Figure 1. Flowchart of work A flow diagram of load input, structural models, and response outputs in low and high-frequency systems. The flow diagram shows load represented by acceleration and jerk over time, followed by low and high-frequency structural systems. These systems are modelled usi... More about this image found in Flowchart of work A flow diagram of load input, structural models, and r...

Figure 2. Diagram of base parameters in linear and curvilinear equations A diagram shows displacement, velocity, and acceleration relationships over time connecting points A, C, and B with interpolated curves. More about this image found in Diagram of base parameters in linear and curvilinear equations A diagram...

Figure 3. Comparison of the acceleration responses in the forced mode of the second- and third-order equations with experimental results: (a) ζ = 0.015; (b) ζ = 0.200 Two graphs show acceleration responses over time comparing second order, third order, and experimental results for damping ratios 20 and 0.15. The graphs present acceleration in centimetres per second squared over time from 0 to 0.6 seconds for two damping ratios. Part a shows damping ratio 20 where acceleration varies between negative 2 and 2 centimetres per second squared with periodic oscillations, reaching peaks near 2 and troughs near negative 2. Part b shows damping ratio 0.15 where acceleration varies between negative 3 and 3 centimetres per second squared with larger oscillations, reaching peaks near 2.5 and troughs near negative 2.5. Second order, third order, and experimental responses closely overlap in both parts. More about this image found in Comparison of the acceleration responses in the forced mode of the second- ...

Figure 4. Effects of acceleration and jerk of the support in the third-order equation compared with the total displacement of the second- and third-order equations (sin 1t, ζ = 0.15; β = 0.8) A graph shows displacement response over time comparing total third order and jerk component contributions. The graph presents displacement in centimetres over time from 0 to 20 seconds. Displacement varies between negative 0.35 and 0.35 centimetres with oscillations increasing to peaks near 0.3 centimetres around 10 seconds and then decreasing. The total third order response shows larger oscillations, while the jerk component remains within a smaller range near negative 0.05 to 0.05 centimetres. More about this image found in Effects of acceleration and jerk of the support in the third-order equation...

Figure 5. Comparison of acceleration and displacement responses of the second- and third-order equations for forced vibration of 600 rpm, β = 30 and different damping ratios (ζ) Two graphs show displacement and acceleration responses over time for multiple damping ratios using second and third order equations. The graphs present displacement in centimetres and acceleration in centimetres per second squared over time. Part a shows displacement where values vary between negative 0.02 and 0.02 centimetres, with oscillations reaching about 0.015 centimetres and negative 0.015 centimetres, and amplitude decreasing as damping ratio increases from 0.005 to 0.150. Part b shows acceleration where values vary between negative 2 and 2 centimetres per second squared, with peaks near 1.8 and troughs near negative 1.8, and higher damping ratios produce slightly reduced amplitudes. Second order and third order responses closely overlap across all damping ratios. More about this image found in Comparison of acceleration and displacement responses of the second- and th...

Figure 6. Comparison of acceleration and displacement responses of the second- and third-order equations for forced vibration of 600 rpm, β = 3 and different damping ratios (ζ) Two graphs show displacement and acceleration responses over time comparing second and third order equations for damping ratios 0.40 and 0.05. The graphs present displacement in centimetres and acceleration in centimetres per second squared over time from 0 to 1 second. Part a shows displacement where values vary between negative 0.001 and 0.001 centimetres, with oscillations reaching about 0.0008 and negative 0.0008 centimetres, and lower damping ratio 0.05 produces slightly larger variations than 0.40. Part b shows acceleration where values vary between negative 1.5 and 1.5 centimetres per second squared, with peaks near 1.3 and troughs near negative 1.3, and both damping ratios show closely matching second and third order responses. More about this image found in Comparison of acceleration and displacement responses of the second- and th...

Figure 7. Comparison of the responses of the second- and third-order equations for the first oscillation in the structure: (a) natural frequency of 1.87 rad/s; (b) natural frequency of 18.7 rad/s Two graphs show displacement responses over time comparing second and third order equations for damping ratios 0.05 and 0.25. The graphs present displacement in centimetres over time. Part a shows values varying between negative 2.5 and 1.5 centimetres, with peaks near 1.2 centimetres and troughs near negative 2.4 centimetres, and lower damping ratio 0.05 produces larger oscillations than 0.25. Part b shows values varying between negative 0.1 and 0.1 centimetres, with peaks near 0.05 centimetres and troughs near negative 0.06 centimetres, and higher damping ratio 0.25 results in reduced oscillation amplitude. Second order and third order responses follow similar trends in both parts. More about this image found in Comparison of the responses of the second- and third-order equations for th...

Figure 8. Amplitudes of the Fourier spectra of acceleration and jerk of North California, Anza, Mammoth Lake and Morgan Hill earthquakes Eight graphs show Fourier amplitude versus frequency for acceleration and jerk at Imperial Valley, North California, Anza, Mammoth Lake, and Morgan Hill. The... More about this image found in Amplitudes of the Fourier spectra of acceleration and jerk of North Califor...

Figure 9. Amplitudes of the Fourier spectra of acceleration and jerk of Imperial Valley, Cape Mendocino, Northridge-Sylmar and Erzican and earthquakes Eight graphs show Fourier amplitude versus frequency for acceleration and jerk across Imperial Valley, Cape Mendocino, Sylmar, and Erzican. The... More about this image found in Amplitudes of the Fourier spectra of acceleration and jerk of Imperial Vall...

Figure 10. Comparison of displacement in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10%, under the Imperial Valley earthquake accelerogram: (a) T = 1 s; (b) T = 10 s Two graphs show displacement responses over time for T 10 and T 1 systems with analytical, true, and jerk based curves. The graphs present displacement in centimetres over time from 0 to 20 seconds for T 10 and T 1 systems. Part a shows T 10 system where displacement oscillates between minus 0.01 and 0.01 centimetres with smooth increasing and decreasing cycles, while the jerk-based response remains close to 0 throughout. Part b shows T 1 system where displacement oscillates between minus 0.001 and 0.001 centimetres with rapid fluctuations, and the jerk-based response remains near 0. Analytical and true responses closely overlap in both parts. More about this image found in Comparison of displacement in the second- and third-order equations for two...

Figure 11. Comparison of acceleration in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10%, under the Imperial Valley earthquake accelerogram: (a) T = 1 s; (b) T = 10 s Two graphs show acceleration responses over time for T 1 and T 10 systems comparing analytical, true, and jerk based curves. The graphs present acceleration in g over time from 3.1 to 3.5 seconds for T 1 and T 10 systems. Part a shows T 1 system where acceleration fluctuates between minus 0.6 and 0.6 g with repeated increasing and decreasing oscillations, and the jerk based response remains within a smaller range near minus 0.2 to 0.2 g. Part b shows T 10 system where acceleration fluctuates between minus 2 and 2 g with larger oscillations, and the jerk based response varies within a narrower band around minus 1 to 1 g. Analytical and true responses align closely in both parts. More about this image found in Comparison of acceleration in the second- and third-order equations for two...

Figure 12. Comparison of acceleration in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10% (Cape Mendocino earthquake accelerogram): (a) T = 1 s; (b) T = 10 s Two graphs show acceleration responses over time for T 1 and T 10 systems with analytical, true, and jerk based comparisons. The graphs present acceleration in g over time from 2.5 to 3.5 seconds for T 1 and T 10 systems. Part a shows T 1 system where acceleration varies between minus 0.6 and 0.6 g with peaks rising near 0.5 g around 3 seconds and then decreasing. Part b shows T 10 system where acceleration varies between minus 0.8 and 0.8 g with sharp peaks reaching about 0.7 g near 3 seconds followed by decreasing oscillations. Analytical and true responses closely match, while the jerk-based response follows similar trends with slightly reduced amplitude. More about this image found in Comparison of acceleration in the second- and third-order equations for two...

Figure 13. Comparison of displacement in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10% (Sylmar earthquake accelerogram): (a) T = 1 s; (b) T = 10 s Two graphs show displacement responses over time for T 10 and T 1 systems with analytical, true, and jerk based curves. The graphs present displacement in centimetres over time from 0 to 15 seconds for T 10 and T 1 systems. Part a shows T 10 system where displacement varies between negative 0.05 and 0.05 centimetres, with peaks rising to about 0.03 centimetres and troughs dropping to about negative 0.05 centimetres, followed by decreasing oscillations. Part b shows T 1 system where displacement varies between negative 0.02 and 0.02 centimetres, with rapid oscillations reaching about 0.015 centimetres and negative 0.015 centimetres, then gradually decreasing. Analytical and true responses closely overlap, while the jerk-based response remains near 0 in both parts. More about this image found in Comparison of displacement in the second- and third-order equations for two...

Figure 14. Comparison of acceleration in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10% (Sylmar earthquake accelerogram): (a) T = 1 s; (b) T = 10 s Two graphs show acceleration responses over time for T 1 and T 10 systems comparing analytical, true, and jerk based curves. The graphs present acceleration in g over time from 2.5 to 4 seconds for T 1 and T 10 systems. Part a shows T 1 system where acceleration varies between negative 1 and 1.5 g, with a sharp increase reaching about 1.2 g near 3.5 seconds followed by a rapid decrease to about negative 0.8 g. Part b shows T 10 system where acceleration varies between negative 0.8 and 1.2 g, with peaks rising to about 1 g near 3.5 seconds and troughs dropping to about negative 0.7 g, followed by decreasing oscillations. Analytical and true responses align closely, while the jerk-based response shows smoother and lower amplitude variations. More about this image found in Comparison of acceleration in the second- and third-order equations for two...

Figure 15. Comparison of acceleration in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10% (Erzincan earthquake accelerogram): (a) T = 1 s; (b) T = 10 s Two graphs show acceleration responses over time for T 1 and T 10 systems with analytical, true, and jerk based comparisons. The graphs present acceleration in g over time from 10 to 25 seconds for T 1 and T 10 systems. Part a shows T 1 system where acceleration varies between negative 1.5 and 1.5 g, with increasing oscillations reaching about 1.2 g near 18 seconds and then decreasing. Part b shows T 10 system where acceleration varies between negative 1 and 1 g, with peaks rising to about 0.9 g near 17 seconds followed by gradual reduction in oscillation amplitude. Analytical and true responses closely match, while the jerk-based response remains closer to 0 with reduced variations. More about this image found in Comparison of acceleration in the second- and third-order equations for two...

Figure 16. Comparison of acceleration in the second- and third-order equations for two structures with periods of 1 s and 10 s, with a damping ratio of 10% (North California, Anza, Mammoth Lake and Morgan Hill earthquake accelerograms): (a) T = 1 s; (b) T = 10 s Six graphs show acceleration responses over time for North California, Anza, and Mammoth Lake for T 1 and T 10 systems. The graphs present acceleration in g over time for three locations, each with T 1 and T 10 systems. Part a shows North California T 1 system where acceleration varies between negative 100 and 100 g with peaks near 90 g and troughs near negative 90 g, showing irregular oscillations. Part b shows North California T 10 system where acceleration varies between negative 100 and 100 g with frequent rapid oscillations reaching about 60 g and negative 60 g. Part c shows Anza T 1 system where acceleration varies between negative 60 and 60 g with peaks near 40 g and troughs near negative 50 g around 6 seconds. Part d shows Anza T 10 system where acceleration varies between negative 60 and 60 g with smoother oscillations reaching about 40 g and negative 40 g. Part e shows Mammoth Lake T 1 system where acceleration varies between negative 150 and 150 g with peaks near 140 g and troughs near negative 120 g. Part f shows Mammoth Lake T 10 system where acceleration varies between negative 150 and 150 g with frequent oscillations reaching about 120 g and negative 120 g. Analytical and true responses closely overlap in all parts, while the jerk based response remains near 0 or shows reduced amplitude. More about this image found in Comparison of acceleration in the second- and third-order equations for two...

Figure 16. continued Two graphs show acceleration responses over time for Morgan Hill T 1 and T 10 systems with analytical, true, and jerk based curves. The graphs present acceleration in g over time from 11 to 12 seconds for Morgan Hill T 1 and T 10 systems. Part a shows T 1 system where acce... More about this image found in continued Two graphs show acceleration responses over time for Morgan Hi...