# How is equilibrium possible here? (motor + linkage) During my engineering course we were discussing systems in equilibrium and I came up with a situation that I don't understand.

Say we have an electric motor which drives a linkage consisting of 'disc 1' with radius r, 'link 1' and 'disc 2'. Disc 2 is attached to the link via an arm. Disc 1 and link 1 are connected with a hinge (red dot). The geometry allows for the motor to complete a full circle, and it spins with a constant velocity.

The motor gives disc 1 a moment/torque M. The effect of M on the hinge is the force F, which equals M/r. When the hinge is right above the center of disc 1, the force F on the hinge is completely horizontal, there are no y-components.

Since we are in a situation of equilibruium (no net moments or forces) and link 1 is a two-force-member, the forces on the link have the same direction. If these assumptions are true, then how can the hinge be in equilibrium?

Which part of my thought process is wrong here? Is it just impossible for the given situation to be in equilibrium?

A follow-up question is: what happens when the direction of the link is completely vertical, where does the horizontal component of force F go?

EDIT: The links and bodies are assumed to be massless and disc 2 turns around an axle (shown as a black point in the middle of the disc).

## 3 Answers

Is it just impossible for the given situation to be in equilibrium?

Yes. Imagine building your setup in a lab and then letting go of it. Would it stay in that configuration? (clearly not...)

• I don't understand your answer. Do you mean the motor wouldn't spin with a constant velocity? – Herbert van Even Feb 17 '18 at 19:38
• What is holding up disc 2? Or is there no gravity? The only way there could be no net forces given the geometry you've drawn is if the scenario is stationary without gravity or if the links are massless and there is not gravity. In both cases all forces would be zero, and therefore sum to zero. But those are pretty boring scenarios so I assumed that was not what you meant. – Duncan Harris Feb 17 '18 at 19:48
• I should have clarified that the bodies and links are asumed massless and disc 2 turns around an axle. – Herbert van Even Feb 17 '18 at 20:09
• If the links are massless then all forces are zero... – Duncan Harris Feb 17 '18 at 20:11
• I'll rephrase it like this: the links and bodies do have masses but their weight is neglected in comparison to force generated by the motor. – Herbert van Even Feb 17 '18 at 20:17

Linkages do not always carry both a vertical and horizontal force component. One of them can be zero.

The forces diagram for a link mechanism, such as the one you have for link 1 in your question, represents the general situation, not the situation at every moment. There is no contradiction to the generalized axial force on link 1 (F2) having a vertical component of 0 at a particular time (such as when the hinge on disc 1 is right above the centre of disc 1).

Force transmission between disc 1 and 2 is not uniform over time, but the instantaneous sum of forces is always balanced - in equilibrium.

When the hinge is right above the center of disc 1, the force F on the hinge is completely horizontal, there are no y-components.

No, the force F provided by the disc is not completely horizontal. Because the hinge is fixed to the disc, the disc also provides a vertical downwards force which prevents the hinge from moving upwards. This downward component balances the upward component of the tension F2 in the link.