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The rotating-frame acceleration identity for a uniform rotation about an axis

Example

Let $ν > 0$ and define $R : R \to S O (3)$ by the rotation about the third standard axis through angle $ν t$ :

$R (t) = (\begin{array}{ccc} \cos ν t & - \sin ν t & 0 \\ \sin ν t & \cos ν t & 0 \\ 0 & 0 & 1 \end{array}) (t \in R) .$

Then $R (t) \in S O (3)$ for every $t$ , and $Ω = R^{T} R^{'}$ is

$Ω (t) = ν (\begin{array}{ccc} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}) .$

The angular-velocity vector of Angular velocity of a moving frame and the rotating-frame acceleration identity is $ω (t) = (0, 0, ν)$ , independent of $t$ . For any twice continuously differentiable $y : R \to R^{3}$ , put $x (t) : = R (t) y (t)$ and let $c : R \to R^{3}$ be identically zero. That lemma gives

$R (t)^{T} x^{''} (t) = y^{''} (t) + 2 ω \times y^{'} (t) + ω \times (ω \times y (t)) (x = R y) .$

Facts & Assumptions

Given: A number $ν > 0$ and the rotation matrix $R (t)$ displayed above, defined for $t \in R$ ; write $c_{t} : = \cos ν t$ and $s_{t} : = \sin ν t$ .

[L1]

The identity $c_{t}^{2} + s_{t}^{2} = 1$ holds for every real $t$ ; sine and cosine have derivatives $(\sin)^{'} = \cos$ and $(\cos)^{'} = - \sin$ , so repeated differentiation gives continuous derivatives of all orders. Vector-valued formulas are differentiated componentwise (Sine and cosine defined by their real power series, Parity and the Pythagorean identity for sine and cosine, The derivatives of sine and cosine are cosine and minus sine, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L2]

For twice continuously differentiable $R : I \to S O (3)$ and $y$ , with $Ω = R^{T} R^{'}$ and $ω (t)$ the unique vector with $Ω z = ω \times z$ for all $z$ , Angular velocity of a moving frame and the rotating-frame acceleration identity gives $R^{T} x^{''} = y^{''} + 2 ω \times y^{'} + ω^{'} \times y + ω \times (ω \times y) + R^{T} c^{''}$ when $x = R y + c$ .

[L3]

The cross product in $R^{3}$ of the standard basis vectors is $e_{1} \times e_{2} = e_{3}$ , $e_{2} \times e_{3} = e_{1}$ , $e_{3} \times e_{1} = e_{2}$ , and it is bilinear and alternating (The cross product in $R^{3}$ ).

Verification

technique · direct; compute $\Omega$ and check the identity on two explicit curves defined on $\mathbb R$

1.1L1givenalgebra

By [L1], $R^{'} = ν (\begin{array}{ccc} - s_{t} & - c_{t} & 0 \\ c_{t} & - s_{t} & 0 \\ 0 & 0 & 0 \end{array})$ , and the product $R^{T} R^{'}$ equals $ν (\begin{array}{ccc} c_{t} & s_{t} & 0 \\ - s_{t} & c_{t} & 0 \\ 0 & 0 & 1 \end{array}) (\begin{array}{ccc} - s_{t} & - c_{t} & 0 \\ c_{t} & - s_{t} & 0 \\ 0 & 0 & 0 \end{array}) = ν (\begin{array}{ccc} - c_{t} s_{t} + s_{t} c_{t} & - c_{t}^{2} - s_{t}^{2} & 0 \\ s_{t}^{2} + c_{t}^{2} & s_{t} c_{t} - c_{t} s_{t} & 0 \\ 0 & 0 & 0 \end{array}) = ν (\begin{array}{ccc} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array})$ , using $c_{t}^{2} + s_{t}^{2} = 1$ ; hence reading off the entries of $Ω$ as in clause 2 of [L2] gives $ω (t) = (0, 0, ν)$ and $Ω z = ω \times z$ for every $z$ . Also $R^{T} R = I_{3}$ and $\det R = c_{t}^{2} + s_{t}^{2} = 1$ , so $R (t) \in S O (3)$ ; [L1] gives its required $C^{2}$ regularity.

2.1step 1.1L1L2L3algebra

Take $y (t) = e_{1}$ for $t \in R$ , so $y^{'} = y^{''} = 0$ , $x (t) = R (t) e_{1} = (c_{t}, s_{t}, 0)$ and $x^{''} = - ν^{2} (c_{t}, s_{t}, 0)$ by [L1]. Then $R^{T} x^{''} = - ν^{2} R^{T} (c_{t}, s_{t}, 0) = - ν^{2} e_{1}$ ; the right-hand side is $ω \times (ω \times e_{1}) = (0, 0, ν) \times (0, ν, 0) = (- ν^{2}, 0, 0)$ by [L3] and step 1.1. The two sides agree.

3.1step 1.1step 2.1L1L2L3algebra∎

Take $y (t) = t e_{1}$ for $t \in R$ , so $y^{'} = e_{1}$ , $y^{''} = 0$ , $x (t) = t (c_{t}, s_{t}, 0)$ and $x^{''} = (- 2 ν s_{t} - t ν^{2} c_{t}, 2 ν c_{t} - t ν^{2} s_{t}, 0)$ by [L1]. Multiplying by $R^{T}$ gives $R^{T} x^{''} = (- t ν^{2}, 2 ν, 0)$ ; the right-hand side is $2 (0, ν, 0) + t (- ν^{2}, 0, 0) = (- t ν^{2}, 2 ν, 0)$ by [L3] and step 1.1, using $2 ω \times e_{1} = 2 (0, ν, 0)$ and $ω \times (ω \times t e_{1}) = t (- ν^{2}, 0, 0)$ . The two sides agree, exhibiting the two rotational terms separately.

Remarks

Both checks are finite computations with the convention $Ω = R^{T} R^{'}$ and angular velocity $ω = (0, 0, ν)$ . The other convention $R^{'} R^{T}$ gives the same vector in this fixed-axis example, so the example does not test the distinction between conventions.
The identity is kinematic. The terms are named "Coriolis" and "centrifugal" only in a mechanical interpretation, which requires the force and mass hypotheses of the consuming physical items.

The rotating-frame acceleration identity for a uniform rotation about an axis

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources