MDOF Structural Dynamics Calculator — Modal & Newmark
Analyze a multi-story shear building: natural frequencies, mode shapes, and seismic response by Newmark-β integration and modal superposition.
- Direct Newmark
- Modal Superposition
- Direct Newmark
- Modal Superposition
- Ground accel
- normalized shape
- Effective mass %
| Floor | u_direct (m) | u_modal (m) | Gap (m) | v_direct (m/s) | a_abs_direct (m/s²) |
|---|---|---|---|---|---|
| 1 | 0.000796 | -0.000491 | 0.001287 | 0.002569 | -0.003962 |
| 2 | 0.001591 | 0.001390 | 0.000201 | 0.005201 | -0.008002 |
| 3 | 0.002358 | 0.002114 | 0.000243 | 0.007868 | -0.012060 |
| 4 | 0.003069 | 0.002569 | 0.000500 | 0.010499 | -0.016024 |
| 5 | 0.003694 | -0.002702 | 0.006396 | 0.012973 | -0.019712 |
| 6 | 0.004204 | 0.002510 | 0.001694 | 0.015118 | -0.022883 |
| 7 | 0.004569 | 0.002090 | 0.002479 | 0.016732 | -0.025253 |
| 8 | 0.004762 | 0.001887 | 0.002875 | 0.017616 | -0.026545 |
| Mode | f (Hz) | ω (rad/s) | ζ | Γ | Eff. Mass % | Residual |
|---|---|---|---|---|---|---|
| 1 | 0.3432 | 2.1563 | 5.000% | -745.175943 | 31.550 | 9.91e-1 |
| 2 | 0.9934 | 6.2415 | 5.000% | 80.589181 | 0.369 | 9.39e-1 |
| 3 | 1.6145 | 10.1444 | 6.830% | -515.414465 | 15.094 | 7.36e-1 |
| 4 | 2.1811 | 13.7045 | 8.744% | 163.187114 | 1.513 | 5.87e-1 |
| 5 | 2.6731 | 16.7955 | 10.477% | 678.269831 | 26.139 | 7.09e-1 |
| 6 | 3.0729 | 19.3074 | 11.910% | 137.284949 | 1.071 | 6.38e-1 |
A clear path from inputs to results.
Model a multi-story shear building as a chain of masses and springs and find how it vibrates and how it responds to shaking at its base. The solver extracts the natural frequencies and mode shapes, then integrates the full equations of motion two independent ways — direct Newmark-β time stepping and modal superposition — so you can compare roof drift, floor accelerations, and base shear.
How to use
- Set the number of floors (degrees of freedom), the mass per floor, and the base story stiffness.
- Set the stiffness gradient, which tapers the story stiffness linearly from the base toward the top to mimic a real building.
- Set the target damping ratio; the tool builds a Rayleigh damping matrix tuned to the first two modes so each mode is damped close to that target.
- Choose the ground-motion input (Ricker wavelet, sine burst, half-sine, or a custom series) and its amplitude in g, frequency, and envelope.
- Set the analysis duration, the number of time steps, and how many modes to keep in the modal superposition.
- Read the natural frequencies and mode shapes, then compare the direct and modal roof-displacement, acceleration, and base-shear histories, along with orthogonality and residual diagnostics.
The formula
Reference: Modal analysis and Newmark-β direct integration of MDOF systems; Chopra, Dynamics of Structures; Bathe, Finite Element Procedures (Newmark average-acceleration method, β = 1/4, γ = 1/2); shear-building closed-form frequencies ω_i = 2√(k/m)·sin((2i−1)π/(2(2n+1))).
Worked example
Eight-story shear building — natural frequencies
- Floors n = 8; floor mass m = 220,000 kg each; base story stiffness k = 32 MN/m; stiffness gradient = 0.2 (story stiffness tapers linearly to 25.6 MN/m at the top); target damping 5%.
- 1
The tool assembles a tridiagonal shear-building stiffness matrix K from the eight story stiffnesses (32.0, 31.09, 30.17, 29.26, 28.34, 27.43, 26.51, 25.60 MN/m) and a diagonal mass matrix M with 220,000 kg on each floor. Because all floor masses are equal, the generalized eigenproblem (K − ω²M)φ = 0 reduces to ω² = eig(K)/m, solved here with a mass-normalized Jacobi rotation sweep. The eight eigenvalues give natural frequencies f = ω/2π of 0.343, 0.993, 1.614, 2.181, 2.673, 3.073, 3.370, and 3.600 Hz. The fundamental mode has ω₁ = 2.156 rad/s, so f₁ = 0.343 Hz and the fundamental period is T₁ = 1/f₁ = 2.91 s. As a check, setting the gradient to zero reproduces the exact uniform-shear-building formula ω₁ = 2√(k/m)·sin(π/34) = 2.226 rad/s.
Eight natural frequencies from 0.343 Hz (fundamental, period 2.91 s) up to 3.600 Hz for the highest mode. The mode shapes are mass-orthonormal (ΦᵀMΦ ≈ I to machine precision), and the direct and modal time histories agree closely when enough modes are retained.
Interpreting the result
The fundamental frequency and period characterise how the building sways; a taller or heavier structure has a lower frequency and longer period. The mode shapes show the deflected pattern of each mode — the first mode leans the whole building one way, higher modes introduce reversals up the height. Keeping only the lowest few modes usually captures most of the response because they carry the bulk of the effective mass; the direct-vs-modal gap shown by the solver tells you whether your modal truncation is adequate. The base shear and roof drift from a given excitation scale with how close the input frequency content is to these natural frequencies.
Assumptions & limitations
- The structure is idealised as a lumped-mass shear building: rigid floor diaphragms, mass concentrated at floor levels, and lateral stiffness provided by story springs (one translational DOF per floor).
- Behaviour is linear elastic; stiffness and mass are constant and there is no yielding, geometric nonlinearity, or soil-structure interaction.
- Damping is classical Rayleigh damping (C = αM + βK) calibrated to the target ratio at the first two modes, so other modes are damped somewhat more or less than the target.
- The ground motions provided are synthetic analytic wavelets (Ricker, sine burst, half-sine) or a user series for demonstration, not recorded earthquake accelerograms; peak time-history responses depend on the chosen input.
- The Newmark average-acceleration scheme (β = 1/4, γ = 1/2) is unconditionally stable but still needs a time step fine enough to resolve the highest retained mode.
- Base shear is computed as the inertial-force surrogate Σ m_i(a_i + a_g), consistent with the lumped-mass idealisation.
Frequently asked questions
What is the fundamental period and why does it matter?
The fundamental period T₁ is the time for one full sway of the lowest vibration mode; for the example building it is about 2.91 s (f₁ = 0.343 Hz). It matters because seismic and wind demand depend strongly on how the building’s period compares with the dominant period of the loading — responses amplify when they coincide. It also sets the scale for the whole modal spectrum.
How are the natural frequencies found?
The tool assembles the stiffness matrix K and mass matrix M, forms the symmetric matrix M^(-1/2)KM^(-1/2), and extracts its eigenvalues with Jacobi rotations; the square roots of those eigenvalues are the circular frequencies ω_i, and f_i = ω_i/2π. For a uniform shear building this matches the closed-form result ω_i = 2√(k/m)·sin((2i−1)π/(2(2n+1))) exactly.
What is Rayleigh damping and why only two modes?
Rayleigh damping writes the damping matrix as C = αM + βK, a convenient classical form that keeps the modes decoupled. Two coefficients can match the target damping ratio exactly at only two frequencies — here the first two modes — so modes between them are slightly underdamped and modes far above are overdamped. The solver reports the actual damping ratio of every mode so you can see the spread.
What is the difference between direct integration and modal superposition?
Direct integration (Newmark-β here) steps the coupled equations of motion forward in time without decomposing them, capturing all modes. Modal superposition first solves the eigenproblem, then integrates each decoupled modal equation and recombines a chosen number of modes. The two should agree closely; the solver reports the roof and all-DOF gaps so you can see the error from truncating modes.
What does the stiffness gradient do?
It tapers the story stiffness linearly from the base to the top, so upper stories are more flexible than lower ones — a gradient of 0.2 makes the top story 80% as stiff as the base. This lowers the natural frequencies slightly compared with a uniform building and shifts the mode shapes, better representing buildings whose columns reduce in size with height.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.