Forced vibration on mass between two pull springs

In summary, the equation of motion for a mass with a pre-stretched pull spring is: L = \frac{1}{2}m v^2 - \frac{1}{2}k_2 (x_2-x)^2 - \frac{1}{2} k_1 (x-x_1)^2. If x=0 is the equilibrium position, then k_1 x_1 + k_2 x_2 = 0.
  • #1
Kurt Couffez
5
0
Consider a mass m with a prestressed pull spring on either end, each attached to a wall. Let k1 and k2 be the pull spring constants of the springs. A displacement of the mass by a distance x results in the first spring k1 lengthening by a distance x(and pulling in the - direction), while the second spring k2 is abbreviating by a distance x (and pulls in the positive direction).
upload_2016-10-19_19-57-6.png


The equation of motion then becomes:
upload_2016-10-19_19-57-38.png


Three questions:

1. Is it correct that
upload_2016-10-19_19-58-18.png


2. If the answer on question (1) is yes, what is ω if k2>k1

3. Is is possible to resonate with a force F=A.cos(ωt) en what would be the amplitude of this force.
 
Physics news on Phys.org
  • #2
What is the resting equilibrium position for each spring individually?
The way you have it written is at x=0, but you say something about prestressed in your description so it isn't clear.
 
  • #3
Lets say the resting equilibrium position for the springs are x1 and x2. The reason why they have to be prestressed is because they are pull springs. They can only have a force in one direction. They both are pulling at any time at the mass. When they are not prestressed then, at the position x=0 (with the springs attached) they would be pressed together.
upload_2016-10-19_21-42-52.png
 
  • #4
Kurt Couffez said:
Lets say the resting equilibrium position for the springs are x1 and x2. The reason why they have to be prestressed is because they are pull springs. They can only have a force in one direction. They both are pulling at any time at the mass. When they are not prestressed then, at the position x=0 (with the springs attached) they would be pressed together.
View attachment 107716

I do not think it even matters. The forces of pre-stretched springs eliminate each other at all times.
 
  • #5
It should be
##ma = (k_1+k_2)x##
not
##ma = (k_1-k_2)x##

You can show it using the Euler-Lagrange equation.
##L = \frac{1}{2}m v^2 - \frac{1}{2}k_2 (x_2-x)^2 - \frac{1}{2} k_1 (x-x_1)^2##
##\frac{\partial L}{\partial x} = k_2 (x_2-x) - k_1 (x-x_1) = x (k_1 + k_2) + k_1 x_1 + k_2 x_2##
If x=0 is the equilibrium position, then ##k_1 x_1 + k_2 x_2 = 0##
 
  • #6
Why is in this the enegery of both springs negative? Spring 1 is pulling upwards, spring 2 is pulling downwards.
 
  • #7
Ok, I have discovered the error. The restoring force F2 should also be upwards. Spring 2 also wants to bring the mass to its equilibrium position even if it is a pull spring.
The question around the negative energy of the spring is also solved.

Thanks.
 

FAQ: Forced vibration on mass between two pull springs

1. What is forced vibration on a mass between two pull springs?

Forced vibration on a mass between two pull springs refers to the oscillation or movement of a mass that is constrained between two pull springs and is subjected to an external force or input. This type of vibration occurs when the frequency of the external force matches the natural frequency of the system.

2. What factors affect the forced vibration on a mass between two pull springs?

The forced vibration on a mass between two pull springs is affected by several factors, including the stiffness of the pull springs, the mass of the object, and the frequency and amplitude of the external force.

3. How does the amplitude of the external force affect the forced vibration?

The amplitude of the external force directly affects the amplitude of the forced vibration on the mass between two pull springs. A larger amplitude of the external force will result in a larger amplitude of the forced vibration.

4. Can the natural frequency of the system be changed?

Yes, the natural frequency of the system can be changed by altering the stiffness of the pull springs or by changing the mass of the object. This can be done to avoid resonance, which can cause excessive vibration and potential damage to the system.

5. How can forced vibration on a mass between two pull springs be controlled?

Forced vibration on a mass between two pull springs can be controlled by adjusting the stiffness of the pull springs, ensuring that it does not match the natural frequency of the system. Additionally, damping techniques can be used to reduce the amplitude of the vibration and prevent potential damage to the system.

Back
Top