I've got a mechanism that uses a bimetallic strip; for aesthetic reasons I'd like to electroplate it with a thin layer of gold - how does this affect the relative displacement of a bimetallic strip in response to temperature? My intuition tells me that if the plating covers the entire assembly, it should all be subject to the same expansion, and shouldn't affect the relative displacement of the strip (accounting for the obvious stiffening effect of adding more material), right? Any advice is appreciated, I'm having trouble thinking about this intuitively.
Why a bimetallic strip heating up is a bending problem
When a bimetallic strip is heated by $\Delta T$, from its original temperature, what happens is that one strip tends to grow longer and the other less. because they are bonded together, they end up to bend and rotate.
Figure 1: Diagram of a bimetallic strip showing how the difference in thermal expansion in the two metals leads to a much larger sideways displacement of the strip (Source: wikiwand)
The bend happens because a moment couple is generated, because one strip applies a constraint to the other (i.e. the one with the higher expansion tries to pull the other one with the lower $\alpha$, and vice versa)
Figure 1: Moment Couple on a bimetallic (Source: wikiwand)
So this ends up being a problem being a bending problem with a bending moment which can be calculated from the diagram above.
Bending problem considerations.
The main issues are how much does the second moment of area changes, and also how does the stresses change.
Just for reference, a bimetallic strip is between 0.5 and 10 mm (sometimes more), and gold electroplating is in the 0.00025 – 0.005 mm .
The second moment of area is affected by the plating -albeit not significantly) because the the gold plating will be further away from the neutral axis, therefore it will have the highest contibution for a layer with that thickness. However, overall because the thickness of the plating is at best 100 less, its effect would be at most about 5% of the overall contribution. If you take into consideration that the steel modulus is also greater than gold, that will probably reduce even further the overall effect.
The other part, is the distribution of the stresses. The presence of the gold plating would impose again another constrain (i.e. try to expand equally both top and bottom layers). That will create an opposite bending moment and the overall bending moment will be slightly less. However, again that will depend on the ratio of thickness of the electroplating and the bimetallic strip.
Plating will decrease the bending by a small nominal amount because the gold will expand equally on either side holding back the metal with highier thermal expansion index and encouraging the one with lower expansion index to expand.
But at the typical thicknesses of 0.00001" to o.ooo1" it's effect is negligible.