19.2: B.2- Mechanical Work, Energy, and Power

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If a force accelerates in translation a body with mass \(m\), \(f_{x}=m a_{x}=m \dot{v}_{x}\), then the work done by \(f_{x}\) is conserved as kinetic energy of the body, from Equation 19.1.3:

\[W \equiv E_{K}=\int_{t=t_{1}}^{t=t_{2}} f_{x} v_{x} d t=\int_{t=t_{1}}^{t=t_{2}} m \dot{v}_{x} v_{x} d t=\frac{1}{2} m \int_{t=t_{1}}^{t=t_{2}} \frac{d}{d t}\left(v_{x}^{2}\right) d t=\frac{1}{2} m\left[v_{x}^{2}\left(t_{2}\right)-v_{x}^{2}\left(t_{1}\right)\right]\label{eqn:B.7} \]

If a force stretches or compresses a standard ideal linear translational spring (assumed here to have negligible mass), \(f_{x}(x)=k x\), then the work done by \(f_{x}\) is stored as strain energy (a form of potential energy) within the spring, from Equation 19.1.2:

\[W \equiv E_{S}=\int_{x=x_{1}}^{x=x_{2}} f_{x}(x) d x=\int_{x=x_{1}}^{x=x_{2}} k x d x=\frac{1}{2} k\left(x_{2}^{2}-x_{1}^{2}\right)\label{eqn:B.8} \]

At any instant, therefore, the total mechanical energy present in an ideal mass-spring system, relative to an initially stationary and unstrained state, is

\[E_{M e}=E_{K}+E_{S}=\frac{1}{2} m v_{x}^{2}+\frac{1}{2} k x^{2}\label{eqn:B.9} \]

In an ideal conservative mass-spring system, without any agent of energy augmentation or energy dissipation, \(E_{M e}\) remains constant in time, oscillating between the kinetic energy of the mass and the strain energy within the spring.

An ideal translational viscous damper dissipates mechanical energy by exerting a force in opposition to velocity: \(f_{x}=-c v_{x}=-c \dot{x}\). Thus, from Equation 19.1.4, the rate of energy dissipation by the damper is

\[P_{c}=f_{x} v_{x}=-c v_{x}^{2}\label{eqn:B.10} \]

Suppose that a mass-damper-spring system is initially stationary and unstrained, and that an independent, externally applied force \(f_{x}(t)\) is imposed upon the mass. This force is a source of power, \(P_{f}=f_{x}(t) v_{x}\). The damper, on the other hand, is a sink of mechanical energy. Therefore, total mechanical energy \(E_{M e}\) varies in time:

\[\frac{d}{d t} E_{M e}=\frac{d}{d t}\left(\frac{1}{2} m v_{x}^{2}+\frac{1}{2} k x^{2}\right)=P_{c}+P_{f} \quad \Rightarrow \quad m v_{x} \dot{v}_{x}+k x \dot{x}=-c v_{x}^{2}+f_{x}(t) v_{x}\label{eqn:B.11} \]

Canceling \(v_{x}=\dot{x}\) out of Equation \(\ref{eqn:B.11}\) and re-arranging terms leads to the general ODE of motion for a mass-damper-spring system¹:

\[m \ddot{x}+c \dot{x}+k x=f_{x}(t)m \ddot{x}+c \dot{x}+k x=f_{x}(t)\label{eqn:B.12} \]

The following relations for rotational motion can be derived from the basic definitions in the manner used above for translational motion. If a moment \(M\) accelerates a body with rotational inertia \(J\) about the axis of rotation, then the work done by \(M\) is conserved as kinetic energy of the body:

\[W \equiv E_{K}=\int_{t=t_{1}}^{t=t_{2}} M \dot{\theta} d t=\frac{1}{2} J\left[\dot{\theta}^{2}\left(t_{2}\right)-\dot{\theta}^{2}\left(t_{1}\right)\right]\label{eqn:B.13} \]

If a moment \(M\) twists an ideal linear torsion (rotational) spring (assumed here to have negligible rotational inertia) with spring constant \(k_{\theta}\), then the work done by \(M\) is stored as strain energy within the spring:

\[W \equiv E_{S}=\int_{\theta=\theta_{1}}^{\theta=\theta_{2}} M d \theta=\frac{1}{2} k_{\theta}\left(\theta_{2}^{2}-\theta_{1}^{2}\right)\label{eqn:B.14} \]

A rotational viscous damper with constant \(c_{\theta}\) dissipates mechanical energy by exerting a moment in opposition to velocity: \(M=-c_{\theta} \dot{\theta}\). Thus, the rate of energy dissipation by the damper is

\[P_{c}=M \dot{\theta}=-c_{\theta} \dot{\theta}^{2}\label{eqn:B.15} \]

For our final example of mechanical energy, consider translation in the \(y\) direction of a mass \(m\) that is within a field of constant gravitational field strength \(g\) (with SI units newton/kilogram). The force required to sustain the mass without acceleration (either stationary or at constant velocity) against gravity is

\[f_{y}=m g\label{eqn:B.16} \]

Therefore, the work required to raise the mass without acceleration against gravity is stored conservatively as gravitational potential energy:

\[W \equiv E_{G}=\int_{y=y_{1}}^{y=y_{2}} f_{y} d y=m g\left(y_{2}-y_{1}\right)\label{eqn:B.17} \]

For reference in the next section, we define also the gravitational potential difference, \(g\left(y_{2}-y_{1}\right)\). The following is an application of Equation \(\ref{eqn:B.17}\). Suppose that we shoot a projectile of mass \(m\) straight up against Earth’s gravity from surface elevation \(y_{1}\), with initial velocity \(v_{1}\) sufficiently low that \(g\) remains essentially constant over the entire trajectory. Let us assume that atmospheric drag is viscous with damping constant c.² Drag force \(-c v_{y}\) dissipates energy, so the total mechanical energy \(E_{M e}\) varies in time:

\[\frac{d}{d t} E_{M e}=\frac{d}{d t}\left(\frac{1}{2} m\left(v_{y}^{2}-v_{1}^{2}\right)+m g\left(y-y_{1}\right)\right)=P_{c} \Rightarrow m v_{y} \dot{v}_{y}+m g \dot{y}=-c v_{y}^{2}\label{eqn:B.18} \]

Canceling out \(v_{y}=\dot{y}\) from Equation \(\ref{eqn:B.18}\) and re-arranging terms leads to the 1^st order ODE describing projectile velocity:

\[m \dot{v}_{y}+c v_{y}=-m g\label{eqn:B.19} \]

1The direct power-balance method giving Equation \(\ref{eqn:B.11}\) and leading to ODE of motion Equation \(\ref{eqn:B.12}\) is also used in this appendix to derive governing ODEs (B-19) and (B-34). Each of these applications is for a one-degreeof-freedom (1-DOF) system, i.e., a system that has only one time-dependent variable. (See Chapters 11 and 12 for more detailed definitions of degrees of freedom and examples of multiple-DOF systems.) Unfortunately, this direct approach fails for systems with more than one DOF, as is observed and illustrated by Cannon, 1967, p. 166. For deriving the governing ODEs of multiple-DOF systems, a more general energy method was developed by Joseph Louis Lagrange (French-Italian mathematician and mechanician, 1736-1813). Lagrange’s equations are derived and illustrated in detail by most textbooks on classical mechanics and structural dynamics, e.g., Bisplinghoff, et al., 1955; Cannon, 1967; Craig, 1981; Greenwood, 1965; and Meirovitch, 1967 and 2001.