Determining the Effects of Gear Ratio on the Effective Motor Torque and Inertia

The parameters of the gearbox shown in Figure 14.7 are $B_1 = 0.025 Nm/rad/s, ω_1 = 210 rad/s,$ $B_m = 0.045 kg-m^2, J_m = 0.32 kg-m^2, T_2 = 20 Nm$ , and $ω_2 = 21 rad/s$ . Determine (a) the gear ratio $GR = N_2/N_1$ , (b) the effective motor torque $T_1$ , (c) the effective inertia J, and (d) the effective
friction coefficient B.

Question

Determining the Effects of Gear Ratio on the Effective Motor Torque and Inertia

The parameters of the gearbox shown in Figure 14.7 are [latex]B_1 = 0.025 Nm/rad/s, ω_1 = 210 rad/s,[/latex] [latex]B_m = 0.045 kg–m^2, J_m = 0.32 kg–m^2, T_2 = 20 Nm[/latex], and [latex]ω_2 = 21 rad/s[/latex]. Determine (a) the gear ratio [latex]GR = N_2/N_1[/latex], (b) the effective motor torque [latex]T_1[/latex], (c) the effective inertia J, and (d) the effective
friction coefficient B.

Accepted Answer

[latex]B_1 = 0.025 Nm/rad/s, ω_1 = 210 rad/s, B_m = 0.045 kg–m^2, J_m = 0.32 kg–m^2, T_2 = 20 Nm[/latex], and [latex]ω_2 = 21 rad/s[/latex].

a. Using Eq. (14.14),

[latex]\frac{\omega _1}{\omega _2} =\frac{N_2}{N_1}[/latex] (14.14)

[latex]GR=\frac{N_1}{N_2}=\frac{\omega _1}{\omega _2}=\frac{210}{21}=10 [/latex]

b. Using Eq. (14.15),

[latex]T_2=\left(\frac{N_2}{N_1} \right)^2T_1[/latex] (14.15)

[latex]T_1=\frac{T_2}{GR^2}=\frac{20}{10^2}=0.2 Nm[/latex]

c. Using Eq. (14.16),

[latex]J=J_m+\left(\frac{N_1}{N_2} \right)^2J_1[/latex] (14.16)

[latex]J=J_m+\frac{J_1}{GR^2}=0.32+\frac{0.25}{10^2}=0.323 Kg –m^2[/latex]

d. Using Eq. (14.17),

[latex]B=B_m+\left(\frac{N_1}{N_2} \right)^2B_1[/latex] (14.17)

[latex]B=B_m+\frac{B_1}{GR^2} = 0.045+\frac{0.025}{10^2}=0.045 Nm/rad/s[/latex]

Question 14.2: Determining the Effects of Gear Ratio on the Effective Motor...

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