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2014竞赛试题[1]

第五届四川省孙训方大学生力学竞赛试题

考试时间:3小时

本试题共6题,总分:100分

一(13分)、3个半径相同重量均为P的匀质圆盘相互之间用光滑铰链联接,并将其置于倾角为θ 的粗糙斜面上,如图所示。设圆盘与斜面之间的静滑动摩擦因素为0.7,铰链几何尺寸和滚动摩阻忽略不计。求:

1、平衡时斜面倾角θ 的最大值;

2、平衡时圆盘O1与O2之间铰链C沿圆盘公切线方向的约束力与倾角θ 的关系。

二(12分)、半径为r的轮子在水平直线轨道上作无相对滑动的纯滚动,角加速度为常数α ,设时间t =0时轮子的角速度为零。现建立一个随轮心C作平行移动的动坐标系Cxy,如图所示。假设一动点M始终与轮C上的加速度瞬心(即加速度为零的点)重合。求:

1、该动点的相对轨迹和相对运动方程;

2、该动点的相对速度的大小(表示为时间t的函数)。

三(25分)、圆盘半径为r 0,绕垂直于盘面且过圆盘中心O 的铅垂轴Oz 作定轴

转动,对转轴Oz 的转动惯量为2

02mr J z =;圆盘上沿径向有一直槽OB ,质量为

m 的质点A 可在槽内自由滑动;如初始时质点A 位于距圆盘中心O 为2

r 处且相对圆盘静止,此时圆盘的角速度为ω 0,不计所有摩擦。

1、如图(a)所示,当质点A 运动到圆盘边沿B 时,求圆盘的角加速度和圆盘作用于质点A 的水平力大小;

2、如图(b)所示,若在圆盘面内作用力偶M ,使得不论质点A 位于直槽中何位置,圆盘均以角速度ω 0作匀速转动,求该力偶矩M 与质点A 相对圆盘位置r 之间的关系,以及质点A 运动到圆盘边沿B 所需要的时间。

四(10分)、一弯曲刚度为EI 等直梁,各段长如图示;CD 为一长为、横截面面积为A 、弹性模量为E 的细长杆(如受压,其长度因素取为=1);梁在CB 段受均布载荷 q 的作用,另外还受一移动载荷 F=qa 作用。求当杆DE 达到其临界力时,q 的最大值。

a)(

b)

(

五(20分)、将一索拉桥简化如图,已知梁AB的弯曲刚度为EI,滑轮G位于支座D的正上方,∠AGD=30度(桥的右边与左边对称),长度a为已知,梁AB 上有方向向下的均布载荷q。

1)图I,如钢索在A、B处直接与地基连接,不与梁连接,可预紧钢索使梁跨中C处的挠度为零,求钢索中的预紧拉力(8分);并求此时梁中的最大弯矩和最大剪力(6分)。

2)图II,如钢索在A、B处直接与梁连接,能否通过预紧钢索使梁跨中C处的挠度为零?请说明理由(6分)。

六(20分)、如图所示上端刚性板与刚性地基之间有四根直杆(长为l、直径为d、弹性模量为E),位置参数a为已知,每根杆上端与刚性板铰接。在刚性板上作用有3个集中力(大小均为F),力的方向(见图):杆3顶端处向下、杆2顶端处向右、杆4顶端处向左。求:

1)杆中的最大弯矩、轴力、最大正应力(15分);

2)试设计一最简单的电测实验,根据应变计读数ε,测定F的大小(要求:贴片方式、大致的测量步骤、计算简式)(5分)。

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