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we want to model vibrations caused by a particle getting caught up in a centrifugal pump with mass, m, in the impeller. the particle acts as a rotating unbalance mass, M_un, and this causes the pump to vibrate. assuming the pump sits on a vibration isolator(made with springs and dampers) and that m=10 kg, M_un=0.05 kg, damping ratio=0.01, e=10cm.
- Plot bode diagram (frequency response plot) for various spring constant (start from k=10 kN/m to k=100kN/m by increments of 10kN/m).
- use [x,y] =ginput commamd of matlab to assess what is the displacement at the peak of each case. you are allowed to use other program.
- which rotational speed (in terms of rpm) should be avoided for each spring stiffness?

Respuesta :

Answer:

I'm unable to create visual plots or execute code directly, but I can provide detailed instructions and guidance for you to accomplish the tasks using MATLAB or a similar program.

Explanation:

Modeling the Vibration System:

Define the system's parameters:

Mass of the pump (m) = 10 kg

Unbalance mass (M_un) = 0.05 kg

Damping ratio (ζ) = 0.01

Eccentricity (e) = 0.1 m

Create a function representing the system's transfer function:

H(s) = (M_un * e * s^2) / (m * s^2 + c * s + k), where:

s is the Laplace variable

c is the damping coefficient (c = 2 * ζ * √(m * k))

k is the spring constant (variable)

2. Plotting Bode Diagrams for Varying Spring Constants:

Use a loop to vary k from 10 kN/m to 100 kN/m in increments of 10 kN/m:

Within the loop:

Calculate the damping coefficient c for each k.

Obtain the frequency response using bode(H(s)).

Plot the magnitude and phase responses.

3. Assessing Displacement Peaks using ginput:

Pause each Bode plot using pause on.

Use ginput(1) to select the peak point with the mouse.

Store the selected frequency (x) and magnitude (y) values for each case.

4. Determining Rotational Speeds to Avoid:

Convert the peak frequencies from rad/s to RPM using RPM = (60 * frequency) / (2 * pi).

Identify these RPMs as the resonant frequencies to avoid for each spring stiffness.

Additional Considerations:

Adjust plot formatting and labels for clarity.

Consider using findpeaks to automatically detect peaks if manual selection is not feasible.

Explore alternative vibration isolation techniques if resonant frequencies cannot be avoided.

Software and Libraries:

MATLAB or compatible software with Control Systems Toolbox or similar capabilities.