Simultaneous equation of springs in series

• hallic
In summary, the conversation discusses a static spring system with 3 springs and 2 masses. The spring stiffnesses and natural lengths are provided, as well as the total length of the compressed springs. The speaker is looking for a way to find the individual values of the compressed springs using simultaneous equations and the iterative or elimination method. A summary of the equations and forces involved is provided.
hallic
I have a static spring system with 3 springs holding 2 masses in place I have the spring stiffnesses (K1=1N/m k2=3N/m k3=2N/m) and natural lengths (L1=3m L2=1m L3=2m) and the total length of the compressed springs (x1+x2+x3=4m)
I know that I derive simultanious equations and use the iterative or elimination method to solve what the x individual values are but I don't know how to get the equations for 3 springs in series and I can't find a relavent reference in my textbook other than the total of sthe spring constant

hallic said:
I have a static spring system with 3 springs holding 2 masses in place I have the spring stiffnesses (K1=1N/m k2=3N/m k3=2N/m) and natural lengths (L1=3m L2=1m L3=2m) and the total length of the compressed springs (x1+x2+x3=4m)
I know that I derive simultanious equations and use the iterative or elimination method to solve what the x individual values are but I don't know how to get the equations for 3 springs in series and I can't find a relavent reference in my textbook other than the total of sthe spring constant
Your question is rather vague. For one thing, what are you trying to find? The force each spring is exerting? The actual length of each of the compressed springs?

Let x1, x2, and x3 be the compressed length of the three springs, with x1 the length of the spring on the "left", corresponding to K1 and L1, x2 the length of the spring between the two masses, corresponding to K2 and L2, and x3 the length of the spring on the "right", corresponding to K3 and L3. The "compression" on the first spring is L1- x and so the force it is exerting is K1(L1- x1)= 1(3- x1). The compression on the second spring is L2- y and so the force it is exerting is K2(L2- x2)= 3(1- x2). The compression on the third spring is L3- x3 and so the force it is exerting is K3(L3- x3)= 2(2- x3).

The total force on the left mass is 3(1- x2)- 1(3- x1)= x1- 3x2, where I have taken the positive direction to be to the right. The total force on the right mass is 2(2- x3)- 3(1- x2)= -1+ 3x2- 2x3. Because the masses are not moving, those forces must be 0:
x1- 3x2= 0, -1+ 3x2- 2x3= 0, and, of course, x1+x2+ x3= 4.

what I was looking for was a set of simultaneous equations to use iterative and elimination methods on, thanks for the help

1. What is the concept behind simultaneous equations of springs in series?

Simultaneous equations of springs in series is a mathematical concept used to calculate the total displacement and force of multiple springs that are connected in a series. This involves using simultaneous equations to solve for the individual displacements and forces of each spring, and then combining them to determine the overall displacement and force of the system.

2. How do you set up simultaneous equations for springs in series?

To set up simultaneous equations for springs in series, you need to first identify all the springs in the system and assign variables to represent their individual displacements and forces. Then, using Hooke's Law, you can write an equation for each spring, setting the sum of the individual forces equal to the total force exerted on the system. Finally, you can solve the equations simultaneously to determine the values of each variable.

3. What is the significance of solving simultaneous equations for springs in series?

Solving simultaneous equations for springs in series allows us to accurately predict the behavior of a complex spring system. By calculating the individual displacements and forces of each spring, we can understand how the system will respond to an external force and determine the overall displacement and force of the system. This is crucial in designing and analyzing various mechanical and engineering systems that involve multiple springs.

4. Are there any limitations to using simultaneous equations for springs in series?

One limitation of using simultaneous equations for springs in series is that it assumes the springs are ideal, meaning they follow Hooke's Law perfectly and do not experience any deformations or energy losses. In reality, most springs have some level of imperfection that can affect the accuracy of the calculations. Additionally, this method may become more complex and time-consuming when dealing with a large number of springs in a system.

5. Can simultaneous equations for springs in series be applied to real-world situations?

Yes, simultaneous equations for springs in series can be applied to real-world situations. In fact, it is commonly used in various engineering and physics applications, such as designing suspension systems, analyzing the forces in a trampoline, and predicting the behavior of a car's suspension system. However, it is important to consider the limitations and assumptions of this method when applying it to real-world scenarios.

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