Example 5 If the equation $x^{3}-27 x+m=0$ has three distinct real roots, find the range of real number $m$.

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Define the auxiliary function $F(x)=27 x-x^{3}$.
Differentiate the function $F(x)$ to get
$$
F^{\prime}(x)=27-3 x^{2} \text {. }
$$

Let $F^{\prime}(x)=0$. Then $x= \pm 3$.
Therefore, $F(x)_{\text {max }}=F(3)=54$,
$$
F(x)_{\text {min }}=F(-3)=-54 \text {. }
$$

From the graph, we know that when $-54<m<54$,
$$
F(x)=m \text { and } F(x)=27 x-x^{3}
$$

have three intersection points.
Thus, the range of real number $m$ is $-54<m<54$.