrator blocks to build this system.
For F(t), we need to use two step functions:
• At t = 1 second, the force increases to 100N.
• At t = 10 seconds, the force decreases back to 0N.
Here is the Simulink model:
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c. Plotting Displacement and Velocity Diagrams
Theabovemodelalreadyincludesthepartforplottingthedisplacementand
velocity curves.
These graphs will show the behavior of the system within the 200-second
simulation time, allowing us to observe the transient response and steady-
state behavior of the system. Since the external force starts at t = 1 second
and disappears at t = 10 seconds, we can see the system’s response to this
impulse.
Problem 3
Case Study 1: Analysis of a Customer Service Center Queuing System
Currently, the office processes license plates by assigning customers to
one of three separate processing lines based on their place of residence. Cus-
tomerarrivalsateachlinefollowanexponentialdistributionwithanaverage
interval of 10 minutes. Each customer type is served by dedicated staff, with
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service times uniformly distributed between 8 and 10 minutes. After service,
all customers proceed to a single checkout counter for form verification and
licenseplateissuance, whichtakesbetween2.65and3.33minutes(uniformly
distributed).
Aconsultanthasproposedasingle-queuesystemwiththreecross-trained
service staff capable of handling any customer type.
I built two Arena models to simulate the two different queuing systems.
The first model represents the current three-queue system, while the second
implements the proposed single-queue, multi-server approach. Both models
wererunfor5,000minutestoensurethesystemreachedasteadystateandto
collect sufficient data samples. Key performance indicators include average
system time (total time from entry to exit) and maximum system time.
In the first model, I created three independent Create modules, each
with an arrival time set to an exponential distribution (10) minutes. Each
Create module feeds into a dedicated Process module with service times set
to UNIF(8,10) minutes. All entities then flow to a single Process module for
the checkout process, with a service time of UNIF(2.65,3.33) minutes.
Inthesecondmodel, IusedasingleCreatemodulefeedingintoaProcess
module with three resources, followed by the same single checkout counter.
**
Simulationresultsshowsignificantdifferencesinaverageandmaximumwait-
ingtimesbetweenthetwosystems. Detaileddataarepresentedintextboxes
within the Arena model files. The single-queue system generally exhibits
lower average waiting times and more balanced resource utilization, aligning
with queuing theory expectations. While the multi-queue system can offer
more personalized service in some cases, it tends to cause uneven resource
allocation.
**
Analysis of waiting time distributions reveals that the single-queue sys-
tem more effectively reduces "unfairness" in the system, where some cus-
tomers experience exceptionally long waits. This improvement is critical for
enhancing customer satisfaction.
Problem 4
The manufacturing system introduces one part every 10 minutes. It consists
of three workstations (A, B, and C), each equipped with one machine. The
system processes four types of parts, each occurring with equal probabil-
ity. Different part types follow distinct processing routes, with processing
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times at each workstation following triangular distributions. Transfer times
between workstations are uniformly set to 3 minutes.
I used Arena’s Sequence function to define the processing routes for the
four part types and the Sets function to collect cycle time data for each
type. Themodelincludesfullanimationtovisualizepartmovementthrough
the system. The simulation was run for 10,000 minutes to ensure adequate
sampling.
In the model, I first set up a Create module with a constant inter-arrival
time of 10 minutes. A Decide module then equally distributes parts into
four types. For each part type, I defined specific processing routes using the
Sequence function:
Part Type 1: Processed at Workstations A and C with processing times
Triangular(5,8.5,11.5) and (8,13.1,18.5) respectively.
Part Type 2: Processed at Workstations A, B, and C with times Triangu-
lar(8.9,13.5,18.1), (7,14,21), and (4.2,8.2,12.5).
Part Type 3: Processed at Workstations A and B with times Triangu-
lar(8.3,11.6,15.5) and (5.3,9.5,13.7).
Part Type 4: Processed at Workstations B and C with times Triangu-
lar(9.2,12.4,16) and (8.5,11.4,14).
Transfers between workstations are handled by Route modules with a
uniform transfer time of 3 minutes. I created four separate statistical sets
using the Sets function to track cycle times for each part type.
**
Simulation results show significant variations in average cycle times among
parttypes, directlyrelatedtotheirprocessingroutesandtimes. PartType2
has the longest cycle time due to processing at all three workstations. Work-
station utilization data indicates potential bottlenecks, particularly when
processing specific part types.
Regarding the number of simulation replications, I recommend conduct-
ing multiple independent runs to reduce the impact of random variability.
Typically, 30 replications provide sufficient statistical reliability, but the ex-
act number should be determined based on the variability observed in initial
results. Using a 95% confidence interval analysis, the required number of
replications can be calculated from the standard deviation of initial runs to
achieve the desired precision.
**
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