12.3: Mechanical Energy Conversion
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- 19015
The previous section summarized how the language of calculus of variations can be applied to electrical and electromagnetic energy conversion devices. Similarly, this language can be used to describe energy conversion processes occurring in linear springs, torsion springs, moving masses, and flywheels. We can convert energy to and from spring potential energy by compressing and releasing a spring. Similarly, we can store or release energy from a moving mass by changing its velocity. A flywheel is a device that stores energy in a spinning mass. Flywheels are used, in addition to batteries, in some electric and hybrid vehicles because storing rotational kinetic energy in a flywheel requires fewer energy conversion processes than storing energy in a battery. All of these energy conversion devices can be described in the language of calculus of variations with some parameter chosen as the generalized path.
Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) summarize the parameters resulting from describing mechanical energy conversion processes in the language of calculus of variations. While electromagnetic systems are described by four vector fields, mechanical systems are described by eight possible vector fields, and they are listed along with their units in Table \(\PageIndex{3}\). Each column of Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) describes the case of choosing a different vector field from Table \(\PageIndex{3}\) as the generalized path. By comparing across the rows of these tables as well as the electrical tables, comparisons can be made between the different energy conversion processes.
| Energy storage device | Linear Spring | Linear Spring | Flywheel | Flywheel |
|---|---|---|---|---|
| Generalized Path | \(\overrightarrow{F}\) Force in \(\frac{J}{m}\) = N | Displacement \(\overrightarrow{x}\) in m | \(\overrightarrow{\omega_{ang}}\) Angular Velocity in \(\frac{rad}{s}\) | \(\overrightarrow{L_{am}}\) Angular Momentum in \(J \cdot s\) |
| Generalized Potential | Displacement \(\overrightarrow{x}\) in m | \(\overrightarrow{F}\) Force in \(\frac{J}{m}\) = N | \(\overrightarrow{L_{am}}\) Angular Momentum in \(J \cdot s\) | \(\overrightarrow{\omega_{ang}}\) Angular Velocity in \(\frac{rad}{s}\) |
| Generalized Capacity | \(K\) in \(\frac{J}{m^2}\) | \(\frac{1}{K}\) in \(\frac{m^2}{J}\) | \(\frac{1}{\mathbb{I}}\) in \(\frac{1}{kg \cdot m^2}\) | \(\mathbb{I}\) in \(kg \cdot m^2\) |
| Constitutive relationship | \(\overrightarrow{F} = K \overrightarrow{x}\) | \(\overrightarrow{x} = \frac{1}{K} \overrightarrow{F}\) | \(\overrightarrow{\omega_{ang}} = \frac{1}{\mathbb{I}} \overrightarrow{L_{am}}\) | \(\overrightarrow{L_{am}} = \mathbb{I} \overrightarrow{\omega_{ang}}\) |
| Energy | \(\frac{1}{2} K|\overrightarrow{x}|^2 = \frac{1}{2}\frac{1}{K} |\overrightarrow{F}|^2\) | \(\frac{1}{2}\mathbb{I} |\overrightarrow{\omega_{ang}}|^2 = \frac{1}{2}\frac{1}{\mathbb{I}} |\overrightarrow{L_{am}}|^2\) | ||
| Law for potential | Newton's Second Law \(\overrightarrow{F} = m\overrightarrow{a}\) | Newton's Second Law \(\overrightarrow{F} = m\overrightarrow{a}\) | Conservation of Angular Momentum | Conservation of Angular Momentum |
In Section 11.4, energy conversion in a linear spring was discussed in the language of calculus of variations. That example considered the displacement of a point mass \(m\) in kg where the generalized path was chosen to be displacement \(x\) in m. The resulting Euler-Lagrange equation was Newton's second law. Section 11.4 concluded with Table 11.4.1 summarizing the resulting parameters. The third column of the Table \(\PageIndex{1}\) repeats that information.
| Energy storage device | Moving Mass | Moving Mass | Torsion Spring | Torsion Spring |
|---|---|---|---|---|
| Generalized Path | \(\overrightarrow{M}\) Momentum in \(\frac{kg \cdot m}{s} = \frac{J \cdot s}{m}\) | \(\overrightarrow{v}\) Velocity in \(\frac{m}{s}\) | \(\overrightarrow{\tau}\) torque in\( \frac{N \cdot m}{\text{rad}} = \frac{J}{\text{rad}}\) | Angular Displacement \(\overrightarrow{\theta}\) in radians |
| Generalized Potential | \(\overrightarrow{v}\) Velocity in \(\frac{m}{s}\) | \(\overrightarrow{M}\) Momentum in \(\frac{kg \cdot m}{s} = \frac{J \cdot s}{m}\) | Angular Displacement \(\overrightarrow{\theta}\) in radians | \(\overrightarrow{\tau}\) torque in\( \frac{N \cdot m}{\text{rad}} = \frac{J}{\text{rad}}\) |
| Generalized Capacity | \(m\) in kg | \(\frac{1}{m}\) in \(\frac{1}{kg}\) | \(\mathbb{K}\) in \(\frac{J}{\text{rad}^2}\) | \(\frac{1}{\mathbb{K}}\) in \(\frac{\text{rad}^2}{J}\) |
| Constitutive relationship | \(\overrightarrow{M} = m \overrightarrow{v}\) | \(\overrightarrow{v} = \frac{1}{m} \overrightarrow{M}\) | \(\overrightarrow{\tau} = \mathbb{K} \overrightarrow{\theta}\) | \(\overrightarrow{\theta} = \frac{1}{\mathbb{K}} \overrightarrow{\tau}\) |
| Energy | \(\frac{1}{2}m|\overrightarrow{v}|^2 = \frac{1}{2}\frac{|\overrightarrow{M}|^2}{m}\) | \(\frac{1}{2}m|\overrightarrow{v}|^2 = \frac{1}{2}\frac{|\overrightarrow{M}|^2}{m}\) | \(\frac{1}{2}\mathbb{K}|\overrightarrow{\theta}|^2 = \frac{1}{2}\mathbb{K}|\overrightarrow{\tau}|^2\) | \(\frac{1}{2}\mathbb{K}|\overrightarrow{\theta}|^2 = \frac{1}{2}\mathbb{K}|\overrightarrow{\tau}|^2\) |
| Law for potential | Conservation of Momentum | Conservation of Momentum | Conservation of Torque | Conservation of Torque |
Circuit devices are often assumed to be point-like while electromagnetic properties of materials, like permittivity and permeability, are specified as functions of position. Similarly, mechanical devices can be treated as pointlike or as functions of position. For example, mass is used to describe a point-like device while density is used to describe a device that varies with position. Researchers studying aerodynamics and fluid dynamics typically prefer the latter description. However, in Tables \(\PageIndex{1}\) and \(\PageIndex{2}\), point-like devices of mass \(m\) are assumed. Ideas in these tables can be generalized to situations where energy conversion devices are not treated as point-like and instead mass and other material properties vary with position.
| Symbol | Quantity | Units |
|---|---|---|
| \(\overrightarrow{F}\) | Force | N |
| \(\overrightarrow{M}\) | Momentum | \(\frac{kg \cdot m}{s}\) |
| \(\vec{v}\) | Velocity | \(\frac{m}{s}\) |
| \(\overrightarrow{x}\) | Positional displacement | m |
| \(\overrightarrow{L_{am}}\) | Angular momentum | \(J \cdot s\) |
| \(\overrightarrow{\theta}\) | Angular displacement vector | rad |
| \(\overrightarrow{\tau}\) | Torque | \(N \cdot m\) |
| \(\overrightarrow{\omega_{ang}}\) | Angular velocity | \(\frac{rad}{s}\) |
The vector fields listed in Table \(\PageIndex{3}\) are related by constitutive relationships:
\[\overrightarrow{M} = m \overrightarrow{v} \label{12.3.1} \]
\[\overrightarrow{F} = K \overrightarrow{x} \label{12.3.2} \]
\[\overrightarrow{\tau} = \mathbb{K} \overrightarrow{\theta} \label{12.3.3} \]
\[\overrightarrow{L_{am}} = \mathbb{I} \overrightarrow{\omega_{ang}} \label{12.3.4} \]
Equation \ref{12.3.2} is more familiarly known as Hooke's law. By analogy to the capacitance of Equation 12.2.1, the coefficients in these equations are referred to in Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) as generalized capacity, and they represent the ability to store energy in the device. The constant \(m\) in Equation \ref{12.3.1} is mass in kg. The constant \(K\) in Equation \ref{12.3.2} is spring constant in \(\frac{J}{m^2}\). The constant \(\mathbb{K}\) in Equation \ref{12.3.3} is torsion spring constant in \(\frac{J}{\text{radians}^2}\). The constant \(\mathbb{I}\) in Equation \ref{12.3.4} is moment of inertia in units \(kg \cdot m^2\). A point mass rotating around the origin has a moment of inertia \(\mathbb{I} = m|\overrightarrow{r}|^2\) where \(|\overrightarrow{r}|\) is the distance from the mass to the origin. A solid shape has moment of inertia
\[\mathbb{I}=\int d \mathbb{I}=\int_{0}^{m}|\overrightarrow{r}|^{2} d m \nonumber \]
Interestingly, there is a close relationship between the quantities in Tables 12.2.3 and \(\PageIndex{2}\). Maxwell's equations, first introduced in Section 1.6.1, relate the four electromagnetic field parameters. Assuming no sources, \(\overrightarrow{J} = 0\) and \(\rho_{ch} = 0\), Maxwell's equations can be written:
\[\overrightarrow{\nabla} \times \overrightarrow{E} =-\frac{\partial \overrightarrow{B}}{\partial t} \nonumber \]
\[\overrightarrow{\nabla} \times \overrightarrow{H} =-\frac{\partial \overrightarrow{D}}{\partial t} \nonumber \]
\[\overrightarrow{\nabla} \cdot \overrightarrow{D} =0 \nonumber \]
\[\overrightarrow{\nabla} \cdot \overrightarrow{B} =0 \nonumber \]
The last two relationships, Gauss's laws, result directly from using calculus of variations to set up the Euler-Lagrange equation and solving for the corresponding equation of motion. We can replace electromagnetic vector fields in the source-free version of Maxwell's equations by mechanical fields according to the transformation:
\[\overrightarrow{D} \rightarrow \overrightarrow{M} \nonumber \]
\[\overrightarrow{E} \rightarrow \overrightarrow{v} \nonumber \]
\[\overrightarrow{B} \rightarrow \overrightarrow{\tau} \nonumber \]
\[\overrightarrow{H} \rightarrow \overrightarrow{\theta} \nonumber \]
The transformation of Equations 12.3.10 - 12.3.13 leads to set of equations accurately describing relationships between these mechanical fields.
\[\overrightarrow{\nabla} \times \overrightarrow{v} =-\frac{\partial \overrightarrow{\theta}}{\partial t} \nonumber \]
\[\overrightarrow{\nabla} \times \overrightarrow{\tau} =-\frac{\partial \overrightarrow{M}}{\partial t} \nonumber \]
\[\overrightarrow{\nabla} \cdot \overrightarrow{M} =0 \nonumber \]
\[\overrightarrow{\nabla} \cdot \overrightarrow{\theta} =0 \nonumber \]
The last rows of Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) list the relationship that results when an energy conversion device is described in the language of calculus of variations, the Euler-Lagrange equation is set up, and the EulerLagrange equation is solved for the equation of motion. The laws that result, Newton's second law, conservation of momentum, conservation of angular momentum, and conservation of torque, are fundamental ideas of mechanics.