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Approximate Methods for Multiple Degree of Freedom Systems: Dunkerley’s Formula

Dunkerley’s Formula is another method of estimating the lowest (fundamental) natural frequency of a system without having to solve an eigenvalue problem. Rather than using the stiffness matrix, Dunkerley’s Method makes use of the flexibility matrix \bigl[a\bigr] which is the inverse of the stiffness matrix.

Starting with the equations of motion for a general N degree of freedom system

    \begin{equation*}         \bigl[m\bigr]\!\bigl\{\ddot{x}\bigr\} + \bigl[k\bigr]\!\bigl\{x\bigr\} = \bigl\{0\bigr\},  \end{equation*}

and assuming simple simultaneous harmonic motion of the form

    \begin{equation*}         \bigl\{x\bigr\} = \bigl\{\!\mathbb{A}\!\bigr\} \sin\left(p t + \phi\right), \end{equation*}

results in

(9.9)   \begin{equation*}  	\  	-p^2 \bigl[m\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} + \bigl[k\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\}.  \end{equation*}

Premultiplication of (9.9) by \bigl[k\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1} results in

    \begin{equation*}         -\ensuremath{p}^2 \underbrace{\bigl[k\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}}_{\bigl[a\bigr]}\bigl[m\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} + \underbrace{\bigl[k\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}\bigl[k\bigr]}_{\bigl[1\bigr]}\!\bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\}, \end{equation*}

so that

    \begin{equation*}         \bigl[a\bigr]\!\bigl[m\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} - \frac{1}{\ensuremath{p}^2} \bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\},  \end{equation*}

or

    \begin{equation*}         \biggl[\bigl[a\bigr]\!\bigl[m\bigr]- \frac{1}{\ensuremath{p}^2} \bigl[1\bigr] \biggr] \bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\}. \end{equation*}

As we have seen previously, for a non-trivial solution to exist it is required that

(9.10)   \begin{equation*}  	\  	\biggl| \bigl[a\bigr]\!\bigl[m\bigr] - \frac{1}{p^2} \bigl[1\bigr] \biggr| = 0.  \end{equation*}

For simplicity we will assume that the mass matrix is diagonal. In that case, the determinant in (9.10) has the form

    \begin{equation*} 	\left| \left[\begin{matrix} 			a_{{1}{1}}	&  a_{{1}{2}}	& \dots 	& a_{{1}{N}}		\\ 			a_{{2}{1}}	&  a_{{2}{2}}	& \dots		& a_{{2}{N}}		\\ 			\vdots		&  \vdots		& \ddots	& \vdots		\\ 			a_{{N}{1}}	&  a_{{N}{2}}	& \dots		& a_{{N}{N}}		\\ 		\end{matrix}\right] 	\left[\begin{matrix} 			m_1			&  0        	& \dots 	& 0       		\\ 			0        	&  m_2     		& \dots		& 0        		\\ 			\vdots		&  \vdots		& \ddots	& \vdots		\\ 			0        	&  0         	& \dots		& m_N			\\ 		\end{matrix}\right] 	- 	\left[\begin{matrix} 			\dfrac{1}{p^2}	&  0        		& \dots 	& 0       			\\ 			0        			&  \dfrac{1}{p^2}	& \dots		& 0        			\\ 			\vdots				&  \vdots			& \ddots	& \vdots			\\ 			0        			&  0         		& \dots		& \dfrac{1}{p^2}	\\ 		\end{matrix}\right]	\right| 	= 0, \end{equation*}

or

(9.11)   \begin{equation*} 	 \  	\left|\begin{matrix} 		\Bigl(a_{{1}{1}} m_1 - \dfrac{1}{p^2}\Bigr)	 	&  a_{{1}{2}} m_2	&	\dots 	& a_{{1}{N}}	m_N	\\ 		a_{{2}{1}} m_1	&  \Bigl(a_{{2}{2}} m_2 - \dfrac{1}{p^2}\Bigr) & \dots		& a_{{2}{N}}	m_N		\\ 			\vdots		&  \vdots			& \ddots	& \vdots										\\ 		a_{{N}{1}} m_1	&  a_{{N}{2}} m_2 	& \dots		& \Bigl(a_{{N}{N}} m_N - \dfrac{1}{p^2}\Bigr)	\\ 		\end{matrix}\right| 	= 0. \end{equation*}

Expanding (9.11) produces the characteristic equation which has the form

(9.12)   \begin{equation*} \begin{split} 	 \  	\biggl( \dfrac{1}{p^2} \biggr)^{N} - 	\Bigl( a_{{1}{1}} m_1 + a_{{2}{2}} m_2 + \dots + a_{{N}{N}} m_N\Bigr) \biggl( \dfrac{1}{p^2} \biggr)^{N-1} 	+ C_{N-2} \biggl( \dfrac{1}{p^2} \biggr)^{N-2} \\ 		+ C_{N-3} \biggl( \dfrac{1}{p^2} \biggr)^{N-3} +\ \dots\ + C_{1} \biggl( \dfrac{1}{p^2} \biggr)^{1} + C_0 = 0, \end{split} \end{equation*}

where C_{N-2}, C_{N-3}, \dots, C_{1} and C_0 are constants (which will depend on \bigl[a\bigr] and \bigl[m\bigr]) which we do not need to consider here.

Let the N roots of (9.12) be given by

    \begin{equation*}         \frac{1}{\ensuremath{p}_1^2}, \: \frac{1}{\ensuremath{p}_2^2}, \:\dots, \frac{1}{\ensuremath{p}_N^2}.  \end{equation*}

Since these N values are the roots, the characteristic equation could also be written in the form

(9.13)   \begin{equation*} 	\  	\biggl(\dfrac{1}{p^2}-\dfrac{1}{p_1^2}\biggr) 	\biggl(\dfrac{1}{p^2}-\dfrac{1}{p_2^2}\biggr) 	\dots 	\biggl(\dfrac{1}{p^2}-\dfrac{1}{p_N^2}\biggr) = 0, \end{equation*}

which is a completely equivalent polynomial to (9.12). Expanding (9.13) we get

(9.14)   \begin{equation*} \begin{split} 	\ 	\biggl( \dfrac{1}{p^2} \biggr)^{N} - 	\Bigl( \dfrac{1}{p_1^2} + \dfrac{1}{p_2^2} +\dots + \dfrac{1}{p_N^2}\Bigr) \biggl( \dfrac{1}{p^2} \biggr)^{N-1} 	+ C_{N-2} \biggl( \dfrac{1}{p^2} \biggr)^{N-2} \\ 		+ C_{N-3} \biggl( \dfrac{1}{p^2} \biggr)^{N-3} +\ \dots\ + C_{1} \biggl( \dfrac{1}{p^2} \biggr)^{1} + C_0 = 0, \end{split} \end{equation*}

where C_{N-2}, C_{N-3}, \dots, C_{1} and C_0 are the same constants which appeared in (9.12). Comparing the \bigl( \frac{1}{\ensuremath{p}^2} \bigr)^{N-1} terms in equations (9.12) and (9.14) shows that

(9.15)   \begin{equation*} 	\  	\dfrac{1}{p_1^2} + \dfrac{1}{p_2^2} + \dots +\dfrac{1}{p_N^2} =  	a_{{1}{1}} m_1 + a_{{2}{2}} m_2 + \dots + a_{{N}{N}} m_N. \end{equation*}

To this point no approximations have been made. Now however, we note that for many systems the higher natural frequencies are significantly larger than the fundamental natural frequency \ensuremath{p}_1 so that

    \begin{equation*}         \frac{1}{\ensuremath{p}_2^2} \ll \frac{1}{\ensuremath{p}_1^2},\qquad         \frac{1}{\ensuremath{p}_3^2} \ll \frac{1}{\ensuremath{p}_1^2},\qquad         \dots,\qquad         \frac{1}{\ensuremath{p}_N^2} \ll \frac{1}{\ensuremath{p}_1^2}. \end{equation*}

As a result, ignoring all but the first term on the LHS of (9.15) results in

    \begin{equation*} 	\dfrac{1}{p_2^2} \ll \dfrac{1}{p_1^2},\qquad 	\dfrac{1}{p_3^2} \ll \dfrac{1}{p_1^2},\qquad 	\dots,\qquad 	\dfrac{1}{p_N^2} \ll \dfrac{1}{p_1^2}. \end{equation*}

As a result, ignoring all but the first term on the LHS of equation (9.15) results in

    \begin{equation*} 	\dfrac{1}{p_1^2} + \dots =  	a_{{1}{1}} m_1 + a_{{2}{2}} m_2 + \dots + a_{{N}{N}} m_N, \end{equation*}

or

    \begin{equation*} 	\dfrac{1}{p_1^2} = a_{{1}{1}} m_1 + a_{{2}{2}} m_2 + \dots + a_{{N}{N}} m_N, \end{equation*}

or more generally

(9.16)   \begin{equation*} 	\  	\dfrac{1}{p_1^2} = \sum_{i=1}^{N} a_{{i}{i}} m_i. \end{equation*}

Equation (9.16) is known as Dunkerley’s Formula and provides an estimate of the fundamental natural frequency of a system. Due to the terms that have been neglected, the natural frequency obtained using Equation (9.16) will be lower than the actual fundamental natural frequency. Dunkerley’s Formula therefore provides a lower bound on the lowest natural frequency of the system (in contrast to Rayleigh’s Quotient which provided an upper bound on the lowest natural frequency).

Note that as long as the mass matrix is diagonal, Equation (9.16)

    \begin{equation*}         \frac{1}{\ensuremath{p}_1^2} = a_{{1}{1}} m_1 + a_{{2}{2}} m_2 + \dots + a_{{N}{N}} m_N \end{equation*}

can be written as

    \begin{equation*}         \frac{1}{\ensuremath{p}_1^2} = \frac{1}{\ensuremath{p}_{11}^2} + \frac{1}{\ensuremath{p}_{22}^2} + \dots + \frac{1}{\ensuremath{p}_{NN}^2} \end{equation*}

where \frac{1}{\ensuremath{p}_{ii}^2} is defined as the natural frequency of the single degree of freedom system that results when mass m_i is the only mass in the system (i.e. all other masses in the system are assumed to be zero).
This is because of the fact that if m_i is the only mass in the system then a_{{i}{i}} represents the deflection of the mass (in the positive i^{th} coordinate direction) when a unit load is applied to the mass m_i (again in the positive i^{th} coordinate direction). The effective stiffness of the (single degree of freedom) system is then

    \begin{equation*}         k_{\mathrm{EFF}} = \frac{1}{a_{{i}{i}}}     \Longrightarrow  a_{{i}{i}} = \frac{1}{k_{\mathrm{EFF}}} \end{equation*}

so that

    \begin{equation*}         a_{{i}{i}} m_i =  \frac{m_i}{k_{\mathrm{EFF}}} = \frac{1}{\ensuremath{p}_{ii}^2} \end{equation*}

as defined above. This interpretation can be useful in applying Dunkerley’s Formula in some instances.

Dunkerley’s Formula to Estimate The Highest Natural Frequency

The ideas behind Dunkerley’s Formula, with a slightly different formulation, can also be used to estimate the highest natural frequency in a system. To see this start with (9.9)

    \begin{equation*} -\ensuremath{p}^2 \bigl[m\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} + \bigl[k\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\}. \end{equation*}

Premultiplication by \bigl[m\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1} produces

    \begin{equation*} -\ensuremath{p}^2 \underbrace{\bigl[m\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}\bigl[m\bigr]}_{\bigl[1\bigr]}\!\bigl\{\!\mathbb{A}\!\bigr\} + \bigl[m\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}\bigl[k\bigr]\!\bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\} , \end{equation*}

or

    \begin{equation*} \biggl[\bigl[m\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}\!\bigl[k\bigr]- \ensuremath{p}^2 \bigl[1\bigr] \biggr] \bigl\{\!\mathbb{A}\!\bigr\} = \bigl\{0\bigr\}. \end{equation*}

This again requires, for non-trivial solutions,

(1)   \begin{equation*}         \biggl| \bigl[m\bigr]\!\!\rule[5mm]{0pt}{0pt}^{-1}\!\bigl[k\bigr]- \ensuremath{p}^2 \bigl[1\bigr] \biggr| = 0.  \end{equation*}

If we again assume that the mass matrix is diagonal, then the inverse is simply

    \begin{equation*}         \left[\begin{matrix}                         m_1             & 0                     & \dots         & 0             \\                         0               & m_2           & \dots         & 0             \\                         \vdots  & \vdots        & \ddots        & \vdots        \\                         0               &  0            & \dots         & m_N           \\         \end{matrix}\right]\!\!\!\rule[15mm]{0pt}{0pt}^{-1}         =           \left[\begin{matrix}                         \frac{1}{m_1}  &  0                    & \dots         & 0                             \\                         0                       &  \frac{1}{m_2}       & \dots         & 0                             \\                         \vdots                  &  \vdots                       & \ddots        & \vdots                        \\                         0                       &  0                    & \dots         & \frac{1}{m_N}                \\                 \end{matrix}\right]. \end{equation*}

If we again assume that the mass matrix is diagonal, then the inverse is simply

    \begin{equation*} \left[\begin{matrix} m_1 & 0 & \dots & 0 \\ 0 & m_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & m_N \\ \end{matrix}\right]\!\!\!\rule[15mm]{0pt}{0pt}^{-1} = \left[\begin{matrix} \frac{1}{m_1} & 0 & \dots & 0 \\ 0 & \frac{1}{m_2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \frac{1}{m_N} \\ \end{matrix}\right]. \end{equation*}

As a result, the determinant in (9.17) has the form

(9.18)   \begin{equation*} \left| \left[\begin{matrix} \  \frac{1}{m_1} & 0 & \dots & 0 \\ 0 & \frac{1}{m_2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \frac{1}{m_N} \\ \end{matrix}\right] % \left[\begin{matrix} k_{{1}{1}} & k_{{1}{2}} & \dots & k_{{1}{N}} \\ k_{{2}{1}} & k_{{2}{2}} & \dots & k_{{2}{N}} \\ \vdots & \vdots & \ddots & \vdots \\ k_{{N}{1}} & k_{{N}{2}} & \dots & k_{{N}{N}} \\ \end{matrix}\right] - \left[\begin{matrix} \ensuremath{p}^2 & 0 & \dots & 0 \\ 0 & \ensuremath{p}^2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \ensuremath{p}^2 \\ \end{matrix}\right] \right| = 0, \end{equation*}

or

(9.18)   \begin{equation*} 	 \  	\left|\begin{matrix} 		\Bigl(\dfrac{k_{{1}{1}}}{m_1} - p^2 \Bigr)	 	&  \dfrac{k_{{1}{2}}}{m_1}	&	\dots 	 			& \dfrac{k_{{1}{N}}}{m_1}	\\[5mm] 		\dfrac{k_{{2}{1}}}{m_2}	&  \Bigl(\dfrac{k_{{2}{2}}}{m_2} - p^2\Bigr) & \dots	&  			  \dfrac{k_{{2}{N}}}{m_2}			\\ 			\vdots		&  \vdots			& \ddots	& \vdots										\\ 		\dfrac{k_{{N}{1}}}{m_N}	&  \dfrac{k_{{N}{2}}}{m_N} 	& \dots		 		& \Bigl(\dfrac{k_{{N}{N}}}{m_N} - p^2\Bigr)	\\ 		\end{matrix}\right| 	= 0. \end{equation*}

Expanding (9.18) produces a characteristic equation with the form

(9.19)   \begin{equation*} \begin{split} 	 \  	\bigl( p^2 \bigr)^{N} - 	\Bigl( \dfrac{k_{{1}{1}}}{m_1} + \dfrac{k_{{2}{2}}}{m_2} + \dots + \dfrac{k_{{N}{N}}}{m_N}\Bigr) \bigl( p^2 \bigr)^{N-1} 	+ D_{N-2} \bigl( p^2 \bigr)^{N-2} \\ 	+ D_{N-3} \bigl( p^2 \bigr)^{N-3} +\ \dots\  	+ D_{1} \bigl(   p^2 \bigr)^{1} + D_0 = 0, \end{split} \end{equation*}

where D_{N-2}, D_{N-3}, \dots, D_{1} and D_0 are again constants which depend on \bigl[m\bigr] and \bigl[k\bigr].
The N roots of (9.19) correspond to the natural frequencies of the system given by

    \begin{equation*} \ensuremath{p}_1^2, \; \ensuremath{p}_2^2, \; \dots, \; \ensuremath{p}_N^2. \end{equation*}

Once again, since these N values are the roots of the characteristic equation, we could equivalently write the characteristic equation as

(9.20)   \begin{equation*} 	 \  	\bigl(p^2-p_1^2\bigr) 	\bigl(p^2-p_2^2\bigr) 	\dots 	\bigl(p^2-p_N^2\bigr) = 0, \end{equation*}

which expanded gives

(9.21)   \begin{equation*} \begin{split} 	 \  	\bigl( p^2 \bigr)^{N} - 	\Bigl( p_1^2 + p_2^2 + \dots + p_N^2 \Bigr) \bigl( p^2 \bigr)^{N-1} 	+ D_{N-2} \bigl( p^2 \bigr)^{N-2} \\ 		+ D_{N-3} \bigl( p^2 \bigr)^{N-3} +\ \dots\ + D_{1} \bigl( p^2 \bigr)^{1} + D_0 = 0 \end{split} \end{equation*}

where D_{N-2}, D_{N-3}, \dots, D_{1} and D_0 are the same constants as in (9.19) Comparing the \bigl( \ensuremath{p}^2 \bigr)^{N-1} terms in equations (9.19) and (9.21) shows that

(9.22)   \begin{equation*} 	 \  	p_1^2 + p_2^2 + \dots + p_N^2 =  	\dfrac{k_{{1}{1}}}{m_1} + \dfrac{k_{{2}{2}}}{m_2} + \dots + \dfrac{k_{{N}{N}}}{m_N}. \end{equation*}

Since \ensuremath{p}_N is the largest natural frequency in the system, we ignore all of the other terms on the LHS of (9.22) to get

    \begin{equation*} \ensuremath{p}_N^2 = \frac{k_{{1}{1}}}{m_1} + \frac{k_{{2}{2}}}{m_2} + \dots + \frac{k_{{N}{N}}}{m_N}, \end{equation*}

or more generally

(9.23)   \begin{equation*}          \          \ensuremath{p}_N^2 = \sum_{i=1}^{N} \frac{k_{{i}{i}}}{m_i}. \end{equation*}

Due to the terms that have been ignored in (9.22), the frequency obtained using (9.23) will be larger than the highest natural frequency in the system. As a result, this version of Dunkerley’s Formula provides an upper bound on the largest natural frequency of the system.

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