ME 370 Fall 2025
Due (12/18/2025)
Instructions:
1- You are allowed to work in groups of up to four students.
2- Groups can be formed from students in different sections. Please ensure you list each student’s section in the report.
3- Only one report and one m-file per group should be submitted.
4- List all members' names in the report and the m-file.
5- Include a statement in your report clearly describing the contribution of each member. (-10 points if not included)
6- Thereport should be typed in Word—NO HANDWRITTEN REPORTS ALLOWED.
7- Include your plots in the report.
8- Make sure your code is working before uploading it to Canvas. IF CODE DOES NOT RUN, YOU'LL AUTOMATICALLY LOSE 50% OF THE PROBLEM'S GRADE.
9- Documents submitted past the deadline will not be accepted.
10- Do not include your MATLAB code in the report
NOTE: For questions about MATLAB, please contact your class TA. For any other inquiries, feel free to ask me.
Problem 1:
The shear building structure is a mechanical system with an infinite number of DOF, but it can be modeled as an equivalent spring-mass system, thereby creating a lumped mass system. This is commonly done to facilitate analysis, since in some engineering applications, the parameters of interest are the frequency and vibration modes. The minimum number of coordinates necessary to describe the motion of the lumped masses and rigid bodies defines the number of degrees of freedom of the system. The system can be modeled as unidimensional due to its vibration characteristic (i.e., the horizontal vibration is more representative than the others). A three-story shear building is studied and modeled as a 3-DOF spring-mass system. Figure 1 shows a schematic of the shear building structure.
Figure 1. a) Experimental model ofa three-story shear building; and b) Equivalent spring-mass model.
a. Use Newton’s second law of motion to derive the system’s EOM. (Show FBDs and all
derivations to receive full credit) (10 points)
b. Assume:
m1 = m2 = m3 = 5,000 kg
k1 = k2 = k3 = k = 2 kN/m
Using modal analysis, calculate the steady-state responses for:
Note: Both forces are applied to the first floor
1) F(t) = 300 cos 20t N (10 points)
2) F(t) = 300δ(t) N (10 points)
• For each case, clearly show your steps in your report and MATLAB code (your code should be organized similarly to “Example_431.m”).
• Include your MATLAB code with your submission (as a separate m-file). If your code doesn’t work, you will automatically lose 50%.
• Your code should plot both modal and physical solutions.
c. List the system’s natural frequencies in order and draw conceptually the three mode shapes ofthe equivalent lumped mass system. (5 points)
d. In reality, the material will exhibit some damping, preventing unbounded resonance responses. Assume a small damping ratio of ξ = 0.02 for all three modes. Redo the steady-state analysis, including damping, for the same input forces from part b. (15 points)
Problem 2:
With the development of social economy and construction technology, tall buildings are increasingly built in large cities, which are sensitive to earthquake and wind excitations. A tuned mass damper (TMD) is one of the most traditional vibration control devices, usually consisting of mass, stiffness, and damping elements. A pendulum TMD (PTMD) is a kind of horizontal TMD usually used to protect a tall building against horizontal vibration, where the pendulum provides the stiffness element ofTMD. The natural frequency of PTMD is a single value corresponding to the pendulum's length. You are helping design vibration control for a 3-story building in downtown Philadelphia (usually used for high buildings, but we will make an exception in this project). Your goal is to prevent excessive sway during storms and small seismic events. Using the shear-building model and a pendulum TMD mounted on the top floor, your team must tune the PTMD to ensure the upper floor motion stays below safe thresholds
Figure 2. a) A simplified building model including a pendulum swinging from the top lumped mass as a tuned mass damper for motion reduction. b) The equivalent spring-mass-pendulum system used to tune the tuned-mass-damper.
a. Derive the system’s EOM using the Euler-Lagrange method. Clearly show the system's kinetic and potential energies. Show all your derivations to receive full credit. (15 points)
b. Choose the values for m4 and L that will keep the oscillations ofthe third floor around 7 cm peak-to-peak in response to the impulse force from problem 1 hitting the first floor. Keep in mind that 0 < m! ≤ 3000 kg and 0 < L≤ 1 m. (15 points)
c. Repeat part b, including a damping ratio ξ = 0.02 for all three modes. Comment on the results and how they differ from part b. (15 points)
d. In your own words, describe what makes the PTMD reduce the oscillations ofall stories and discuss how the oscillations are affected by the values of m4 and L. (5 points)