Free Vibration- Viscous Damping

[rmunoz]

Homework Statement

Scenario 1: Mass suspended from dasphot (damper) and spring
Mass=M
Damping Coefficient=c
Spring Constant=K

Scenario 2: Mass supported by dashpot (damper) and spring
Mass=M
Damping Coefficient=c
Spring Constant=K

In both cases, derive the equations of motion assuming that each mass is at some point displaced from static equilibrium.

Justify the use of positive or negative signs in the problem

2. Relevant Concepts
D'almbert's principle
Fspring= -kx
Fdamping= cx'

The Attempt at a Solution

Sign Conventions are throwing me for a loop. The equation will in both cases be of the form:
mx"+cx'+k(deltastatic+x)=weight

solutions will be of the form:
x(t)=Ae^(a+lamda*i)t + Be^(a-lambda*i)t


BUT HOW DO I JUSTIFY THE SIGN CONVENTIONS!? Free body diagrams are really confusing me in what is a relatively simple problem

 

[KataKoniK]

.

Thank you for your post. It seems like you are struggling with understanding the significance of sign conventions in this problem. Let me explain it to you in detail.

Firstly, in both scenarios, the mass is displaced from its static equilibrium position. This means that the displacement, represented by "x" in the equations, will be a positive value. This is because the mass is moving away from its equilibrium point in the positive direction. Therefore, the weight of the mass will also have a positive value in the equations since it is acting in the opposite direction of the displacement.

Secondly, let's consider the force of the spring and the damping force. According to Hooke's law, the force of the spring is given by F_spring = -kx, where "k" is the spring constant and "x" is the displacement of the mass from its equilibrium position. The negative sign in this equation signifies that the force of the spring is always acting in the direction opposite to the displacement. This is because the spring is trying to restore the mass back to its equilibrium position. Similarly, the damping force is given by F_damping = cx', where "c" is the damping coefficient and "x'" is the velocity of the mass. The negative sign in this equation signifies that the damping force is always acting in the direction opposite to the velocity of the mass. This is because the damping force opposes the motion of the mass and tries to bring it to rest.

Now, let's apply these sign conventions to the equations of motion. In both scenarios, the equation of motion will be of the form mx'' + cx' + kx = weight. As mentioned earlier, the displacement and weight will have positive values in the equations. The damping force, being in the direction opposite to the velocity, will have a negative sign. Similarly, the spring force, being in the direction opposite to the displacement, will also have a negative sign. This will result in a negative value for the spring force term in the equations.

In summary, the positive and negative signs in the equations of motion signify the direction of the forces acting on the mass. It is important to follow these conventions to ensure that the equations accurately represent the physical situation.

I hope this explanation helps you in understanding the sign conventions in this problem. If you have any further questions, please do not hesitate to ask. Best of luck with your work.
Scientist