How to Determine the Force of a Spring-Damper in 3D
If you often have to calculate forces between objects and you want to have a systematic procedure for determining the forces, you have come to the right place. In this article, we will explain how to determine the force of a spring-damper in 3D. We will explain the general case with the distance r0, as well as the case where r0=0.
The Set-Up of the Problem
Imagine two point masses with mass m1 and m2 which are connected by a linear spring (relaxed if the distance between both masses is r0; spring constant c) and a linear viscous damper (damping constant d). The positions of both masses are given by the position vectors r1 and r2. The forces F1 and F2=-F1 are the internal forces that result from creating the free body diagram. Our goal is to write down an expression for the force F1 as a function of r1,r2, r˙1,r˙2, r0 as well as the parameters c and d.
The Damping Force Fd
The damping force Fd is given by:
This force is proportional to the velocity difference between the two masses and opposes the relative motion of the masses. The damping force is always a dissipative force, which means it does negative work on the system.
The Spring Force Fs
The spring force Fs is given by:
This force is proportional to the displacement of the spring from its equilibrium position and tries to restore the spring to its equilibrium position. If the displacement from the equilibrium position is positive, the spring force is negative and vice versa. The spring force is always a conservative force, which means it does zero net work on the system over one complete cycle.
The Total Internal Force F1
The total internal force F1 is the result of the addition of the spring force Fs and the damping force Fd. Hence,
Expression for the Force F1 with r0=0
If we assume r0=0, the spring is unstretched in its equilibrium condition. Therefore, the spring force is zero, and only the damping force acts on the system. The internal force acting on mass 1 can be written as:
Expression for the Force F1 with r0 ≠ 0
In this case, both the spring and the damper forces are acting on the system. If we attach the spring to mass 2, the direction of the spring force can be written as:
The direction of the damping force is the same as before. Therefore, the internal force acting on mass 1 can be written as:
Conclusion
We have explained how to determine the force of a spring-damper in 3D. The damping force and spring force are the two forces that act on the system. The damping force opposes the relative motion of the masses, while the spring force tries to restore the spring to its equilibrium position. We have given expressions for the force F1 with r0=0 and r0≠0.
How to Determine the Force of a Spring-damper In 3d
How to Determine the Force of a Spring-Damper in 3D
If you often have to calculate forces between objects and you want to have a systematic procedure for determining the forces, you have come to the right place. In this article, we will explain how to determine the force of a spring-damper in 3D. We will explain the general case with the distance r0, as well as the case where r0=0.
The Set-Up of the Problem
Imagine two point masses with mass m1 and m2 which are connected by a linear spring (relaxed if the distance between both masses is r0; spring constant c) and a linear viscous damper (damping constant d). The positions of both masses are given by the position vectors r1 and r2. The forces F1 and F2=-F1 are the internal forces that result from creating the free body diagram. Our goal is to write down an expression for the force F1 as a function of r1,r2, r˙1,r˙2, r0 as well as the parameters c and d.
The Damping Force Fd
The damping force Fd is given by:
This force is proportional to the velocity difference between the two masses and opposes the relative motion of the masses. The damping force is always a dissipative force, which means it does negative work on the system.
The Spring Force Fs
The spring force Fs is given by:
This force is proportional to the displacement of the spring from its equilibrium position and tries to restore the spring to its equilibrium position. If the displacement from the equilibrium position is positive, the spring force is negative and vice versa. The spring force is always a conservative force, which means it does zero net work on the system over one complete cycle.
The Total Internal Force F1
The total internal force F1 is the result of the addition of the spring force Fs and the damping force Fd. Hence,
Expression for the Force F1 with r0=0
If we assume r0=0, the spring is unstretched in its equilibrium condition. Therefore, the spring force is zero, and only the damping force acts on the system. The internal force acting on mass 1 can be written as:
Expression for the Force F1 with r0 ≠ 0
In this case, both the spring and the damper forces are acting on the system. If we attach the spring to mass 2, the direction of the spring force can be written as:
The direction of the damping force is the same as before. Therefore, the internal force acting on mass 1 can be written as:
Conclusion
We have explained how to determine the force of a spring-damper in 3D. The damping force and spring force are the two forces that act on the system. The damping force opposes the relative motion of the masses, while the spring force tries to restore the spring to its equilibrium position. We have given expressions for the force F1 with r0=0 and r0≠0.