Problem 8.4. Frequency of a beam supported by springs

Problem a)

Spring constant, c [kN/mm]=

Problem b)

Spring constant, c [kN/mm]=

Problem c)

Spring constant, c [kN/mm]=

Follow steps of Example 55.

Step 1.  Calculate natural frequency of the beam built-in at both ends with uniform mass.

Show frequency, fm,1 Hide frequency, fm,1

See Eq.(8-49) and the previous problem (Problem 8.3 b)) .

$f_{m,1}^{}=\sqrt{5.1}{f}_m^{ss}=\sqrt{5.1}\sqrt{\frac{{\pi }^{2}EI}{4m{L}^4}}=\sqrt{5.1}\sqrt{\frac{{\pi }^2\times 18640\times {10}^3}{4\times 42.2\times 7.0^4}}=48.11 \mathrm{Hz}$

Step 2.  Determine natural frequency of the beam supported by springs with uniform mass.

Show frequency, fm,2Hide frequency, fm,2

Equation (8-15)

$f_{m,2}^{}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{2c}{mL}}$

Step 3.  Approximate fundamental frequency of the beam with uniform mass applying Föppl’s summation theorem.

Show frequency, fmHide frequency, fm

See Table 30 and Eq.(8-59).

$\frac{1}{f_m^2}=\frac{{\displaystyle 1}}{{\displaystyle f_{m,1}^2}}+\frac{{\displaystyle 1}}{{\displaystyle f_{m,2}^2}}=\frac{1}{48.{11}^2}+\frac{4{\mathrm{\pi }}^2}{2c}42.2\times 7.0$

Step 4.  Express spring constant from the condition that the natural frequency must be reduced by 20%.

Show spring constant, cmHide spring constant, cm

Problem b)

Step 1.  Calculate natural frequency of the beam built-in at both ends with concentrated mass.

Show frequency, fM,1 Hide frequency, fM,1

See Eq.(8-53) and the previous problem (Problem 8.3 b)) .

$f_{M,1}^{}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{k}{M}}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{192EI}{M{L}^3}}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{192\times 18640\times {10}^3}{150\times 7.0^3}}=41.98 \mathrm{Hz}$

Step 2.  Determine natural frequency of the beam supported by springs with concentrated mass.

Show frequency, fM,2Hide frequency, fM,2

Eq.(8-15)

$f_{M,2}^{}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{2c}{M}}$

Step 3.  Approximate fundamental frequency of the beam with concentrated mass applying Föppl’s summation theorem.

Show frequency, fMHide frequency, fM

See Table 30 and Eq.(8-59).

Step 4.  Express spring constant from the condition that the natural frequency must be reduced by 20%.

Show spring constant, cMHide spring constant, cM

Problem c)

Step 1.  Determine natural frequency of the beam built-in at both ends with uniform and additional concentrated masses.

Show frequency, f1Hide frequency, f1

Eigenfrequency with both masses is approximated with Dunkerley’s summation theorem.

See Table 30 and the previous problem (Problem 8.3 b)).

$\frac{1}{f_1^2}=\frac{{\displaystyle 1}}{{\displaystyle f_{m,1}^2}}+\frac{{\displaystyle 1}}{{\displaystyle f_{M,1}^2}} \to f_1^{}={\left(\frac{1}{48.{11}^2}+\frac{1}{41.{98}^2}\right)}^{–0.5}=31.63 \mathrm{Hz}$

 

Step 2.  Determine natural frequency of the beam supported by springs with uniform and additional concentrated masses.

Show frequency, f2Hide frequency, f2

Eq.(8-15).

Step 3.  Approximate fundamental frequency of the beam with both masses applying Föppl’s summation theorem.

Show frequency, fHide frequency, f

See Table 30 and Eq.(8-59).

Step 4.  Express spring constant from the condition that the natural frequency must be reduced by 20%.

Show spring constant, cHide spring constant, c

See Table 30 and Eq.(8-59).

$$\begin{gathered} f=0.8f_1^{}= \to {\left(\frac{{\displaystyle 1}}{{\displaystyle f_1^2}}+\frac{{\displaystyle 1}}{{\displaystyle f_2^2}}\right)}^{–1}=0.8^2{f}_1^2 \to f_2^2=f_1^2{\left(\frac{{\displaystyle 1}}{{\displaystyle 0.8^2}}–1\right)}^{–1}=1.78f_1^2 \\ \to \mathit{c}=\frac{4{\mathrm{\pi }}^2(mL+M)}{2} f_2^2=2{\mathrm{\pi }}^2\times (42.2\times 7.0+150)\times 1.78\times 31.{63}^2=15.63\times {10}^6\frac{\mathrm{N}}{\mathrm{m}}\mathbf{=}\mathbf{15}\mathbf{.}\mathbf{63}\frac{\mathrm{kN}}{\mathrm{mm}} \end{gathered}$$

Follow steps of Example 55.

First the natural frequency is calculated assuming that the beam is built-in at both ends (only the uniform mass is considered).

See Eq.(8-49) and the previous problem (Problem 8.3 b)) .

Then the natural frequency of the beam is calculated, when it is supported by springs (with uniform mass).

Eq.(8-15)

$f_{m,2}^{}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{2c}{mL}}$

The fundamental frequency of the beam with both supports (and with uniform mass) is approximated by Föppl’s summation theorem.

See Table 30 and Eq.(8-59).

Spring constant can be expressed from the condition that the natural frequency must be reduced by 20%:

$$\begin{gathered} f_m^{}=0.8f_{m,1}^{}= \to {\left(\frac{{\displaystyle 1}}{{\displaystyle f_{m,1}^2}}+\frac{{\displaystyle 1}}{{\displaystyle f_{m,2}^2}}\right)}^{–1}=0.8^2{f}_{m,1}^2 \to f_{m,2}^2=f_{m,1}^2{\left(\frac{{\displaystyle 1}}{{\displaystyle 0.8^2}}–1\right)}^{–1}=1.78\times f_{m,1}^2 \\ \to {\mathit{c}}_{\mathbf{m}}=\frac{4{\mathrm{\pi }}^{2}mL}{2} f_{m,2}^2=2{\mathrm{\pi }}^2\times 42.2\times 7.0\times 1.78\times 48.11=24.0\times {10}^6\frac{\mathrm{N}}{\mathrm{m}}\mathbf{=}\mathbf{24}\mathbf{.}\mathbf{0}\mathbf{ }\frac{\mathrm{kN}}{\mathrm{mm}} \end{gathered}$$

Problem b)

Natural frequency of the beam assuming built-in ends (with concentrated mass) is

See Eq.(8-53) and the previous problem (Problem 8.3 b)) .

Assuming springs supports (and concentrated mass) the natural frequancy results in:

Eq.(8-15)

$f_{M,2}^{}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{2c}{M}}$

The fundamental frequency of the beam (with concentrated mass) is approximated by Föppl’s summation theorem.

See Table 30 and Eq.(8-59).

Spring constant is expressed from the condition that the natural frequency must be reduced by 20%.

$$\begin{gathered} f_M^{}=0.8f_{M,1}^{}= \to {\left(\frac{{\displaystyle 1}}{{\displaystyle f_{M,1}^2}}+\frac{{\displaystyle 1}}{{\displaystyle f_{M,2}^2}}\right)}^{–1}=0.8^2{f}_{M,1}^2 \to f_{M,2}^2=f_{M,1}^2{\left(\frac{{\displaystyle 1}}{{\displaystyle 0.8^2}}–1\right)}^{–1}=1.78f_{M,1}^2 \\ \to {\mathit{c}}_{\mathbf{M}}=\frac{4{\mathrm{\pi }}^{2}M}{2} f_{M,2}^2=2{\mathrm{\pi }}^2\times 150\times 1.78\times 41.97=9.27\times {10}^6\frac{\mathrm{N}}{\mathrm{m}}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{27}\frac{\mathrm{kN}}{\mathrm{mm}} \end{gathered}$$

Problem c)

Eigenfrequency with both masses is approximated with Dunkerley’s summation theorem. First built-in ends are assumed.

See Table 30 and the previous problem (Problem 8.3 b)).

The natural frequency of the beam supported by springs with uniform and additional concentrated masses is

Eq.(8-15).

Fundamental frequency of the beam with both masses can be approximated by Föppl’s summation theorem.

See Table 30 and Eq.(8-59).

Spring constant is expressed from the condition that the natural frequency must be reduced by 20%.

See Table 30 and Eq.(8-59).