Physics · Ch 10 — Oscillations

Combinations of springs

10.4.3

Combinations of springs

The spring (or stiffness) constant measures how stiff a spring is: a larger $k$ needs more force to stretch or compress the spring by a given amount, and a smaller $k$ means the spring stretches easily. Springs can be combined end-to-end (series) or side-by-side (parallel). Series: if two springs $k_1,k_2$ are connected in series to a mass, the same force $F$ acts through both, but each stretches by its own amount, $x_1$ and $x_2$ , with total displacement $x=x_1+x_2$ . Since $-k_1x_1=-k_2x_2=F$ , substituting $x_1=-F/k_1$ and $x_2=-F/k_2$ into $x_1+x_2=-F/k_s$ gives $\dfrac{1}{k_s}=\dfrac{1}{k_1}+\dfrac{1}{k_2}$ , i.e. $k_s=\dfrac{k_1k_2}{k_1+k_2}$ ; for $n$ springs, $\dfrac{1}{k_s}=\sum_{i=1}^n\dfrac{1}{k_i}$ . A series combination is always softer than the softest individual spring; if all $n$ are identical ( $=k$ ), $k_s=k/n$ . The ratio of elastic PE stored in the two springs works out to $U_1/U_2=k_2/k_1$ . The reciprocal of stiffness, $C=1/k$ (SI unit m N $^{-1}$ ), is called flexibility or compliance; series compliances add, $C_s=\sum C_i$ . Parallel: if two springs are attached side by side to the same mass, both stretch by the same displacement $x$ under a force $F$ , so $F=-k_1x-k_2x=-k_px$ , giving $k_p=k_1+k_2$ ; for $n$ springs, $k_p=\sum_{i=1}^n k_i$ ; if all are identical, $k_p=nk$ -- a parallel combination is always stiffer than any individual spring. Cutting a spring: because the spring constant of a length of spring is inversely proportional to its length, cutting an ori …

Figure 10.16Combination of springs as a shock-absorber in a motorcycle

What this figure shows. A motorcycle's rear suspension unit is shown, in which a coil spring is combined with a hydraulic damper to form the shock-absorber assembly that connects the rear wheel to the frame. It is used here as a real, everyday example of an engineered spring combination -- the figure motivates why the mathematics of series and parallel spring combinations developed in this sub-section, worked out for idealised point masses, actually matters for the design of real mechanical systems that must …

Figure 10.17Springs connected in series

What this figure shows. Two springs of stiffness constants k1 and k2 are connected end to end (in series) between a fixed wall and a mass m resting on a horizontal frictionless surface at equilibrium position x0. Because the springs are joined only at one flexible connection point rather than rigidly, each spring is free to stretch by its own different amount when a force is applied, which is exactly the geometric fact used in the following derivation to combine k1 and k2 …

Figure 10.18Effective spring constant in series connection

What this figure shows. The two-spring series arrangement (with force F pulling the mass to the right) is shown on the left, and on the right an equivalent single spring of effective constant k_s attached to the same mass under the same applied force F is shown producing the identical net displacement. The side-by-side comparison is the visual statement of the derivation's conclusion: any series combination of two (or more) springs behaves, from the point of view of the attached mass, exactly like one single spring whose cons …

Figure 10.19Springs connected in parallel

What this figure shows. Two springs of stiffness constants k1 and k2 are connected side by side, both joining the same fixed wall to the same mass m at its equilibrium position x0, so that any displacement of the mass stretches or compresses both springs by exactly the same amount simultaneously. This shared-displacement geometry, different from the series case where each spring could stretch independently, is what leads to the springs' individual forces simply ad …

Figure 10.20Effective spring constant in parallel connection

What this figure shows. The two-spring parallel arrangement (force F applied to the mass, both springs stretching by the same displacement) is shown on the left, and an equivalent single spring of effective constant k_p under the same force F, producing the same displacement, is shown on the right. This is the visual statement of the parallel-combination result: the two springs acting together are equivalent to one spring whose constant is the simple sum k_p = k1 + k2, always stiffe …

Misc Example 10.9Effective spring constant of two springs in series

Worked out. Two springs of force constants 1 N/m and 2 N/m are connected in series, and the task is to compute the effective spring constant and comment on it. Using k_s = k1 k2/(k1+k2) with k1 = 1 and k2 = 2, k_s = (1 x 2)/(1+2) = 2/3 N/m. Since 2/3 is less than both 1 N/m and 2 N/m, this numerically confirms the general rule that a series combination is always softer (has a smaller effective spring constant) than either individual spring ma …

Misc Example 10.10Effective spring constant of two springs in parallel

Worked out. Two springs of force constants 1 N/m and 2 N/m are connected in parallel, and the task is to compute the effective spring constant and comment on it. Using k_p = k1 + k2 = 1 + 2 = 3 N/m. Since 3 N/m is greater than both 1 N/m and 2 N/m individually, this numerically confirms the general rule that a parallel combination is always stiffer (has a larger effective spring constant) than either individual spring making it up -- the opposite behaviour to the series case in the pr …

Misc Example 10.11Equivalent spring constant of mixed series-parallel networks

Worked out. Two more elaborate spring networks are given, each connecting a mass m to a wall through several springs arranged partly in parallel and partly in series, with the task of finding the overall equivalent spring constant, including the special case where all individual spring constants are equal to a common value k. In network (a), springs k1 and k2 are first combined in parallel to give k_u = k1+k2, and springs k3 and k4 are combined in parallel to give k_d = k3+k4; these two parallel sub-combinations then sit in series with each other, giving k_eq = (k_u k_d)/(k_u+k_d), which reduces to k_eq = k when all four springs equal k. In network (b), k1 and k2 combine in parallel to kA, and k4 and k5 combine in parallel to kB; then kA, k3, kB and k6 all sit together in series, giving 1/k_eq = 1/kA + 1/k3 + 1/kB + 1/k6, which reduces to k_eq = k/3 when all six springs equal k. Both cases show the general …

Misc Example 10.12Maximum spring compression from a moving mass

Worked out. A mass m moving with speed v on a smooth horizontal surface collides with a nearly massless spring of spring constant k and comes to rest, and the task is to find the maximum compression of the spring. By the law of conservation of energy, the entire kinetic energy the mass had just before impact must convert completely into the spring's elastic potential energy at the instant the mass is momentarily brought to rest (zero velocity) at maximum compression, since no energy is lost elsewhere in this idealised collision. Setting (1/2) m v^2 equal to (1/2) k x^2 and solving for x gives the maximum compression as x = v sqrt(m/k), a compact result obtained purely from energy conserv …

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