Section 13.3 Power Coefficients
Recall that the power generated by the action screw \(\{ \cA_{\coB \to \cB} \}\) exerted on a rigid body \(\cB\) relative to a referential \(\cE\) is given by
\begin{equation*}
\Pow_{\coB \to \cB/ \cE} = \{ \cA_{\coB \to \cB} \} \cdot \{\cV_{\cB /\cE} \}
\end{equation*}
Assuming that body
\(\cB\) belongs to a system
\(\Sigma\) whose configuration is defined by
\((n+1)\) coordinates
\((\bq, t)\text{,}\) we can use the expansion
(13.1.4) of the kinematic screw
\(\{\cV_{\cB /\cE} \}\) into its partial kinematic screws (under assumption
(13.1.2) to find the expression
\begin{equation*}
\Pow_{\coB \to \cB/ \cE} = \sum_{i=1}^n \dq_i \{ \cA_{\coB \to \cB} \} \cdot \{\cV_{\cB/\cE}^{q_i}\}
+ \{ \cA_{\coB \to \cB} \} \cdot \{\cV_{\cB/\cE}^{t}\}
\end{equation*}
This result leads to the following definition:
Definition 13.3.1. Power Coefficient.
We call
power coefficient of the action
\(\{ \cA_{\coB \to \cB} \}\) with respect to coordinate
\(q_i\) \((i=1, \ldots , n)\) and variable
\(t\) satisfying assumption
(13.1.2) the following scalar quantities:
\begin{equation}
\qQ_{\coB\to\cB /\cE} ^{q_i} = \{ \cA_{\coB \to \cB} \} \cdot \{\cV_{\cB/\cE}^{q_i}\}\tag{13.3.1}
\end{equation}
\begin{equation}
\qQ_{\coB\to\cB /\cE} ^{t} = \{ \cA_{\coB \to \cB} \} \cdot \{\cV_{\cB/\cE}^{t}\}\tag{13.3.2}
\end{equation}
Theorem 13.3.4.
The power \(\Pow_{\coB \to \cB/ \cE}\) can be expressed in terms of the corresponding power coefficients according to the equation
\begin{equation}
\Pow_{{\coB} \to \cB/ \cE} = \sum_{i=1}^n \dq_i \qQ_{{\coB}\to\cB /\cE} ^{q_i} + \qQ_{{\coB}\to\cB /\cE} ^{t}\tag{13.3.4}
\end{equation}
\(\danger\) In view of equation
(13.3.4) it is tempting to find each
\(q_i\)-power coefficient from the corresponding power according to
\begin{equation*}
\qQ_{{\coB}\to\cB /\cE} ^{q_i} = \frac{\partial}{\partial \dq_i} \Pow_{{\coB} \to \cB/ \cE}
\end{equation*}
However, this is risky and often incorrect since the power term may have been determined without consideration of assumption
(13.1.2). For instance,
\(\Pow_{\coB \to \cB/ \cE}\) may be found to vanish, yet the corresponding power coefficients may not be zero!
The following examples illustrate the calculation of power coefficients.
Example 13.3.7.
The plate
1 of
Example 13.1.4 is subject to a force
\(F_A \bx_0\) applied at point
\(A\) where the contact is friction-free. The contact of the plate with the corner point
\(Q\) of the step is assumed with friction and characterized by (kinetic) friction coefficient
\(\mu\text{.}\) The position of mass center
\(G\) is given by
\(\br_{AG} = l\bx_1\text{.}\)
Find the total power \(\Pow_{\bar{1}\to 1/0}\) of external action on body
Find the power coefficients \(\qQ_{\bar{1}\to 1/0}^q\) for \(q=x,\theta\text{.}\)
Solution.
a. We start by giving the expression of the total action screw \(\{\cA_ {\bar{1}\to 1/0}\}\) (denoting by \(N_Q\by_1\) the normal force and \(F_Q \bx_1\) the friction force at \(Q\text{.}\)
\begin{equation*}
\{\cA_ {\bar{1}\to 1/0}\} =
\begin{Bmatrix}F_A \bx_0 + N_A \by_0 \\\\ \bze \end{Bmatrix}_A
+
\begin{Bmatrix}-mg \by_0 \\\\ \bze \end{Bmatrix}_G
+
\begin{Bmatrix}N_Q \by_1 +F_Q \bx_1 \\\\ \bze \end{Bmatrix}_Q
\end{equation*}
and of the kinematic screw \(\{\cV_{1/0}\}\)
\begin{equation*}
\{\cV_{1/0}\} = \begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0 \end{Bmatrix}_A =
\begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0 + l \dte \by_1 \end{Bmatrix}_G =
\begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0+ \frac{h}{\sin\te} \dte \by_1 \end{Bmatrix}_Q
\end{equation*}
Power \(\Pow_{\bar{1}\to 1/0}\) is then found by treating each action screw separately
\begin{align*}
\Pow_{\bar{1}\to 1/0} \amp =
\begin{Bmatrix}F_A \bx_0 + N_A \by_0 \\\\ \bze \end{Bmatrix}_A
\cdot
\begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0 \end{Bmatrix}_A
+
\begin{Bmatrix}-mg \by_0 \\\\ \bze \end{Bmatrix}_G
\cdot
\begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0 + l \dte \by_1 \end{Bmatrix}_G\\
\amp
\quad +\begin{Bmatrix}N_Q \by_1 +F_Q \bx_1 \\\\ \bze \end{Bmatrix}_Q
\cdot
\begin{Bmatrix} \dte \bz_0 \\\\ \dx \bx_0+ \frac{h}{\sin\te} \dte \by_1 \end{Bmatrix}_Q
\end{align*}
giving
\begin{align*}
\Pow_{\bar{1}\to 1/0} \amp = F_A \dx - mg l\dte \cos\te + N_Q (-\dx \sin\te + \frac{h\dte}{\sin\te})+ F_Q \dx \cos\te\\
\amp = F_A \dx - mg l\dte \cos\te + F_Q \dx \cos\te
\end{align*}
where the power of normal force \(N_Q \by_1\) vanishes due to the constraint \(\vel_{Q\in 1/0}\cdot \by_1 =0\text{.}\)
b. We find the power coefficients by using the expressions of
\(\{\cV_{1/0}^q\}\) found in
Example 13.1.4. Each power coefficient
\(\qQ_{\bar{1}\to 1/0}^q\) is found by treating each contribution of the total action screw individually.
\begin{align*}
\qQ_{\bar{1}\to 1/0}^x \amp =
\begin{Bmatrix}F_A \bx_0 + N_A \by_0 \\\\ \bze \end{Bmatrix}_A
\cdot
\begin{Bmatrix}\bze \\\\ \bx_0 \end{Bmatrix}_A
+
\begin{Bmatrix}-mg \by_0 \\\\ \bze \end{Bmatrix}_G
\cdot
\begin{Bmatrix}\bze \\\\ \bx_0 \end{Bmatrix}_G\\
\amp \quad +\begin{Bmatrix} N_Q \by_1 +F_Q \bx_1\\\\ \bze \end{Bmatrix}_Q
\cdot
\begin{Bmatrix}\bze \\\\ \bx_0 \end{Bmatrix}_Q\\
\amp = F_A - N_Q \sin\te +F_Q \cos\te
\end{align*}
\begin{align*}
\qQ_{\bar{1}\to 1/0}^\te \amp =
\begin{Bmatrix}F_A \bx_0 + N_A \by_0 \\\\ \bze \end{Bmatrix}_A
\cdot
\begin{Bmatrix}\bz_0 \\\\ \bze \end{Bmatrix}_A
+
\begin{Bmatrix}-mg \by_0 \\\\ \bze \end{Bmatrix}_G
\cdot
\begin{Bmatrix}\bz_0 \\\\ l\by_1 \end{Bmatrix}_G\\
\amp \quad +
\begin{Bmatrix} N_Q \by_1 +F_Q \bx_1\\\\ \bze \end{Bmatrix}_Q
\cdot
\begin{Bmatrix}\bz_0 \\\\ \frac{h}{\sin\te}\by_1 \end{Bmatrix}_Q \\\\
\amp = -mgl \cos\theta + \frac{h}{\sin\te} N_Q
\end{align*}
We can verify here that the power coefficients cannot be found by using
\(\frac{\partial}{\partial \dq} \Pow_{\bar{1}\to 1/0}\) from the expression
\(\Pow_{\bar{1}\to 1/0} =F_A \dx - mg l\dte \cos\te -\mu \dx N_Q \cos\te\text{.}\) Indeed, this result was found by taking into account the constraint between coordinates
\(x\) and
\(\theta\) in violation
1 of assumption
(13.1.2).
Example 13.3.8.
The sphere (of mass
\(m\))
1 of
Example 13.1.6 is subject to the total external action
\begin{equation*}
\{\cA_ {\bar{1}\to 1/0}\} =
\begin{Bmatrix}-mg\bz_0 \\\\ \bze \end{Bmatrix}_G
+
\begin{Bmatrix}F_u \bu + F_v \bv + N \bz_0 \\\\ \bze \end{Bmatrix}_I
\end{equation*}
where \(I\) is the contact point and \(F_u \bu + F_v \bv\) is the friction force acting on the sphere. Hence, rolling and pivoting friction are neglected.
Find the total power \(\Pow_{\bar{1}\to 1/0}\) of the actions exerted on body 1.
Find the power coefficients \(\qQ_{\bar{1}\to 1/0}^q\) for \(q=x,y,\psi,\theta,\phi\text{.}\)
Solution.
a. Since \(\vel_{I\in 1/0} = \bze\text{,}\) we easily find \(\Pow_{\bar{1}\to 1/0} = 0\text{.}\)
b. The variables
\((q=x,y,\psi,\theta,\phi)\) are now required to satisfy the assumption
(13.1.2): we must ignore the kinematic condition
\(\vel_{I\in 1/0} = \bze\text{.}\) Instead we must use the expressions of the partial kinematic screws
\(\{\cV^q_{1/0}\}\) found in
Example 13.1.6. We then find
\begin{equation*}
\qQ_{\bar{1}\to 1/0}^x =
\begin{Bmatrix}F_u \bu + F_v \bv + (N-mg) \bz_0 \\\\ \bze \end{Bmatrix}_I
\cdot
\begin{Bmatrix} \bze \\\\ \bx_0 \end{Bmatrix}_I
=
F_u \cos\psi - F_v \sin\psi
\end{equation*}
\begin{equation*}
\qQ_{\bar{1}\to 1/0}^y =
\begin{Bmatrix}F_u \bu + F_v \bv + (N-mg) \bz_0 \\\\ \bze \end{Bmatrix}_I
\cdot
\begin{Bmatrix} \bze \\\\ \by_0 \end{Bmatrix}_I
=
F_u \sin\psi + F_v \cos\psi
\end{equation*}
\begin{equation*}
\qQ_{\bar{1}\to 1/0}^\psi =
\begin{Bmatrix}F_u \bu + F_v \bv + (N-mg)\bz_0 \\\\ \bze \end{Bmatrix}_I
\cdot
\begin{Bmatrix} \bz_0 \\\\ \bze \end{Bmatrix}_I
= 0
\end{equation*}
\begin{equation*}
\qQ_{\bar{1}\to 1/0}^\te =
\begin{Bmatrix}F_u \bu + F_v \bv + (N-mg) \bz_0 \\\\ \bze \end{Bmatrix}_I
\cdot
\begin{Bmatrix} \bu \\\\ r \bv \end{Bmatrix}_I
= r F_v
\end{equation*}
\begin{equation*}
\qQ_{\bar{1}\to 1/0}^\phi =
\begin{Bmatrix}F_u \bu + F_v \bv + (N-mg) \bz_0 \\\\ \bze \end{Bmatrix}_I
\cdot
\begin{Bmatrix} \bz_1 \\\\ r\sin\te \bu \end{Bmatrix}_I
= r F_u \sin\te
\end{equation*}
Note that we have to resolve the partial kinematic screws \(\{\cV^q_{1/0}\}\) at contact point \(I\text{.}\)
In each of the previous examples, it appears that the power coefficient
\(\qQ_{\coB\to\cB/\cE}^q\) could have been found by finding
\(\frac{\partial}{\partial \dq} \Pow_{\coB\to\cB/\cE}\text{,}\) had the expression of the power been determined in accordance with the assumption
(13.1.2) imposed on the coordinates. However, it is straightforward to construct an example of an action for which the power is zero, yet the corresponding power coefficients are non-zero without violating the assumption
(13.1.2): consider the notional action exerted on a rigid body
\(\cB\) in the form of couple
\begin{equation*}
\{\cA_{\cB_1 \to \cB}\} = \begin{Bmatrix} \bze \\\\ c(\dq_1 \bv - \dq_2 \bu) \end{Bmatrix}
\end{equation*}
where \(c\) is a constant and \(\bu\) and \(\bv\) are two unit vectors satisfying \(\bu\cdot \bv=0\text{.}\) The kinematic screw \(\{\cV_{\cB/\cE} \}\) is assumed of the form
\begin{equation*}
\{\cV_{\cB/\cE}\} = \begin{Bmatrix} \dq_1 \bu+ \dq_2 \bv \\\\ \vel_A \end{Bmatrix}
\end{equation*}
(the expression of velocity
\(\vel_A\) is unimportant). The coordinates
\((q_1, q_2)\) satisfy the assumption
(13.1.2). It is easy to find
\(\Pow_{\cB_1\to\cB/\cE} =0\) yet
\(\qQ_{\cB_1 \to\cB/\cE}^{q_1}= -c \dq_2\) and
\(\qQ_{\cB_1 \to\cB/\cE}^{q_2}= c \dq_1\text{.}\)
In conclusion, it is best to find each power coefficient
\(\qQ_{\coB\to\cB/\cE}^q\) by a careful determination of the partial kinematic screws
\(\{\cV^q_{\cB/\cE}\}\) without violating assumption
(13.1.2).
Another type of power coefficient can be defined, corresponding to an interaction between two rigid bodies \(\cB_1\) and \(\cB_2\) of a system \(\Sigma\text{.}\) Recall that if two bodies are interacting, a power of interaction \(\Pow_{\cB_1 \leftrightarrow \cB_2}\) can be defined as
\begin{equation*}
\Pow_{\cB_1 \leftrightarrow \cB_2 } = \{\cA_{\cB_1 \to \cB_2}\} \cdot \{\cV_{\cB_2/\cB_1}\} =
\{\cA_{\cB_2 \to \cB_1}\} \cdot \{\cV_{\cB_1/\cB_2}\}
\end{equation*}
Suppose that the set of variables
\((q_1, q_2 , \cdots , q_n,t)\) satisfies assumption
(13.1.2): we can then write
\begin{equation*}
\Pow_{\cB_1 \leftrightarrow \cB_2} =\{\cA_{\cB_1 \to \cB_2}\} \cdot \left(
\dq_1 \{\cV_{\cB_2/\cB_1}^{q_1}\}
+
\dq_2 \{\cV_{\cB_2/\cB_1}^{q_2}\}
+
\cdots
+
\dq_n \{\cV_{\cB_2/\cB_1}^{q_n}\}
+
\{\cV_{\cB_2/\cB_1}^t\}
\right)
\end{equation*}
with \(\{\cV_{\cB_2/\cB_1}^{q_i}\} =\{\cV_{\cB_2/\cE}^{q_i}\} -\{\cV_{\cB_1/\cE}^{q_i}\}\text{.}\) This result leads to the following definition:
Definition 13.3.9. Power Coefficient of Interaction.
The power coefficient of interaction between two rigid bodies \(\cB_1\) and \(\cB_2\) of a system \(\Sigma\) whose configuration is defined coordinates \((q_1, \ldots, q_n)\) and time \(t\) are the following scalar quantities:
\begin{equation}
\qQ_{\cB_1\leftrightarrow \cB_2} ^{q_i} = \{ \cA_{\cB_1 \to \cB_2} \} \cdot \{\cV_{\cB_2/\cB_1}^{q_i}\}=
\{ \cA_{\cB_2 \to \cB_1} \} \cdot \{\cV_{\cB_1/\cB_2}^{q_i}\}\tag{13.3.8}
\end{equation}
where the variables
\((\bq, \dbq, t)\) satisfy assumption
(13.1.2).
