Fig. 1

Schematic of the nonlinear vibration isolator, where k(xm) denotes the nonlinear stiffness, and F(xm) denotes the nonlinear restoring force (color online)"

Fig. 2

Schematic of MPINN (color online)"

Fig. 3

Steady-state module based on harmonic balance network (color online)"

Fig. 4

Transient module with exponential and Fourier layers (color online)"

Fig. 5

Restoring-force module in a polynomial form (color online)"

Fig. 6

Feasibility validation of the MPINN: (a) complete responses comprising steady-state and transient components at ω¯F=0.5; (b) steady-state responses at ω¯F=0.5; (c) complete responses comprising steady-state and transient components at ω¯F=1.0; (d) steady-state responses at ω¯F=1.0; (e) complete responses comprising steady-state and transient components at ω¯F=2.0; (f) steady-state responses at ω¯F=2.0. The linear isolator parameters m, k1, c, H, and X¯F are 1 kg, 1 N/m, 0.2 kg/s, 1 m, and 0.1, respectively, and the initial displacement and velocity are both set to zero when calculating the complete response. The dotted blue, solid red, and dashed black lines represent the analytical solution, MPINN prediction, and excitation, respectively (color online)"

Table 1

Coefficient of determination (R2) and root mean square error (ERMSE) of the responses predicted by the MPINN in Fig. 6"

Fig. 7

Comparative analysis of five MPINN cases: (a) complete responses predicted by the MPINN; (b) corresponding residual errors (eres). Case 1 (solid orange line) employs MLPs for all three modules; Cases 2–4 (solid gray, blue, and green lines, respectively) use an MLP only for the steady-state, transient, and restoring-force module, respectively; Case 5 (solid red line) adopts the customized architectures for all three modules. The solid cyan line is the analytical solution. The linear isolator parameters m, k1, c, H, X¯F, and ω¯F are 1 kg, 1 N/m, 0.2 kg/s, 1 m, 0.1, and 1.5, respectively, where the initial displacement and velocity are zero (color online)"

Table 2

Coefficients of determination (R2), root mean square error (ERMSE), and mean of residual error (eres) of the responses predicted by the MPINN in Fig. 7"

Fig. 8

Dynamic responses of a cubic Duffing-type isolator predicted by the MPINN with or without the energy-conservation loss: (a) force-displacement of the cubic Duffing-type isolator; (b) complete responses; (c) learning curves for complete response predictions; (d) steady-state responses; (e) learning curves for steady-state response predictions. The cubic Duffing-type isolator parameters m, k1, k3, c, H, X¯F, and ω¯F are 1 kg, 0.1 N/m, 1 N/m3, 0.2 kg/s, 1 m, 0.1, and 2.0, respectively, and the initial displacement and velocity are zero for complete response predictions. The dashed black and dotted cyan lines are the base excitation and the responses by the Runge-Kutta method, whose detailed settings are provided in Section S4 of the electronic supplementary material. The solid red and gray lines represent results by the MPINN with or without the energy-conservation loss, respectively (color online)"

Fig. 9

Steady-state responses of a quintic Duffing-type isolator predicted by the MPINN under different excitation amplitudes: (a) force-displacement of the quintic Duffing-type isolator; (b) X¯F is 0.1; (c) X¯F is 0.12; (d) X¯F is 0.15; (e) mean of residual errors. The quintic Duffing-type isolator parameters m, k1, k3, k5, c, H, and ω¯F are 1 kg, 0.1 N/m, 1 N/m3, 10 N/m5, 0.2 kg/s, 1 m, and 2.0, respectively. The dashed black lines are the base excitations. The dotted cyan and solid red lines are the results by the Runge-Kutta method and the MPINN, respectively. The cyan and red boxes are means of residual errors of results by the Runge-Kutta method and the MPINN, respectively. The green points are the response amplitudes (color online)"

Fig. 10

Steady-state responses of a metamaterial-based isolator predicted by the MPINN under different excitation frequencies: (a) chiral multi-curved shell metamaterial, where the dotted orange and solid green lines are the force-displacement curves obtained from Ref. [74] and predicted by the MPINN’s restoring-force module, respectively; the parameters m, c, H, and XF are 2.39 kg, 0.2 kg/s, 0.027 6 m, and 0.001 m, respectively; (b) excitation frequency is 1 Hz; (c) excitation frequency is 2 Hz; (d) excitation frequency is 3 Hz; (e) excitation frequency is 4 Hz. The dashed black lines are the base excitations, and the solid red lines are the results by the MPINN (color online)"

Fig. 11

Inverse design for a three-spring isolator: (a) design target and designed results by the MPINN; (b) designed force-displacement curves obtained from the approximation relation in Eq. (S24) (dashed black line), the exact geometry-based restoring-force relation in Eq. (S23) (dotted orange line), and the restoring-force module (solid green line), respectively; (c) steady-state response of the designed result, where the dashed black, solid red, and dotted green lines are the base excitation, response by the MPINN, and maximum of target response, respectively. The prescribed parameters of the three-spring isolator m, c, H, kS, f, and XF are 1 kg, 0.2 kg/s, 1 m, 10 N/m, 0.5 Hz, and 0.1 m, respectively (color online)"

Fig. 12

Inverse design for isolators with a fifth-order polynomial restoring-force module: (a) designed force-displacement curves, where corresponding stiffness weights are provided in Table 3; (b) steady-state response of Design 1; (c) steady-state response of Design 2; (d) steady-state response of Design 3. The prescribed parameters of the isolator m, c, H, f, and XF are 1 kg, 0.2 kg/s, 1 m, 0.5 Hz, and 0.1 m, respectively. The solid red, magenta, and violet lines are the results of Designs 1, 2, and 3, respectively. The dashed and dotted green lines are the base excitation and maximum of target response, respectively (color online)"

Table 3

Designed stiffness weights of the restoring-force module for Designs 1, 2, and 3 in Fig. 12"