Objective: (3 sentences)
The goal of this experiment is to study the dynamic response of a two-mass system under different loading conditions. By analyzing the system’s behavior with and without a dashpot, the experiment will determine key parameters such as natural frequencies, damping ratios, and spring constants. This will provide insights into how the system's masses, damping, and spring elements influence its vibrational characteristics and overall stability.

1. Theory Overview

Mechanical systems subjected to dynamic forces often exhibit vibrational behavior that can be characterized by key parameters such as natural frequency and damping ratio. Understanding these parameters is essential for predicting the system's response to external forces and for designing systems that operate safely within desired conditions. In this context, the study of free vibrations in a mass-spring-damper system provides foundational insights into how systems oscillate under their own natural dynamics without any external input.

A mass-spring-damper system is a classic model used to describe oscillatory motion in many physical systems. The equation of motion for such a system is derived from Newton’s second law, incorporating contributions from inertia (mass), restoring force (spring), and resistance to motion (damping). In the unforced case, where no external force is applied, the system’s motion is governed solely by its initial displacement and velocity.

The natural frequency ω_n​, which is a function of the spring stiffness k and mass m, represents the frequency at which the system would oscillate if there were no damping. Damping, quantified by the damping ratio ζ, describes how the system's oscillations decay over time due to energy dissipation. In lightly damped systems, oscillations persist for a while before eventually decaying, whereas in heavily damped systems, oscillations are rapidly attenuated.

The solution to the unforced system's motion is harmonic in nature and can be used to identify important parameters such as the natural frequency, period of oscillation, and damping behavior. These parameters are crucial in analyzing the vibrational response of mechanical systems.

Suggested reading:

Inman, Daniel J. - Engineering Vibrations - (2013)

2. Pre-Lab Quiz

• Quiz Date: xxxxxx

• The quiz will cover the mass spring damper system and system control .

• Review system theory.

3. Experimental Procedure

• Step-by-Step Instructions:

  • Procedure:

    First, the relevant parameters such as the mass, spring stiffness and damping coefficients will be determined in various configurations shown in Fig. 4.

    

    Before starting, identify the way the mass carriages are labeled: one, two and three from left to right in Fig. 4.

    1. Clamp the second mass to put the mechanism in the configuration shown in Figure 4a above using a shim (e.g. 1/4 inch nut) between the stop tab and stop bumper. Connect the first and second mass carriages by a spring.

    2. Secure four 500g masses on the first and second mass carriages. The 2^nd mass carriage is fixed not to move in this part of the experiment. 

    3. Set up the data acquisition. With the controller powered up, enter the Control Algorithm box via the Set-up menu and set Ts = 0.00442 s. Enter the Command menu, go to Trajectory and select Step, Set-up.  Select Open Loop Step and input a step size of 0, a duration of 3000 ms and 1 repetition.  Exit to the background screen by consecutively selecting OK.  This puts the controller in a mode for acquiring 6 sec of data on command but without driving the actuator.  This procedure may be repeated and the duration adjusted to vary the data acquisition period.

    4. Go to Set up Data Acquisition in the Data menu and select Encoder 1 as data to acquire and specify data sampling every 2 (two) servo cycles  (i.e. every 2 Ts's).  Select OK to exit.  Select Zero Position from the Utility menu to zero the encoder positions.

    5. Select Execute from the Command menu.  Manually displace the first mass carriage approximately 2.5 cm in either direction. select Run from the Execute box and release the mass approximately 1 second later. The mass will oscillate and attenuate while encoder data is collected to record this response. Select OK after data is uploaded.

    6. Select Set-up Plot from the Plotting menu and choose Encoder #1 Position then select Plot Data from the Plotting menu. You will see the first mass time response.

    7. Choose several consecutive cycles (say ~5) in the amplitude range between 5500 and 1000 counts (This is representative of oscillation amplitudes during later closed loop control maneuvers. Much smaller amplitude responses become dominated by nonlinear friction effects and do not reflect the salient system dynamics). Divide the number of cycles by the time taken to complete them being sure to take beginning and end times from the same phase of the respective cycles.  Convert the resulting frequency from Hz to rad/sec.  This damped frequency, ω_d, approximates the natural frequency, ω_n, according to:

    

    where the "m₁₁" subscript denotes mass carriage #1, trial #1.

    8. Next, remove the four masses from the first mass carriage and repeat Steps 5 through 7 to obtain w_{n_m12} (natural frequency for mass carriage #1, trial #2) for the unloaded carriage. Shorten the test duration set in Step 3 if needed.

    9. Measure the initial cycle amplitude Xo and the n^th cycle amplitude Xn for the n cycles of the plot generated in Step 8 (choose any n cycles from the plot).  Determine the damping ratio for this case using the following equation of logarithmic decrement. Find the damping ratio ζ_m12 and show that for this small value the approximations of Eq's (1-1, -2) are valid.

    

    10.    Repeat Steps 5 through 9 for the second mass carriage. Here in Step 6 you will need to remove Encoder #1 position and add Encoder #2 position to the plot set-up. Hence obtain ω_{n_m21} , ω_{n_m22} and ζ_m22. How does this damping ratio compare with that for the first mass?

    11. Connect the mass carriage extension bracket and dashpot to the second mass as shown in Figure 6.1-2c. Open the damping (air flow) adjustment knob 2.0 turns from the fully closed position. Repeat Steps 5, 6, and 9 with four 500 g masses on the second carriage and using only amplitudes ≥ 500 counts in your damping ratio calculation. Hence obtain ζ_d where the "d" subscript denotes "dashpot".

    12.  The mass of each brass weight is 0.5kg.  Let m_w be the total mass of the four weights combined. Use the following relationships to solve for the mass of the unloaded carriage, m_c2, and spring constant, k:                       

    

    Find the damping coefficient c_m2 by equating the first order terms in the following equation:

                             

    where ζ is ζ_m22 , ω_n is ω_{n_m22}, and m is m_c2 in this case.  Repeat the above for the  first mass carriage, spring and damping m_c1, c_m1 and k respectively. Calculate the damping coefficient of the dashpot, c_d.

4. Data Collection

• During the lab, observe the TA during experiment, and keep track of the experiments & recorded data.

5. Post-Lab Report

• Due Date: xxxxxx

The report for this part is expected to include: 

·        Abstract and Introduction

·        Procedures and Steps

·        Experiment Related (Plots & Calculations)

·        Discussion and Conclusion

Experiment Related (Plots & Calculations)

Plots used to determine the natural frequency of the system, along with titles, labels to clearly show which plot corresponds to which situation. Plots include: 

• Plot of Mass 1 loaded

• Plot of Mass 1 unloaded

• Plot of Mass 2 loaded

• Plot of Mass 2 unloaded

• Plot of Mass 2 loaded with dashpot

Calculations showing how you found the following values, along with units for every quantity.           

• Mass 1 loaded natural frequency, w_nm11

• Mass 1 unloaded natural frequency, w_nm12

• Mass 1 unloaded 2 damping ratio, z_m12 

• Mass 2 loaded natural frequency, w_nm21

• Mass 2 unloaded natural frequency, w_nm22

• Mass 2 unloaded damping ratio, z_m22

• Mass 2 damping ratio loaded with dashpot connected, z_d 

• Spring constant, k

• Mass of Carriage 1 plus driving unit, m_c1

• Mass of Carriage 2, m_c2

• Mass 1 & driving unit damping coefficient, c_m1

• Mass 2 damping coefficient, c_m2

• Dashpot damping coefficient, c_d

Additional Resources:

• [Video Tutorial on Vibration Analysis] (Link)

• [Sample Report Format] (Link)

Responsible TAs:

• Alp Toymus, atoymus21@ku.edu.tr