November 6, 2019

6.002 Recitation Notes - Fall 2019

Translational Mechanics

Farnaz Niroui, Rm 13-3005B, [email protected]

The behavior of various physical systems, for example those that are electrical, mechanical, or thermal in nature, can be modeled based on topologies consisting of discrete entities in an approach that helps simplify their analysis. There are also analogies between these systems that can be made to help gain intuition about a particular system based on its equivalent counterpart in another domain. The analogy that we are in particular interested in is between the electrical and mechanical systems. There are two valid ways of modeling mechanical systems with electrical circuit components. In the first method, current-velocity, and voltage-force are considered analogous. Alternatively, in the second method, voltage-velocity, and current-force are considered analogous. The latter, which is referred to as the mobility (or admittance) analogy, is advantageous in that it helps us maintain the topology of the system that we are working with as we move between the mechanical and electrical domains. This is the approach that we will focus on here.

METHOD 1 METHOD 2 Direct/Impedance Analogy Indirect/Mobility/Admittance Analogy Force à Voltage Force à Current Velocity à Current Velocity à Voltage

1 November 6, 2019

Mechanical System Electrical Analogy

dx dv d� d� F = m = m i = m = C dt dt dt dt

C ≈ m

dx dv F = m + kx = m + k v dt dt dt d� i = m + k � dt dt d� 1 i = C + � dt dt L

C ≈ m 1 L ≈ k

dx dx dv d� F = m + b = m + bv i = m + b� dt dt dt dt d� 1 i = C + � dt R

C ≈ m 1 R ≈ b

2 November 6, 2019

� = L v = i = C F = m � = Ri v = F

Mechanical System Electrical Analogy Power P = Fv P = i� KVL v = 0 � = 0 KCL F = 0 i = 0

Consider the mechanical system shown below and find its electrical analogy.

Using KCL: d� 1 1 i = C + � + � dt F = F + F + F dt R L dx dx Note that � = �, � = �, � = , � = F = m + b + kx dt dt d� dv F = m + b� + k � dt F = m + bv + k v dt dt dt

As we covered in the last recitation, the RLC parallel circuit behaves as shown below.

3 November 6, 2019

Apply KCL to the top node in the impedance model. � � � � = + + �� � 1 �� 1 1 � = ( + + ��)� �� � �� � � = = � 1 1 � 1 + + �� � + + �� � �� �� ��� � = � �� 1 −� + + �� ��

The resonance frequency � = . By analogy the mechanical system will have a resonant frequency of � = .

Determine the electrical circuit analogy of the following mechanical model.

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