Derive independent modes before interpreting energy exchange
Two equal 1 kg masses each connect to a fixed wall with a 4 N/m spring and to each other with a spring of stiffness kc. Their equations are x1ddot = −4x1 − kc(x1 − x2) and x2ddot = −4x2 − kc(x2 − x1). The coupling acts on both recipients with opposite forces.
Learning outcome: Derive sum and difference equations, reconstruct the physical coordinates and audit every energy term at two times.
Starting recap
Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.
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Try
At kc = 2, the antisymmetric frequency is sqrt(8) while the symmetric frequency remains 2. Initial energy becomes 0.03 J because the initially stretched connector stores more energy.
Open the experimentPredict
Increasing only the coupling stiffness changes which normal frequency?
Experiment
Derive sum and difference equations, reconstruct the physical coordinates and audit every energy term at two times.
Decompose two coupled coordinates into normal modes
Worked teaching model · synthetic values · Two equal1 kg masses each attached to a fixed wall by a4 N/m spring, connected by a selected coupling spring. Initial positions(.1,0) m and velocities zero. Linear conservative one-dimensional model with no apparatus.
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Mode superposition in both physical coordinates
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: first mass (solid).2: second mass (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Mode superposition in both physical coordinates
first mass
| time (s) | displacement (m) |
|---|---|
| 0 | 0.1 |
| 0.125 | 0.09612012 |
| 0.25 | 0.08479346 |
| 0.375 | 0.06693276 |
| 0.5 | 0.04397441 |
| 0.625 | 0.01775885 |
| 0.75 | -0.009622335 |
| 0.875 | -0.03599936 |
| 1 | -0.05930263 |
| 1.125 | -0.07773137 |
| 1.25 | -0.08989834 |
| 1.375 | -0.09493853 |
| 1.5 | -0.09257305 |
| 1.625 | -0.08312323 |
| 1.75 | -0.06747449 |
| 1.875 | -0.04699351 |
| 2 | -0.0234067 |
| 2.125 | 0.001349329 |
| 2.25 | 0.02529187 |
| 2.375 | 0.04655667 |
| 2.5 | 0.06354876 |
| 2.625 | 0.07506703 |
| 2.75 | 0.08039233 |
| 2.875 | 0.07933249 |
| 3 | 0.07222131 |
| 3.125 | 0.05987292 |
| 3.25 | 0.04349654 |
| 3.375 | 0.02458025 |
| 3.5 | 0.004754741 |
| 3.625 | -0.01435036 |
| 3.75 | -0.03124473 |
| 3.875 | -0.04469812 |
| 4 | -0.05383362 |
| 4.125 | -0.05818592 |
| 4.25 | -0.05772069 |
| 4.375 | -0.05281473 |
| 4.5 | -0.0442002 |
| 4.625 | -0.03287935 |
| 4.75 | -0.02001881 |
| 4.875 | -0.006833997 |
| 5 | 0.005525124 |
| 5.125 | 0.01607625 |
| 5.25 | 0.02408573 |
| 5.375 | 0.02912136 |
| 5.5 | 0.03107319 |
| 5.625 | 0.03014143 |
| 5.75 | 0.02679407 |
| 5.875 | 0.02169971 |
| 6 | 0.01564307 |
| 6.125 | 0.009432735 |
| 6.25 | 0.003810667 |
| 6.375 | -0.0006270301 |
| 6.5 | -0.003490094 |
| 6.625 | -0.004629014 |
| 6.75 | -0.00414167 |
| 6.875 | -0.002351562 |
| 7 | 0.0002395854 |
| 7.125 | 0.003019327 |
| 7.25 | 0.005344566 |
| 7.375 | 0.006625276 |
| 7.5 | 0.006402631 |
| 7.625 | 0.004412796 |
| 7.75 | 0.0006291677 |
| 7.875 | -0.004721991 |
| 8 | -0.01117478 |
| 8.125 | -0.01806504 |
| 8.25 | -0.02459605 |
| 8.375 | -0.02992241 |
| 8.5 | -0.03324339 |
| 8.625 | -0.03389644 |
| 8.75 | -0.03144079 |
| 8.875 | -0.0257221 |
| 9 | -0.01691058 |
| 9.125 | -0.005507679 |
| 9.25 | 0.007680796 |
| 9.375 | 0.02160377 |
| 9.5 | 0.03505177 |
| 9.625 | 0.04676407 |
| 9.75 | 0.05554661 |
| 9.875 | 0.06038954 |
| 10 | 0.06057318 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
second mass
| time (s) | displacement (m) |
|---|---|
| 0 | 0 |
| 0.125 | 0.0007711177 |
| 0.25 | 0.002964797 |
| 0.375 | 0.00623613 |
| 0.5 | 0.01005582 |
| 0.625 | 0.01377338 |
| 0.75 | 0.01669606 |
| 0.875 | 0.01817476 |
| 1 | 0.01768794 |
| 1.125 | 0.01491401 |
| 1.25 | 0.00978398 |
| 1.375 | 0.002508296 |
| 1.5 | -0.006426202 |
| 1.625 | -0.01628974 |
| 1.75 | -0.02617118 |
| 1.875 | -0.03506243 |
| 2 | -0.04195766 |
| 2.125 | -0.04595808 |
| 2.25 | -0.04637145 |
| 2.375 | -0.04279645 |
| 2.5 | -0.03518254 |
| 2.625 | -0.02385848 |
| 2.75 | -0.009525356 |
| 2.875 | 0.006786755 |
| 3 | 0.02379572 |
| 3.125 | 0.04007203 |
| 3.25 | 0.05416222 |
| 3.375 | 0.06472038 |
| 3.5 | 0.07063548 |
| 3.625 | 0.07114278 |
| 3.75 | 0.06590826 |
| 3.875 | 0.05507755 |
| 4 | 0.03928361 |
| 4.125 | 0.01961113 |
| 4.25 | -0.002480502 |
| 4.375 | -0.02526984 |
| 4.5 | -0.04691282 |
| 4.625 | -0.06559716 |
| 4.75 | -0.0796984 |
| 4.875 | -0.08792398 |
| 5 | -0.08943228 |
| 5.125 | -0.08391564 |
| 5.25 | -0.07163942 |
| 5.375 | -0.0534327 |
| 5.5 | -0.03063062 |
| 5.625 | -0.004972461 |
| 5.75 | 0.0215364 |
| 5.875 | 0.04678733 |
| 6 | 0.06874233 |
| 6.125 | 0.08560435 |
| 6.25 | 0.09596916 |
| 6.375 | 0.09894577 |
| 6.5 | 0.09423477 |
| 6.625 | 0.08215756 |
| 6.75 | 0.06363374 |
| 6.875 | 0.04010822 |
| 7 | 0.01343414 |
| 7.125 | -0.01427871 |
| 7.25 | -0.04083699 |
| 7.375 | -0.064144 |
| 7.5 | -0.08237142 |
| 7.625 | -0.09410828 |
| 7.75 | -0.09847451 |
| 7.875 | -0.09518967 |
| 8 | -0.08459116 |
| 8.125 | -0.06760094 |
| 8.25 | -0.04564365 |
| 8.375 | -0.02052386 |
| 8.5 | 0.005727052 |
| 8.625 | 0.03102087 |
| 8.75 | 0.05338479 |
| 8.875 | 0.07112128 |
| 9 | 0.08294225 |
| 9.125 | 0.08806631 |
| 9.25 | 0.08627169 |
| 9.375 | 0.07790107 |
| 9.5 | 0.06381869 |
| 9.625 | 0.04532473 |
| 9.75 | 0.02403489 |
| 9.875 | 0.001736664 |
| 10 | -0.01976498 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Symmetric angular frequency | 2 | rad/s |
| Antisymmetric angular frequency | 2.44949 | rad/s |
| First mass displacement | -0.05383362 | m |
| Second mass displacement | 0.03928361 | m |
| Total kinetic energy | 0.01178207 | J |
| Wall-spring energy | 0.008882522 | J |
| Coupling-spring energy | 0.00433541 | J |
| Total conserved model energy | 0.025 | J |
| Initial energy | 0.025 | J |
Adding and subtracting the coupled equations isolates two independent normal coordinates.
Equations x1ddot=−4x1−kc(x1−x2),x2ddot=−4x2−kc(x2−x1). Their sum has frequency2 and difference sqrt(4+2kc). Equal initial mode amplitudes reconstruct the disclosed one-sided displacement. Slow modulation for nearby frequencies resembles beats, but displacement zeros alone do not certify complete transfer of all kinetic/spring energy.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Adding and subtracting the coupled equations isolates two independent normal coordinates.
Equations x1ddot=−4x1−kc(x1−x2),x2ddot=−4x2−kc(x2−x1). Their sum has frequency2 and difference sqrt(4+2kc). Equal initial mode amplitudes reconstruct the disclosed one-sided displacement. Slow modulation for nearby frequencies resembles beats, but displacement zeros alone do not certify complete transfer of all kinetic/spring energy.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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Why does a displacement node not by itself locate all of the energy?
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