Calculation
det(K−ω²M)=0; f1,2=ω1,2/(2π); μ1,2=A2/A1
Two masses can move together or against each other. Each mode has its own natural frequency and ratio of mass displacements.
Freie Schwingung einer translatorischen Zwei-Massen-Drei-Federn-Kette
Calculate Two-Mass Natural Frequencies and Mode Shapes
Two masses can move together or against each other. Each mode has its own natural frequency and ratio of mass displacements.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- First natural frequency f1 — Lower natural frequency of the coupled chain; f1=0 represents free rigid-body translation rather than an oscillation.
- Second natural frequency f2 — Higher natural frequency, usually with opposite mass motions; use in resonance assessment.
- First-mode amplitude ratio μ1 — Ratio A2/A1 of mass-2 to mass-1 displacement amplitudes; positive means the same direction.
- Second-mode amplitude ratio μ2 — Ratio A2/A1 for the higher natural frequency; negative means opposite motion directions.
- First moving mass m1 — First concentrated moving mass from parts list or measurement; spring and support inertia are neglected.
- Second moving mass m2 — Second concentrated moving mass, for translational motion only.
- Left spring stiffness c1 — Stiffness between fixed left support and mass 1, force per displacement from spring data or measurement; zero means no spring.
- Coupling spring stiffness c2 — Stiffness of the spring between the two masses; must be positive for two coupled modes.
- Right spring stiffness c3 — Stiffness between mass 2 and fixed right support; zero means no spring.
Example
With m1=m2=1 kg and c1=c2=c3=1,000 N/m, f1=√1000/(2π)=5.033 Hz and f2=√3000/(2π)=8.717 Hz. Amplitude ratios are μ1=+1 and μ2=−1.
Assumptions and limits
Two translational concentrated masses, three linear massless springs, frictionless support and no damping. Outer spring ends are fixed. Amplitude ratios describe mode shapes, not actual motion amplitudes; those depend on initial conditions. With c1=c3=0 the first frequency is zero and denotes rigid-body motion.
Understand Natural frequencies of a two-mass spring chain
Two moving masses and three springs can vibrate in two distinct ways. This calculator gives both natural frequencies and the displacement ratio for each mode.
What does this quantity describe?
Outer springs c1 and c3 connect one mass each to a fixed support; c2 couples the masses. Newton's law gives m1ẍ1+(c1+c2)x1−c2x2=0 and m2ẍ2−c2x1+(c2+c3)x2=0. Free harmonic motion requires det(K−ω²M)=0, with stiffness matrix K, mass matrix M and circular natural frequency ω in rad/s. The two solutions give f1 and f2 in Hz. Ratio μ=A2/A1 describes the mode shape: positive μ means motion together, negative μ opposite motion.
Formula and variables
det(K−ω²M)=0; f1,2=ω1,2/(2π); μ1,2=A2/A1
Eigenvalue condition: det(K−ω²M)=0K=[[c1+c2,−c2],[−c2,c2+c3]]; M=diag(m1,m2)Frequencies: f1,2=ω1,2/(2π)Mode ratio: μ=(c1+c2−m1ω²)/c2
| Symbol / input | Meaning |
|---|---|
| First natural frequency f1 | Lower natural frequency of the coupled chain; f1=0 represents free rigid-body translation rather than an oscillation. |
| Second natural frequency f2 | Higher natural frequency, usually with opposite mass motions; use in resonance assessment. |
| First-mode amplitude ratio μ1 | Ratio A2/A1 of mass-2 to mass-1 displacement amplitudes; positive means the same direction. |
| Second-mode amplitude ratio μ2 | Ratio A2/A1 for the higher natural frequency; negative means opposite motion directions. |
| First moving mass m1 | First concentrated moving mass from parts list or measurement; spring and support inertia are neglected. |
| Second moving mass m2 | Second concentrated moving mass, for translational motion only. |
| Left spring stiffness c1 | Stiffness between fixed left support and mass 1, force per displacement from spring data or measurement; zero means no spring. |
| Coupling spring stiffness c2 | Stiffness of the spring between the two masses; must be positive for two coupled modes. |
| Right spring stiffness c3 | Stiffness between mass 2 and fixed right support; zero means no spring. |
Choose the inputs correctly
m1 and m2 are moving masses in kg from a parts list or weighing and must be positive. c1 and c3 are outer spring stiffnesses in N/m from data sheets or force-displacement tests; zero means that outer spring is absent. c2 is positive coupling stiffness between masses. f1 and f2 are lower and upper natural frequency in Hz; μ1 and μ2 are their dimensionless amplitude ratios A2/A1. Select the desired output above the fields. Zero frequency denotes joint rigid-body motion, not periodic vibration.
How to use the calculator
Compare the arrangement fixed–c1–m1–c2–m2–c3–fixed with the actual machine. Determine the moving masses and spring rates and enter them. Select f1 and f2 in turn; μ1 and μ2 show relative direction and displacement for each mode. Compare operating excitation with both frequencies.
Worked example
With m1=m2=1 kg and c1=c2=c3=1,000 N/m, the normalized stiffness matrix has eigenvalues 1,000 s⁻² and 3,000 s⁻². Therefore f1=√1000/(2π)=5.033 Hz and f2=√3000/(2π)=8.717 Hz. Masses move together in the first mode (μ1=1) and oppositely in the second (μ2=−1).
Understand the result and units
With positive outer springs, the lower mode is in phase and upper mode out of phase. Amplitude ratios are not actual displacements; their scale depends on excitation and initial conditions. If both outer springs are absent, the whole chain can translate freely and f1=0.
Internally, masses use kg and stiffnesses N/m. This gives ω² in s⁻², ω in rad/s and f=ω/(2π) in Hz. μ is dimensionless. The existing unit registry converts available input units.
Typical applications
Resonance screening of coupled machinery masses, vehicle and plant models, and laboratory rigs with two dominant translational degrees of freedom.
Assumptions, limits and common mistakes
Linear massless springs, two concentrated masses moving only in translation, fixed outer supports, small displacements, frictionless guidance and no damping. Spring/support mass, nonlinear stiffness, rotational degrees of freedom, forced response and actual amplitudes are excluded. c2 must be positive.
Common mistake: Do not assume the upper natural frequency is twice the lower. μ is A2/A1 and may be negative; negative means opposite phase, not a negative physical amplitude. Check N/m versus N/mm for spring rates.
Frequently asked questions
What is “Natural frequencies of a two-mass spring chain” used for?
Resonance screening of coupled machinery masses, vehicle and plant models, and laboratory rigs with two dominant translational degrees of freedom.
Where do the input values come from?
m1 and m2 are moving masses in kg from a parts list or weighing and must be positive. c1 and c3 are outer spring stiffnesses in N/m from data sheets or force-displacement tests; zero means that outer spring is absent. c2 is positive coupling stiffness between masses. f1 and f2 are lower and upper natural frequency in Hz; μ1 and μ2 are their dimensionless amplitude ratios A2/A1. Select the desired output above the fields. Zero frequency denotes joint rigid-body motion, not periodic vibration.
What does the result not cover?
Linear massless springs, two concentrated masses moving only in translation, fixed outer supports, small displacements, frictionless guidance and no damping. Spring/support mass, nonlinear stiffness, rotational degrees of freedom, forced response and actual amplitudes are excluded. c2 must be positive.