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The forces F_k1 and F_k2 are the forces applied by the two springs. Notice that the free-body diagram is drawn so that F_k2 acts in opposite directions for body 1 and body 2, which ensures that Newton’s third law is satisfied. The force F_d is the force applied by the damper. The force F_f is the friction force applied to body 2 by the ground, which causes it to rotate in the clockwise direction.

Notice that there are no forces in the y or z directions for either body. This is because, from the constraints, there is no motion in those directions, and so the forces and moments in those directions are not necessary to develop the equations of motion. Next, the equations of motions are developed using Newton’s second law.

1.1 Body 1

Note that, as the motion for this body is constrained to the î direction only, one only needs to write the equation of motion in that direction. Applying Newton’s second law for a body with constant mass, written with the î-direction taken as positive:

(1) Mx₁̈ = F_k2 − F_k1 − F_d

It is now necessary to relate the forces to the generalized variables. In this step, it is necessary to be consistent in using the same sign convention from the free-body diagram and in Eq. 1↑. This results in:

(2) F_k1 = k₁x₁

(3) F_k2 = k₂(x₂ − x₁)

The spring forces given above are for a linear spring. For a linear spring, the force equation is spring constant × stretched length. In this problem, the generalized variables x₁ and x₂ are defined so that both springs are unstretched when x₁ = x₂ = 0. The stretched length of spring 1 depends only on x₁, while the stretched length of spring 2 depends on both generalized variables. For spring 2, the stretched length increases when the difference between x₂ and x₁ increases.

(4) F_d = cx₁̇

Substituting these relations into Eq. 1↑ yields

(5) Mx₁̈ = k₂(x₂ − x₁) − k₁x₁ − cx₁̇

(6) Mx₁̈ + cx₁̇ + (k₁ + k₂) − k₂x₂ = 0

Equation 6↑ is the equation of motion for body 1. However, there are still two unknown variables in the equation (x₁ and x₂) but only one equation. To get the other equation, Newton’s laws are applied to body 2. Note that we didn’t apply Newton’s laws for rotation of body 1 were not applied, because the constraints specify that body 1 does not rotate.

1.2 Body 2

Body 2 both translates in the î direction and rotates about the axis out of the plane. Therefore, Newton’s law for translation and rotation for this body are needed. From the free-body diagram, Newton’s second law for a constant-mass body is used, again being careful of the sign convention:

(7) mx₂̈ = − F_k2 − F_f

Substituting in the expression developed previously for F_k2:

(8) mx₂̈ = − k₂(x₂ − x₁) − F_f

There is currently no expression for F_f, but that is okay because we still need to write Newton’s law for rotation about the z axis. Newton’s second law for rotation should be applied at the center of mass. Considering a clockwise rotation to be positive (note that the constraint relation between x₂ and θ was written with this sign convention), the rotation equation is

(9) Iθ̈ = F_fR

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