Normal Modes and Normal Frequencies

In summary, According to the homework statement, the frequencies of the normal modes of oscillation for the system must be determined. The differential equations were solved and found that there are four solutions of the form \omega=\pm \sqrt{\frac{-(-4\omega_0^2 -2\tilde{\omega_0})\pm \sqrt{(-4\omega_0^2 -2\tilde{\omega_0})^2 -4(-4\omega_0^2 -2\tilde{\omega_0})(
  • #1
RicardoMP
49
2

Homework Statement


I have to determine the frequencies of the normal modes of oscillation for the system I've uploaded.

Homework Equations


[/B]
I determined the following differential equations for the coupled system:
[tex]\ddot{x_A}+2(\omega_0^2+\tilde{\omega_0}^2)x_A-\omega_0^2x_B = 0[/tex]
[tex]\ddot{x_B}+2\omega_0^2x_B-\omega_0^2x_A = 0[/tex]
which I confirmed are correct.

The Attempt at a Solution


I assumed that the usual normal modes solutions are:
[tex]x_A=Ccos(\omega t)[/tex]
[tex]x_B=C'cos(\omega t)[/tex]
where C and C' are the amplitudes for each of the oscillating masses and [tex]\omega[/tex] is the associated normal mode frequency. Therefore, I proceeded by determining the ratio [tex]\frac{C}{C'}[/tex] for each equation, after substituting the solutions in the differential equations.
[tex]\frac{C}{C'}=\frac{\omega_0^2}{-\omega^2 +2\omega_0^2+2\tilde{\omega_0}^2}[/tex] and [tex]\frac{C}{C'}=\frac{-\omega^2+2\omega_0^2}{\omega_0^2}[/tex]
My problem now is that, when I try to determine the solutions for [tex]\omega[/tex], I arrive to 4 solutions of the form:
[tex]\omega=\pm \sqrt{\frac{-(-4\omega_0^2 -2\tilde{\omega_0})\pm \sqrt{(-4\omega_0^2 -2\tilde{\omega_0})^2 -4(-4\omega_0^2 -2\tilde{\omega_0})(3\omega_0^3 +4\tilde{\omega_0}\omega_0)}}{2}}[/tex]
which I think it's wrong, since with these solutions I don't even know how to schematically draw the normal modes with these kind of solutions. So is my solving method wrong or is this method only applicable to symmetric systems? Did I get the calculations wrong? Is there another way to solve this problem?
Thank you in advance!
 

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  • #2
RicardoMP said:

Homework Statement


I have to determine the frequencies of the normal modes of oscillation for the system I've uploaded.

Homework Equations


[/B]
I determined the following differential equations for the coupled system:
[tex]\ddot{x_A}+2(\omega_0^2+\tilde{\omega_0}^2)x_A-\omega_0^2x_B = 0[/tex]
[tex]\ddot{x_B}+2\omega_0^2x_B-\omega_0^2x_A = 0[/tex]
which I confirmed are correct.

The Attempt at a Solution


I assumed that the usual normal modes solutions are:
[tex]x_A=Ccos(\omega t)[/tex]
[tex]x_B=C'cos(\omega t)[/tex]
where C and C' are the amplitudes for each of the oscillating masses and [tex]\omega[/tex] is the associated normal mode frequency. Therefore, I proceeded by determining the ratio [tex]\frac{C}{C'}[/tex] for each equation, after substituting the solutions in the differential equations.
[tex]\frac{C}{C'}=\frac{\omega_0^2}{-\omega^2 +2\omega_0^2+2\tilde{\omega_0}^2}[/tex] and [tex]\frac{C}{C'}=\frac{-\omega^2+2\omega_0^2}{\omega_0^2}[/tex]*
My problem now is that, when I try to determine the solutions for [tex]\omega[/tex], I arrive to 4 solutions of the form:
[tex]\omega=\pm \sqrt{\frac{-(-4\omega_0^2 -2\tilde{\omega_0})\pm \sqrt{(-4\omega_0^2 -2\tilde{\omega_0})^2 -4(-4\omega_0^2 -2\tilde{\omega_0})(3\omega_0^3 +4\tilde{\omega_0}\omega_0)}}{2}}[/tex]
which I think it's wrong, since with these solutions I don't even know how to schematically draw the normal modes with these kind of solutions. So is my solving method wrong or is this method only applicable to symmetric systems? Did I get the calculations wrong? Is there another way to solve this problem?
Thank you in advance!
What are the data? The masses of A and B, the spring constants. What do the ω 0-s mean?
Assuming your derivation is correct up to *, the last equation perhaps is not. First, ω should not be negative. Solve for ω2. You get two ω2-s, that is, two normal-mode frequencies, which correspond two kinds of motion of A and B, (in the same direction and in opposite directions). Check the powers and simplify the expression for ω2. Show your work. One can not find your mistakes without seeing it.
 
  • #3
ehild said:
What are the data? The masses of A and B, the spring constants. What do the ω 0-s mean?
Assuming your derivation is correct up to *, the last equation perhaps is not. First, ω should not be negative. Solve for ω2. You get two ω2-s, that is, two normal-mode frequencies, which correspond two kinds of motion of A and B, (in the same direction and in opposite directions). Check the powers and simplify the expression for ω2. Show your work. One can not find your mistakes without seeing it.
The relevant information for this problem is that the masses are the same [tex]m_A=m_B[/tex]. The [tex]\omega_0[/tex] and [tex]\tilde{\omega_0}[/tex] are the natural frequencies associated with each spring. The long springs have constant [tex]\tilde{k}[/tex], so [tex]\tilde{\omega_0}=\sqrt{\frac{\tilde{k}}{m}}[/tex]. The short springs have constant k, so [tex]\omega_0=\sqrt{\frac{k}{m}}[/tex]. Sorry for the lack of information.
So, assuming that up to * I'm correct (which I verified again), my calculations are the following:
[tex](-\omega^2 + 2\omega_0^2+2\tilde{\omega_0^2})(-\omega^2+2\omega_0^2)=(\omega_0^2)^2[/tex]
I call my \omega^2 = a, so [tex]a^2-2\omega_0^2a-2\omega_0^2a+4\omega_0^4-2\tilde{\omega_0}a+4\tilde{\omega_0}\omega_0=(\omega_0^2)^2[/tex].
And that's it. I solve the quadratic equation for [tex]\omega^2[/tex] and get the last equation in my original post without the large square root.
 
  • #4
I found at least one mistake, when solving the quadratic equation, dumb me!
The final equation I get, solving the quadratic equation for [tex]\omega^2[/tex] is :
[tex]\omega^2=\frac{(4\omega_0^2+2\tilde{\omega_0}^2)\pm \sqrt{(4\omega_0^2+2\tilde{\omega_0}^2)^2-4(3\omega_0^4+4\omega_0^2 \tilde{\omega_0^2}})}{2}[/tex]
and therefore:
[tex]\omega^2=\frac{(4\omega_0^2+2\tilde{\omega_0}^2)\pm 2\sqrt{\omega_0^4+\tilde{\omega_0^4}}}{2}[/tex]
 
  • #5
RicardoMP said:
So, assuming that up to * I'm correct (which I verified again), my calculations are the following:
[tex](-\omega^2 + 2\omega_0^2+2\tilde{\omega_0^2})(-\omega^2+2\omega_0^2)=(\omega_0^2)^2[/tex]
I call my \omega^2 = a, so [tex]a^2-2\omega_0^2a-2\omega_0^2a+4\omega_0^4-2\tilde{\omega_0}a+4\tilde{\omega_0}\omega_0=(\omega_0^2)^2[/tex].
.
You have mistakes again with powers and tilde.
 
  • #6
ehild said:
You have mistakes again with powers and tilde.
Yes, just corrected it in the reply above. Is my solving method correct at least? Won't I have 4 solutions for [tex]\omega[/tex]?
 
  • #7
RicardoMP said:
Yes, just corrected it in the reply above. Is my solving method correct at least? Won't I have 4 solutions for [tex]\omega[/tex]?

RicardoMP said:
I found at least one mistake, when solving the quadratic equation, dumb me!
The final equation I get, solving the quadratic equation for [tex]\omega^2[/tex] is :
[tex]\omega^2=\frac{(4\omega_0^2+2\tilde{\omega_0}^2)\pm \sqrt{(4\omega_0^2+2\tilde{\omega_0}^2)^2-4(3\omega_0^4+4\omega_0^2 \tilde{\omega_0^2}})}{2}[/tex]
and therefore:
[tex]\omega^2=\frac{(4\omega_0^2+2\tilde{\omega_0}^2)\pm 2\sqrt{\omega_0^4+\tilde{\omega_0^4}}}{2}[/tex]
It looks correct. Simplify by 2.
You get two values for ω2, and ω can not be negative, so it is 2 solutions for ω.
Remember, the solutions are of the form C*cos(ωt). Do you get different solution with -ω?
 
Last edited:

Related to Normal Modes and Normal Frequencies

1. What are normal modes and normal frequencies?

Normal modes refer to the different ways in which a system can vibrate or oscillate at a specific frequency. Normal frequencies, also known as eigenfrequencies, are the frequencies at which the system will naturally vibrate.

2. How are normal modes and normal frequencies calculated?

Normal modes and frequencies are calculated using mathematical models and equations, such as the mode shapes and eigenvalue equations. These equations take into account the physical properties and boundary conditions of the system to determine the possible modes and frequencies.

3. What is the significance of normal modes and normal frequencies?

Normal modes and frequencies are important in understanding the behavior and characteristics of a system. They can help identify potential resonant frequencies and predict how the system will respond to external forces.

4. How do normal modes and normal frequencies affect structural design?

In structural design, normal modes and frequencies are used to ensure that the structure can withstand natural oscillations and vibrations without experiencing excessive stress or damage. They are also considered in designing for specific performance requirements, such as minimizing noise or maximizing stability.

5. Can normal modes and normal frequencies change over time?

Yes, normal modes and frequencies can change over time due to factors such as changes in the system's physical properties, external forces, and environmental conditions. This is why it is important to regularly analyze and monitor these modes and frequencies in certain systems, such as bridges and buildings.

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