How do I correctly analyze the mass balance of a tractor seat?

[Sunwoo Bae]

Homework Statement

The system under consideration is the driver seat of a tractor. Research has shown that excessive
vibrations of the seat during driving adversely affect the driver’s health. In order to reduce
vibrations, the seat is equipped with a shock absorber. This shock absorber can be modelled as a
spring-mass-damper system as shown in Figure 3.
Consider that the driver seat has a certain mass m. Assume that an external input force Fe
acts vertically downwards on the driver’s seat. Denote the spring constant with k and damping
coefficient with c. Write a mathematical model for the driver seat-shock absorber system. What
are the parameters and constants in the model?

Relevant Equations

F =ma

Given is the diagram shown in the context:

My solution:

However, the correct solution is

I am confused with the direction and signs of the system. How would I derive the solution?
Thank you for your help.

 

[erobz]

I am confused with the direction and signs of the system. How would I derive the solution?
Thank you for your help.

I think for starters you must clearly label/make a coordinate system.

 

[haruspex]

Homework Statement: The system under consideration is the driver seat of a tractor. Research has shown that excessive
vibrations of the seat during driving adversely affect the driver’s health. In order to reduce
vibrations, the seat is equipped with a shock absorber. This shock absorber can be modelled as a
spring-mass-damper system as shown in Figure 3.
Consider that the driver seat has a certain mass m. Assume that an external input force Fe
acts vertically downwards on the driver’s seat. Denote the spring constant with k and damping
coefficient with c. Write a mathematical model for the driver seat-shock absorber system. What
are the parameters and constants in the model?
Relevant Equations: F =ma

Given is the diagram shown in the context:
View attachment 353564
My solution:
View attachment 353565

However, the correct solution is
View attachment 353566

I am confused with the direction and signs of the system. How would I derive the solution?
Thank you for your help.

Which way are you taking as positive for y? The book is taking down as positive.
When the spring is extended beyond its relaxed position, does it act to increase or reduce that extension?

Btw, it is very odd to consider the mass of the seat but ignore the mass of the occupant.

 

[Steve4Physics]

My solution:
View attachment 353565

A secondary point, but worth mentioning, is that the second derivative of y(t) with respect to t is written as $\frac {d^2y}{dt^2}$ and not as $\frac {d^2y}{dt}$. Or, more compactly, you could use $\ddot y$.